Water Resources Planning and Management

Daene C. McKinney

Multipurpose Water Resource Systems

Reservoirs in Series

• B1, B2 – benefits from various purposes– Municipal water supply– Agricultural water supply– Hydropower– Environmental– Recreation– Flood protection

Q1,t

K2K1

Q2,t

R1,t

S1,t S2,t R2,t

ttttt LRQSS ,1,1,1,11,1

tttttt LRRQSS ,2,2,1,2,21,2

1,1 KS t

2,2 KS t

T

ttt BBMaximize

1,2,1

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,tR1_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

Highland Lakes

Buchannan

Inks

LBJ

Marble Falls

Travis

Lake Austin

.

918,000 acre-feet

1,170,000 acre-feet

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.

R1,t

R5,t

Q1,t

Q2,t

Q3,t

R2,t =R1,t

R3,t =R2,t + Q2,t

R4,t =R3,t

S1,t

S2,t

S3,t

S4,t

S5,t

K1 = 918 kaf

K5 = 1,170 kaf

0

50,000

100,000

150,000

200,000

250,000

300,000

0 10 20 30 40 50 60

Month

Flo

w,

Dem

and

(AF

/mon

th)

Flow

Total Demand

Austin

Irrigation

Downstream

Highland Lakestitititititi LRRQSS ,,,1,,1,

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)

Continuity iti KS ,

Ki Capacity of lake i

Capacity

ItItI

AtAtA

TfX

TfX

,,

,,

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 (%)

tBttItAtt XCLXXIR ,,,,5 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)

ReleaseHi,t elevation of lake i

Head vs Storage

2

)()( 1,,,

tiitiiti

SHSHH

E,t Energy (kWh)ei efficiency (%)

Energy

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

TA,t

ZA

XA,t

minimum

penalty for missing target in month t

target release

Irrigation

minimum

penalty for missing target in month t

target releaseXI,tTI,t

ZI

Recreation

minimum

penalty for missing target in month t

target elevationhi,tTR,i,t

ZR

wA weight for Austin demandwT weight for Austin demandwR weight for Lake levelsTA,t monthly target for Austin demandTI,t monthly target for irrigation demandTR,i,t monthly target for lake levels,

i = Buchanan, Travis

Results K1 = Buchannan = 918 kaf

K5 = Travis = 1,170 kaf

No Storage Deficits

0

200

400

600

800

1000

1200

0 12 24 36 48 60Month

Sto

rag

e (1

000

AF

)

S_Buch

S_Trav

No Storage Deficits

0

10

20

30

40

0 12 24 36 48 60Month

Rel

ease

Def

icit

(10

00 A

F)

Austin

Irrigation

No Release Deficits

0

200

400

600

800

1000

1200

0 12 24 36 48 60Month

Sto

rag

e (1

000

AF

)

S_Buch

S_Trav

1170

No Release Deficits

0

100

200

300

400

500

600

700

0 20 40 60Month

Sto

rag

e D

efic

it (

1000

AF

) Buchanan

1,000 acre feet = 1,233,482 m3

Results

Water Supply vs Recreation Tradeoff

y = -5.271x + 4409.6

0

1000

2000

3000

4000

5000

0 200 400 600 800

Release Deficit (1000 AF)

Sto

rag

e D

efi

cit

(1

00

0 A

F)

Weights Release Storage100-100-1 1.61 4423.7910-10-1 16.41 4341.561-1-1 162.45 3531.821-1-2 307.62 2747.641-1-10 752.12 465.86

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

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 Alternative

4 2

2 3

3 1

1 4

Alternative Net Benefit

1 $120k

2 $700k

3 $650k

4 $800k

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

Example

4)4(

6)3(

8)2(

3)1(

4)(

25)(

tosubject

)](),([)(Maximize

2

1

21

21

212

211

21

x

x

xx

xx

xxZ

xxZ

ZZ

x

x

xxxZ

x1 x2

A 0 0

B 6 0

C 6 2

D 4 4

E 1 4

F 0 3

Evaluate the extreme points in decision space (x1, x2) and get objective function values in objective space (Z1, Z2)

AB

C

DE

F

1 2

4

3

x1

x2

Feasible Region

Decision Space

Cohen & Marks, WRR, 11(2):208-220, 1973

Example

• 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?

A

B

C

D

EF

Feasible Region

Noninferior set

Z2

Z1

Objective Space

Z1 Z2

A 0 0

B 30 -6

C 26 2

D 12 12

E -3 15

F -6 12

Example

4)4(

6)3(

8)2(

3)1(

4)(

25)(

tosubject

)](),([)(Maximize

2

1

21

21

212

211

21

x

x

xx

xx

xxZ

xxZ

ZZ

x

x

xxxZ

x1 x2

A 0 0

B 6 0

C 6 2

D 4 4

E 1 4

F 0 3

Evaluate the extreme points in decision space (x1, x2) and get objective function values in objective space (Z1, Z2)

AB

C

DE

F

1 2

4

3

x1

x2

Feasible Region

Noninferior set

Z2

Z1

Decision Space

Cohen & Marks, WRR, 11(2):208-220, 1973

A

B

C

DF

E

Slope = -(1/w) = -(1/5)

Z = 5

Z = 10

Z = 20

A

B

C

DF

E

w = 0

w = ∞

Tradeoffs

)(

)(

x

x

j

i

Z

Z

A

B

C

D

E

Z2

Z1

• 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

0),...,,(

)(

)(

tosubject

)(...)()()(Mazimize

21

11

2211

n

mm

pp

xxx

bg

bg

ZwZwZwZ

x

x

x

xxxx

0),...,,(

)(

)(

tosubject

)}(,),(),({)(Mazimize

21

11

21

n

mm

p

xxx

bg

bg

ZZZ

x

x

x

xxxxZ

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

Constraint Method

0),...,,(

)(

)(

,,...,1)(

tosubject

)(Mazimize

21

11

n

mm

ii

k

xxx

bg

bg

kipiLZ

Z

x

x

x

x

x

0),...,,(

)(

)(

tosubject

)}(,),(),({)(Mazimize

21

11

21

n

mm

p

xxx

bg

bg

ZZZ

x

x

x

xxxxZ

– 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

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2driest dry

normal wet

wettest

Flow to Aral Sea from Amu Darya (km^3/year)

Sup

ply/

Dem

and

Results

Objective weight matrixObjective

Weight Set Supply/Demand Flow toAral Sea

Saltconcentration.

1 1 0 02 0 1 03 0 0 1

WeightSet

1 2 3Objective Value

Supply/Demand 0.99 0.32 0.50Flow to Aral Sea (km^3) 4.71 55.3 46.3Salt concentration in river (g/L) 0.56 0.36 0.30Salt concentration in groundwater (g/L) 0.82 0.38 0.31

Results