Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow: a Case Study at Joe Pool Lake texaswaterjournal.org An online, peer-reviewed journal published in cooperation with the Texas Water Resources Institute Volume 13 Number 1 | 2022 http://texaswaterjournal.org Volume 13, Number 1 2022 ISSN 2160-5319 texaswaterjournal.org THE TEXAS WATER JOURNAL is an online, peer-reviewed journal devoted to the timely consideration of Texas water resources management, research, and policy issues. The journal provides in-depth analysis of Texas water resources management and policies from a multidisciplinary perspective that integrates science, engineering, law, planning, and other disciplines. It also provides updates on key state legislation and policy changes by Texas administrative agencies. For more information on on the Texas Water Journal as well as our policies and submission guidelines, please visit texaswaterjournal.org. 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Votteler, Ph.D. Collaborative Water Resolution LLC Managing Editor Chantal Cough-Schulze Texas Water Resources Institute Layout Editor Sarah Richardson Texas Water Resources Institute Staff Editor Cierra George Texas Water Resources Institute Cover photo: A view of the Milky Way over Phoinix Ranch in Jim Wells and Live Oak counties. ©2022 Rey Garza and Jim Quisenberry The Texas Water Journal is indexed by Scopus, Google Scholar, and the Directory of Open Access Journals. http://texaswaterjournal.org https://twri.tamu.edu/ http://texaswaterjournal.org https://twj.media/donate/ http://texaswaterjournal.org http://texaswaterjournal.org https://www.scopus.com/home.uri https://scholar.google.com/scholar?hl=en&as_sdt=0%2C44&q=source:%22Texas+Water+Journal%22&btnG= https://scholar.google.com/scholar?hl=en&as_sdt=0%2C44&q=source:%22Texas+Water+Journal%22&btnG= https://doaj.org/toc/2160-5319 https://doaj.org/apply/seal/ Texas Water Journal, Volume 12, Number 1 Texas Water Resources Institute Texas Water Journal Volume 13, Number 1, February 22, 2022 Pages 1-12 Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow: a Case Study at Joe Pool Lake Abstract: Maintaining an adequate water supply is one of the key challenges faced by the Dallas-Fort Worth Metroplex, where increasing population and rising water demand have elevated the vulnerability of the communities to water shortages. We conducted a preliminary study exploring the possibility of converting flood storage in the Joe Pool Lake to improve water supply reliability and achieve better cost efficiency. This study employs a mixed integer linear programming approach that considers the costs of using flood storage conversion to meet water needs over the northern portion of the Trinity River Basin. It includes tradeoffs between capturing and storing runoff versus return flow from the wastewater treatment facilities of the Trinity River Authority. A set of hypothetical prices and demand figures with the period of record of 1940–1996 was considered to test the mixed integer linear programming model. Results from the mixed integer linear programming produce an optimal strategy that increases the firm yield of Joe Pool Lake and associated storage-diversion on an annual basis. Also, the outcomes of the analyses of the results suggest that while the conversion would have a positive impact on water availability, a lower expansion cost of $20 per acre-foot per year would be required to produce sufficient cost savings. Keywords: Trinity River Basin, Joe Pool Lake, mixed integer linear programming, optimization, cost efficiency 1 Department of Industrial, Manufacturing, and Systems Engineering, University of Texas at Arlington, Arlington, Texas 2 Iowa Department of Transportation, Grimes Resident Construction Engineer’s office 3 Trinity River Authority of Texas, Arlington, Texas 4 Department of Civil Engineering, The University of Texas at Arlington, Arlington, Texas * Corresponding author: srividya.sekar@mavs.uta.edu Received 23 October 2020, Accepted 9 November 2021, Published online 22 February 2022. Citation: Sekar S, Daghighi A, Chen VCP, Clingenpeel G, Zhang Y, Rosenberger JM, Boskabadi A. 2022. Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow: a Case Study at Joe Pool Lake. Texas Water Journal. 13(1):1-12. Available from: https://doi.org/10.21423/twj.v13i1.7124. © 2022 Srividya Sekar, Amin Daghighi, Victoria CP Chen, Glenn Clingenpeel, Yu Zhang, Jay M Rosenberger, Azam Boskabadi. This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit https://creativecom- mons.org/licenses/by/4.0/ or visit the TWJ website. Srividya Sekar1*, Amin Daghighi2, Victoria CP Chen1, Glenn Clingenpeel3, Yu Zhang4, Jay M Rosenberger1, Azam Boskabadi1 mailto:srividya.sekar%40mavs.uta.edu?subject= https://doi.org/10.21423/twj.v13i1.7124 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://journals.tdl.org/twj/index.php/twj/about#licensing Texas Water Journal, Volume 12, Number 1 Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow2 Terms used in paper Acronym/Initialism Descriptive Name ac-ft acre-feet CRWS Central Regional Wastewater System JPL Joe Pool Lake LLP Lake Livingston MILP Mixed Integer Linear Programming TCEQ Texas Commission on Environmental Quality TRA Trinity River Authority TWDB Texas Water Development Board USACE United States Army Corps of Engineers WAM Water Availability Model WRAP Water Rights Analysis Package The population of Texas is projected to increase by 82% between 2010 and 2060 according to the Texas Water Devel- opment Board (TWDB; TWDB 2017). During the same time period, TWDB notes that current water resources are expected to decline by 10%, from 17.0 million acre-feet (ac-ft) to 15.3 million ac-ft (TWDB 2012), as siltation consumes reservoir conservation capacity. In addition, the state experiences con- secutive periods of droughts and floods that are generally asso- ciated with the El Niño–Southern Oscillation cycle (Pu et al. 2016; Naden and Platt n.d.). Extreme examples of droughts and floods can be seen in the meteorological records of 2011 and 2015, respectively. The driest and hottest year was in 2012 (Nielsen-Gammon 2012), the wettest year was in 2015 (National Centers for Environmental Information 2022). Just 3 years later, Hurricane Harvey and Tropical Storm Imelda set records for precipitation in 2017 and 2019, respectively, with both events producing in excess of 40 inches of rain (LeComte 2020). Texas water managers must therefore balance the needs of water supply with flood control. For this reason, many reservoirs in Texas have both conservation and flood pools. Throughout the 20th century, the United States invested heavily in flood INTRODUCTION Population and economic growth intensify the demand for water sources. Reservoirs are important in addressing this grow- ing demand, especially in regions like Texas that experience both significant variability in inter-seasonal precipitation and inter-annual precipitation. Specifically, the state experiences a bimodal precipitation pattern with strong frontal systems pro- ducing significant rainfall in the spring and fall. This is bifur- cated by hot and dry summers that can last for several months. During this time, surface flow in major river systems is habit- ually and significantly reduced. For these reasons, reservoirs, which catch runoff from wet months and store them for use during dry months, have been paramount to the development of Texas. Reservoirs are also critical in providing adequate water supplies through multi-year periods of low rainfall relative to normal precipitation. Many of the state’s reservoirs were built in the 1950s following the dustbowl years of the 1930s (Reyn- olds et al. 1999). Relatively few reservoirs have been built in the last few decades, while existing storage capacity has become increasingly stressed by the state’s growing municipal, agricul- tural, industrial, and energy needs (Cervellera et al. 2006). Texas Water Journal, Volume 12, Number 1 3Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow Figure 1. Joe Pool Lake Reservoir and Location Characteristics. control infrastructure in response to devastating floods along major rivers across the nation. Between 1901 and 1991, 51 major reservoirs were constructed in Texas for flood control or with flood storage capacity (TWDB 2019). In addition, the Natural Resources Conservation Service constructed approx- imately 2,000 smaller reservoirs to also provide flood control throughout the state (TWDB 2019). Typically, a multipurpose reservoir consists of a sediment or inactive pool at the bottom, a water conservation pool in the middle, and a flood control pool above the water conservation pool. Reservoir operations entail maintaining conservation pools as full as possible, while simultaneously supplying water demands and maintaining flood control pools as empty as possible (Yeh 1985; Yang et al. 2015; Slade 2020; Yaghoubi et al. 2020). Conserving emp- ty flood pool capacity is essential for allowing flooding oper- ations to catch and temporarily store flood waters to reduce downstream flooding. In some situations, agreements are made between the operators and water suppliers to allow the desig- nated conservation storage to be raised or lowered permanent- ly, seasonally, or as a function of other changing conditions (Cromarty et al. 1982). The state thus finds itself with the ever-growing demands for reliable water supplies frustrated by the challenges of new res- ervoir construction due to economic, financial, environmen- tal, and institutional considerations. However, reallocation of flood storage capacity and related modifications in the opera- tion of existing reservoirs is difficult because it is contentious, and the authors recognize that it might not be possible if the U.S. Army Corps of Engineers (USACE) deems it inappropri- ate. But the studies on reallocated storage can be used in opti- mization strategies involving the movement of water between reservoirs, rivers, and other water supply infrastructure. The emergence and widespread adoption of sophisticated models such as RiverWare are enabling entities to view their systems in ways that were heretofore impossible and explore cooperative efforts between entities with disparate water supply infrastruc- tures. This has the possibility of significantly increasing the efficiency of water supply systems, offsetting the need to con- struct new reservoirs, and providing in-stream environmental benefits.1 The Trinity River Basin is located in east-central Texas. The watershed begins to the north within a few miles of the Red River and the Oklahoma border. From there it flows to the south and east some 700 miles before emptying into the Gulf of Mexico. The river is an important source of water, supply- ing or supplementing supplies for approximately half of the 1 Please note that co-author Glenn Clingenpeel is currently working with the USACE on a long-term project that would allow use of the flood pool for water supply. He also co-presented the work on this project at Texas Water Conservation Association’s conference in June 2021 under the topic “Fore- cast Informed Reservoir Operations (FIRO).” This methodology is being implemented successfully in California. state’s population. This is due almost exclusively to the net- work of reservoirs located throughout the 18,000-square-mile watershed. These reservoirs are owned and operated by several different entities, most notably, USACE, the Tarrant Regional Water District, the North Texas Municipal Water District, and the Trinity River Authority (TRA). The Joe Pool Lake (JPL) reservoir is located in the Dallas-Fort Worth Metroplex. It is owned and operated by USACE and provides both water supply and flood control benefits. Com- pleted in 1989, JPL reservoir has a capacity of 176,900 ac-ft and collects runoff from a watershed area of 232 square miles, as shown in Figure 1 (Ghimire 2014). Technical information about reservoir and reservoir storage data in daily time steps is available on the USACE Fort Worth District website (USACE n.d.). Table 1 describes technical fea- tures of Joe Pool Dam (Demirel and Wurbs 2015). TRA is exploring the possibility of expanding the con- servation, or water supply, pool capacity of JPL reservoir to address growing water demand in the region along with alle- viating potential shortages during droughts. One possibility is to convert flood pool storage into water supply storage and to capture and store treated wastewater from the TRA Cen- tral Regional Wastewater System (CRWS). Treated wastewater is increasingly becoming an important source of water in the region containing the upper portion of the Trinity River (the Region C regional water planning area, as defined by the Tex- as Water Development Board). Economic and environmen- tal issues together are one of the main factors for reusing the wastewater treatment (Haghiri et al. 2018; Gheytaspour and Texas Water Journal, Volume 12, Number 1 Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow4 Table 1. Technical details of Joe Pool Lake storage. Figure 2. System design of JPL reservoir diversion and sale of stored return flows. River Basin, water demand in the northern portion of the Trin- ity River Basin (henceforth referred to as the northern mar- ket), water supply from outside the Trinity River Basin, and the operating strategy. The operating strategy will consist of a set of dynamic decisions regarding the amount of return flow to be sold to the neighboring cities or stored at JPL reservoir, the capture and release of runoff, and the amount of stored water to be sold at a given year. The term runoff is used to distinguish between runoff (sometimes called natural flow) and inflows, which include return flows and imported water. These deci- sions will be formed by considering hydroclimate conditions, water demand, water rights, and contractual and regulatory requirements. Thus, the objective of this project is to deter- mine optimal conversion capacity and associated operational Habibzadeh Bigdarvish 2018; Asgari-Motlagh et al. 2019); however, the reuse of treated wastewater also greatly depends on public agreement, awareness, and support (Daghighi et al. 2020). TRA holds both the water rights for treated wastewa- ter discharged from their facilities, henceforth referred to as return flow, and the water stored in JPL reservoir. Increasing the conservation capacity of JPL reservoir will provide opera- tional flexibility to store and sell a portion of the return flows to adjacent cities. Figure 2 provides schematic design details of this research design. The cost-effectiveness of the JPL reservoir conversion project is determined by a number of factors, among which are hydro- climate conditions (future precipitation amounts and inflow volumes) over both the upper and lower portions of the Trinity Feature Elevation (feet) Accumulative (acre-feet) Incremental (acre-feet) Spillway capacity (cubic feet per second) Top of dam 564.5 --- --- --- Maximum design water 559.5 642,400 279,700 11,900 Spillway crest 541 362,700 58,700 --- Top of flood control pool 536 304500 127600 --- Top of conservation pool 522 176,900 142,900 --- Sediment reserve --- --- 38,000 --- Streambed 456 --- --- --- Texas Water Journal, Volume 12, Number 1 5Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow strategies that would minimize an objective function that fac- tors in conversion cost, revenue from water sales, and demand/ supply gap, while meeting contractual and regulatory require- ments. Mixed integer linear programming (MILP) is one of the most common and user-friendly techniques for water res- ervoir optimization (Eusuff and Lansey 2003; Ghahraman and Sepaskhah 2004; Daghighi et al. 2017; Samani and Mottaghi 2006). We will proceed by formulating a MILP problem that superimposes realistic runoff and return flow on known histor- ical hydro-climate conditions over multiple years. The Texas Commission on Environmental Quality (TCEQ) is charged with issuing water rights in Texas based upon a legal framework known as the prior appropriation doctrine. This requires an accounting and allocation of available water based on seniority, with the oldest water rights being the most senior. When there is insufficient water to satisfy all water rights, water is allocated first to the most senior rights. The model that TCEQ uses to account for and allocate water among water rights holders is called the Water Rights Analysis Package, or WRAP (Wurbs 2001). WRAP can be used to evaluate the impact of various demand and permit scenarios on water rights, based on historic hydro-climate conditions. These scenario-based model runs are called Water Availability Model runs, or WAM runs for short. The hydrology upon which they are based represents a docu- mented and accepted set of hydrologic conditions that include the drought of record and is used to evaluate the reliability of water rights. This set of hydrologic conditions will be used as the basis of evaluation for MILP. WAM Run 3, last updated on October 7, 2014, will be used in the study. MATERIALS AND METHODS Definition of input variables Table 2 specifies the input data that will be required to pop- ulate the MILP problem. Most values will change from year to year and are represented as a function of time. But values that are assumed to be static across all years are represented by themselves. Abbreviations are used for Joe Pool Lake (JPL), Lake Livingston (LLP), and CRWS. For the first tier return flow volume (V1(t)), MILP requires the available volume after accounting for channel or evaporative losses. In addition, losses can be incorporated in the fractions β and γ. The MILP formu- lation allows input parameters to vary by year except for those values pertaining directly to the JPL reservoir expansion. Definition of decision variables Table 3 defines the MILP formulation by the decision vari- ables, objective function, and constraints. The primary deci- sion is the expansion of JPL reservoir, the secondary decisions are year-to- year volumes of water sold, and slack variables are used in a penalty approach for demand and reservoir overflow. The expansion of JPL reservoir is assumed static across all years. The resolution of the MILP is annual, so it is assumed that every day is identical and that there are 365 days per year. Under this assumption, the annual volume of first tier return flow diverted to storage in JPL reservoir and the volume of water from JPL reservoir sold to northern region values are Input Detail CJPL Cost to expand one-unit volume of Joe Pool Lake (JPL) reservoir CD (t) Penalty cost per volume of demand not met rJPL (t) Revenue selling from JPL reservoir to northern region per volume sold rN (t) Revenue selling from the Central Regional Wastewater System (CRWS) to northern region per volume sold rS (t) Revenue selling from CRWS to southern region per volume sold β (t) Fraction of JPL reservoir first tier flow sold to northern market that returns via CRWS (adds to second tier flow) γ (t) Fraction of diverted first tier flow that reaches storage into JPL reservoir (loss is channel and evaporative loss) V1 (t) Volume of available first tier water (different for wet vs. dry years) V2 (t)* Volume of available second tier water RLLP (t) Salable runoff volume into Lake Livingston (can vary by year, but for current study fixed at 351,600 acre-feet). It is inflow, but it is only that fraction of inflows that resulted from runoff in the watershed. DN (t) Demand volume for the northern region DS(t) Demand volume for the southern region (constant across different years) W Upper limit on JPL reservoir expansion *The volume of available second tier water can be calculated via V2 = β(yN + 365 yJPL). Table 2. Linear programming problems inputs. Texas Water Journal, Volume 12, Number 1 Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow6 distributed equally per day. Finally, penalty costs per volume of unmet demand for the northern and southern regions are assumed the same. Formulation of linear program The objective function is formulated to minimize the costs minus the revenue obtained. The costs involved in the equation include the cost to expand JPL reservoir (CJPL x), a penalty cost for not meeting demand (CD (sN+ sS)). The penalty coefficient CD can be set to zero if no penalty is desired. Relevant JPL reservoir storage per day consists of the diverted first tier return flow that reaches JPL reservoir (γ xd). Revenue comes from three basic sources (JPL reservoir, CRWS, and LLP), which are permitted to have different revenue rates (rJPL, rN, rS) in the MILP. The revenue from JPL reservoir is calculated for the volume of water sold from JPL reservoir for the year (365 yJPL). The revenue on first tier return flow from CRWS that is sold to the northern region is denoted by yN. The revenue on volume sold to the southern region (yS) only comes from LLP but is a mix of return flows and runoff. The objective in Equation 1 is obtained by summing up the costs and subtracting the revenues. (1) In Equation 2, the JPL reservoir expansion cannot exceed a user-specified maximum (W). In Equation 3, the annual vol- ume of first tier return flow that is diverted to JPL reservoir cannot exceed the annual volume of first tier return flows (V1). Also, in Equation 4, the diverted flow cannot exceed the avail- able JPL reservoir expansion capacity, where the capacity of the JPL reservoir expansion (x) is reduced by the volume of runoff stored in the JPL reservoir expansion (RJPL (x, t)) and the vol- ume of carryover stored in JPL reservoir from the previous year (IJPL(t)). Currently, the carryover in year 1 is set to 0. But the value for the carry-over in year 1 is formulated as an input and can be set to any starting value. In Equation 5, the volume of carryover for the current year is represented as a state transition equation, which is the difference between the aggregate of the volume of water carried over from the previous year, the firm yield of JPL reservoir, and the diverted flow reduced by channel losses and the volume of water sold from JPL reservoir. The firm yield is the minimum amount of water that can be divert- ed from a reservoir on an annual basis under the hydrologic conditions of the drought of record (1952–1957). It is more loosely defined as the reliable amount of water that a reservoir can produce through a severe drought. The runoff stored in the JPL reservoir expansion (RJPL(x, t)) depends on the available JPL reservoir expansion capacity and the year under consideration. The runoff stored in the JPL res- ervoir expansion is obtained by using a piecewise linear approx- imation on the lookup table provided by TRA (impoundment targets from WAM). (2) (3) (4) (5) In the MILP model, we formulate decision variables that consider the expansion of JPL reservoir, which will enable sell- ing from JPL reservoir to the northern region (yJPL). Some frac- tion (β) of the volume sold to the northern region (365yJPL + yN) will return via CRWS and become second tier return flow. At that point, this flow must now go to LLP. Min CJPL x + CD(t) [sN(t) + sS(t)] – rJPL(t) [365 yJPL(t)] – rN(t) yN(t) – rs(t) ys(t) x ≤ W 365 xd(t) ≤ V1(t) 365 xd(t) ≤ 365 x – RJPL(x, t) – IJPL(t) IJPL(t+1) = IJPL(t) + (0.1x) + 365 γ(t) xd(t) - 365 yJPL(t) Input Detail x Increased volume of Joe Pool Lake (JPL) reservoir xd (t) Volume/day of first tier return flow diverted to storage in JPL reservoir yJPL (t) Volume/day of water from JPL reservoir (mix of return flows and runoff) sold to northern region yN (t) Volume of first tier return flow sold to northern region yS (t) Volume from LLP (mix of return flows and runoff) sold to southern region sN (t) Slack variable for northern region demand sS (t) Slack variable for southern region demand sJPL (t) Slack variable for overflow in JPL reservoir IJPL (t) Volume of water available at the start of the year in JPL reservoir (carryover) RJPL (x,t) Salable runoff volume into JPL reservoir (table values from the Water Availability Model are a function of year t and the size of the JPL reservoir expansion x) Table 3. Mixed integer linear programming formulation variables. 7 To prevent the MILP from going unbounded and to make sure that only the available amount of water is sold, selling con- straints are specified in Equations 6, 7, and 8 for the amount sold to the northern region from CRWS (yN), the amount sold to the northern region from JPL reservoir (yJPL), and the amount sold to the southern region (yS). It is required that at least 30% of first tier return flows from CRWS must flow to LLP. The first tier return flow volume from CRWS is the input V1. The first tier return volume that does not flow to LLP con- sists of the volume from CRWS that is sold to the northern region (yN) and the volume diverted to JPL reservoir storage (365 xd). Hence, the numerator in Equation 6 is the first tier return flow that reaches LLP. The 30% requirement in Equa- tion 6 subsequently restricts the volume that can be sold to the northern region annually (yN), as well as the volume that can be diverted to JPL reservoir per day (xd) (6) Equation 7 presents Equation 6 with a different format in MILP form. (7) In Equation 8, the volume sold to the southern region (yS) is limited by the first tier return flow that reaches LLP, the vol- ume available second tier return flows, and the amount of run- off in LLP. (8) In Equation 9, the volume sold to the northern region from JPL reservoir (yJPL) is limited by the firm yield of JPL reservoir, which is a function the expansion, the volume diverted to JPL reservoir storage (γ xd), and the volume of carryover stored in JPL reservoir from the previous year (IJPL(t)). (9) Given water volume demand requirements for the northern and southern regions (DN and DS), constraints are needed to meet demand. These demand values are based on what TRA has been able to sell historically or based on existing contracts. This is formulated as a hard constraint because TRA cannot sell to more than the available pool of customers. The constraints are stated separately for the two regions, as in Equations 10 and 11: (10) (11) In addition, it is desirable to satisfy customer demand. To avoid the instance of an infeasible solution, we have incorpo- rated a penalty approach to encourage that demand is met. The penalty cost (CD) on failing to meet demand was seen earlier in the objective function in Equation 1. The constraints to satisfy demand are stated separately for the two regions as Equations 12 and 13, respectively: (12) (13) Finally, bounds must be specified on nonnegative decision vari- ables, as is presented by Equation 14: (14) RESULTS AND DISCUSSION The code was tested with different inputs to check whether the MILP appropriately adjusts its optimal solution for differ- ent inputs. Because the purpose of the MILP is a hypothet- ical exploration, unrealistic parameter settings in the current system may be used in the MILP. All runs were executed for the years 1940 through 1996, with this period of hydrological record chosen because it covers a scenario where there was at one time excess water in JPL reservoir, followed by a drought of record. This sequence of events could help determine wheth- er the MILP performs according to the purpose for which it was designed, i.e., storing water in JPL reservoir storage when excess water is available and selling it in years when there is deficit in supply. The cost for expansion per acre feet of JPL res- ervoir was identified by TRA’s board and is an important input for JPL reservoir expansion. Another input was the demand function based on the volume of available water to store in JPL reservoir (supply) and designed to mimic the demand function in real life. Slack variables in Equations 12 and 13 were used to track whether customer demand was met. If slack variables (sN(t),sS(t)) are positive, customer demand was not fully met, with the value of the slack variable equal to the volume of unmet demand. If the slack variables are zero, then customer demand was fully satisfied, but there may also be the possibility of surplus volume from JPL reservoir, CRWS, or LLP. The MILP was created using Matlab, and the Matlab run- time executable code has been made freely available to TRA for its own use. yN (t)≤ 0.70 V1(t) – 365 xd(t) 365 yJPL(t) + yN (t) + sN(t) ≥ DN(t) x, xd(t), yN(t), yS(t), yJPL(t), sN(t), sS(t), IJPL(t) ≥ 0 yS(t) + sS(t) ≥ Ds(t) 365 yJPL(t) ≤ (0.1x) + 365 γ(t) xd(t) + IJPL(t) 365 yJPL(t) + yN(t) ≤ DN(t) yS(t) ≤ DS (t) yS(t) ≤ V1(t) – [yN(t) + 365 xd(t)] + β(t) [yN(t) + 365 yJPL(t)] + RLLP(t) Texas Water Journal, Volume 12, Number 1 Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow8 Table 4. Mixed integer linear programming Scenario 1 inputs. Input Detail CJPL $88.82/acre-foot (ac-ft) for the first 20 years and no cost for the remaining years CD (t) $100/ac-ft/year rJPL (t) $95/ac-ft/year rN (t) $95/ac-ft/year rS (t) $95/ac-ft/year β (t) 0.50 γ (t) 0.89 RLLP (t) Fixed at 351,600 ac-ft/year DN (t) Represented by black line in Figure 3 W 123,100 ac-ft Scenario 1 Scenario 1 was a scheme devised by TRA’s board. The inputs used for this scenario are specified in Table 4. Figure 3 shows the hydrology in JPL reservoir lake reservoir and the demand/supply scenario in the northern market. Our results showed that expanding the JPL reservoir is not worthwhile because the cost of conversion is very high when compared to the revenue from JPL reservoir. Even though there is unmet demand and penalty on the unmet demand, the MILP chose not to convert JPL reservoir. Figure 4 shows the cost and revenue for the entire system, including the northern and southern market. Another result is that the payback period, assuming a cost of conversion per acre-foot of $88.82 and a revenue of $95 per acre-foot, was 170 years, as shown in Figure 5. Figure 3. Joe Pool Lake (JPL) reservoir hydrology for Scenario 1. Texas Water Journal, Volume 12, Number 1 9Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow Figure 4. Cost and revenue for entire system for Scenario 1. Figure 5. Payback period calculation for Scenario 1 (AF – acre-feet). Figure 6. Joe Pool Lake (JPL) reservoir hydrology for Scenario 2. Texas Water Journal, Volume 12, Number 1 Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow10 Scenario 2 Scenario 2 was performed as a theoretical exercise using a cost of expansion per acre-foot of JPL reservoir from the result of a break-even analysis and to test the functionality of the MILP model. The break-even analysis identified the cost of expan- sion per acre-foot of JPL reservoir at which it would be feasible to expand the reservoir given the demand/supply scenario and revenue from expansion per acre-foot of JPL reservoir. The cost of expansion per acre-foot of JPL reservoir was determined to be around $20, so the MILP was run with this input to check whether the MILP performed as expected. Table 5 shows the inputs used for Scenario 2. The MILP solution recommends expansion of the JPL reser- voir to around 40,000 ac-ft, and Figure 6 shows the hydrology in JPL reservoir and the demand/supply scenario in the north- ern market. Our results showed that the excess water available when there was sufficient rainfall in the earlier years after meeting the demand was stored and carried over to the later years when there was a rainfall deficit. The results also showed that the demand was met using the additional carryover water when there was not enough water to meet the demand with just the first tier flowing. Despite this compensation, there was unmet demand in a few years because there was not enough water available to carry over, due to most of the surplus water from earlier years Figure 7. Cost and revenue for entire system for Scenario 2. Figure 8. Payback period calculation for Scenario 2 (AF – Acre Feet). Table 5. Linear programming Scenario 2 inputs. Input Detail CJPL $20/acre-foot (ac-ft) for the first 20 years and no cost for the remaining years CD (t) $100/ac-ft/year rJPL (t) $95/ac-ft/year rN (t) $95/ac-ft/year rS (t) $95/ac-ft/year β (t) 0.50 γ (t) 0.89 ARLLP (t) Fixed at 351,600 ac-ft/year DN (t) Represented by black line in Figure 6 W 123,100 ac-ft Texas Water Journal, Volume 12, Number 1 11Optimizing Water Supply through Reservoir Conversion and Storage of Return Flow being used to compensate for the drought of record. Another reason for this result is that there was an effective evaporation coefficient of 0.89 (or 89%) applied to the water stored in the reservoir, which in turn impacts the amount of water available to be carried over the water to the following years. Figure 7 shows the cost and revenue for the entire system, including the northern and southern market. Another result is that the payback period for Scenario 2, assuming a cost of conversion per acre-foot of $20 and a rev- enue of $95 per acre-foot, was 40 years, as shown in Figure 8. CONCLUSION Because of growing water demand and water shortage in the Dallas-Fort Worth Metroplex, TRA is considering expanding the capacity of JPL reservoir. The two main sources for JPL res- ervoir are the natural stream flow and the return flow from the CRWS. Based on these inputs, TRA tries to meet the demand in dry years by creating additional storage and diverting excess water to the storage during wet years. The optimal conversion capacity of JPL reservoir was determined by modeling the hydrology of the different sources as an MILP with the objec- tive of minimizing the costs of expansion. Several scenarios with different penalty costs and costs of expansion per acre foot of JPL reservoir were run to determine whether the designed MILP could store sufficient water in JPL reservoir during wet years and sell it during a drought period. We presented two scenarios that yielded contrasting results in this paper. Based on the data availability and scenario coverage, years 1940 through 1996 were chosen because there were wet years followed by a drought record for the example and testing timeline. In Scenario 1, the expansion cost for JPL reservoir was taken as $88.82 per acre foot for the first 20 years with no cost for the remaining years. This model did not meet the requirements for expanding JPL reservoir storage because the expansion cost was too high in comparison to the penalty of unmet demand and selling revenue. In Scenario 2, the cost of expansion per acre-foot of JPL reservoir, based on the payback period calculation, was set to be about $20. The model chose to expand the JPL reservoir to 40,000 ac-ft, implying that the model chose to save water during wet years and store it in the JPL reservoir to meet the demand during drought. Even though the model attempted to meet water demand during dry years, there was still some unmet demand in the later years because there was not suffi- cient water to meet the demand. Testing the MILP with differ- ent scenarios demonstrates that the MILP works and takes into consideration the different hydrology and inputs. FUNDING SOURCES This research was funded by TRA and a National Science Foundation grant (CMMI #1926792). DATA AVAILABILITY Some data, models, or code that support the findings of this study are available from the corresponding author, Sriv- idya Sekar, upon reasonable request. Data or code available on request are: • Input data for Scenario 1 and Scenario 2; • Output data for Scenario 1 and Scenario 2; • MATLAB code (.m) file; and • Executable MATLAB file (.exe). 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