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

2. 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

3. 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

4. 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

5. 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

6. 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 (%)

7. 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

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

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 Zj(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
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

13. 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

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)
feasible region
Vilfredo Federico Damaso Pareto

15. 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): , 1973

16. 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

17. 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): , 1973

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

19. E
w = ∞ F D
w = 0 C A B

20. 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

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 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

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 (Lk) are used

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

26. Results

27. Results