Stochastic Dynamic Programming with Risk Consideration for Transbasin Diversion System

Tawatchai Tingsanchali, F.ASCE1; and Thana Boonyasirikul2

Abstract: An optimization procedure called SDPR that includes dynamic programming, stochastic dynamic programming ͑SDP͒, simulation, and trial and error adjustment of risk coefficient is developed and applied to determine the optimal operation policy of the proposed Kok-Ing-Nan transbasin diversion system in Thailand. Subject to hydrologic uncertainty, transition probabilities of inflows and its related uncertainty were considered. Due to dimensionality problems, the system is decomposed into three serially linked subsystems: two for the proposed upstream Kok and Ing diversion storages and one for the existing Sirikit reservoir. Optimization of each subsystem is done sequentially from upstream to downstream with specified sets of hydrologic state variables and diversion/release targets. The targets of the three subsystems are interrelated and link the subsystems together. From the derived optimal operation policies, simulation results show that the transbasin diversion increases the Sirikit reservoir release, irrigation reliability and net benefit of the system each by about 50–60%. Compared to SDP, the SDPR optimal operation policy increases both the maximum irrigation reliability and maximum system net benefit by about 10%. DOI: 10.1061/͑ASCE͒0733-9496͑2006͒132:2͑111͒ CE Database subject headings: Stochastic models; River basins; Irrigation; Hydroelectric power generation; Risk management; Optimization.

Introduction to the Nan River. The transbasin diversion will increase inflow to the Sirikit Reservoir and hence its releases for irrigation, hydro- The Chao Phraya River Basin, the largest and most important power generation and other domestic uses. river basin in Thailand, covers approximately 180,000 km2. The The study presented in this paper is conducted to determine an main stream, the Chao Phraya River, and its four major tributar- optimal operation policy for the transbasin diversion system. In ies, Ping, Wang, Yom, and Nan Rivers, yield an annual river such tropical regions as the Southeast Asian peninsula, hydrologic runoff of 30,300 million cubic meters ͑mcm͒ on average. There conditions have large seasonal fluctuations and high uncertainty are two regulating reservoirs: the Bhumibol Reservoir which has that significantly affect reservoir operation and increase risk of an active storage of 9,662 mcm on the Ping River and the Sirikit water shortage. Therefore, inflow uncertainty must be incorpo- Reservoir of 6,660 mcm on the Nan River. The increase of up- rated into optimization when determining the optimal operation stream water use has reduced runoff from the upstream catchment policy. areas whereas the downstream water requirement has increased Many related studies have been reported in the literature. For ͑ ͒ due to growth in irrigation area, population, and urbanization. example, Stedinger et al. 1984 developed a stochastic dynamic ͑ ͒ Therefore, water shortages have occurred frequently in the last programming SDP model employing the best forecast of current decade, especially in the downstream irrigation area of the Chao period inflows to define a reservoir operation policy. Karamouz ͑ ͒ Phraya river basin. In order to solve this water shortage, the and Vasiliadis 1992 used Bayesian decision theory to develop Kok-Ing-Nan transbasin diversion system, as shown in Fig. 1, Bayesian stochastic dynamic programming for reservoir optimi- was proposed by the Electricity Generating Authority of Thailand zation for Bayesian interpretation of transition probabilities. ͑ ͒ ͑EGAT͒͑1983͒ and by the Royal Irrigation Department of Loaiciga and Marino 1986 introduced the interaction between Thailand ͑1993͒ to divert water from the Kok and Ing river basins the expected value and variance of the objective function as a means of analyzing the risk averse nature of decision making in reservoir operations. Yeh ͑1985͒ and Simonovic ͑1992͒ provided 1Professor, School of Civil Engineering, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani, Thailand 12120. an extensive literature review and evaluation of various models E-mail: [email protected] used in reservoir management and operations. Several of these 2Senior Water Resources Engineer, Electricity Generating Authority studies applied stochastic optimization to examine the perfor- of Thailand, Nonthaburi, Thailand; formerly, Doctoral Graduate, Asian mance or to determine the operational policy of an existing Institute of Technology. system. Some incorporated reliability constraints to evaluate an Note. Discussion open until August 1, 2006. Separate discussions existing system operation or to determine the heuristic operating must be submitted for individual papers. To extend the closing date by rule based on deterministic approaches. However, fully incorpo- one month, a written request must be filed with the ASCE Managing rating the risk due to inflow uncertainty in stochastic optimization Editor. The manuscript for this paper was submitted for review and pos- sible publication on August 6, 2001; approved on July 15, 2005. This in determining optimal operational policies for water resource paper is part of the Journal of Water Resources Planning and Manage- systems continues to be a challenging problem. ment, Vol. 132, No. 2, March 1, 2006. ©ASCE, ISSN 0733-9496/2006/ In this study, stochastic dynamic programming with a multi- 2-111–121/$25.00. objective function and consideration of risk due to inflow uncer- Fig. 2. Schematic diagram of Kok-Ing-Nan transbasin diversion system

Fig. 1. Map of proposed Kok-Ing-Nan transbasin diversion system SDPR to determine the optimal operation policy of Kok-Ing-Nan transbasin diversion system. The optimization is based on maxi- mizing system net benefits. The performance of the transbasin tainty is applied to determine optimal operational policies for the diversion system is evaluated under the optimal operational poli- proposed Kok-Ing-Nan transbasin diversion system in Thailand. cies obtained from both SDPR and SDP by simulation using input Since the multistorage transbasin diversion system is large scale time series of generated monthly inflows. Finally, the simulation and complex, the SDP algorithm is inhibited by the large state- results with and without the proposed transbasin diversion system space dimensionality of the problem. Many attempts have been based on SDPR and SDP policies are compared to determine the made to simplify this problem by using combined optimization improvement in the system performance. techniques such as linear programming, dynamic programming ͑DP͒, simulation, regression, and search techniques to evaluate system performance. A DP model was sequentially applied by Stochastic Dynamic Programming with Risk Boehle et al. ͑1982͒ to optimize reservoir system operation. Consideration Hla ͑1991͒ formulated an SDP model for optimizing annual hydropower energy generation from Ubol Ratana reservoir in Uncertainty due to random characteristics and stochastic pro- Thailand. Braga et al. ͑1991͒ developed a DP-SDP model for cesses of hydrologic parameters is involved in optimization of optimization of hydropower production of a multiple storage- operation policies of diversion systems. Some policies may offer reservoir system with correlated inflows and applied it to the a relatively high expected benefit but at the expense of an in- Brazilian hydroelectric system. Karamouz et al. ͑1992͒ extended creased variability or risk on outcomes. Therefore, the expected implicit stochastic optimization to a three-step cyclic procedure value and variability of the objective function are to be carefully including DP, regression and simulation to improve initial oper- considered in the optimization formulation. Since the stochastic ating rules of a multiple reservoir system. Niedda and Sechi processes of inflow are also reflected in the objective function, the ͑1996͒ developed a mixed optimization procedure based on net- Markov chain approximation model is applied to determine the work linear programming and subgradient method to deal with expected value of the objective function. The variability of the large design problems for water resources systems. However, objective function due to random inflows introduces uncertainty most of these studies emphasized deterministic approaches to in the system output. For a single reservoir subsystem, the objec- evaluate system performance rather than to determine an explicit tive function of expected utility function u in terms of net mon- ͑ ͒ operation stochastic policy for the system. To overcome the high etary benefit b rt can be written as dimensionality of multistorage operation problems, the transbasin m diversion system is decomposed into three sequentially linked Max Eͭu͚ͫ b͑r ͒ͬͮ ͑1͒ single reservoir subsystems, namely Kok-Ing, Ing-Nan, and t t=1 Sirikit ͑Fig. 2͒, with each subsystem then individually optimized sequentially from upstream to downstream. The total net benefit is where rt =a decision variable, i.e., the release of a storage dam or ͑ ͒ ͑ ͒ the sum of the net benefits of the three subsystems. Although the diverted flow of a diversion dam in month t mcm ; b rt =net decomposition method does not fully guarantee the overall benefit b as a function of rt due to implementation and operation optimal result of the entire system, it provides an approximate of the transbasin diversion system ͑in millions of U.S. dollars͒; ͕ ͓͚m ͑ ͔͖͒ solution for practical purposes. E u t=1b rt =expected value of the utility function ͓͚m ͑ ͔͒ The purpose of this study is to develop an optimization proce- u t=1b rt ; and m=number of months for the SDP to converge dure called SDPR that involves the use of DP, SDP, simulation, to the annual optimal operation policy. No discounting of the and trial and error adjustment of risk coefficient; and to apply future benefits is considered in this study. * ͑ The optimal regulated release rt from a single storage dam or inflow uncertainty or risk is not considered. The optimization diverted flow of a diversion dam͒ is related to the followingj model is simply SDP. In the case of risk aversion ͑␾Ͼ0͒,an parameters as follows: increase in ␾ implies a higher risk aversion or a more conserva- tive system operation, e.g., u will increase asymptotically from 0 * ͑ ͒͑͒ rt = f st,it−1 2 to unity. Conversely, when ␾Ͻ0, a more negative ␾ means more ͑ ͒ risk or a less conservative system operation; i.e., u decreases where st =storage at the beginning of month t mcm , and, below zero. In the present study, ␾ is adjusted by trial and error it−1 =sum of inflow volumes to storage in previous month t−1 ͑mcm͒. from −5 to +5 taking one value at a time in each optimization to For a single reservoir, the continuity equation of reservoir determine the maximum system net benefit. ͑ ͒ ͑ ͒ storage, inflow and release can be written as The subsystem net benefit b rt in Eq. 4 can be determined by the following equation: s = s + i − r ͑3͒ t+1 t t t ͑ ͒ ␤ ͑ ͒ ͑ ͒ b rt = n rt − x/12 5 In Eq. ͑3͒ the reservoir evaporation and losses are assumed to be ␤ ͑ ͒ ͑␤ ͒ small compared to inflow and hence neglected. In a particular where n rt =sum of benefits due to irrigation i and hydro- ͑␤ ͒ calendar month of a year, the reservoir inflow for that month in power generation h minus construction, operation, and main- ͑ ͒ different years is considered to be a random variable due its sto- tenance cost c , and x=average annual uncertainty cost due to ͑ ͒ ␤ ͑ ͒ chastic nature. The reservoir inflow is assumed to be normally inflow uncertainty. In Eq. 5 , the total benefit n rt is given by distributed with respect to its expected value, which is equal to its ␤ ͑ ͒ ␤ ͑ ͒ ␤ ͑ ͒ ͑ ͒͑͒ n rt = i rt + h rt,stav − c rt 6 long-term average. Hence, the linear combination of reservoir in- flows, i.e., sum of inflows is also assumed normally distributed. where stav =average storage in month t, other parameters are as This assumption has been used by previous investigators such as previously defined. The irrigation benefit and hydropower benefit ͑ Loaiciga and Marino ͑1986͒. The assumption of log-normal dis- are calculated from the given benefit functions Boonyasirikul ͒ tribution for inflow could be another alternative; however it is not 1998 . The subsystem irrigation benefit is a function of diversion tried in this study. flow for Kok-Ing and Ing-Nan subsystems and a function of irri- Due to risk aversion nature in the decision making process, a gation release of the Sirikit Reservoir. The hydropower benefit is negative exponential utility function with a constant risk coeffi- a function of the Sirikit Reservoir release and average storage or cient which has a concave property is considered. A negative head. The construction cost cc of the hydraulic diversion struc- exponential utility function is a subset of a family of hyperbolic tures is a function of the design diversion capacity, whereas the absolute risk aversion linear risk tolerance utility functions with a operation and maintenance cost, com is a function of the diversion constant risk aversion coefficient. It is widely accepted for mod- discharge of each diversion dam or of the Sirikit Reservoir eling risk aversion. Similar to that proposed by Loaiciga and release. Marino ͑1986͒, the following negative exponential utility function u is introduced: Inflow Uncertainty and its Cost m m u͚ͫ b͑r ͒ͬ =1−expͭ− ⌽͚ͫ b͑r ͒ͬͮ ͑4͒ t t From the linear assumptions and the derivation of Loaiciga and t=1 t=1 Marino ͑1986͒, the annual average inflow uncertainty cost x in where ␾=given risk aversion coefficient reflecting preference in this study is expressed as follows: decision making under hydrologic uncertainty in the optimization, x͑␾,⌳,⍀,R ͒ = ␾͓ 1 ͑RЈ⌳R +2RЈ⍀͔͒0.5 ͑7͒ and the other parameters are already defined. In the case of risk o 2 o o o ͑␾Ͼ ͒ ␾ ͑ ͒ aversion 0 , u value increases asymptotically to 1 when where Ro =column matrix ro1 ,ro2 ,...,ro12 of monthly regulated ͑ ͒ ␾ Ј increases. In the case of risk challenge risk prone , when be- releases according to the derived policy; Ro =row matrix which is ⌳ comes more and more negative, the value of u changes much the transpose of Ro; =serial covariance matrix of observed ͑ ͒ ⍀ more rapidly toward a large negative value. This is desirable for monthly inflow matrix Io io1 ,io2 ,...,io12 of a year; and =the water resource problems in which risk of a water shortage should covariance matrix of observed inflow matrix Io and its annual be avoided and water resources availability should be assured. random component. By writing an expression similar to the one ͑ ͒ ␴ Compared to other utility functions, such as the S-shaped utility derived by Loaiciga and Marino 1986 , the component tj at the function, the negative exponential utility function emphasizes tth row and jth column of the serial covariance matrix ⌳ of more of risk averse nature due to its concavity property. Therefore observed inflow matrix Io is given by it is more suitable in this study in which water shortage is unac- n ceptable. The effect of risk aversion and risk challenge is more 1 ␴ = ͚ͫ ik ik − ni¯ ¯i ͬv2 for t and j =1,2, ...,12 pronounced in the months of highly fluctuating inflows such as tj ot oj ot oj n −1 k=1 May, June, and July in Thailand. ͑ ͒ SDPR does not explicitly account for risk but indirectly ac- 8 counts for by trial and error adjustment of a risk coefficient. The Similarly, the component zt at the tth row of the random covari- risk coefficient ␾ is introduced as a given parameter rather than a ance matrix ⍀ between the mean inflow and its random compo- decision variable in the optimization. The maximum utility func- nent is given by tion u can be achieved when ␾͓͚m b͑r ͔͒ in Eq. ͑4͒ is maximized t=1 t n for a specific value of ␾. The optimization is performed for vari- 1 ͫ k k ¯ ¯ͬ ͑ ͒ ous values of coefficient ␾ reflecting aversion to risk in relation to zt = ͚ ioth − nioth vg for t = 1,2,3, ... ,12 9 n −1 k=1 the hydrologic inflow uncertainty. The risk aversion coefficient is categorized into three levels reflecting: No risk, risk aversion, and where n=number of years ͑50 years͒ for observed data; ␾ ͑ ͒ k risk challenge. When =0, Eq. 4 yields u=0, which means the iot =observed inflow in month t and year k in which the monthly ¯ k inflow calculated from the cumulative transition probability of the average value is iot ;h =annual average random number of ob- served monthly inflows in year k in which the long-term average 100-year synthetic inflows agrees reasonably well with that computed from the gamma distribution. value for n years is ¯h; superscript k=year index in a period of observed data of n years. The term g in Eqs. ͑8͒ and ͑9͒ is the slope of the damage cost curve due to water shortage or overspill- age with respect to release ͑or diversion flow͒. The term v is the Representative Inflow change of g with respect to the release. In this study due to lim- ited data of damage cost, average values of g and v are assumed. In this study the Ing diversion dam receives natural inflow from The SDPR procedure of the transbasin diversion system maxi- upstream and the diversion flow from the Kok diversion dam. The mizes the economic net benefit. By separating the terms ␾ and Kok river basin and Ing river basin are adjacent and hence their ͚m ͑ ͒ ͑ ͒ ͚m ͑ ͒ ␾ inflows are spatially and temporally related. To incorporate the t=1b rt in Eq. 4 , the term t=1b rt is optimized for each value. Instead of maximizing the utility function u in Eq. ͑1͒, the bivariate temporal and spatial relationships of inflows in full, the optimization, the following objective function is considered computational dimensions in SDP will be excessively large. Therefore a representative inflow is used to represent the Kok m x inflow and Ing inflow. In this study, the representative inflow itt is Max E͚ͭ ͫ␤ ͑r ͒ − ͬͮ ͑10͒ n t taken as the sum of the inflows ikt to the Kok diversion dam and t=1 12 iit to the Ing diversion dam, that is where E=expectation, and other parameters are as previously ͑ ͒ defined. itt = ikt + iit 11 The criterion of accepting the representative inflow is that the minimum correlation of the representative inflow itt to either ikt Inflow Classification and Transition Probabilities or iit is higher than the maximum correlation between ikt and iit. That is Five series of 100-year monthly inflows of Kok, Ing, and Sirikit dams were generated based on the statistical characteristics of ͓ ͑ ͒ ͑ ͔͒ Ͼ ͓ ͑ ͔͒ Min corr itt,ikt ,corri itt,iit Max corr ikt,iit monthly inflow data in the past 50 years and by using HEC-4 model developed Hydrologic Engineering Center ͑1989͒. They The actual correlations were checked and found to satisfy ͑ ͒ were used to derive the inflow transition probabilities. For each the above-mentioned criterion Boonyasirikul 1998 . The tran- subsystem, the diversion flow is restricted by the downstream sition probability of the representative inflow is used in the water requirement of the diversion dam. The monthly inflow to optimization. diversion storage of each subsystem for the entire 100-year period was discretized into five classes based on inflow mean and stan- dard deviation. The discrete classes are concentrated around the Diversion Targets and Sequential Optimization mean with class size related to the dispersion of the inflow data. The range of these five inflow classes are categorized as follows: As shown in the flow chart in Fig. 3, the optimization procedure is processed sequentially from the most upstream subsystem, i.e., Inflow classes Range of inflow the Kok-Ing diversion subsystem to the Ing-Nan diversion sub- Very high InflowϾmean+0.6 s system and then to the most downstream subsystem that is Sirikit Reservoir. The initial estimated diversion discharges of the Kok- High mean+0.6 sജinflowϾmean+0.2 s Ing and Ing-Nan diversion subsystems and the release of the Siri- Medium mean+0.2 sജinflowϾmean−0.2 s kit Reservoir are determined from deterministic DP of the entire Low mean−0.2 sജinflowϾmean−0.6 s system in which, diversion targets are not required in DP. From Ͼ Very low mean−0.6 s inflow DP optimization, the diversion targets are estimated and used in sequential SDP to determine the optimal operation policy of each where mean=mean monthly inflow and s=standard deviation. subsystem from upstream to downstream. The individual SDP The inflow transition probability was estimated from the optimization of Kok-Ing and Ing-Nan subsystems is subjected to frequencies of occurrence of inflows in each discrete class of the the diversion target of the subsystem. The most downstream sub- current month and the previous month. However, the relative system, the Sirikit Reservoir which collects upstream diversion values of the transition probabilities may change with the number flow and natural inflow has its release target determined from its of generated data considered. To eliminate possible errors and to existing actual irrigation release. The actual release data of the mathematically describe the distribution, various probability Sirikit reservoir over the past decade until now shows that the distribution functions were considered such as normal, log annual release trend approaches asymptotically a maximum value normal, and gamma distribution function for representation of the due to the full utilization of the maximum storage capacity of the frequency of occurrence of inflows to the transbasin diversion reservoir. When the transbasin diversion system is completed, system. The two-parameter gamma distribution is selected as it Sirikit reservoir is expected to yield a monthly release not less gives a better fit compared to the normal or log normal than its existing release. Therefore the existing monthly release of distributions. The scale parameter and the shape parameter of the Sirikit reservoir is considered to be the minimum release require- gamma distribution function were estimated based on the ment or the release target trnt that should be satisfied. Any release maximum likelihood method using the current month inflow data from the Sirikit reservoir above the release target is considered to of each class of inflow of previous month. By using the derived be an additional benefit due to overachievement and any below gamma distribution function, the cumulative transition probability the release target is considered to be a loss due to underachieve- of the current month inflow was computed for each class of inflow ment. Monthly release targets are used in the optimization and of previous month. The transition probability of the current month simulation of the whole system for future operation. The optimization and simulation is done for various values of ␾ between +5 and −5. The relationship of the maximum net ben- efit, the irrigation reliability and the values of ␾ can be obtained. The value of ␾ that gives the highest maximum net benefit can be identified.

Application for Kok-Ing-Nan Transbasin Diversion System

Based on the SDPR formulation, the objective function of each subsystem in the entire system can be expressed as follows: For the Kok-Ing Diversion Sub-system, the objective function is ͓ ͑ ͔͒ Max E bt IKt,DKIt,SKt The SDP backward recursive equation is f ͑IT ,SK ͒ = Max͕⌺ ͓⌺ ͓b ͑IK ,DKI ,SK ͔͒P ͑IK ͉IT ͒ t t−1 t ITt IKt t t t t K t t ͑ ͔͒ ͑ ͉ ͖͒ ͑ ͒ + ft+1 ITt,SKt+1 PT ITt ITt−1 14 where ͓ ␤ ͑ ͒ ͑ ͔͒ bt = pirr wo irrk DKIt,tdkit − wulirrk DKIt,tdkit ͑␾ ⌳ ⍀ ͒ ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ − pxx , , ,DKIt − pc com DKIt + cc ddki 15 For the Ing-Nan diversion subsystem, the objective function is ͓ ͑ ͔͒ Max E bt IIt,DKIt,DINt,SIt The SDP backward recursive equation is f ͑IT ,SK ,SI ͒ = Max͕⌺ ͓⌺ ͓b ͑II ,DKI ,DIN ,SI ͔͒P ͑II ͉IT ͒ t t−1 t t ITt IIt t t t t t I t t Fig. 3. Computational flow chart of SDPR for Kok-Ing-Nan + f ͑IT ,SK ,SI ͔͒P ͑IT ͉IT ͖͒ ͑16͒ transbasin diversion system t+1 t t+1 t+1 T t t−1 where ͓ ␤ ͑ ͒ ͑ ͔͒ bt = pirr wo irri DINt,tdint − wulirri DINt,tdint The monthly diversion targets for irrigation of the Kok-Ing ͑␾ ⌳ ⍀ ͒ ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ and Ing-Nan diversion subsystems are specified in proportion to − pxx , , ,DINt − pc com DINt + cc ddin 17 the release target of the Sirikit reservoir multiplied by their inflow For the Sirikit Reservoir subsystem, the objective function is ratio ͓ ͑ ͔͒ Max E bt INt,DINt,RNt,SNt ik t ͑ ͒ The SDP backward recursive equation is tdkit = trnt 12 int + dint ͑ ͒ ft INt−1,ITt−1,DINt,SNt = Max͕͚ ͓͚ ͓b ͑IN ,DIN ,RN ,SN ͒ ii + dki INt ITt t t t t t t t ͑ ͒ tdint = trnt 13 ͑ ͔͒ int + dint + ft+1 INt,ITt,DINt+1,SNt+1 ϫ ͑ ͉ ͔͒ ͑ ͉ ͖͒ ͑ ͒ where trnt =specified monthly Sirikit Reservoir release target in PT ITt ITt−1 PN INt INt−1 18 ͑ ͒ month t mcm ; tdkit, tdint =monthly diversion targets for Kok- ͑ ͒ where Ing and Ing-Nan diversion subsystems in month t mcm ; dkit, ͑ ͒ ͓ ␤ ͑ ͒ ͑ ͒ ␤ ͑ ͒ dint =Kok-Ing and Ing-Nan diversion flows in month t mcm bt = pirr wo irr RNt,trnt − wulirr RNt,trnt + phyp hyp RNt,SNt obtained by DP optimization in the first trial and later by simula- ͑␾ ⌳ ⍀ ͒ ͑ ͔͒ ͑ ͒ − pxx , , ,RNt − pccom RNt 19 tion after SDP optimization process; and ikt, iit, int =inflows to ͑ ͒ ͑ ͒ ͑ ͒ Kok, Ing, and Nan Reservoirs in month t, respectively mcm . In Eqs. 14 – 19 , IKt,DKIt,SKt =matrices of monthly inflow, di- Based on the optimal operation policy, the actual diversion version flow, and beginning storage of Kok diversion dam of discharges of the two diversion subsystems and the release of the month t, respectively. For example, the inflow matrix IKt has five Sirikit Reservoir are determined by simulation using generated components according to the five inflow classes. IIt,DINt,SIt monthly inflow time series. The actual diversion discharges and =matrices of monthly inflow, diversion flow, and beginning stor- the actual storage releases are used in recalculating the diversion age of Ing diversion dam, respectively; INt,RNt,SNt =matrices of targets according to Eqs. ͑12͒ and ͑13͒. The revised diversion monthly inflow, regulated release, and beginning storage of the targets are used in SDP optimization to find the revised optimal Sirikit Reservoir, respectively; ITt =matrices of representative in- operation policy. The procedure is repeated iteratively until the flow of Kok and Ing diversion dams. DKIt, DINt, and RNt are the optimal policy is unchanged. decision variables; SK, SI, and SN are the state variables. ment and loss due to underachievement of the Sirikit Reservoir are given respectively as follows: ␤ ͑ ͒ Benefit due to overachievement = wo irr RNt,trnt Ͼ ͑ ͒ for RNt trnt 20

͑ ͒ Loss due to underachievement = wulirr RNt,trnt Ͻ ͑ ͒ for RNt trnt 21 ␤ ͑ ͒ ͑ ͒ When RNt =trnt, irr =lirr =0. In Eqs. 20 and 21 , wo and wu =weights for over- and underachievement of irrigation target, respectively, wo and wu are assigned to be 0.1 and 0.9, respec- tively, for the base case. The benefit and losses due to over- and underachievement of the upstream diversion subsystems, namely the Kok-Ing diversion subsystem and the Ing-Nan diversion sub- system, are determined following a procedure similar to that of the above-mentioned calculation procedures. ␤ Fig. 4. Irrigation benefit function irr and loss function lirr versus the In Eq. ͑19͒, a fixed tariff rate of hydropower Sirikit Reservoir release ␤ hyp =0.0375 per kW h is used according to the EGAT. This fixed tariff rate is the official rate according to the power purchase agreement between EGAT and its customer. Other parameters are as previously defined. The cost com depends on actual discharges ͑ ͉ ͒ ͑ ͉ ͒ϭ PK IKt ITt and PI IIt ITt spatial transitional probabilities of whereas the cost cc depends on the design discharge. Boonyasir- ͑ ͒ Kok inflow matrix and Ing inflow matrix, respectively, ikul 1998 provided unit costs for com and cc. ͑ ͉ ͒ ͑ ͉ ͒ϭ ͑ PT ITt ITt−1 and PN INt INt−1 temporal transitional probabili- The constraints of state transformation function storage ties of the representative inflow IT matrix and of the natural in- balance equation͒, the flow capacity and the storage capacity are flow matrix IN to the Sirikit Reservoir. as follows: ͑ ͒ ͑ ͒ ddki ddin drn In Eqs. 14 – 19 , , , and =design hydraulic ca- ͑ ͒ pacities of Kok-Ing diversion channel, Ing-Nan diversion channel, skt+1 = skt + ikt − rkt − dkit 22 and downstream channel of the Sirikit Reservoir, respectively; si = si + ii + dki − ri − din ͑23͒ pirr, px, phyp, and pc =priority weights of irrigation benefit, uncer- t+1 t t t t t tainty cost, hydropower benefit and construction and maintenance ͑ ͒ cost, respectively. The priorities are assigned as a multiplication snt+1 = snt + int + dint − rnt 24 ͑ ͒ factor to each component. The first priority pirr is assigned to the ഛ ഛ ഛ ͑ ͒ increase in irrigation benefit above the irrigation target, which is dkit ddki, dint ddin, rnt drn 25 the existing release target ͑without project͒. This is the main ob- ഛ ഛ ഛ ഛ ഛ ഛ jective of the Thai government, i.e., to provide more irrigation skmin skt skmax, simin sit simax, snmin snt snmax benefit to farmers even if its economic return is less than that of ͑ ͒ ͑ ͒ 26 the hydropower. The second priority px is assigned to the uncer- ͑ ͒ ϭ tainty cost. The third priority phyp is assigned to hydropower where sk, ik, and rk storage, inflow, and release of the Kok benefit increase which depends on the release for irrigation of the Reservoir; si, ii, and riϭstorage, inflow, and release of the Ing ͑ ͒ ϭ Sirikit Reservoir. The fourth priority pc is assigned to the con- Reservoir; and sn, in, and rn storage, inflow, and release of the struction, operation, and maintenance cost because the project Sirikit Reservoir. Other parameters are as previously defined. definitely has to be built according to the given mandate of the The representative inflow is used to represent the Kok inflow policy makers to satisfy the need of farmers in the irrigation and Ing inflow so that a univariate transitional probability of the project area for their irrigation benefit and their well being. The representative inflow can be applied in SDP. The lag-1 Markov values of the first, second, third, and fourth priorities are specified process in SDP is generally assumed in many previous studies such that they are significantly different, i.e., 1,000, 100, 10, and such as Stedinger et al. ͑1984͒ and hence in this study. According ͑ ͒ ͑ ͒ ␤ ␤ ␤ 1, respectively. In Eqs. 14 – 19 , irrk, irri, and irr =irrigation to the Bayes theorem as explained by Karamouz and Vasiliadis benefit functions of Kok-Ing, Ing-Nan, and Sirikit subsystems, ͑1992͒, the bivariate transitional probability of the Kok inflow ͑ ͉ ͒ respectively; lirrk, lirri, and lirr =loss functions of the same sub- PK IKt IKt−1 ,IIt−1 can be approximated by the spatial ␤ ͑ ͉ ͒ systems. The relationships of irr and lirr with the Sirikit Reservoir univariate transitional probability of Kok inflow PK IKt ITt release are given by straight lines as shown in Fig. 4. and the temporal univariate transition probability of the represen- ͑ ͉ ͒ For the irrigation objective, the release target may not always tative inflow PT ITt ITt−1 . Similarly for the Ing inflow, the bi- ͑ ͉ ͒ be met due to the hydrologic uncertainty of inflows. Such a con- variate transition probability PI IIt IKt−1 ,IIt−1 can be approxi- dition can cause an unrealistic solution during optimization. In mated by the univariate transition probability of Ing inflow ͑ ͉ ͒ order to provide flexibility in optimization, SDPR is applied con- PI IIt ITt and the univariate transition probability of the repre- ͑ ͉ ͒ ͑ ͒ sidering over- and underachievement of the irrigation release tar- sentative inflow PT ITt ITt−1 . For example, in Eq. 14 , ͚ ͓͚ ͓ ͑ ͔͒ ͑ ͉ ͒ ͑ ͔͒ get. For the Sirikit Reservoir, overachievement of the target oc- IT IK b IKt ,DKIt ,SKt PK IKt ITt + ft+1 ITt ,SKt+1 is multi- t t ͑ ͉ ͒ curs when the Sirikit Reservoir release RNt is larger than the plied by PT ITt ITt−1 . Note that the Kok storage matrix SKt+1 in ͑ ͒ release target trnt. On the other hand, underachievement of the Eq. 14 is not random but it is regulated. Also SKt+1 is already target occurs when the release from the Sirikit Reservoir is less optimized in month t+1, therefore it is not multiplied by ͑ ͉ ͒ than the release target. The irrigation benefit due to overachieve- PK IKt ITt . Similarly, the same explanation can be given to Fig. 6. Monthly releases of the Sirikit Reservoir obtained from simulation based on SDPR policy for various ␾ values

t. Additionally, SDP optimization ͑␾=0͒ was performed to deter- mine the operation policy for the system in comparison with the SDPR policy. Based on the operation policies obtained from SDPR and SDP, the Kok-Ing-Nan transbasin diversion subsystem was simulated by using five series of monthly synthetic inflows, each for a 100-year period. The monthly Sirikit Reservoir release Fig. 5. Optimum monthly release policy of the Sirikit Reservoir obtained from the simulation for various ␾ values is compared ␾ obtained from SDPR at =1.5 for medium class of total Kok and with the release target for irrigation as shown in Fig. 6. In the Ing inflow, Ing-Nan diversion flow and Nan inflow simulation, the irrigation reliability, the irrigation benefit, the hy- dropower benefit, the net benefit, the construction, operation, and maintenance cost, and the uncertainty cost of the transbasin di- ͑ ͒ ͑ ͒ Eq. 16 .InEq. 18 , the Ing-Nan diversion flow matrix DINt is version system were computed to measure the performance of the independent of the Sirikit Reservoir natural inflow matrix INt. SDPR in deriving the optimal operation policy. The irrigation Therefore no spatial transition probability is involved. The total reliability is determined by comparing the percentage of time that inflow to the Sirikit Reservoir is the sum of the Ing-Nan diversion the Sirikit Reservoir release can meet the irrigation release target flow and the natural inflow of the Sirikit reservoir. Therefore the within the simulation period. ͑ ͉ ͒ ͑ ͉ ͒ temporal transition probabilities PR IRt IRt−1 and PN INt INt−1 Fig. 7 shows the simulation results on the system net benefit, are multiplied to the recursive function of the Sirikit Reservoir at the hydropower and irrigation benefits, the total cost ͑i.e., sum of time t. construction cost, operation, and maintenance cost and uncer-

Results

The Kok-Ing-Nan transbasin diversion system is an extension of the existing Sirikit Reservoir system. The implementation of the Kok-Ing-Nan transbasin diversion system will provide better uti- lization of the Sirikit Reservoir, downstream irrigation supply and hydropower generation. The additional water gained from the transbasin diversion from the Kok and Ing rivers will consider- ably increase downstream irrigation water supply, hydropower generation at the Sirikit dam and irrigation reliability. The optimal operation policy of the Kok-Ing-Nan transbasin diversion was obtained by using SDPR after 60 months of iteration to achieve convergence. The optimal values of system net benefit, monthly values of ͑ ͒ the Kok-Ing diversion flow DKIt ITt−1 ,SKt , the Ing-Nan diver- ͑ ͒ sion flow DINt ITt−1 ,SKt ,SIt , and the Sirikit Reservoir release ͑ ͒ RNt INt−1 ,ITt−1 ,DINt ,SNt are obtained. The operation policy is expressed by contour charts of optimal diversion or release deci- sions as a function of time and the associated state variables. Fig. 5 shows the SDPR optimal policy diagram at ␾=1.5 for the Siri- kit Reservoir for the medium class of the total Kok and Ing inflow and the Sirikit natural inflow in the previous month t−1, and the Fig. 7. Reliability, benefits, and total cost of proposed transbasin medium class of the Ing-Nan diversion flow in the current month diversion system under SDPR policy for the base case tainty cost͒, and the irrigation reliability over a wide range of ␾ Table 1. Optimal Sirikit Reservoir Release, Reliability, Various Benefits, values which are varied in increments of 0.5 from −5 to 5 in the Annual Net Benefit, and Associated Costs of Transbasin Diversion base case. The effective range of ␾ is defined as the range in System under Existing, SDP, and SDPR Policies in Base Case which SDPR gives higher annual net benefits than SDP. The Without With range of −0.5ഛ␾ഛ1.5 is found to be the effective range in this transbasin transbasin case. The system net benefit and irrigation reliability increase diversion diversion with ␾ and both reach their maximum values when ␾=1.5. The ͑existing system͒ ͑proposed system͒ irrigation benefit has a maximum at ␾=−0.4. Outside the effec- ͑ ␾Ͻ ␾Ͼ ͒ EGAT SDP SDP SDPR tive range when −0.5 or when 1.5 , the net benefit, irri- policy policy policy policy gation and hydropower benefits, and irrigation reliability decrease System output ͑existing͒ ͑␾=0͒ ͑␾=0͒ ͑␾=1.5͒ continuously as ␾ deviates from the effective range in both direc- ͑ ͒ tions. The total cost is minimum at ␾=1.5 and slowly increases as Release mcm/year 5,481 5,481 8,187 8,231 ͑ ͒ ␾ deviates from the effective range in both directions. The maxi- Reliability % 53.6 54.50 77.92 85.25 mum irrigation benefit which is equal to $160ϫ106 at ␾=−0.4 is Hydropower 1,224 1,287 2,116 2,358 only slightly reduced to $157.43ϫ106 when ␾=1.5. On another Production ͑ ͒ hand, when ␾ is increased from −0.4 to 1.5, the system net benefit GW h/year is increased from $342.55ϫ106 to a maximum of $390.25ϫ106. Irrigation benefit 103.75 103.77 156.83 157.43 ͑$ϫ106͒ The hydropower benefit increases from $235.61ϫ106 to a maxi- mum of $271.17ϫ106 and the irrigation reliability from 76% to a Increase in irrigation 0 0.02 53.08 53.68 benefita ͑$ϫ106͒ maximum of 85.25%, respectively. The irrigation benefit is Hydropower benefit 140.76 148.00 243.34 271.17 $157.43ϫ106 at ␾=1.5, which is only slightly less than its maxi- ͑$ϫ106͒ mum of $160ϫ106 at ␾=−0.4. Considering the maximum system ␾ Increase in 0 7.24 102.58 130.41 net benefit, the value of which yields the tradeoff between net hydropower benefits and reliability is selected at 1.5. The maximum system benefita ͑$ϫ106͒ net benefit and its corresponding irrigation and hydropower ben- Increase in 0 0 35.75 35.75 efits as well as costs of construction, operation, maintenance, and construction, uncertainty are shown in Table 1 for the cases with and without operation, and the transbasin diversion system when ␾=0 ͑SDP͒ and 1.5 maintenance costsa ͑SDPR͒. ͑$ϫ106͒ For the purpose of comparison of system performance, the Uncertainty cost 3.8 3.42 3.38 2.6 operation of the existing Sirikit Reservoir without the proposed ͑$ϫ106͒ transbasin diversion is also studied under the existing operating Net benefit 240.71 248.35 361.04 390.25 rule of the EGAT. From the results of the simulation, the optimal ͑$ϫ106͒ values of irrigation benefit, hydropower benefit, and net benefit Increase in net benefita 0 7.26 119.91 148.34 are determined and shown in Table 1. The annual Sirikit Reser- ͑$ϫ106͒ voir release based on EGAT’s existing operation rule without the aWith respect to existing system under EGAT policy. diversion system is 5,481 mcm. With the transbasin diversion, the release increases to 8,187 mcm for the SDP policy ͑␾=0͒ and to 8,231 mcm for the SDPR policy ͑␾=1.5͒, which is approximately 6 50% more than the existing releases from the Sirikit Reservoir $390.25ϫ10 or 62.12% increase for SDPR policy. without the transbasin diversion. The transbasin diversion system can increase the annual hydropower production of the Sirikit dam from the existing 1,224 to 2,116 GW h under the SDP policy and Discussions to 2,358 GW h or 93% increase equivalent to $130.41ϫ106 under the SDPR policy. The hydropower benefit is increased from In the Kok-Ing-Nan transbasin system, the net benefit is mainly the existing $140.76ϫ106 $243.34ϫ106 and $271.17ϫ106 for obtained from hydropower and irrigation subject to irrigation SDP and SDPR policies, respectively. The annual irrigation diversion/release targets. The negative exponential utility function benefit is increased from the existing $103.75ϫ106 to with a constant risk coefficient which has a risk aversion nature is $156.83ϫ106 for SDP policy and $157.43ϫ106 or 51.73% in- used in the optimization based on monetary net benefits. The crease for SDPR policy. The irrigation reliability is increased stochastic nature of inflow and its uncertainty are considered in from the existing 53.6 to 77.92% for SDP and to 85.25% for deriving various probabilities of system operation. To determine SDPR ͑an increase of 59.04%͒. The dry season irrigation down- the level of performance of the Kok-Ing-Nan transbasin system, stream of the Sirikit Reservoir is important and its benefit is con- sensitivity analysis has been carried out to determine the effects siderably higher than in the wet season. Of the optimal annual of the following parameters on the performance of the transbasin release of 8,231 mcm from the Sirikit Reservoir, 3,849 mcm is diversion system under the SDPR policy: ͑1͒ The priority ranks of used in wet season irrigation and 4,382 mcm is used in dry season various objective components in the objective function; ͑2͒ the irrigation. The annual uncertainty cost is reduced from the exist- weighting factor for achievement of irrigation target; and ͑3͒ the ing $3.8ϫ106 to $3.38ϫ106 or 11.05% decrease for SDP policy effective storages of Kok and Ing diversion dams. It was found and to $2.6ϫ106 or 33.33% decrease for SDPR policy. The an- that the change in the weights of over- and underachievement of nual cost of construction, maintenance and operation of the trans- the irrigation target has the most significant effects on the irriga- basin diversion system is $35.75ϫ106. The annual net benefit of tion reliability and the system net benefit. The changes in the the transbasin diversion is increased from $240.71ϫ106 to storages of Kok and Ing Reservoirs have a comparatively less $361.04ϫ106 or 50% increase for SDP policy and to significant effect while the changes in the priority ranks in the Table 2. Parameters considered and results of sensitivity analysis under SDPR policy

Priority ranking Results coefficients for Storages of irrigation, uncertainty Weights for over- and Kok and Ing Irrigation Annual net cost, hydropower, operation, underachieve-ment of reservoirs reliability benefit Case and maintenance cost irrigation target ͑mcm͒ ͑%͒ ͑$ϫ106͒ Base case 1,000;100;10;1 0.1;0.9 2,400; 85.25 390.25 3,500 Study Case A1 ͑priority rank͒ 1;1;1;1 0.1;0.9 2,400; 83.42 400.38 3,500 Study Case A2 ͑priority rank͒ 100;10;1,000;1 0.1;0.9 2,400; 79.50 399.43 3,500 Study Case A3 ͑priority rank͒ 100;1,000;10;1 0.1;0.9 2,400; 77.92 359.78 3,500 Study Case B1 ͑achievement͒ 1,000;100;10;1 0.5;0.5 2,400; 78.58 428.01 3,500 Study Case B2 ͑achievement͒ 1,000;100;10;1 0.9;0.1 2,400; 27.08 292.19 3,500 Study Case C1 ͑Kok and Ing storages͒ 1,000;100;10;1 0.1;0.9 1,200; 78.17 374.86 1,750 Study Case C2 ͑Kok and Ing storages͒ 1,000;100;10;1 0.1;0.9 4,800; 89.17 411.24 7,000

objective function have the least effect. Discussion of the sensi- process yields releases too large in the months of high irrigation tivity analysis as presented in Table 2 follows: target and too small in the months of low irrigation release target. ͑1͒ Priority ranks of various objective components in the ob- ͑3͒ Effective storages of Kok and Ing diversion dams: The jective function: These components are irrigation benefit, hydro- storage capacity of the existing Sirikit Reservoir, which is the power benefit, uncertainty cost, and cost of construction, opera- main regulating reservoir, is fixed due to limitation in the dam tion, and maintenance, respectively. In the base case, the first, height. The maximum design capacities of the proposed Kok and second, third, and fourth priorities are assigned to irrigation ben- Ing diversion storages in the base case were given by the Royal efit, uncertainty cost, hydropower benefit, construction, , opera- Irrigation Department, i.e., 2,400 and 3,500 mcm, respectively. tion, and maintenance cost, respectively. In the sensitivity analy- Two study cases were considered: Case C1—in which the effec- sis, three study cases are considered: Case A1—in which tive storages of Kok and Ing are equal to half of the base case and priorities of all components are equal; Case A2—in which the Case C2—in which the effective storages of both dams are twice first priority is for hydropower benefit, second priority for irriga- that of the base case. In Case C1, the effective range of ␾ is found tion benefit, third priority for uncertainty cost, and fourth priority to be from 0 to 1.0. The simulation results based on the SDPR for construction, operation, and maintenance cost; and, Case show that in case C1, the irrigation reliability and the annual net A3—in which the first priority is for uncertainty cost, second benefit are lower compared to the base case. The small effective priority for irrigation benefit, third priority for hydropower ben- storages of the Kok and Ing dams are not sufficient to provide efit, and fourth priority for construction, operation, and mainte- efficient operation of the diversion system. In Case C2, the effec- ␾ nance cost. In Case A1, the effective range of is found to be tive range of ␾ is found to be from 0 to 2.0. The irrigation reli- from −2.0 to 2.0. The irrigation reliability is lower but the annual ability and annual net benefit are higher than in the base case. The net benefit is higher compared to the base case. In Case A2 the SDPR is very effective in providing the optimal operation policy ␾ effective range of is the same as in Case A1. Compared to the for the transbasin diversion system. base case, the irrigation reliability is slightly lower but the net annual benefit is higher. In Case A3, the effective range of ␾ is found to be from 0 to 0.5, the maximum reliability and the annual benefit are lower than the base case. Conclusions ͑2͒ Weighting factor for achievement of irrigation release tar- get: In the base case, a weight wo of 0.1 is assigned to the over- The main contribution of this study is the development of an achievement of irrigation release target and a weight wu of 0.9 is optimization technique called SDPR that involves the use of DP, assigned to the underachievement of irrigation release target. Two SDP, simulation, and trial and error of risk coefficient of inflow study cases are considered: Case B1—which considers an equal uncertainty. The risk is included in term of uncertainty cost in the weight of 0.5 for target overachievement and target underachieve- economic objective function. ment and Case B2—which considers a weight of 0.9 for over- SDPR was applied to the Kok-Ing-Nan transbasin diversion achievement and of 0.1 for underachievement. In Case B1, the system in Thailand. The system receives inflows from three sepa- effective range of ␾ is found to be from 0 to 2.0. The simulation rated rivers: Kok, Ing, and Nan. Parts of the flows in Kok and Ing results show lower irrigation reliability compared to the base rivers are diverted to Nan river where the flows combine as inflow case, but a higher annual net benefit is obtained due to hydro- to the main storage reservoir, Sirikit. Satisfying the minimum power. In Case B2, the effective range of ␾ is found to be from downstream requirement of Kok and Ing rivers, it was found that −2.0 to 1.5. Both irrigation reliability and annual net benefit are about 60% of the annual flows of the Kok and Ing rivers can be very low compared to the base case. This is because the SDPR diverted to the Nan river which flow into the Sirikit reservoir. The ϭ transbasin diversion increases the annual inflow to the Sirikit ITt ,itt representative inflow matrix and its component reservoir from 5,481 to 8,231 mcm or 50% increase in inflow. value; ϭ SDPR determines the optimal system operation policy by maxi- lirr irrigation loss function due to mizing the system net benefit. In the objective function, different underachievement of irrigation release target of weights or priorities are assigned to irrigation benefit, hydropower Sirikit reservoir; ϭ benefit, and costs due to construction, operation, maintenance, lirrk ,lirri irrigation loss functions due to and uncertainty. underachievement of diversion targets of Kok Due to the large dimensionality problem, the multireservoir and Ing reservoirs, respectively; system is decomposed into three sequentially linked reservoir m ϭ number of months to converge in SDPR; subsystems in this study. The optimization was performed for n ϭ number of years; each subsystem sequentially from upstream to downstream. The P ϭ probability; coupling between the upstream subsystem and the downstream p ϭ priority; ϭ subsystem was specified by an irrigation diversion or release tar- Ro column matrix of monthly regulated release; Ј ϭ get. Sensitivity analysis was carried out to determine the signifi- Ro row matrix which is transpose of Ro; ϭ cance of priority ranking of each component in the objective RNt ,rnt matrix of Sirikit reservoir release in month t function. It was found that the irrigation targets have more effects and its component value; * ϭ on the irrigation reliability and the system net benefit than the rt ,rt diversion flow or release and its optimal priority ranking. Due to the individual optimization, the algorithm value in month t; ϭ developed for optimization may not fully guarantee rigorous over- SKt ,SIt ,SNt matrices of storages of Kok, Ing, and Sirikit all optimal result of the entire system but an approximate optimal reservoirs; ϭ solution for practical purposes. st storage at beginning of month t; ϭ In the trial and error procedure of SDPR, specific values of the skt ,sit ,snt component values of SKt,SIt, and SNt; risk coefficient ␾ were tried in the range from −5 to 5. The value tdin ϭ Ing-Nan diversion target; of ␾ of 1.5 was found to yield the overall optimal policy, i.e., the tdki ϭ Kok-Ing diversion target; ϭ tradeoff between net benefit and irrigation reliability. Compared trnt Sirikit reservoir release target in month t; to SDP optimization ͑␾=0͒, SDPR yielded a higher maximum u ϭ utility function; system net benefit by about 10%. Compared to the existing sys- v ϭ derivative of g; ͑ ͒ ϭ tem without transbasin diversion , the proposed transbasin diver- wo ,wu weights for over- and underachievement of sion increases the annual system net benefit by 62.12% or from irrigation release target; $240.71ϫ106 to $390.25ϫ106. The increase in the system net x ϭ annual average uncertainty cost; ϭ ⍀ benefit is attributed to a 93% increase in annual hydropower ben- zt component of covariance column matrix ; ϫ 6 ␤ ␤ ϭ efit or $130.41 10 , a 51.73% increase in annual irrigation h , i benefit function of hydropower and irrigation, benefit or $53.68ϫ106, the annual construction, operation, and respectively; ϫ 6 ␤ ϭ maintenance cost of $35.75 10 and a reduction of 31.5% in the irr irrigation benefit function due to annual uncertainty cost from $3.8ϫ106 to $2.6ϫ106. overachievement of irrigation release target of Sirikit reservoir; ␤ ␤ ϭ irrk , irri irrigation benefit functions due to overachievement of diversion targets of Kok Notation and Ing reservoirs, respectively; ␤ ϭ n sum of benefits; The following symbols are used in this paper: ⌳ϭserial covariance matrix of inflow; ϭ ␴ ϭ ⌳ b net benefit function; tj component at tth row and jth column in c ϭ construction, operation, and maintenance cost; matrix; d ϭ change rate of unit damage cost per unit ␾ϭrisk aversion coefficient; and inflow; ⍀ϭcovariance matrix between inflow and random ϭ DINt ,dint Ing-Nan diversion flow matrix in month t and component. its component value; ϭ DKIt ,dkit Kok-Ing diversion flow matrix in month t and its component value; References ddin ϭ design capacity of Ing-Nan diversion channel; ddki ϭ design capacity of Kok-Ing diversion channel; Boehle, W., Harboe, R., and Schultz, G. 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