1. Water Resources Planning and Management Daene C. McKinney Multipurpose Water Resource Systems

2. Reservoirs in Series B 1, B 2 – benefits from various purposes – Municipal water supply – Agricultural water supply – Hydropower – Environmental – Recreation – Flood protection Q 1,t K2K2 K1K1 Q 2,t R 1,t S 1,t S 2,t R 2,t

3. Reservoirs in Series Reservoir 1 Reservoir 2 Reservoir 3 Reservoir 4 Reservoir 5 R1,t R2,t R3,t R4,t R5,t R1_Hydro,t R1_Spill,t R2_Spill,t R3_Spill,t R4_Spill,t R2_Hydro,t R3_Hydro,t R4_Hydro,t R5_Spill,t R5_Hydro,t Sometimes, cascades or reservoirs are constructed on rivers Some of the reservoirs may be “pass- through” Flow through turbines may be limited Cascade of Reservoirs

4. Highland Lakes Buchannan Inks LBJ Marble Falls Travis Lake Austin. 918,000 acre-feet 1,170,000 acre-feet

5. Highland Lakes Austin M&I Incremental Flow Channel Losses Bay & Estuary Lake Buchannan Inks Lake Lake LBJ Lake Marble Falls Lake Travis Rice Irrigation Pedernales R. Llano R. Colorado R. R 1,t R 5,t Q 1,t Q 2,t Q 3,t R 2,t =R 1,t R 3,t =R 2,t + Q 2,t R 4,t =R 3,t S 1,t S 2,t S 3,t S 4,t S 5,t K 1 = 918 kaf K 5 = 1,170 kaf

6. Highland Lakes tTime period (month) iLake (1 = Buchannan, 2 = Inks, 3 = LBJ, 4 = Marble Falls, 5 = Travis, 6=Austin) S t,l Storage in lake i in period t (AF) Q t,l Inflow to lake i in period t (AF) L l Loss from lake i in period t (AF) R t,l Release from lake i in period t (AF) Continuity K i Capacity of lake i Capacity T A target for Austin water demand (AF/year) T I target for irrigation water demand (AF/year) f A,t monthly Austin water demand (%) f I,t monthly irrigation water demand (%) R 5,t Release from Lake Travis in period t (AF) X A,t Diversion to Austin (AF/month) X I,t Diversion to irrigation (AF/month) CL t Channel losses in period t (AF/month) X B,t Bay & Estuary flow requirement (AF/month) Release H i,t elevation of lake i Head vs Storage E,t Energy (kWh)  i efficiency (%) Energy

7. Objective Municipal Water Supply –Benefits: Try to meet targets Irrigation Water Supply –Benefits: Try to meet targets Recreation (Buchanan & Travis) –Benefits: Try to meet targets Municipal T A,t ZAZA X A,t minimum penalty for missing target in month t target release Irrigation minimum penalty for missing target in month t target release X I,t T I,t ZIZI Recreation minimum penalty for missing target in month t target elevation h i,t T R,i,t ZRZR w A weight for Austin demand w T weight for Austin demand w R weight for Lake levels T A,t monthly target for Austin demand T I,t monthly target for irrigation demand T R,i,t monthly target for lake levels, i = Buchanan, Travis

8. Results K 1 = Buchannan = 918 kaf K 5 = Travis = 1,170 kaf 1,000 acre feet = 1,233,482 m 3

9. Results

10. What’s Going On Here? Multipurpose system –Conflicting objectives –Tradeoffs between uses: Recreation vs. irrigation –No “unique” solution Let each use j have an objective Z j (x) We want to

11. Multiobjective Problem Single objective problem: –Identify optimal solution, e.g., feasible solution that gives best objective value. That is, we obtain a full ordering of the alternative solutions. Multiobjective problem –We obtain only a partial ordering of the alternative solutions. Solution which optimizes one objective may not optimize the others –Noninferiority replaces optimality

12. Example Flood control project for historic city with scenic waterfront AlternativeNet Benefit MethodEffects 1$120kIncrease channel capacity Change riverfront, remove historic bldg’s 2$700kConstruct flood bypass Create greenbelt 3$650kConstruct detention pond Destroy recreation area 4$800kConstruct leveeIsolate riverfront

13. Example Does gain in scenic beauty outweigh $100k loss in NB? (Alt 4  2) Alternative 2 is better than Alternatives 1 and 3 with respect to both objectives. Never choose 1 or 3. They are inferior solutions. Alternatives 2 and 4 are not dominated by other alternatives. They are noninferior solutions. Objective 1Objective 2 Maximize Net Benefit Maximize Scenic Beauty Alternative 42 23 31 14 Net Benefit 1$120k 2$700k 3$650k 4$800k

14. Noninferior Solutions (Pareto Optimal) A feasible solution is noninferior –if there exists no other feasible solution that will yield an improvement in one objective w/o causing a decrease in at least one other objective –(A & B are noninferior, C is inferior) All interior solutions are inferior –move to the boundary by increasing one objective w/o decreasing another –C is inferior Northeast rule: –A feasible solution is noninferior if there are no feasible solutions lying to the northeast (when maximizing) Vilfredo Federico Damaso Pareto Noninferior Solutions feasible region

15. Example x1x1 x2x2 A00 B60 C62 D44 E14 F03 Evaluate the extreme points in decision space (x 1, x 2 ) and get objective function values in objective space ( Z 1, Z 2 ) A B C D E F 1 2 4 3 x1x1 x2x2 Feasible Region Decision Space Cohen & Marks, WRR, 11(2):208-220, 1973

16. Example Noninferior set contains solutions that are not dominated by other feasible solutions. Noninferior solutions are not comparable: C: 26 units Z 1 ; 2 units Z 2 D: 12 units Z 1 ; 12 units Z 2 Which is better? Is it worth giving up 14 units of Z 2 to gain 10 units of Z 1 to move from D to C? A B C D E F Feasible Region Noninferior set Z2Z2 Z1Z1 Objective Space Z1Z1 Z2Z2 A00 B30-6 C262 D12 E-315 F-612

17. Example x1x1 x2x2 A00 B60 C62 D44 E14 F03 Evaluate the extreme points in decision space (x 1, x 2 ) and get objective function values in objective space ( Z 1, Z 2 ) A B C D E F 1 2 4 3 x1x1 x2x2 Feasible Region Noninferior set Z2Z2 Z1Z1 Decision Space Cohen & Marks, WRR, 11(2):208-220, 1973

18. A B C D F E Slope = -(1/w) = -(1/5) Z = 5 Z = 10 Z = 20

19. A B C D F E w = 0 w = ∞

20. Tradeoffs A B C D E Z2Z2 Z1Z1 Tradeoff = Amount of one objective sacrificed to gain an increase in another objective, i.e., to move from one noninferior solution to another Example: Tradeoff between Z 1 and Z 2 in moving from D to C is 14/10, i.e., 7/5 unit of Z 1 is given up to gain 1 unit of Z 2 and vice versa

21. Multiobjective Methods Information flow in the decision making process –Top down: Decision maker (DM) to analyst (A) Preferences are sent to A by DM, then best compromise solution is sent by A to DM Preference methods –Bottom up: A to DM Noninferior set and tradeoffs are sent by A to DM Generating methods

22. Methods Generating methods –Present a range of choice and tradeoffs among objectives to DM Weighting method Constraint method Others Preference methods –DM must articulate preferences to A. The means of articulation distinguishes the methods Noninterative methods: Articulate preferences in advance –Goal programming method, Surrogate Worth Tradeoff method Iterative methods: Some information about noninferior set is available to DM and preferences are updated –Step Method

23. Weighting Method Vary the weights over reasonable ranges to generate a wide range of alternative solutions reflecting different priorities.

24. Constraint Method –Optimize one objective while all others are constrained to some particular bound –Solutions are noninferior solutions if correct values of the bounds (L k ) are used