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

Reservoirs in Series Q2,t Q1,t S1,t S2,t R1,t R2,t K2 K1 B1, B2 – benefits from various purposes Municipal water supply Agricultural water supply Hydropower Environmental Recreation Flood protection

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

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

Highland Lakes Colorado R. Lake Buchannan Inks Lake Llano R. Lake LBJ Q1,t Lake Buchannan S1,t K1 = 918 kaf R1,t Inks Lake S2,t Llano R. Q2,t R2,t =R1,t Lake LBJ S3,t R3,t =R2,t + Q2,t Lake Marble Falls S4,t Pedernales R. R4,t =R3,t Q3,t Lake Travis S5,t K5 = 1,170 kaf R5,t Austin M&I Incremental Flow Channel Losses Rice Irrigation Bay & Estuary

Highland Lakes Continuity Capacity Head vs Storage Release Energy t Time period (month) i Lake (1 = Buchannan, 2 = Inks, 3 = LBJ, 4 = Marble Falls, 5 = Travis, 6=Austin) St,l Storage in lake i in period t (AF) Qt,l Inflow to lake i in period t (AF) Ll Loss from lake i in period t (AF) Rt,l Release from lake i in period t (AF) Ki Capacity of lake i Head vs Storage Hi,t elevation of lake i Release Energy R5,t Release from Lake Travis in period t (AF) XA,t Diversion to Austin (AF/month) XI,t Diversion to irrigation (AF/month) CLt Channel losses in period t (AF/month) XB,t Bay & Estuary flow requirement (AF/month) E,t Energy (kWh) ei efficiency (%) TA target for Austin water demand (AF/year) TI target for irrigation water demand (AF/year) fA,t monthly Austin water demand (%) fI,t monthly irrigation water demand (%)

Objective Municipal Water Supply Irrigation Water Supply Benefits: Try to meet targets Irrigation Water Supply Recreation (Buchanan & Travis) wA weight for Austin demand wT weight for Austin demand wR weight for Lake levels TA,t monthly target for Austin demand TI,t monthly target for irrigation demand TR,i,t monthly target for lake levels, i = Buchanan, Travis TA,t ZA XA,t minimum penalty for missing target in month t target release minimum penalty for missing target in month t target release XI,t TI,t ZI minimum penalty for missing target in month t target elevation hi,t TR,i,t ZR Municipal Irrigation Recreation

Results K1 = Buchannan = 918 kaf K5 = Travis = 1,170 kaf 1,000 acre feet = 1,233,482 m3

Results

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 Zj(x) We want to

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

Example Flood control project for historic city with scenic waterfront Alternative Net Benefit Method Effects 1 $120k Increase channel capacity Change riverfront, remove historic bldg’s 2 $700k Construct flood bypass Create greenbelt 3 $650k Construct detention pond Destroy recreation area 4 $800k Construct levee Isolate riverfront

Maximize Scenic Beauty Alternative Net Benefit 1 $120k 2 $700k 3 $650k 4 $800k 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 1 Objective 2 Maximize Net Benefit Maximize Scenic Beauty Alternative 4 2 3 1

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) feasible region Vilfredo Federico Damaso Pareto

Example A B C D E F 1 2 4 3 x1 x2 Feasible Region Decision Space x1 x2 A B 6 C 2 D 4 E 1 F 3 Evaluate the extreme points in decision space (x1, x2) and get objective function values in objective space (Z1, Z2) Cohen & Marks, WRR, 11(2):208-220, 1973

Example Z1 Z2 A B 30 -6 C 26 2 D 12 E -3 15 F Noninferior set contains solutions that are not dominated by other feasible solutions. Noninferior solutions are not comparable: C: 26 units Z1; 2 units Z2 D: 12 units Z1; 12 units Z2 Which is better? Is it worth giving up 14 units of Z2 to gain 10 units of Z1 to move from D to C? Objective Space Z2 E F D Noninferior set Feasible Region C A Z1 B

Example A B C D E F 1 2 4 3 x1 x2 Feasible Region Noninferior set Z2 Z1 Decision Space x1 x2 A B 6 C 2 D 4 E 1 F 3 Evaluate the extreme points in decision space (x1, x2) and get objective function values in objective space (Z1, Z2) Cohen & Marks, WRR, 11(2):208-220, 1973

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

E w = ∞ F D w = 0 C A B

Tradeoffs 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 Z1 and Z2 in moving from D to C is 14/10, i.e., 7/5 unit of Z1 is given up to gain 1 unit of Z2 and vice versa A B C D E Z2 Z1

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

Methods Generating methods Preference 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

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

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

Example – Amu Darya River Multiple Objectives Maximize water supply to irrigation Maximize flow to the Aral Sea Minimize salt concentration in the system

Results

Results