Multipurpose Water Resource Systems. Water Resources Planning and Management Daene C. McKinney. Q 2, t. Q 1, t. S 1, t. S 2, t. R 1, t. R 2, t. K 2. K 1. Reservoirs in Series. B 1 , B 2 – benefits from various purposes Municipal water supply Agricultural water supply Hydropower
Multipurpose Water Resource Systems Water Resources Planning and Management Daene C. McKinney
Q2,t Q1,t S1,t S2,t R1,t R2,t K2 K1 Reservoirs in Series • 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_Hydro,t R1_Spill,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 918,000 acre-feet Inks LBJ Marble Falls . 1,170,000 acre-feet Travis Lake Austin
Austin M&I Channel Losses Incremental Flow Highland Lakes Colorado R. Q1,t Lake Buchannan K1 = 918 kaf S1,t R1,t Inks Lake S2,t Llano R. Q2,t R2,t =R1,t S3,t Lake LBJ R3,t =R2,t + Q2,t S4,t Lake Marble Falls Pedernales R. R4,t =R3,t Q3,t K5 = 1,170 kaf Lake Travis S5,t R5,t Rice Irrigation Bay & Estuary
Highland Lakes Continuity Capacity 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 (%)
ZA penalty for missing target in month t ZI ZR penalty for missing target in month t penalty for missing target in month t minimum minimum minimum TI,t TR,i,t XI,t hi,t TA,t XA,t target target release elevation target release 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 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 Municipal Irrigation Recreation
Results K1 = Buchannan = 918 kaf K5 = Travis = 1,170 kaf 1,000 acre feet = 1,233,482 m3
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
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.
Z2 Noninferior solutions A 14 C Feasible region 10 B 5 16 5 3 Z1 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
Decision Space Noninferior set x2 Z2 E D 4 1 2 Z1 F C Feasible Region 3 B A x1 Example Evaluate the extreme points in decision space (x1, x2) and get objective function values in objective space (Z1, Z2)
Z2 E F Objective Space D Noninferior set Feasible Region C A Z1 B Example (continued) • 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?
Z2 E D C A Z1 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
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 • 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