LINEAR PROGRAMMING FOR FLOOD CONTROL IN THE IOWA AND DES MOINES RIVERS
By Jason T. Needham,1 David W. Watkins Jr.,2 Jay R. Lund,3 Associate Members, ASCE, and S. K. Nanda,4 Fellow, ASCE
ABSTRACT: This study addresses questions related to ¯ood-control operating procedures followed by the U.S. Army Corps of Engineers, Rock Island District. Application is presented of a mixed integer linear programming model for a reservoir system analysis of three U.S. Army Corps of Engineers' projects on the Iowa and Des Moines rivers. A strategy for evaluating the value of coordinated reservoir operations is developed. Results of this study suggest that operating Coralville Reservoir, on the Iowa River, for ¯ood control on the Mississippi River does not provide appreciable bene®ts and, therefore, an operation plan coordinating releases from Cor- alville Reservoir with the two reservoirs on the Des Moines River may be unnecessary. Damage-minimizing results were obtained by operating the three reservoirs independently for 8 of the 10 largest ¯ood events on record. Also, a review of the operating procedures for the ¯ood of 1993 illustrates how much damage could have been reduced if in¯ows could be predicted months in advance or if the existing operating rules were more averse to extreme ¯ood events.
INTRODUCTION logic Engineering Center (HEC) Prescriptive Reservoir Model, which uses a 1-month time step, limiting its capabilities for Record ¯oods in the past decade have caused enormous ec- assessing ¯ood-control operations because decisions need to onomic damage and human suffering. In particular, the Great be made on a daily or even hourly basis during ¯ood events. Midwest Flood of 1993 along the Upper Mississippi River and Optimization models for reservoir ¯ood-control operations its tributaries caused an estimated 48 fatalities and $15±$20 have not been applied as vigorously. One approach that has billion in economic damages, surpassing all ¯oods in the been applied is dynamic programming (Glanville 1976; Beard United States in modern times (U.S. Department of Commerce and Chang 1979). Dynamic programming is popular because 1994). With increasing environmental concern and decreasing it can directly accommodate the nonlinear and stochastic fea- public support for large-scale ¯ood-control structures, a grow- tures that characterize many water resources systems. How- ing number of engineers and hydrologists are concentrating on ever, discretization of state, input, and decision variables, es- developing computer models for optimizing the operation of pecially for multiple reservoirs, causes dimensionality existing systems rather than proposing/designing new ¯ood- problems. Wasimi and Kitanidis (1983) avoided dimensional- control projects. Better forecasting methods are being devel- ity problems through the application of linear quadratic Gauss- oped to provide the most accurate data possible for these mod- ian control for the optimization of real-time daily operation of els. Along these lines, this paper describes an application of a multireservoir system under ¯ood conditions. Their ap- deterministic optimization to assess ¯ood-control operations proach, however, is valid only under moderate ¯ood conditions for the Iowa/Des Moines River Reservoir System and to pro- when capacity constraints are not likely to become binding. vide insight for possible modi®cations to the current operating Windsor (1973) formulated a linear programming (LP) model plan [U.S. Army Corps of Engineers (USACE) 1999]. that includes storage and release capacity constraints, along Developing optimization models for analyzing operating with some theoretical discussion of how it could be used with policies of multiple reservoir systems has been a popular area the latest forecast information to adjust reservoir operations of research for >30 years. Yeh (1985) and Wurbs (1993) pre- during a ¯ood event. The model presented in this work is sented in-depth reviews of reservoir management and opera- similar to that of Windsor. tions models that contain extensive references in this area. La- badie (1997) presented a thorough discussion and formulation IOWA/DES MOINES RIVER RESERVOIR SYSTEM of reservoir optimization models and comments on reasons for the gap between theoretical developments and real-world im- The Iowa/Des Moines River Reservoir System consists of plementation. The USACE has applied optimization methods three reservoirs, one on the Iowa River main stem and two on in studies of reservoir system operations on both the Missouri the Des Moines River main stem, as shown in Fig. 1. The and Columbia rivers (USACE 1994, 1996). These studies fo- reservoirs are operated and maintained by the USACE, with cused on seasonal and long-term operations using the Hydro- the Rock Island District responsible for day-to-day decision making. Operators follow guidelines described in the reservoir 1Assoc. Engr., David Ford Consulting Engineers, Sacramento, CA; for- regulation manuals that have been prepared as part of the de- merly, Intern, Hydrologic Engrg. Ctr., Davis, CA 95616. E-mail: need- sign of the system (Master 1983, 1988, 1990). [email protected] Authorized purposes for these reservoirs include ¯ood con- 2Asst. Prof., Dept. of Civ. and Envir. Engrg., Michigan Technol. Univ., trol, low-¯ow augmentation, ®sh/wildlife, water supply, and Houghton, Mich. 49931; formerly, Hydr. Engr., Hydrologic Engrg. Ctr., recreation. In each case, access and facilities are provided for Davis, CA. E-mail: [email protected] 3 recreation, but water is not controlled for that purpose Prof., Dept. of Civ. and Envir. Engrg., Univ. of California at Davis, Davis, CA. E-mail: [email protected] (USACE 1992). Total capacities and average in¯ows for the 4Chf., Hydro. and Hydr., Rock Island Dist., U.S. Army Corps of Engrs. three reservoirs are shown in Table 1. Other pertinent char- E-mail: [email protected] acteristics of the Iowa and Des Moines rivers are shown in Note. Discussion open until November 1, 2000. To extend the closing Tables 2 and 3, respectively. date one month, a written request must be ®led with the ASCE Manager Table 2 illustrates that Coralville Reservoir can regulate no of Journals. The manuscript for this paper was submitted for review and more than 25% of the total average annual ¯ow entering the possible publication on July 26, 1999. This paper is part of the Journal of Water Resources Planning and Management, Vol. 126, No. 3, May/ Mississippi from the Iowa River. Because of this, one could June, 2000. ᭧ASCE, ISSN 0733-9496/00/0003-0118±0127/$8.00 ϩ $.50 expect that Coralville Reservoir's ¯ood-control effectiveness per page. Paper No. 21529. below the con¯uence of Cedar River and on the Mississippi
118 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 FIG. 1. Map of Iowa/Des Moines River Reservoir System
TABLE 1. Capacities of, and Average In¯ows to, Three Reservoirs Capacity (acre-ft/year) In¯ows Reservoir (acre-ft/year) Conservation Flood control Total Percenta (1) (2) (3) (4) (5) (6) Coralville (Iowa River) 1,271,800 25,900b 435,300 461,200 18 Saylorville (Des Moines River) 1,540,600 90,000 586,000 676,000 20 Red Rock (Des Moines River) 3,568,000 265,500b 1,494,900 1,760,400 62 aPercent of total federal project ¯ood storage in Des Moines/Iowa system. bVaries seasonally, value is minimum which corresponds to maximum ¯ood storage.
TABLE 2. Iowa River Characteristics Keosauqua are similar because no major tributaries join the Drainage Mean Des Moines downstream of Ottumwa. area in¯ow Coralville Reservoir was completed and placed in operation Location (sq mi) (cfs) during 1958 as a unit in the general ¯ood-control plan for the (1) (2) (3) Upper Mississippi River Basin. Under current operations, Cor- alville Reservoir is to be operated for ¯ood control at Lone Coralville Reservoir 3,115 1,760 Iowa River (con¯uence with Cedar River) 4,770 2,360 Tree and Wapello on the Iowa River and Burlington, Iowa, on Cedar River (con¯uence with Iowa River) 7,870 4,230 the Mississippi River (Master 1990). Presumably, when op- Iowa River (con¯uence with Mississippi River) 12,980 7,120 erated in conjunction with the reservoirs on the Des Moines Mississippi River (con¯uence with Iowa River) 89,000 49,000 River, the ¯ood peaks can be offset enough to cause a signif- icant difference in the water levels on the Mississippi River during ¯ooding. TABLE 3. Des Moines River Characteristics Saylorville Reservoir and Lake Red Rock projects also are Drainage Mean associated with the comprehensive ¯ood-control plan for the area in¯ow Upper Mississippi River Basin. Lake Red Rock was completed Location (sq mi) (cfs) in 1969, and Saylorville Dam was completed in 1975. Ac- (1) (2) (3) cording to the reservoir regulation manuals, Saylorville Res- Saylorville Reservoir 5,823 2,200 ervoir is operated not only to reduce ¯ood damage in the city Red Rock Reservoir 12,323 4,928 of Des Moines, but it is also operated in tandem with Red Des Moines River (con¯uence with Mississippi Rock Reservoir to reduce ¯ood damage at Ottumwa and Keo- River) 14,540 8,210 sauqua on the Des Moines River and at Quincy, Ill., on the Mississippi River (con¯uence with Des Moines Mississippi River (Master 1983, 1988). River) 119,000 64,520 MIXED-INTEGER LP MODEL River is limited. Conversely, as illustrated in Table 3, Saylor- A mixed-integer LP model, termed HEC-FCLP, has been ville and Red Rock reservoirs regulate more than half of the developed at HEC to assist with USACE's ¯ood management average ¯ow entering the Mississippi River from the Des studies. This model is based on work done by Ford (1978), Moines River. The main tributaries of the Des Moines River with mixed-integer variables added to the formulation to in- join the main stem upstream of, or at, Lake Red Rock. An corporate nonconvex hydraulic relationships (Watkins et al. important tributary is the Raccoon River, which converges in 1999). The model treats the ¯ood-operation problem as one of the southern part of the city of Des Moines and has a large ®nding a system-wide set of releases that minimize total sys- effect on the stage there. The hydrographs at Ottumwa and tem penalties for too much or too little release, storage, and
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 / 119 ¯ow. A simulation model embedded in the LP model uses Si, j,3 Յ YSi, jjmax,3 (7) given releases to compute storage and downstream ¯ows, ac- ʦ commodates reservoir continuity and linear channel routing Yi, j {0,1} (8) (e.g., Muskingum routing), and accounts for hydraulic limi- These constraints ensure that, for example, storage zones 1 and tations such as reservoir outlet capacities. 2 are ®lled before water is stored in zone 3. HEC-FCLP reads a description of the ¯ood control system The continuity constraint for each control point other than from an ASCII text ®le and generates a set of linear equations a reservoir takes the following general form: that constitute the LP model. Using IBM/OSL, a general-pur- pose, large-scale LP/MIP solver (Optimization 1995), HEC- i Ϫ FCLP calculates the optimal values of decision variables and fi, jtcf,kt,ki= I , j (9) then translates the LP results into terms familiar to hydrologic k,kʦ⍀ t=1 engineers (e.g., release, ¯ows, storage values). HEC-FCLP is where fi, j = average control-point ¯ow during period j; and linked to the HEC Data Storage System (USACE 1995), from Ii, j = local in¯ow during period j. For proper representation of which it reads incremental ¯ow data and to which it writes the the damage function, control-point ¯ow may also be divided model results. into zones. The control-point continuity equation then takes The model constraint set includes continuity constraints for the form each reservoir and control point, along with constraints on res- ervoir release capacity, in each time period. The objective NF i Ϫ function includes penalties for too much or too little storage, fi, j,ltcf,kt,ki= I , j (10) release, or ¯ow in each time period. l=1 k,kʦ⍀ t=1 The general form of the reservoir continuity constraints, for where l = index of discharge zone; and NF = number of dis- reservoir j, time period i,is charge zones. i Penalties for too much or too little storage represent oper- 1 Ϫ Ϫ ϩ Ϫ ators' aversion to storage levels outside of a target range. The ⌬ [Si, jiS 1, ji] f , jtcf,kt,ki= I , j (1) t k,kʦ⍀ t=1 penalties are speci®ed for each reservoir as a piecewise linear convex function of volume of water stored in the reservoir whereSiϪ1, j and Si, j = storage at the beginning and end of ⍀ during the period. The total penalty for storage SP is de®ned period i, respectively; fi, j = total release in period i; = set as of all control points upstream of j from which ¯ow is routed to j; f = average ¯ow at control point k in period t; c = iNLF t,k t,k linear coef®cient to route period t ¯ow from control point k SPjj= AS,li, j,l (11) t=1 l=1 to control point j for period i; and Ii, j = in¯ow to the reservoir. The routing coef®cients are found directly from the Mus- where Aj,l = slope of the storage penalty function in zone l of kingum model coef®cients. reservoir j. To model desired storage-balancing schemes among reser- Penalties for changing release rates too rapidly quantify voirs, the total storage capacity of each reservoir in the system negative impacts such as bank sloughing or inadequate re- is divided into zones. The total storage at any time i is the sponse time to changing conditions downstream. Changes in sum of storage in these zones release rates may also be limited by the equipment available
NLF to change gate or outlet settings. To impose this penalty, the LP model includes a set of auxiliary constraints that segregate Si, ji= S , j,l (2) l=1 the release for each period into the previous period's release, plus or minus a change in release. If the absolute value of this where l = index of storage zone; and NLF = number of storage change in release exceeds a speci®ed maximum, a penalty is zones. Substituting this in the continuity equation yields imposed. NLF NLF i The auxiliary constraints relate the release for each period 1 ͫϪ Ϫ ͬϩ Ϫ to release in the previous period by the equation Si, j,liS 1, j,lif , jtcf,kt,ki= I , j (3) ⌬t ʦ⍀ l=1 l=1 k,k t=1 ϩϩ ϪϪ Ri, ji= R Ϫ1, jiϩ [Ra , jiϩ Re , ji] Ϫ [Ra , jiϩ Re ,j] (12)
The storage in each zone l is constrained as ϩϩ whereRai, ji , Re , j = acceptable and excessive release increase, ϪϪ S Յ S max (4) respectively; andRaij , Re i, j = acceptable and excessive release i, j,lj,l ϩϪ decrease, respectively.Rai, ji and Ra , j are constrained not to The maximum reservoir release physically possible is lim- exceed the user-speci®ed desirable limits, and a penalty, RP, ϩϪ ited by the hydraulic properties of the reservoir outlet works. is imposed onRei, ji and Re , j at reservoir j as follows: This limitation is expressed as a piecewise linear function of ii the storage in the reservoir. That is, the maximum release from ϩϪϩ reservoir j for period i is speci®ed as RPji= BRe, ji, jiDRe, ji, j (13) t=1 t=1 NLF  j,l where Bi, j = penalty per unit ¯ow for a positive change in Յи Ϫ ϩ fi, ji(S 1, j,liS , j,l) (5) l=1 2 release greater than the user-speci®ed limits; and Di, j = penalty per unit ¯ow for a negative change in release greater than the  where j,l = slope of the storage-discharge capacity relation- user-speci®ed limits. ship in storage zone l. To correctly represent nonconvex stor- Flow penalties are speci®ed as a piecewise linear convex age-discharge functions, critical under forced spill conditions, function of downstream ¯ow, which is the sum of local runoff the following binary variable and logical constraints must be and routed reservoir releases. The penalty for ¯ow QP is given added for each reservoir j: by
22 iNF Ն Si, j,liYS, jjmax,l (6) QPkk= Ef,li,k,l (14) l=1 l=1 t=1 l=1
120 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 FIG. 2. System Decomposition
where Ek,l = slope of the penalty function in ¯ow zone l at ANALYSIS STRATEGY control point k. Incorporating penalty terms given by (11), (13), and (14), The ®rst step in the analysis is to select a number of ¯ood the objective function is as follows: events out of the approximately 70 years of record. Since the water year and the calendar year are similar in this region, the 10 years with the largest ¯ood events were identi®ed based on ͫϩ ϩ ͬ min TP = QPkjjRP SP (15) a combination of peak ¯ow and total volume at each gauge. k,kʦ⌿ j, jʦ⌽ j, jʦ⌽ For each of the selected years, beginning and ending dates of where TP = total penalty; ⌿ = set of all control points; and the ¯ood events were estimated visually from hydrographs. = set of all reservoirs. The release schedule that yields the To estimate the bene®ts from operating the reservoirs as a minimum total penalty is the optimal schedule. coordinated system, the larger Iowa/Des Moines/Mississippi It should be noted that HEC-FCLP makes release decisions River System was divided into various smaller subsystems as for all periods simultaneously, with perfect knowledge of the illustrated in Fig. 2. By optimizing the operations of each sub- complete ¯ow hydrographs. Despite their inherent optimism, system independently, the bene®ts from operating the three results from this type of deterministic model have proven use- reservoirs as a system can be evaluated, and the question of ful for inferring general reservoir system operational policies whether or not these bene®ts are signi®cant and obtainable can (Lund and Ferreira 1996). Historical operation of a reservoir be addressed. can be compared with the optimal operation determined by the System A, the most complex, consists of the three reservoirs model to identify possible shortcomings in current procedures, located on the Iowa and Des Moines rivers and all 10 control and questions regarding the operation of multiple reservoirs or points, two of which are on the Mississippi River. System B the effects of changing physical aspects of the system can be isolates the Iowa River, which causes Coralville Reservoir to addressed quickly. operate only for damage locations on the Iowa River, plus
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 / 121 Burlington on the Mississippi River. System C is similar to control pool, emergency ¯ood-control pool, and ¯ood sur- System B except that Burlington is removed from considera- charge pool. Storage-discharge capacity relationships are tion. This illustrates the potential effect of Burlington on the derived from outlet and spillway rating curves. All values are operation of Coralville Reservoir. System D represents the two obtained from the master reservoir regulation manuals (Master reservoirs on the Des Moines River operating in tandem for 1983, 1988, 1990). control at all damage locations on the Des Moines River and Penalties for high ¯ow are based on economic data found Quincy on the Mississippi River. System E is identical to Sys- in the reservoir regulation manuals and subsequent surveys tem D except that Quincy is not considered. Dividing System conducted by the Rock Island District. The penalty functions D just upstream of Red Rock Reservoir to form Systems F used in this study represent the total penalty at each location, and G helps illustrate the effect of operating Saylorville Res- which is a combination of urban, rural, and agricultural dam- ervoir and Red Rock Reservoir independently. Possible com- age. Penalty functions are developed by approximating the binations of these systems include A, BD, CD, BE, CE, BFG, nonlinear ¯ow-damage relationships with convex piecewise and CFG. linear functions. Flows are divided into zones based on ver- tices of the penalty functions, and the same penalties are used MODEL APPLICATION for all ¯ood events studied. Rate of change of release penalties are dif®cult to determine. Application of HEC-FCLP to the Iowa/Des Moines River The reservoir regulation manual for Saylorville (Master 1983) System required the collection of ¯ow data and the estimation states that a maximum change of 3,000 cfs/day is allowable of a number of model parameters. Daily incremental (local) during normal ¯ood operations. This limits bank sloughing in ¯ows and Muskingum routing parameters (e.g., Ponce 1989) the reservoir and along the downstream channel. A relatively for each river reach are estimated from USGS stream gauge large penalty of 0.1 dollars/cfs for rates of change >3,000 cfs/ data. Initial storage levels in each reservoir are set as the top day is set to discourage larger rates of change but still allow of the conservation pool, and reservoir storage pools were di- them when necessary. Maximum desirable rate of change val- vided into ®ve zones: drought pool, conservation pool, ¯ood- ues of 3,000 cfs/day for Coralville and 6,000 cfs/day for Lake Red Rock were determined through discussions with the Rock Island District and comparisons with historical observed res- ervoir storage data. Storage penalties are set to force the model to operate within the ¯ood-control pool when feasible. The penalty prescribed when storage enters the emergency ¯ood-control pool or the surcharge pool represents the risk associated with uncontrolled spills. A small ``persuasion'' penalty is placed on storage within the ¯ood pool so that reservoir levels return to the top of the conservation pool when downstream ¯ows recede below ¯ood stage. Fig. 3 illustrates an example storage-penalty func- tion. A more detailed discussion of parameter and penalty func- tion estimation is provided in USACE (1999). MODEL RESULTS The ¯ood of 1993 is important not only because of the FIG. 3. Example Storage-Penalty Function record-breaking ¯ows, but also because of the time it covered.
FIG. 4. Coralville Reservoir StorageÐFlood of 1993
122 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 FIG. 5. Iowa City HydrographÐFlood of 1993
FIG. 6. Quincy HydrographÐFlood of 1993
To consider the full impact of these ¯ows, the model was plest operating scheme that leads to penalty values within 2% con®gured to run from February 20 through November 25, a of those from System A is considered optimal. For example, total of 282 days, which resulted in a model with >18,000 if Schemes BD and BFG lead to penalty values within 2% of continuous decision variables, exactly 282 integer (0±1) var- Scheme A, then BFG would be selected as the optimal system. iables, and >5,600 constraints. Figs. 4±6 illustrate results from For the 1993 ¯ood, penalty values resulting from optimal the HEC-FCLP model and how they differ from observed data operation of the various Subsystems are listed in Table 4. for the ¯ood of 1993. Since Subsystems BD, CD, BFG, and CFG lead to essentially One would assume that the largest bene®t will come from the same optimal penalty value as System A, there would have operating the reservoirs as one coordinated system. However, been little potential bene®t from operating the reservoirs as a operating as a coordinated system also will lead to the most system. The only noticeable bene®t would have resulted from complex operating procedures, as well as increased hydrologic operating Red Rock Reservoir for ¯ood control at Quincy, uncertainty when reservoirs are operated for points far down- indicated by Subsystems BE and CE having signi®cantly stream. Therefore, upon comparing the model-computed pen- higher penalty values than BD and CD, respectively. In 1993, alties resulting from the different operating schemes, the sim- the reservoir system was simply overwhelmed, and the rela-
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 / 123 TABLE 4. Flood of 1993 Calculated Penalties (Subsystems Shown in Fig. 2) Site Observed A BD CD BE CE BFG CFG (1) (2) (3) (4) (5) (6) (7) (8) (9) Iowa City 69 33 31 31 31 31 31 31 Lone Tree 19 13 13 13 13 13 13 13 Wapello 313 278 278 278 278 278 278 278 2nd Avenue 212 148 148 148 148 148 148 148 14 Street 0 0000000 Tracy 450 422 422 422 422 422 422 422 Ottumwa 854 853 853 853 853 853 853 853 Keosauqua 102 89 89 89 97 97 89 89 Burlington 154 145 145 145 145 145 145 145 Quincy 2,208 1,427 1,430 1,431 2,006 2,008 1,430 1,431 Coralville Ð 10999999 Saylorville Ð 4,414 4,414 4,413 4,414 4,414 4,407 4,407 Red Rock Ð 199 200 201 61 61 208 208 Total without storage 4,381 3,408 3,409 3,410 3,993 3,995 3,409 3,410 Total with storage Ð 8,031 8,032 8,033 8,477 8,479 8,033 8,034
TABLE 5. Optimal Combinations of Subsystems control at Burlington. For all other events, the penalty at Bur- Flood year Optimal system Within 2% of optimal lington was reduced by <2%. (1) (2) (3) Table 6 shows the release priorities for Coralville Reservoir ¯ood-control operations, derived by comparing the penalty 1993 CFG A, BD, CD, BFG function slopes on the Iowa River and at Burlington and ar- 1965 BFG A, BD 1947 CFG All ranging them in descending order. Hydrographs of 1965 model 1973 CFG All results show a release >10,000 cfs from Coralville, which 1991 CE A, BD, CD, BD causes damage at Iowa City, to make space in the reservoir to 1960 CFG All dampen an upcoming peak at Burlington. This operation re- 1990 CFG All duces the ¯ow at Burlington by approximately 2,000 cfs. The 1979 CE A, BD, CD, BE 1993 ¯ood event also recorded a peak ¯ow above 265,000 cfs 1974 CFG All 1944 CFG All at Burlington; in this case however, Coralville Reservoir's ¯ood control space was needed to reduce the ¯ow at Iowa City below 20,000 cfs. From these results, it appears that op- tively small amount of ¯ood storage provided by the three erating Coralville Reservoir for ¯ood control at Burlington is projects could not be coordinated to make an appreciable dif- bene®cial only under very special circumstancesÐwhen ¯ows ference throughout much of the basin. at Iowa City and Wapallo can be maintained below 20,000 and Similar reasoning was used to determine the optimal oper- 48,500 cfs, respectively. ating scheme for each of the other 10 ¯ood events (USACE Table 7 lists the operating priorities for the reservoirs on the 1998). Table 5 summarizes the most basic optimal system (set Des Moines River, again based on penalty function slopes. of subsystems) for each ¯ood event listed, from the most se- According to this list, Saylorville Reservoir's entire ¯ood-con- vere ¯ood to the least severe ¯ood. trol pool should be used to ensure that ¯ow at 2nd Avenue is Table 5 illustrates that during most ¯ood events it is best to <40,000 cfs. Moving down the priority list, if the ¯ow at 2nd operate the three reservoirs independently. Thus, Coralville Reservoir operations are only concerned with ¯ooding on the TABLE 6. Coralville Release Priorities Iowa River, Saylorville Reservoir ¯ood storage is used only for ¯ood control in the city of Des Moines, and Red Rock Keep ¯ow less than Priority (cfs) Location Reservoir is operated to control ¯ooding on the lower Des (1) (2) (3) Moines River and at Quincy, Ill. Model results indicate that this policy would be the easiest to implement while still pro- 1 20,000 Iowa CityÐIowa River 2 48,500 WapelloÐIowa River viding near-optimal results. 3 265,000 BurlingtonÐMississippi River For the ¯ood of 1991, and to a lesser extent the ¯ood of 4 10,000 Iowa CityÐIowa River 1979, potential bene®ts exist from operating Saylorville Res- 5 17,500 Lone TreeÐIowa River ervoir and Red Rock Reservoir in tandem for ¯ood control on 6 30,000 WapelloÐIowa River the Des Moines River and at Quincy. Operating Saylorville 7 150,000 BurlingtonÐMississippi River Reservoir for ¯ood control downstream of Red Rock Reservoir leads to a more complex release policy than the previous one, TABLE 7. Des Moines River Flood Control Priorities but in these cases bene®ts could be realized. Coralville Res- ervoir operates only for damages on the Iowa River for both Keep ¯ow less than of these events. Priority (cfs) Location (1) (2) (3) DISCUSSION 1 40,000 2nd AvenueÐDes Moines River 2 107,000 OttumwaÐDes Moines River Model runs for the ¯ood of 1965 are the only results in 3 335,000 QuincyÐMississippi River which an appreciable difference (14%) was observed at Bur- 4 19,400 2nd AvenueÐDes Moines River lington with and without ¯ood control from Coralville Res- 5 19,000 OttumwaÐDes Moines River ervoir. This is due to the combination of large magnitude ¯ows 6 270,000 QuincyÐMississippi River on the Mississippi River and relatively small ¯ows on the Iowa 7 90,000 KeosauquaÐDes Moines River River. During this event, operators would have been able to 8 13,000 TracyÐDes Moines River 9 28,000 KeosauquaÐDes Moines River use the majority of Coralville Reservoir's storage for ¯ood
124 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 TABLE 8. Effects of Tandem Operation of Des Moines River TABLE 9. Effects of Operating Des Moines Reservoir for Reservoirs Flood Control at Quincy Total PenaltyÐDes Moines River and Quincy, Ill. Total DamageÐDes Moines River Tandem Operation Savings Savings Year Independent Tandem (%) Year Without Quincy With Quincy (%) (1) (2) (3) (4) (1) (2) (3) (4) 1993 2,943 2,943 0.0 1993 3,526 2,942 16.5 1991 248 194 21.7 1973 109 108 0.9 1990 33 33 0.0 1965 205 155 24.4 1979 87 77 11.5 1960 69 67 2.9 1974 27 27 0.0 1973 183 107 41.5 1965 158 154 2.5 lorville Reservoir ¯ood pool into two ``virtual'' poolsÐone 1960 68 67 1.5 for ¯ood control downstream of Lake Red Rock and the other 1947 211 206 2.4 for ¯ood control in the city of Des Moines. 1944 64 64 0.0 The results endorse operating the Des Moines Reservoirs Total 4,022 3,872 3.7 for ¯ood control on the Mississippi River at Quincy, Ill. Ben- e®ts at Quincy are seen in all four of the years studied that had damaging ¯ows on the Mississippi River. Table 9 illus- Avenue is not in danger of surpassing 40,000 cfs, the ¯ood- trates the ¯ow-related penalty reduction for these four events control space of both reservoirs should be utilized to keep the when the Des Moines Reservoirs operate for ¯ood control at ¯ow at Ottumwa <107,000 cfs. The remaining priorities are Quincy. more complicated. For example, if the ®rst two priorities are HEC-FCLP model results and observed 1993 operations met, then both reservoirs should be used to keep the ¯ow at have many signi®cant differences. Although Table 4 shows Quincy <335,000 cfs. However, the question remains as to that the total ¯ow-related penalty could have been reduced by whether the ¯ood-control burden should be placed evenly on nearly 25%, it is incorrect to conclude that current operating the two reservoirs, or should Lake Red Rock control most of procedures are inadequate without ®rst looking at where and the ¯ows and allow Saylorville Reservoir's ¯ood storage to why the differences in penalty occur. remain empty for protection of the city of Des Moines? The most notable difference between observed and model Analysis of results from Subsystems D and FG can help to results during the ¯ood of 1993 is in the operation of Coral- answer this question. As illustrated in Table 8, subsystem re- ville Reservoir. With perfect foresight, HEC-FCLP drew down sults indicate that modest bene®ts can be obtained from op- the reservoir much more in the ®rst few weeks of June than erating the two reservoirs on the Des Moines River in tandem. was recorded, as illustrated in Fig. 4. Historical data show When operating the reservoirs independently, releases from releases were cut back to prepare for the planting season Saylorville Reservoir are regulated only for the city of Des downstream although the reservoir was still relatively full. The Moines. For most events, only a small portion of Saylorville additional HEC-FCLP drawdown allowed Coralville Reservoir Reservoir's ¯ood storage capacity is utilized, since in¯ows to provide more protection from the large in¯ows that occurred into the reservoir are rarely enough to ®ll the ¯ood-control in late July and August. Were this policy of rapidly drawing pool. Model results show that only for the 1965 and 1993 down Coralville Reservoir following a ¯ood event adopted ¯ood events would Saylorville Reservoir reach capacity when every year, it would likely result in greater agricultural losses. operated independently of Red Rock Reservoir. Flooding The operating procedures for Coralville Reservoir should be would be so widespread for these events that Saylorville Res- reviewed with this in mind. ervoir's ¯ood pool would best be used mainly for control in It is impossible to predict hydrologic conditions 2±3 months the city of Des Moines whether operated in tandem or inde- in the future with enough certainty to justify making damaging pendently. When operated in tandem with Red Rock Reservoir, prereleases. Even with much shorter lead times, it is often Saylorville Reservoir's ¯ood-control pool would be ®lled dur- dif®cult to convince the general public that they should be ing every event except for 1974. However, reservoir operators ¯ooded today in order to reduce potential system-wide damage do not have perfect foresight, as HEC-FCLP does, and typi- in the future. As shown in Fig. 7, such a policy was proposed cally it would be imprudent to use the full capacity of Say- by the model for the ¯ood of 1993, when the Lake Red Rock lorville Reservoir's ¯ood-control space with the city of Des ¯ood-control pool was kept empty for 3 months in order to Moines directly downstream. dampen the mid-July ¯ood peak. The optimization results are According to Table 8, a signi®cant bene®t would have been impracticable in this regard, serving mainly to represent the obtained through tandem operation in 1973. During this event, lower bound of ¯ood damage from a ¯ood event. large ¯ows entered the Des Moines River System downstream of the city of Des Moines, while ¯ow into Saylorville Reser- LIMITATIONS voir was low. The rare hydrologic conditions of 1973 would have allowed Saylorville Reservoir's ¯ood pool to be used to As with all reservoir model applications, the implications of reduce damages downstream of Lake Red Rock. If operated the results from this study are limited by the approximations independently, Lake Red Rock would not have had the ¯ood- necessary to model an existing physical system. The most control capacity needed to control this ¯ood. During most widely recognized limitation inherent to LP is that all rela- other events studied, however, the ¯ood pool of Lake Red tionships in the model, such as routing equations and penalty Rock alone would have been large enough to control the ¯ood functions, must be approximated with linear or piecewise lin- ¯ows. ear functions. Nonlinear ¯ood routing techniquesÐthose us- Since the risk assumed is large when ®lling Saylorville Res- ing variable routing coef®cientsÐmay provide more accurate ervoir's ¯ood pool for control downstream of Lake Red Rock, results than do linear techniques over a range of discharges and Lake Red Rock's ¯ood storage is large enough to contain [e.g., Ponce (1989)]. Similarly, a nonlinear and/or discontin- most ¯oods, the majority of Saylorville Reservoir's ¯ood pool uous objective function may be more appropriate than the lin- should probably be reserved for ¯ood control in the city of ear (additive) function used in this study. Des Moines. A possible solution would be to divide the Say- Additional limitations speci®c to HEC-FCLP that should be
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 / 125 FIG. 7. Lake Red Rock StorageÐFlood of 1993 considered during the analysis of practical problems include policy would be to operate each reservoir independently. Re- the following: sults indicate that Coralville Reservoir could be operated for ¯ood control on the Iowa River with little consideration for · Penalties on change in release, ¯ow, and storage are as- Burlington, Saylorville Reservoir's ¯ood capacity could be sessed for each period. This duration-based solution does used mainly for ¯ood protection in the city of Des Moines, not directly account for ¯ood damage, especially urban and Lake Red Rock could be operated for ¯ood control on the ¯ood damage that is primarily a function of peak dis- Lower Des Moines River and at Quincy. charge, but it does capture the desire of reservoir opera- Operating Coralville Reservoir for ¯ood control on the Mis- tors to reduce the ¯ow in ¯ooded areas as soon as pos- sissippi River is risky because ¯ood-control space is consumed sible. The duration-based penalty leads to more realistic that could prove more valuable to Iowa City. It is acceptable results during nonpeak periods than does a penalty func- to operate Coralville Reservoir for ¯ood control on the Mis- tion based solely on peak ¯ows. HEC-FCLP is currently sissippi River as long as current and forecasted ¯ows in the being modi®ed to incorporate a combination of peak- Iowa River are low. Optimization results illustrate that this based and duration-based penalties. scenario occurred only once during the historical record. · HEC-FCLP does not account for the seasonal variation of By dividing the Des Moines River just upstream of Lake damage potential that is often associated with agricultural Red Rock, the effect of operating the two reservoirs, Red Rock areas. Consequently, for a ¯ood event that spans two or and Saylorville, in tandem was illustrated. When operated in more seasons, it cannot properly optimize the seasonal tandem, most of Saylorville Reservoir's ¯ood control capacity allocation of ¯ood storage capacity that is common in was used for protection downstream of Lake Red Rock. When ¯ood-control systems. operated independently, the majority of Saylorville Reservoir's · FCLP is a deterministic model, which means it implicitly ¯ood pool capacity was rarely used, and the resulting ¯ows assumes that future ¯ows are known with certainty. Thus, were regulated by Lake Red Rock. Penalty values obtained as illustrated by this study, HEC-FCLP draws down res- from tandem and independent operation were within 3% for ervoirs in anticipation of future ¯oods, perhaps even most of the ¯ood events studied. Since the city of Des Moines months in advance. While this is not necessarily a limi- is potentially one of the highest damage locations on the river, tation, it is an important feature that should be taken into it would be more risk averse to save a majority of Saylorville account when using results from deterministic optimiza- Reservoir's ¯ood storage for the city of Des Moines and use tion models. Lake Red Rock for ¯ood control downstream. Review of operations during the Great Flood of 1993, a very When used for planning studies, computational time is not rare event, shows that damages could have been reduced if a limitation of HEC-FCLP. In this study, a global optimum in¯ows were known months in advance. Obviously, this is not solution to each mixed-integer LP problem was obtained in possible with current forecasting technology. However, the <15 min using a 200-MHz Pentium II PC. However, this so- damage could also have been reduced during the 1993 ¯ood lution time might limit the use of HEC-FCLP for real-time if current reservoir operation were more averse to extreme decision support. events. Release decisions during the ¯ood of 1993 were made based on knowledge of previous events. With new data and a CONCLUSIONS better understanding of the runoff these drainage areas can The method of dividing the system into various smaller sys- produce, the release rules should be modi®ed to account for tems produces results that quantify the potential bene®ts of events of this magnitude in the future. making reservoir releases based on selected control points. For Deterministic optimization models are useful for evaluating the majority of ¯ood events studied, the optimal operational the potential bene®ts of a reservoir system when analyzing
126 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 2000 operating procedures, but results from these models need to Master reservoir regulation manual: Saylorville Lake. (1983). USACE be kept in perspective. Detailed simulation modeling should Rock Island Dist., U.S. Army Corps of Engineers. always accompany these omniscient optimization procedures Master reservoir regulation manual: Lake Red Rock. (1988). USACE Rock Island Dist., U.S. Army Corps of Engineers. when developing operating rules for a reservoir or set of res- Master reservoir regulation manual: Coralville Lake. (1990). USACE ervoirs, since simulation models can give a more accurate es- Rock Island Dist., U.S. Army Corps of Engineers. timate of the system performance given a set of operating pol- Optimization subroutine library release 2.1, guide and reference. (1995). icies. IBM Corp., Poughkeepsie, N.Y. Optimization models are only as good as their penalty func- Ponce, V. M. (1989). Engineering hydrology: Principles and practices. tions and constraints. Establishing these penalty functions and Prentice-Hall, Englewood Cliffs, N.J. Scienti®c Assessment and Strategy Team. (1994). Science for ¯oodplain producing a standard set of historical in¯ows is an important, management into the 21st century. Interagency Floodplain Manage- though time-consuming task. Not only has this study led to ment Review Committee, Washington, D.C. increased understanding of the Iowa/Des Moines Reservoir U.S. Army Corps of Engineers. (1992). Authorized and Operating Pur- SystemÐthe potential ¯ood control value of each reservoir poses of Corps of Engineers Reservoirs. Department of the Army, and the potential damage at various locationsÐbut it has also Washington, D.C. produced a standardized set of data that will prove invaluable U.S. Army Corps of Engineers. (1994). ``Operating rules from HEC pre- scriptive reservoir model results for the Missouri River System: De- in future studies. velopment and preliminary testing.'' Rep. PR-22, Hydrologic Engrg. Ctr., Davis, Calif. ACKNOWLEDGMENTS U.S. Army Corps of Engineers. (1995). ``HEC-DSS user's guide and util- ity manuals, user's manual.'' CPD-45, Hydrologic Engrg. Ctr., Davis, This work was supported by the USACE Hydrologic Engineering Cen- Calif. ter (HEC), Davis, Calif., and the USACE Rock Island District (CEMVR). Mike Burnham of HEC provided general guidance for the study. Theresa U.S. Army Corps of Engineers. (1996). ``Application of HEC-PRM for Carpenter and Shirley Johnson of CEMVR reviewed and commented on seasonal reservoir operation of the Columbia River System.'' Rep. RD- the HEC report from which this paper was derived. Mike Tarpey of 43, Hydrologic Engrg. Ctr., Davis, Calif. CEMVR was instrumental in the derivation of incremental ¯ow data and U.S. Army Corps of Engineers. (1999). ``Analysis of ¯ood control per- routing parameters. David Ford Consulting Engineers, Sacramento, Calif., ation of the Iowa/Des Moines River Reservoir System using linear developed the original model and provided helpful insights throughout programming techniques.'' Rep. PR-38, Hydrologic Engrg. Ctr., Davis, the study. Calif. U.S. Department of Agriculture Soil Conservation Service. (1994). The Soil Conservation Service responds to the 1993 Midwest ¯oods. Eco- APPENDIX. REFERENCES nomics and Social Sciences Division, Washington, D.C. Beard, L. R., and Chang, S. (1979). Optimizing ¯ood operation rules. U.S. Department of Commerce. (1994). ``The Great Flood of 1993.'' Nat- Ctr. for Res. in Water Resour., University of Texas at Austin, Austin, ural Disaster Survey Rep., Washington, D.C. Tex. Wasimi, S. A., and Kitanidis, P. K. (1983). ``Real-time forecasting and Ford, D. T. 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(1996). ``Operating rule optimization for models,'' J. Water Resour. Plng. and Mgmt., ASCE, 119(4), 455±472. Missouri River Reservoir System.'' J. Water Resour. Plng. and Mgmt., Yeh, W. W-G. (1985). ``Reservoir management and operation models: ASCE, 122(4), 287±295. A state-of-the-art review.'' Water Resour. Res., 21(12), 1797±1818.
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