Risk-aversion optimal hedging scenarios during droughts

The primary purpose of a water supply reservoir is to regular highly fluctuating streamflow for providing reliable water supplies. Reducing water shortage risk for impending droughts is a challenge task in real-time reservoir operation due to future inflow uncertainty. The main aim of this study is to propose risk-aversion optimal hedging scenarios during droughts, which is achieved by a two-stage approach. The water shortage probabilities of future lead times are analytically estimated first, then follows an optimization framework that simultaneously minimizing water shortage probabilities of future lead times and sustaining long-term water supply reliability. With an illustration application of the Nanhua Reservoir located in southern Taiwan, the results indicate that the proposed optimization framework provides an efficient hedging to reduce future water shortage probabilities and mitigate severe water shortages in real-time operation. The proposed optimal hedging scenarios outperforms the rule-curve-based current operation. Further improvements are noted for the time-varying rationing coefficient scenarios than the constant-coefficient scenario and the most favorable scenario is the scenario with the highest time-varying frequency. Using the storage to trigger hedging associated with estimated future lead-time water shortage probabilities as the objective functions, the proposed optimal hedging scenarios are not only risk aversion but also executable in real-time operation during droughts.


Introduction
Difficulties encountered in real-time reservoir operation are induced by multiple sources of uncertainties that affect reservoir operation performance. These uncertainties stem from natural inflow processes and socioeconomic factors since they are unknown with certain in advance (Loucks et al. 1981;Xu and Tung 2009;Gauvin et al. 2017;Alizadeh et al. 2018;Huang et al. 2018). Such inherent and inevitable uncertainties imply sequential decisions made by decision makers subject to risks (Nardini et al. 1992). Pursuing optimal long-term or average performance in reservoir operation models often yields a risk-neutral decision that fails to highlight the extreme events such as low-probability highdamage floods and droughts (Haimes 1998). On the other hand, focusing on minimizing severe water shortages may lead to inefficient water uses for a water supply reservoir.
A traditional approach to take risks into account in reservoir operation is using a utility function to represent decision maker's risk aversion attitude (Keeney and Wood 1977;Loucks et al. 1981;Krzysztofowicz 1986;Loaiciga and Mariño 1986;Bouchart and Goulter 1998;Tingsanchali and Boonyasirikul 2006). However, it is difficult to identify the utility function that actually represents decision maker's risk aversion attitude (Hashimoto et al. 1982;Nardini et al. 1992). Nicolosi et al. (2009) indicated that the vague concept of risk is one of the most difficult tasks to manage water shortage risk caused by droughts in water supply systems. Multidimensional characteristics of risk is the major reason that is difficult to reach a universally agreed definition of risk (Haimes 2009). There are two main broad categories of risk proposed in the literature including the probability of an adverse event and the expected consequence of an adverse event (Nicolosi et al. 2009). The first category defines the water shortage risk as the probability of failure for water resource systems, i.e., water supply insufficient to meet water demand (Hashimoto et al. 1982;Zhang et al. 2016;Zhao et al. 2021). The second category of risk often involves hazard, exposure, and vulnerability to evaluate the outcome of a nature hazard and its corresponding consequences. Many studies used various approaches to assess water shortage risk or drought risk by comprising these three factors (Shahid and Behrawan 2008;Shiau and Hsiao 2012;Carrão et al. 2016;Ahmadalipour et al. 2019;Blauhut 2020;Li et al. 2022).
One common approach to predict or estimate water shortage probability or risk for a water resources system is uses of various statistical methods. For instance, Yu et al. (2014) integrated a seasonal weather outlook, a rainfallrunoff model, and a system dynamic model to forecast water-shortage probabilities of a water-resources system in central Taiwan for coming 3 months. Qian et al. (2016) proposed a double integral model to calculate water shortage risk in Beijing by a simulated probability density function of water shortages and a nonlinear vulnerability function. Zhang et al. (2016) used a SVM (support vector machine) model to estimate water shortage risk indicators including hazard rate, restorability, vulnerability, recurrence period, and risk level in Jilin City, China. Cordão et al. (2020) built a water shortage risk map of the Campina Grande in Brazil based on multi-criteria decision analysis and geographic information systems to evaluate urban water shortage levels. To improve reservoir performance and mitigate negative impacts of water shortages during prolonged and severe droughts, various hedging rules were proposed in the literature (e.g., Bayazit and Unal 1990;Shih and ReVelle 1994;Neelakantan and Pundarikanthan 1999;Draper and Lund 2004;You and Cai 2008;Shiau 2011;Taghian et al. 2014;Ahmadianfar et al. 2017;Ding et al. 2017;Kumar and Kasthurirengan 2018;Adeloye and Dau 2019;Ahmadianfar and Zamani 2020;El Harraki et al. 2021;Zeng et al. 2021;Shiau et al. 2021;Thiha et al. 2022;and Meng et al. 2022) to determine triggering scheme and rationing factors used in hedging policies. However, water shortage risks are rarely addressed in these hedging rules.
Deriving optimal trade-off between risk aversion of water shortages and water supply reliability is still a challenge task in reservoir operation. Orlovski et al. (1984) proposed a deterministic min-max approach that includes two management goals, satisfaction of the water demand and attenuation of storage peaks, in a reservoir to avoid severe failures. Nardini et al. (1992) extended such risk aversion formulation to incorporate average-performance optimization. That is, a two-step optimization formulation first starts a min-max risk aversion problem formulation and then follows a stochastic average-performance optimization. However, these two approaches depend on known or synthetic inflow sequences. Future inflow cannot be precisely predicted and inaccurate inflow prediction would worsen reservoir performance, especially when reservoir operation is under stress.
The main aim of this study is to develop an optimization framework of real-time operation for a water supply reservoir that simultaneously considers risk aversion hedging operation and maintenance of water supply reliability. Such optimized trade-offs are achieved by a two-stage approach. The first stage analytically estimates water shortage probabilities of future lead times in each month for various initial useful storage based on fitted inflow probability distributions. The second stage integrates such future water shortage probability information associated with water shortage indices in a multiobjective optimization framework to derive the risk-aversion optimal hedging rules. The proposed methodology is illustrated with an application of the Nanhua Reservoir located in southern Taiwan. The Nanhua Reservoir is a major surface water resource for domestic demands of Tainan and Kaohsiung metropolitan areas. However, limited capacity associated with highly fluctuating inflow leaded to frequent shortage problems in the past. Facing impending droughts, the Nanhua Reservoir needs more efficient withinyear operation to avoid dramatic water shortages and maintain stable water supplies.
The remainder of this article is organized as follows. Section 2 presents a detailed description of the proposed methodology which includes water shortage probabilities of future lead times and a multiobjective optimization framework to construct hedging operation for pursuing risk aversion and water supply reliability. Section 3 gives an introduction of the illustration example, the Nanhua Reservoir located in southern Taiwan. Results of the current operation and the proposed optimal hedging scenarios for the Nanhua Reservoir are presented in the Sect. 4, which is followed by the Sect. 5 of conclusions.

Unconditional and conditional water shortage probabilities
The first category of risk, the probability of an adverse event, is adopted in this study. Therefore, water shortage risk refers to the probability that the reservoir release is insufficient to meet the demand. Two types of water shortage probabilities, unconditional and conditional, are proposed in this study. The unconditional water shortage probability is the water shortage probability that is not regard to the initial conditions such as reservoir storage. The unconditional water shortage probability can be estimated by long-term reservoir operation simulation under specific operating rules and is expressed in terms of the ratio of number of insufficient releases for a specific time to the total simulation years. That is, where P j u is the unconditional water shortage probability at time j within a year, for example, j = 1, …, 12 represents various calendar months; D j denote the projected demand at time j, which is the target for reservoir release and is known in advance; N j is the number of insufficient releases at time j; N Y represents the total simulation years; R j denotes the reservoir release at time j, which is determined by specific reservoir operating rules.
The unconditional water shortage probability reflects seasonal variations of inflow and provides preliminary information of potential water shortages under specific operating rules for any time period within a year. In contrast, given the specific initial conditions, the conditional water shortage probability reveal real-time water shortage information. Whether water shortages actually occur at time t depends on the water stored in reservoirs at time t and inflow at time t to compensate for the deficits if storage is insufficient to satisfy the demand. Since reservoir inflow is not precisely known in advance and is represented by a probability distribution, the water shortage probability can be estimated based on the initial useful storage (water storage minus dead storage) and inflow distribution, which is called the conditional water shortage probability in this study. That is, where P t c is the conditional water shortage probability at time t; WA t denotes the water availability at time t, which is the sum of useful storage at the beginning of time t and the inflow at time t and is expressed as where S t denotes the useful storage at the beginning of time t, which is defined as the water storage minus dead storage.
Inflow I t is unknown in advance and is represented by a probability distribution which is fitted from historical records. The useful storage at time t S t is a known value when time t is the starting time. Otherwise, S t is also a random variable which is determined by the water balance equation based on the previous-time useful storage and inflow, i.e., S t = min max S t−1 + I t−1 − D t−1 , 0 ,S max , where S max denotes the reservoir capacity. The distribution of WA t needs to be determined by the convolution theorem between random variables of S t and I t .
Water shortages occur when demand exceeds water availability ( WA t <D t ). The water shortage at time t is thus defined as (3) WA t = S t + I t After determining the cumulative distribution function (CDF) of WA t ( F WA t (⋅) ), the conditional water shortage probability at time t ( P t c ) can be determined by Eq.
(2). The CDF of water shortage can also be determined via the CDF of water availability. That is, where F Sh t (⋅) denotes the CDF of water shortage at time t.
The conditional water shortage probability at future lead time t + k (k > 1) can be analytical derived in a similar way. Detailed derivation refers to Shiau (2021).

Hedging rules
Reservoir storage at current time represents the most reliable resources for satisfying future demands. Whether implementing water rationing is determined by the relationship between reservoir storage at current time and future demands in this study. Reservoir capacity is divided into four zones by three future demands (high, moderate, and low) and each zone corresponds to a specific rationing coefficient. The releasing rules are expressed as where D t H , D t M , and D t L denote the total demand of future H, M, and L periods (H > M > L), which are generally defined as t H , t M , and t L in Eq. (6) represent rationing coefficient for storage less than D t H , D t M , and D t L , respectively, and has a constraint of t H ⩾ t M ⩾ t L . Value of rationing coefficients ranges between 0 and 1 and less-than-1 coefficient represents hedging. These rationing coefficients are decision variables that are determined by the optimization model.

Multiobjective optimization framework
Sufficient releases for demands decline reservoir storage, which may increase possibility of future water shortage if insufficient inflow occurs in the future. On the other hand, hedging saves water for future uses can reduce future water shortage probabilities but at the cost of current-period deficits. Construction of optimal hedging operation should simultaneously consider satisfying current demand and reducing future water shortage probabilities. Future water shortage probabilities depend on initial storage, lead time, and inflow distributions. The maximum conditional water shortage probability for future lead time is considered as the target to be minimized in the optimization framework, which is defined by where LT is future lead time.
The risk-aversion hedging operation proposed in this study is therefore to minimize the maximum conditional water shortage probabilities from future 1-period to LTperiod. That is, Satisfying demands and maintaining long-term reliability are evaluated in terms of water shortage indices including the maximum single-period shortage ratio (XSR), the maximum cumulative water shortage for consecutive shortage periods (XCS), and the total shortage ratio (TSR), which are, respectively, defined as where SH t is the actual water shortage at time t, which is defined as SH t = max{D t − R t , 0} ; SR t is the water shortage ratio at time t and is defined as SR t = SH t ∕D t × 100% ; L is the consecutive shortage periods; N is the total operation periods.
These three indices span three different time scales (single, consecutive, and long-term periods) to characterize water shortages. Early hedging is an effective measure to reduce future water shortage probabilities, but it may deteriorate long-term reliability due to unnecessary hedging. Simultaneously considering these conflicting objective functions in the optimization framework constitutes a multiobjective optimization problem, which is formulated as

TOPSIS-based optimal trade-off solution
To find the best trade-off solution among these conflicting objective functions, this multiobjective optimization problem is transformed into a single objective optimization problem and solved by the technique for order performance by similarity to ideal solution (TOPSIS) developed by Hwang and Yoon (1981). Since these objective functions span different range of values, the following normalized formula is used to ensure the values of these objective functions ranged between 0 (the most favorable value) and 1 (the least favorable value).
where OF i and OF i ′ denote the original and normalized values of the ith objective function, respectively; Max(OF i ) and Min(OF i ) are the maximum and minimum values of the ith objective function, respectively.
The multiobjective optimization stated in Eq. (13) is thus rewritten as The basic idea in the TOPSIS to search the best tradeoff solution among conflicting objective functions that has the shortest distance to the most favorable solution and has the longest distance to the least favorable solution simultaneously. The most and least favorable solutions are called the positive ideal solution (PIS) and negative ideal solution (NIS), respectively, with values of 0 and 1 in the normalized scheme defined by Eq. (14). The weighted total distances of these normalized objective functions to the PIS and NIS are thus determined by where D + and D − denote the weighted distances to the PIS and NIS, respectively; w i is the weighting factor of the ith objective function and ∑ i w i = 1. The optimal trade-off solution is then determined by maximizing the relative distance to NIS, which is the overall objective function D * in the optimization model and expressed by (13) Min XSP 1 , Min XSP 2 , … , Min XSP LT , Min XSR, Min XCS, and Min TSR Min XSR � , Min XCR � , and Min TSR � (16) In this study, the optimal trade-off solution is searched by the real coded genetic algorithm (GA) with a population size of 1000, a crossover rate of 0.8, and a mutation rate of 0.05. The simulated binary crossover (SBX) (Deb and Agrawal 1995), the polynomial mutation (Deb and Goyal 1996), and elite-preserving operators are used in GA to improve search efficiency.

An overview
Steadily increasing domestic and industrial water needs of Tainan and Kaohsiung metropolitan areas located in southern Taiwan leaded to construction of the Nanhua Reservoir in 1995 for providing stable water supplies. However, lim- to increase water supply ability of the Nanhua Reservoir. The simplified Nanhua Reservoir-Chiahsien diversion weir operation system is illustrated in Fig. 1.

Operation system
As shown in Fig. 1, inflows to the Nanhua Reservoir include natural streamflow of the Houku Creek and diverted flow from the Chiahsien diversion weir. The water balance equation in terms of volumetric units defines the monthly operation of the Nanhua Reservoir. That is, where S t and S t+1 are reservoir storage at the beginning of times t and t + 1, respectively, which are constrained by S min ≤ S t ≤ S max , here S min and S max denote the dead storage (0 million m 3 ) and capacity (91.45 million m 3 ), respectively; Q t H denote the inflow of the Nanhua Reservoir at time t; DV t denotes the diverted flow from the Chiahsien diversion weir, which is determined by 0 denote the remaining streamflow at time t of the Chishan Creek and Houku Creek, respectively; W t C and W t H represent the reserved amounts of downstream uses at the Chishan Creek and Houku Creek, respectively, which have a higher priority than the projected demand; DV t max denotes the maximum allowable diverted flow.
R t in Eq. (18) denotes the reservoir release for the projected demand at time t and determined by where R t N denotes the reservoir release at time t for the normal condition. If the reservoir release cannot meet the R t N , then it releases the remaining storage, which is the latter term in Eq. (20).
The current operation of the Nanhua Reservoir determines the R t N using the rule-curve-based releasing rules, which are expressed as where URC t and LRC t denote the upper and lower rule curves at time t, respectively; U , M , and L denote the Fig. 1 Schematic operation system of the Nanhua Reservoir and the Chiahsien diversion weir rationing coefficients for various storage zones, respectively. The rule curves of the Nanhua Reservoir is shown in Fig. 2 and the rationing coefficients of U = M = 1 and L = 0.7 are adopted in the current operation.
For the proposed water shortage risk averse hedging operations, R t N is determined by Eq. (6) with decision variables of α H , α M , and α L denoting the rationing coefficients when storage less than D t 4 , D t 2 , and D t 1 (future 4-, 2-, and 1-month demand), respectively, which are also shown in Fig. 2. The multiobjective optimization model searches the optimal hedging rules by simultaneously minimizing XSP 1 , … , XSP 5 , XSR, XCS, and TSR.
Reservoir evaporation loss at time t, E t , is estimated by where 0.7 is the pan coefficient; e t is evaporation rate; A t and A t+1 denotes reservoir surface area at times t and t + 1, respectively.
Reservoir outflow O t H includes releases of downstream uses and spill flow, which is determined by Detailed description of the Nanhua Reservoir-Chiahsien diversion weir operation system refers to Shiau et al. (2018Shiau et al. ( , 2021. Known variables of this system include the streamflow of the Houku Creek ( Q t H ) and Chishan Creek ( Q t C ), reserved amounts of downstream uses of the Houku Creek ( W t H ) and Chishan Creek ( W t C ), the maximum allowed diversion ( DV t max ), the demand ( D t ), and evaporation rate (e t ). The monthly values of these variables are reported in Table 1. The monthly mean streamflows reported in Table 1 reveal common characteristics of southern Taiwan creeks, i.e., exceeding 90% of the mean annual streamflow is clustered within the wet season of May-October, which implicitly implies that reservoirs are needed to regulate highly fluctuating streamflow for providing stable water supplies.

Results and discussion
The unconditional water shortage probabilities for the current operation The monthly streamflow series of the Houku Creek ( Q t H ) and Chishan Creek ( Q t C ) for the period of 1959-2016 are used in this study to simulate the long-term performance of the Nanhua Reservoir-Chiahsien diversion weir system. The rule curves associated with rationing coefficients of U = M = 1 and L = 0.7 and the known values reported in Table 1 are used to determine the unconditional water shortage probabilities of each month for the current operation (Scenario C thereafter), which are demonstrated in Fig. 3a.
Apparently high and low unconditional water shortage probabilities are observed for the first half-year period (January-June) and remaining half-year period (July-December), respectively. Very high water shortage probabilities occurred in January-May are caused by consecutive low-inflow months from November to April that lead to less excess water stored in reservoir in this period. This consecutive low-inflow period also induces moderate water shortage probability in June although monthly mean inflow in June ranks second. On the contrary, consecutive high inflows in June-September lead to the lowest water shortage probability of 0.017 from August to October. Low water shortage probabilities in November-December benefit from previous high-inflow months although low inflows are noted in these 2 months.
The unconditional water shortage probabilities shown in Fig. 3a reveal that the capacity of the Nanhua Reservoir is inadequate for over-year operation. Constructing reliable reservoir operating rules to cope future uncertain inflows and reducing future lead-time water shortage probabilities are important in real-time reservoir operation.

The conditional water shortage probabilities of future 1-to 6-month lead times for various useful storages
The conditional water shortage probabilities of future lead times are analytically estimated based on fitted inflow distributions of various months and relationship between water availability and demand. Since the total inflow to the Nanhua Reservoir includes the natural streamflow of the Houku Creek and the diverted flow from the Chiahsien diversion weir, the distribution of the total inflow is the sum of two fitted streamflow distributions. Commonly used two-parameter distributions in hydrology are used to fit streamflows of the Houku Creek and Chishan Creek. The results indicate that the best-fitted distributions in the Houku Creek are lognormal distribution for January-February and August-December, gamma distribution for June, and Weibull distribution for the remaining months, while the best-fitted distributions in the Chishan Creek are gamma distribution for January-February and December, Weibull distribution for May, and lognormal distribution for the remaining months. The total inflow distribution of the Nanhua Reservoir of each month is then obtained by the convolution theorem with an independence assumption between two fitted distributions of the Houku Creek and Chishan Creek.
The estimated water shortage probabilities of future 1-to 6-month lead times conditioned on useful storages (0-91.45 million m 3 ) for various months are shown in Fig. 4. Clearly different probabilities for various months are caused by inflow distributions and length of lead time. These facts reveal that water shortage probabilities during the lowinflow months (January-May and October-December) are sensitive to the initial useful storage. This means that inflow during the low-inflow months is fail to compensates deficits caused by low useful storages. Abundant inflow during highinflow months provides greater capability to compensates   Fig. 3 a Unconditional water shortage probability of each month and b the maximum conditional water shortage probabilities of various lead times for current operation (scenario C) and various optimal scenarios (Scenarios Y, S, Q, and M) of the Nanhua Reservoir deficits of low useful storages and leads to low water shortage probabilities. Length of lead times is another factor to affect water shortage probabilities. For instance, the water shortage probability of 5-month lead time in February (i.e., June) is lower than the value of 5-month lead time in October (i.e., February) since greater inflow in June.

Constant-parameter optimal hedging scenario
Rule curve-based current operation (Scenario C) of the Nanhua Reservoir reveals that it does not provide efficient hedging and result in severe shortage problems. In additional to the unconditional water shortage probabilities shown in Fig. 3a, the overall performance in terms of objective functions in optimization model is XSP 1 = 0.354, XSP 2 = 0.092, XSP 3 = 0.087, XSP 4 = 0.105, XSP 5 = 0.070, XSR = 100%, XCS = 60.3 million m 3 (MCM), and TSR = 8.56%. Severe shortage events include 1980/1-1980/8 (consecutive 8-month shortage duration, cumulative shortages of 60.3 MCM, and maximum 1-month shortage ratio of 100%), 1993/9-1994/5 (consecutive 9-month shortage duration, cumulative shortages of 50.8 MCM, maximum 1-month shortage ratio of 100%, and XSP 1 , XSP 2 , XSP 3 , XSP 4 , and XSP 5 all occurred within this period), and 2002/11-2003/6 (consecutive 8-month shortage duration, cumulative shortages of 45.3 MCM, and maximum 1-month shortage ratio of 96%). Performance of Scenario C is served as a basis to compare outcomes of other optimal hedging scenarios. Fig. 4 The estimated water shortage probabilities conditioned on initial useful storages for 1-to 6-month lead times in various calendar months The objective functions to derive the risk-aversion hedging rules are to simultaneously minimizing the XSP 1 , XSP 2 , XSP 3 , XSP 4 , XSP 5 , XSR, XCS, and TSR. This multiobjective optimization problem is solved by the TOPSIS with decision variables of t H , t M , and t L . The equal weighting of 0.5 is assigned to future lead-time water shortage probabilities and reliability indices, respectively. Therefore, XSP 1 , XSP 2 , XSP 3 , XSP 4 , and XSP 5 has an equal weighting of 0.1, while the equal weight of 0.5/3 is giving to XSR, XCS, and TSR. Different weightings would result in different optimal solutions. Determining weightings is a decision-making problem which depends on decision maker's preference for objectives. Assigning greater weighting in favor of any objective to prioritize different objectives is not considered in this study. Effects of different weightings on the optimal solutions remains a topic of future studies. The maximum and minimum values of these objective functions are obtained first by optimizing a single objective function and summarized in Table 2.
The first hedging operation considered in this study is the constant-parameter scenario. This scenario, denoted as Scenario Y, has fixed t H , t M , and t L within a year in the releasing rules (Eq. 6). Therefore, three decision variables are involved in the Scenarios Y. The optimal outcomes of the Scenario Y, including objective functions and decision variables, are reported in Table 2. According to the overall objective function D*, the Scenario Y outperforms the current operation (Scenario C) since the D * improved from D * C = 0.575 to D * Y = 0.710. Improvements made by the Scenario Y are attributed to the early hedging (triggered when storage less than future 4-month demand) and smaller rationing coefficients (0.46 and 0.45 in Scenario Y vs. 0.7 in Scenario C). This derived optimal hedging efficiently reduce future lead-time water shortage probabilities (XSP 1 , XSP 2 , XSP 3 and XSP 4 ) and maximum 1-month shortage ratio (XSR) since more water is saved to compensate potential deficits for impendinf droughts. However, such early hedging scheme leads to inefficient uses of water resources and induces greater XCS (= 72.4 MCM) and TSR (= 20.72%) when comparing with the results of Scenario C, XCS = 60.3 MCM and TSR = 8.56%. This fact reveals that constant rationing coefficients in Scenario Y do not vary seasonally in response to the variations of reservoir inflow and result in less reliable operation.

Time-varying-parameter optimal hedging scenarios
Three time-varying-parameter scenarios are proposed in this study. The first time-varying-parameter scenario (Scenario S) specifies two t H , t M , and t L for the high-inflow months (June-September) and low-inflow months (January-May and October-December). The second quarterly varying scenario (Scenario Q) specifies four t H , t M , and t L for the periods of February-April, May-July, August-October and November-January, respectively. The third monthly varying scenario (Scenario M) specifies 12 t H , t M , and t L for each month. The numbers of decision variables involved in Scenarios S, Q, and M are equal to 6, 12, and 36, respectively. The GA iteration numbers to reach the optimal solutions for these three time-varying-parameter scenarios are greater than that of the constant-parameter scenario due to the greater number of decision variables. The approximate iteration numbers for Scenarios Y, S, Q, and M are less than 20, 70, 130, and 1000, respectively.
The outcomes of these time-varying-parameter scenarios are also reported in Table 2. Further improvements, when comparing with Scenarios C and Y, are observed for these time-varying-parameter scenarios according to the overall objective function D * ( D * It is worthy to note that less frequently varying scenarios induce lower D*. That is, the monthly-varying model (Scenario M) is the most favorable optimal hedging and the constant-parameter model (Scenario Y) is the least favorable optimal scenario.
Values of t H , t M , and t L of low-inflow months in Scenario S are identical to those values in Scenario Y and hedging implemented in high-inflow months only when storage less than future 1-month demand. Less hedging in high-inflow months of Scenario S leads to improvement on XCS = 64.82 MCM and TSR = 18.72%, when comparing with XCS = 72.44 MCM and TSR = 20.72% in Scenario Y. Effects of more frequent variation of decision variables on improvements of XCS and TSR are also noted for Scenarios Q and M, which result in XCS = 56.64 MCM, TSR = 12.72% and XCS = 46.35, TSR = 12.08%, respectively. Scenarios Q and M also have improvements on XSP 3 , XSP 4 and XSP 5 , but XSP 1 and XSP 2 in Scenarios Q and M are slightly worse than those in Scenarios Y and S.
The derived optimal decision variables reported in Table 2 reflect inflow fluctuations and variations of estimated future water shortage probabilities. No hedging ( t H , t M , or t L = 1) is implemented in various high-inflow periods for different scenarios when storage exceeds certain levels. For example, no water rationing in June-September, February-July, and February-August for Scenarios S, Q, and M, respectively, when storage exceeds future 2-month demand. Frequent variations of rationing parameters provide flexibleness for coping highly fluctuating inflows. For instance, no hedging is implemented in July-August for Scenario M even when the storage is less than 1-month demand. Besides, smaller rationing coefficients ( t H = 0.35) in low-inflow months of Scenarios Q and M than t H = 0.46  1980/1-1980/8 1993/9-1994/5 2002/11-2003 in Scenarios Y and S efficiently reduce XCS and TSR but at the cost of increasing XSR.

Discussion
The proposed optimal hedging scenarios outperform the current operation (Scenario C) due to the minimizing the worst conditions (the maximum water shortage probabilities, the maximum 1-month shortage ratio, and the maximum consecutive water shortage) during the analysis period. The performances of Scenarios C, Y, S, Q, and M for the severe drought events (1980/1-1980/8, 1993/9-1994/5, and 2002/11 2003/6) are examined in this section, which are reported in Table 3 and shown in Fig. 5.
The results shown in Fig. 5 indicate that the Scenario C provides insufficient hedging and leads to abruptly high deficits. Early and sufficient hedging involved in the proposed optimal scenarios avoid abruptly high deficits and induce more even deficits within the consecutive shortage duration (Fig. 5a, c, and e). For instance, XSR = 100% in Scenario C becomes XSR = 55-65% in these severe shortage event for various optimal scenarios. Such hedging mechanism can save more water for future use and maintain high level of storage, which is shown in Fig. 5b, d, and f. The high-level storages also efficiently reduce future lead-time water shortage probabilities, which are illustrated in Fig. 3b. For example, XSP 1 = 0.354 in Scenario C dramatically reduces to 0.019-0.025 for various optimal scenarios, while XSP 2 = 0.092 and XSP 3 = 0.087 also reduce to 0.010-0.011 and 0.004-0.014, respectively. More frequent variations of parameters such as in Scenarios Q and M can separate long consecutive shortage duration into two short shortage periods. Consecutive 8-month shortage duration of 1993-1994 event in Scenario Y becomes consecutive 4-and 3-month shortage durations in Scenario M, for example. Reduced shortage duration also leads to reduction of consecutive water shortage (XCS). For instance, XCS = 60.3, 50.8, and 45.3 MCM for these three shortage events in Scenario C become 46.3, 40.3, and 30.3 MCM in Scenario M. Figure 3a also demonstrates the unconditional water shortage probabilities of each month for various optimal scenarios. Generally, the water shortage probabilities of the first 6-month period (January-June) for the optimal scenarios are different with those in Scenario C, while they are similar for the remaining 6-month period (July-December). Similar unconditional water shortage probabilities in July-December period for all scenarios are attributed to the consecutive high inflows from June to September that leads to low water shortage probabilities regardless of what types of hedging implemented. Different water shortage probabilities in the period of January-June are caused by different hedging schemes adopted in various scenarios. For example, low rationing coefficients in the period of November-January for the Scenarios Y and M conserve water that induce high storage in February and lead to dramatically low water shortage probability in February for these two scenarios.

Conclusions
Three objectives are pursued in this study. First, the water shortage probabilities of future 1 to 6 months are analytically estimated for various storages of the Nanhua Reservoir in each month in advance. Second, the estimated future water shortage probabilities are served as the objective functions associated with water shortage indices to formulate the multiobjective optimization model for deriving the optimal riskaversion hedging rules. Third, optimal hedging scenarios including constant and three time-varying rationing coefficients are developed to cope with inflow fluctuations and to explore effects on reservoir performance.
The results indicate that the proposed optimal scenarios (Scenarios Y, S, Q, and M) outperform the rule-curve-based current operation (scenario C) according to the overall objective function D * . Greater improvements are also noted for higher time-  1979/10 1979/11 1979/12 1980/1 1980/2 1980/3 1980/4 1980/5 1980/6 1980/7 1980/8 1980/9 Water shortage (MCM)  1979/10 1979/11 1979/12 1980/1 1980/2 1980/3 1980/4 1980/5 1980/6 1980/7 1980/8 1980/9 Reservoir storage (MCM)   1979-1980, 1993-1994, and 2002-2003, respectively That is, the monthly-varying scheme (Scenario M) is the most favorable scenario that simultaneously promotes water shortage risk aversion and maintenance water supply reliability. The water shortage risk aversion operation is made possible by the proposed scenarios, which provide sufficient and early hedging that maintain high level storage and effectively reduce the unconditional and partly conditional water shortage probabilities. Using the known information (storage) to trigger hedging associated with previously estimated future lead-time water shortage probabilities, the proposed optimal hedging scenarios can be used in real-time operation during droughts. However, the estimated future lead-time water shortage probabilities are not the actual future inflow. How to incorporate inflow prediction into the proposed optimal hedging scenarios to improve reservoir performance for mitigating negative impacts during droughts remains as a topic for future studies.