Linear Programming for Flood Control on the Iowa and Des Moines Rivers1

David W. Watkins, Jr.2

Background

The Great Midwest Flood of 1993 along the Upper Mississippi River and its tributaries caused an estimated 48 fatalities and $15-20 billion in economic damages, surpassing all floods in the United States up to that time (Natural Disaster Survey Report 1994). As a result of the flood, the Federal Emergency Management Agency declared 504 counties in nine states eligible for assistance, with the most severe damage occurring in Iowa, Illinois, and Missouri. The flood forced 74,000 people from their homes, disrupted commercial activity along the Mississippi and Missouri rivers and adjacent areas, and destroyed thousands of acres of crops. Many farms also lost facilities and equipment, and an estimated 72,000 private homes either were washed away or suffered major damage. Approximately 40,000 commercial structures were damaged. Virtually all forms of transportation on and across the Mississippi River were interrupted by the flood. Along the length of the Mississippi River that forms the western boundary of Illinois, more than 1,000 miles of roads were closed, and nine of the 25 non-railroad bridges were shut down (USACE, 1996). The Great Flood of 1993 was caused by a highly unusual series of thunderstorms repeatedly forming and moving over the same area, combined with above average precipitation and below average temperatures in the preceding months. Starting in November 1992, precipitation was above normal, and temperatures were below normal throughout much of the upper Midwest. Persistent rains and early snowmelt led to high spring runoff and very high soil moisture levels. Due to an eastward- flowing jetstream that extended from central Colorado northeastward across Kansas to northern Wisconsin, a weather-front convergence zone formed across the upper Midwest during the spring and summer of 1993. Moist, warm air from the Gulf of Mexico was drawn northward along this jetstream, where it collided with cooler air masses from central Canada. This combination of extreme conditions generated frequent occurrences of heavy precipitation over the upper Mississippi River basin, leading to the destructive floods. In January through July 1993, more than 20 inches of rain fell over most of the flood-affected area, with more than 40 inches of rainfall occurring in areas of northeast Kansas and east-central Iowa (USACE, 1996). In the aftermath of this disaster, some concern was voiced that the U.S. Army Corps of Engineers did not operate flood control reservoirs on Upper Mississippi tributaries in an optimal manner. Although there was no evidence of deviations from the reservoir regulation plans, a modeling study was commissioned to provide insight for

1 Based on Needham, J.T, D.W. Watkins, J.R. Lund, and S.K. Nanda (2000). “Linear Programming for Flood Control on the Iowa and Des Moines Rivers,” Journal of Water Resources Planning and Management, ASCE, 126(3): 118-127. 2 Professor, Department of Civil and Environmental Engineering, Michigan Technological University, Houghton, MI 49931. E-mail: [email protected]

possible modifications to the operating plans (USACE, 1999). In particular, a deterministic optimization model was applied to a three-reservoir system on the Iowa and Des Moines Rivers to estimate the best possible operation of these reservoirs (with “perfect foresight”) and to determine whether or not tandem operating rules would provide appreciable benefits. Reservoir System The Iowa/Des Moines River Reservoir System consists of three reservoirs, one on the Iowa River main stem and two on the Des Moines River main stem, as shown in Figure 1. Authorized purposes for these reservoirs include flood control, low-flow augmentation, fish/wildlife, water supply, and recreation. The Rock Island District of the Army Corps is responsible for day-to-day decision making regarding reservoir operations. Operators follow guidelines described in the reservoir regulation manuals that have been prepared as part of the design of the system (USACE 1983, 1988, 1990).

Saylorville C Iowa e Riv da Reservoir er r R iv er IOWA Coralville # DES MOINES CITY # Lake Reservoir Red Rock # LONE TREE # TRACY

# WAPELLO # OTTUMWA De s M oin es KEOSAUQUA # R # BURLINGTON iv IOWA er

MISSOURI

ILLINOIS LA GRANGE # QUINCY #

HANNIBAL#

Figure 1. Map of Iowa/Des Moines River Reservoir System

Total capacities and average inflows for the three reservoirs are shown in Table 1, and other pertinent characteristics of the Iowa and Des Moines Rivers are shown in Tables 2 and 3, respectively. Table 2 illustrates that Coralville Reservoir can regulate no more than 25% of the total average annual flow entering the Mississippi from the Iowa River. Because of this, one could expect that Coralville Reservoir’s flood control effectiveness below the confluence of Cedar River and on the Mississippi River is limited. Conversely, as illustrated in Table 3, Saylorville and Red Rock

reservoirs regulate over half of the average flow entering the Mississippi River from the Des Moines River.

Table 1. Capacities of and Average Inflows to the Three Reservoirs (m3 x 106)

Inflows Capacity (acre-ft/year) Reservoir (acre-ft/year) Conservation Flood Control Total %a (1) (2) (3) (4) (5) (6) Coralville (Iowa River) 1,271,800 25,900* 435,300 461,200 18 Saylorville (D.M. 1,540,600 90,000 586,000 676,000 20 River) Red Rock (D.M. River) 3,568,000 265,500* 1,494,900 1,760,400 62 * Varies seasonally, value is minimum which corresponds to maximum flood storage a Percent of total federal project flood storage in Des Moines/ Iowa system

Table 2. Iowa River Characteristics

Drainage Area Mean Inflow Location (sq. mi.) (cfs) (1) (2) (3) Coralville Reservoir 3,115 1,760 Iowa River (Confluence w/Cedar R.) 4,770 2,360 Cedar River (Confluence w/Iowa R.) 7,870 4,230 Iowa River (Confluence w/Mississippi R.) 12,980 7,120 Mississippi River (Confluence w/Iowa R.) 89,000 49,000

Table 3. Des Moines River Characteristics

Drainage Area Mean Inflow Location (sq. mi.) (cfs) (1) (2) (3) Saylorville Reservoir 5,823 2,200 Red Rock Reservoir 12,323 4,928 Des Moines R. (Confluence w/Mississippi R.) 14,540 8,210 Mississippi R. (Confluence w/Des Moines R.) 119,000 64,520

Under current operations, Coralville Reservoir is to be operated for flood control at Iowa City, Lone Tree and Wapello on the Iowa River; and Burlington, Iowa, on the Mississippi River (USACE 1990). Presumably, when operated in conjunction with the reservoirs on the Des Moines River, the flood peaks can be offset enough to cause a significant difference in the water levels on the Mississippi River during flooding.

Saylorville Reservoir and Lake Red Rock projects also are associated with the comprehensive flood control plan for the Upper Mississippi River Basin. According to the reservoir regulation manuals, Saylorville Reservoir is operated not only to reduce flood damage in the City of Des Moines, but it is also operated in tandem with Red Rock Reservoir to reduce flood damage at Ottumwa and Keosauqua on the Des Moines River and at Quincy, Illinois, on the Mississippi River (USACE 1983; USACE 1988). Flood control priorities for this system are summarized in Tables 4 and 5.

Table 4. Coralville Release Priorities

Priority Keep flow less than (cfs) At Location (1) (2) (3) 1 20,000 Iowa City- Iowa River 2 48,500 Wapello – Iowa River 3 265,000 Burlington – Miss. River 4 10,000 Iowa City – Iowa River 5 17,500 Lone Tree – Iowa River 6 30,000 Wapello – Iowa River 7 150,000 Burlington – Miss. River

Table 5. Des Moines River Flood Control Priorities

Priority Keep flow less than (cfs) at Location (1) (2) (3) 1 40,000 2nd Ave. - Des Moines River 2 107,000 Ottumwa - Des Moines River 3 335,000 Quincy - Mississippi River 4 19,400 2nd Ave. - Des Moines River 5 19,000 Ottumwa – Des Moines River 6 270,000 Quincy - Mississippi River 7 90,000 Keosauqua - Des Moines River 8 13,000 Tracy – Des Moines River 9 28,000 Keosauqua - Des Moines River

Linear Programming Model

A linear programming (LP) model was developed at the Hydrologic Engineering Center of the USACE to assist with Corps’ flood management studies (Figure 2). The model treats the flood-operation problem as one of finding a system-wide set of releases that minimize total system penalties for too much or too little release,

storage, and flow. Essentially embedded in the LP model constraints is a simulation model that computes storage and downstream flows based on reservoir releases. This model accommodates reservoir continuity and linear channel routing (e.g. Muskingum routing) and accounts for hydraulic limitations such as reservoir outlet capacities. The model constraint set includes continuity constraints for each reservoir and control point, along with constraints on reservoir release capacity, in each time period. The objective function includes penalties for too much or too little storage, release, or flow in each time period.

Figure 2. Model Schematic of Iowa/Des Moines River Reservoir System

The general form of the reservoir continuity constraints, for reservoir j, time period i, is 1 i Si, j  Si1, j  fi, j  ct,k ft,k  Ii, j t k ,1k t (1) where Si-1,j and Si,j = storage at the beginning and end of period i, respectively; fi,j = total release in period i; = set of all control points upstream of j from which flow is routed to j; ft,k = average flow at control point k in period t; ct,k = linear coefficient to route period t flow from control point k to control point j for period i; Ii,j = inflow to the reservoir. The routing coefficients are found directly from the Muskingum model coefficients.

To model desired storage-balancing schemes amongst reservoirs, the total storage capacity of each reservoir in the system is divided into zones. The total storage at any time i is the sum of storage in these zones:

NLF Si, j   Si, j,l (2) l1 where l = index of storage zone; and NLF = number of storage zones. Substituting this in the continuity equation yields 1 NLF NLF  i  Si, j,l   Si1, j,l   fi, j  ct,k ft,k  I i, j (3) t  l1 l1  k,1k t The storage in each zone l is constrained as

Si, j,l  SMAX j,l (4) The maximum reservoir release physically possible is limited by the hydraulic properties of the reservoir outlet works. This limitation is expressed as a piecewise linear function of the storage in the reservoir. That is, the maximum release from reservoir j for period i is specified as

NLF  j,l fi, j   Si1, j,l  Si, j,l (5) l1 2 where βj,l is the slope of the storage-discharge capacity relationship in storage zone l. In order to correctly represent non-convex storage-discharge functions, critical under forced spill conditions, the following binary variables and logical constraints must be added for each reservoir j.

2 2  Si, j,l  Yi, j  SMAX j,l (6) l1 l1

Si, j,3  Yi, j SMAX j,3 (7)

Yi, j 0,1 (8) These constraints ensure that, for example, storage zones 1 and 2 are filled before water is stored in zone 3.

The continuity constraint for each control point other than a reservoir takes the following general form:

i fi, j  ct,k ft,k  Ii, j (9) k ,1k t where fi,j = the average control-point flow during period j; Ii,j = local inflow during period j. For proper representation of the damage function, control-point flow may also be divided into zones. The control-point continuity equation then takes the form

NLF i  fi, j,l  ct,k ft,k  Ii, j (10) l1 k ,1k t where l = index of discharge zone; and NF = number of discharge zones. Penalties for too much or too little storage represent operators’ aversion to storage levels outside of a target range. The penalties are specified for each reservoir as a piece-wise linear convex function of volume of water stored in the reservoir during the period. The total penalty for storage, SP , is defined as

i NLF SPj A j,l Si, j,l (11) t11l where Ajl, is the slope of the storage penalty function in zone l of reservoir j .

Penalties for changing release rates too rapidly quantify negative impacts such as bank sloughing or inadequate response time to changing conditions downstream. Changes in release rates may also be limited by the equipment available to change gate or outlet settings. To impose this penalty, the LP model includes a set of auxiliary constraints that segregate the release for each period into the previous period’s release plus or minus a change in release. If the absolute value of this change in release exceeds a specified maximum, a penalty is imposed. The auxiliary constraints relate the release for each period to release in the previous period by the equation

    Ri, j  Ri1, j  Rai, j  Rei, j  Rai, j  Rei, j  (12)

 where Raij,,, Re ij= acceptable and excessive release increase, respectively; and    Raij,,, Re ij= acceptable and excessive release decrease, respectively. Raij, and Raij, are constrained not to exceed the user-specified desirable limits, and a penalty, RP , is  imposed on Reij,,and Re ij at reservoir j as follows:

i i   RPj  Bi, j Rei, j  Di, j Rei, j (13) t1 t1 where Bij, is the penalty per unit flow for a positive change in release greater than the user-specified limits and Dij, is the penalty per unit flow for a negative change in release greater than the user-specified limits. Flow penalties are specified as a piece-wise linear convex function of downstream flow, which is the sum of local runoff and routed reservoir releases. The penalty for flow, QP, is given by

i NF QPk Ek ,l fi,k ,l (14) t11l where Ek,l is the slope of the penalty function in flow zone l at control point k.

Incorporating penalty terms given by equations (11), (13) and (14), the objective function is as follows:   minTP QPk  RPj  SPj  (15) kj,,,k jjj   where TP is the total penalty; Ψ = set of all control points and Φ = set of all reservoirs. The release schedule that yields the minimum total penalty is the optimal schedule. It should be noted that the LP makes release decisions for all periods simultaneously, with perfect knowledge of the complete flow hydrographs. Despite their inherent optimism, results from this type of deterministic model have proven useful for inferring general reservoir system operational policies (Lund 1996). Historical operation of a reservoir can be compared with the “optimal” operation determined by the model to identify possible shortcomings in current procedures; and questions regarding the operation of multiple reservoirs or the effects of changing physical aspects of the system can be addressed quickly.

Model Application

Application of the LP model to the Iowa/Des Moines River system required the collection of flow data and the estimation of a number of model parameters. Daily incremental (local) flows and Muskingum routing parameters (e.g., Ponce, 1989) for each river reach were estimated from U.S.G.S. stream gage data. Initial storage levels in each reservoir were set as the top of the conservation pool, and reservoir storage pools were divided into five zones: drought pool, conservation pool, flood control pool, emergency flood control pool, and flood surcharge pool. Storage-discharge capacity relationships were derived from outlet and spillway rating curves. All values are obtained from the master reservoir regulation manuals (USACE 1983, 1988, 1990). Penalties for high flow were based on economic data found in the reservoir regulation manuals and subsequent surveys conducted by the Rock Island District. The penalty functions represented the total penalty at each location, which is a combination of urban, rural, and agricultural damage. Penalty functions were developed by approximating the nonlinear flow-damage relationships with convex piecewise linear functions. Flows were divided into zones based on vertices of the penalty functions. An example is shown in Figure 3.

Figure 3. Iowa River Flood Penalty Functions

Rate of change of release penalties were difficult to determine. The reservoir regulation manual for Saylorville states that a maximum change of 3000 cfs/day is allowable during normal flood operations. This limits bank sloughing in the reservoir and along the downstream channel. A relatively large penalty of 0.1 $/cfs for rates of change greater than 3000 cfs/day was set to discourage larger rates of change but still allow them when necessary. Maximum desirable rate of change values of 3000 cfs/day for Coralville and 6000 cfs/day for Lake Red Rock were determined through discussions with the Rock Island District and comparisons with historical observed reservoir storage data. Storage penalties were set to force the model to operate within the flood control pool when feasible. The penalty prescribed when storage enters the emergency flood control pool or the surcharge pool represents the risk associated with uncontrolled spills. A small “persuasion” penalty is placed on storage within the flood pool so that reservoir levels return to the top of the conservation pool when downstream flows recede below flood stage. Figure 4 illustrates an example storage-penalty function.

Surcharge pool Drought pool Drought Penalty Conservation pool Conservation Flood control pool control Flood Emergency flood control control pool flood Emergency

Storage

Figure 4. Example Storage-Penalty Function

Activities

The accompanying spreadsheet contains a LP model representing the Coralville Reservoir and downstream operating points on the Iowa River. Flood damage and the reservoir storage-discharge relationships are specified on the sheet Diagram & Data. Incremental flow data for the 1993 flood are provided on a two-day time step on the sheet Flow-inc. These data are used to define the LP model on the sheet Optimization-LP. Solution of the LP requires use of the Premium Solver (available from Frontline Systems, Inc. < http://www.solver.com/>). 1. Model Formulation First, look at the columns and formulas in the spreadsheet. The “adjustable cells” (variables) are shaded green and include the reservoir release variables (Column B), the reservoir storage variables (Columns C-E, representing three storage zones), and the flow variables at Iowa City (Columns I-J, representing three flow zones). The cells shaded blue represent cells associated with model constraints. Upper bounds on the flow and storage variables are in Row 2. Columns F and G are used to represent the reservoir mass balance constraints. Similarly, Columns L and M represent the flow mass balance at Iowa City. Column H represents the upper bound on reservoir releases as a function of reservoir storage. Finally, cells associated with objective function terms are shaded yellow. The Coralville Reservoir storage and Iowa City flow penalties for all time periods are summed in Cell Q2. (Lone Tree and Wapello penalty terms are omitted due to Solver size limits.) Note how the formulas in the reservoir release constraint and penalty function cells relate to the model data in the Diagram & Data sheet. Next, open the Solver (from the Tools menu) and inspect the target cell, adjustable cells, and constraints. Choose the Standard Simplex Solver as shown

in Figure 5. Click on the Options button, and choose “Assume Non-Negative” and “Use Automatic Scaling” as shown in Figure 6. Also, adjust the Precision to 0.0001. (Using a smaller value may lead to scaling problems.) Due to Solver size limits (1000 variables), the change in release constraints (Eqs. 12 and 13) are omitted, as are the flood damage penalties at Lone Tree and Wapello. The binary variables (Eqs. 6-8) are also omitted. The size of the current model can be viewed by clicking on the Problem tab in the Solver Options window (Fig. 6). Q: How many additional variables, constraints, and bounds would be required to include the change in release constraints? Q: How many additional variables, constraints, and bounds would be required to include flood damage penalties at Wapello?

Figure 5. Solver Parameters dialog

Figure 6. Solver Options dialog

2. Model Solution Click on the Solve button to solve the LP problem. Results of the LP model and the historical 1993 storage and flow values are plotted on the worksheets Coralville Storage and Iowa City Flows. Note differences between the observed and “optimal” values. Also note that the observed decreases in reservoir releases in early April and late May are due to attempts to reduce high flows at Wapello (early April) and to allow farmers to plant near the river (late May), which are objectives that are not represented in the Excel LP model. Q: Why are the LP reservoir releases constant at 10,000 cfs for long periods of time? Q: Why does the LP model hold reservoir storage much lower than the observed value until early July? Q: Why does the LP model hold reservoir storage below 462,000 acre-ft, even though total storage penalties are much smaller than total flood damage penalties?

Q: Based on these results, do you believe that the Army Corps could have operated the reservoir more effectively?

3. Sensitivity Analysis and Duality In this activity, we will observe the effects of small changes to model data (objective function coefficients and constraint right-hand sides) and discuss the implications of these changes. The following tasks involve repeatedly running the optimization model, which may be avoided by appropriate interpretation of the Sensitivity Report. (Note: If a Sensitivity Report is generated, you may need to display shadow prices in scientific notation, as they may be small values.) (a) Change the value in Cell I2 to 10,001 and re-solve the model. Q: What is the change in the objective function value? How might this information be useful to planners? (b) With the value in Cell I2 reset to 10,000, change the value in Cell D2 to 435,400 and re-solve the model. (Note that this also changes the penalty function as specified on the Diagram & Data worksheet.) Q: What is the change in the objective function value, and how might this information be useful to reservoir operators? (c) Reset the value in Cell D2 to 435,300, and increase the reservoir release constraint value in Cells H2:H148 by 1.0. Re-solve the model. Q: What is the change in the objective function value, and what are the implications for infrastructure modifications? (d) Reset the release constraint values, and make one additional change to model data that you expect will result in a decrease in the total penalty function. Re- solve the model. Q: What are the implications of the change in model data?

4. Summary Q: What are the main limitations of this optimization approach? Q: What are advantages and disadvantages of optimization as compared to simulation modeling?

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