Multipurpose Water
Resource Systems
Water Resources Planning and
Management
Daene C. McKinney
Reservoirs in Series
Q2,t
Q1,t
S1,t
K1
• B1, B2 – benefits from
various purposes
–
–
–
–
–
–
Municipal water supply
Agricultural water supply
Hydropower
Environmental
Recreation
Flood protection
R1,t
S2,t
K2
R2,t
Maximize  B1,t  B2,t 
T
t 1
S1,t 1  S1,t  Q1,t  R1,t  L1,t
S 2,t 1  S 2,t  Q2,t  R1,t  R2,t  L2,t
S1,t K1
S 2,t K 2
Reservoirs in Series
Cascade of Reservoirs
Reservoir 1
R1,t
R1_Spill,t
Reservoir 2
R2_Spill,t
Reservoir 3
R3_Spill,t
Reservoir 4
R4_Spill,t
Reservoir 5
R5_Spill,t
Sometimes, cascades or reservoirs are
constructed on rivers
R1_Hydro,t
R2,t
R2_Hydro,t
Some of the reservoirs may be “passthrough”
Flow through turbines may be limited
R3,t
R3_Hydro,t
R4,t
R4_Hydro,t
R5,t
R5_Hydro,t
Rti = Ri _ Hydrot + Ri _ Spillt
Pti = kH ti Rti £ P i,Max
R i _ Hydrot £
P i,Max
k' H ti
kW
Highland Lakes
Buchannan
918,000 acre-feet
Inks
LBJ
.
Marble Falls
1,170,000 acre-feet
Travis
Lake Austin
Highland Lakes
Colorado R.
Q1,t
Lake Buchannan
Inks Lake
Llano R. Q2,t
Lake LBJ
S1,t
K1 = 918 kaf
R1,t
S2,t
R2,t =R1,t
S3,t
R3,t =R2,t + Q2,t
S4,t
R4,t =R3,t
300,000
Q3,t
Lake Travis
S5,t K5 = 1,170 kaf
R5,t
Austin M&I
Channel Losses
Rice Irrigation
Bay & Estuary
Incremental
Flow
Flow, Demand (AF/month)
Lake Marble Falls
Pedernales R.
Flow
250,000
Total Demand
Austin
200,000
Irrigation
150,000
Downstream
100,000
50,000
0
0
10
20
30
Month
40
50
60
Highland Lakes
Continuity
t
i
St,l
Qt,l
Ll
Rt,l
Release
R5,t
XA,t
XI,t
CLt
XB,t
Si ,t 1  Si ,t  Qi,t  Ri 1,t  Ri ,t  Li ,t Capacity Si ,t K i
Time period (month)
Lake (1 = Buchannan, 2 = Inks, 3 = LBJ,
4 = Marble Falls, 5 = Travis, 6=Austin)
Storage in lake i in period t (AF)
Inflow to lake i in period t (AF)
Loss from lake i in period t (AF)
Release from lake i in period t (AF)
R5,t  I t  X A,t  X I ,t CLt  X B,t
Release from Lake Travis in period t (AF)
Diversion to Austin (AF/month)
Diversion to irrigation (AF/month)
Channel losses in period t (AF/month)
Bay & Estuary flow requirement (AF/month)
X A,t  f A,tTA
X I ,t  f I ,t TI
TA
TI
fA,t
fI,t
target for Austin water demand (AF/year)
target for irrigation water demand (AF/year)
monthly Austin water demand (%)
monthly irrigation water demand (%)
Ki
Capacity of lake i
Head vs Storage
H i ,t 
Hi,t
H i ( Si,t )  H i ( Si,t 1 )
2
elevation of lake i
Energy
Ei,t = k i H i,t Ri,t
E,t
ei
Energy (kWh)
efficiency (%)
Objective
2
é æ
æ TI,t - X I,t ö2
æ TR,i,t - hi,t öù
TA,t - X A,t ö
Minimize åêw A ç
÷ + wI ç
÷ + wR å ç
÷ú
t=1ê
i=1&5 è TR,i,t
è TI,t ø
øúû
ë è TA,t ø
T
•
Municipal Water Supply
–
•
Benefits: Try to meet targets
Irrigation Water Supply
–
•
wA
wT
wR
TA,t
TI,t
TR,i,t
Benefits: Try to meet targets
Recreation (Buchanan & Travis)
–
Benefits: Try to meet targets
ZA
penalty for missing
target in month t
ZI
minimum
target
TA,t
release
Municipal
weight for Austin demand
weight for Austin demand
weight for Lake levels
monthly target for Austin demand
monthly target for irrigation demand
monthly target for lake levels,
i = Buchanan, Travis
ZR
penalty for missing
target in month t
minimum
XA,t
target
TI,t
release
Irrigation
penalty for missing
target in month t
minimum
XI,t
target
TR,i,t
elevation
Recreation
hi,t
K1 = Buchannan = 918 kaf
Results
K5 = Travis = 1,170 kaf
No Storage Deficits
1200
Release Deficit (1000 AF)
Storage (1000 AF)
Irrigation
40
1000
800
600
400
S_Buch
200
S_Trav
30
20
10
0
0
0
12
24
Month
36
48
0
60
12
1000
600
Storage Deficit (1000 AF)
700
800
600
S_Buch
S_Trav
200
1170
0
0
12
1,000 acre feet = 1,233,482 m3
24
Month
36
36
48
60
No Release Deficits
1200
400
24
Month
No Release Deficits
Storage (1000 AF)
Austin
No Storage Deficits
48
60
Buchanan
500
400
300
200
100
0
0
20
Month
40
60
Release Storage
1.61 4423.79
16.41 4341.56
162.45 3531.82
307.62 2747.64
752.12
465.86
Results
Water Supply vs Recreation Tradeoff
5000
Storage Deficit (1000 AF)
Weights
100-100-1
10-10-1
1-1-1
1-1-2
1-1-10
4000
y = -5.271x + 4409.6
3000
2000
1000
0
0
200
400
Release Deficit (1000 AF)
600
800
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
Maximize Z(x )=[Z1 (x), Z 2 (x),..., Z p (x )]
subject to
x = (x1 , x 2 ,..., x n ) Î X
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.
Maximize Z1 (x)
subject to
x = (x1 , x 2 ,..., x n ) Î X
• 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
Maximize Z(x)= {Z1 (x), Z 2 (x),
subject to
Z p (x)}
x = (x1 , x 2 ,..., x n ) Î X
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
Alternative
Net
Benefit
1
$120k
2
$700k
3
$650k
4
$800k
Example
Objective 1
Objective 2
Maximize
Net Benefit
Maximize
Scenic Beauty
Alternative
Alternative
4
2
2
3
3
1
1
4
•
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.
Noninferior Solutions
(Pareto Optimal)
Noninferior Solutions
40
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
35
Irriga on Supply
•
30
25
20
feasible region
15
10
5
0
0
5
10
15
Energy Produc on
20
Alternative Energy Production Irrigation Supply
A
22
20
B
10
35
C
20
32
D
12
21
E
6
25
25
Example
 


Maximize Z(x)  [ Z1 (x), Z 2 (x)]
subject to

Z1 (x)  5 x1  2 x2

Z 2 (x)   x1  4 x2
(1)
 x1  x2  3
( 2)
x1  x2  8
(3)
x1  6
( 4)
x2  4
x2
E
4
D
1
2
F
C
Feasible Region
3
x1
x2
A
0
0
B
6
0
C
6
2
D
4
4
E
1
4
F
0
3
Cohen & Marks, WRR, 11(2):208-220, 1973
Decision Space
B
A
x1
Evaluate the extreme points in
decision space (x1, x2) and get
objective function values in
objective space (Z1, Z2)
Example
• Noninferior set contains
solutions that are not
dominated by other feasible
solutions.
Z2
Objective Space
Z2
A
0
0
B
30
-6
C
26
2
D
12
12
E
-3
15
F
-6
12
E
F
D
• Noninferior solutions are not
comparable:
Noninferior set
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?
Z1
Feasible Region
C
A
Z1
B
Example
 


Maximize Z(x)  [ Z1 (x), Z 2 (x)]
subject to

Z1 (x)  5 x1  2 x2

Z 2 (x)   x1  4 x2
(1)
 x1  x2  3
( 2)
x1  x2  8
(3)
x1  6
( 4)
x2  4
Noninferior set
x2
Z2
E
4
D
1
2
Z1
F
C
Feasible Region
3
x1
x2
A
0
0
B
6
0
C
6
2
D
4
4
E
1
4
F
0
3
Cohen & Marks, WRR, 11(2):208-220, 1973
Decision Space
B
A
x1
Evaluate the extreme points in
decision space (x1, x2) and get
objective function values in
objective space (Z1, Z2)
30
Maximize Z = Z1 + w Z2
Z2 = (-1/ w)Z1 + Z / w
St. line : Slope = -1 / w
Intercept = Z / w
e.g., w = 5
Slope = -(1/w) = -(1/5)
20
E
D
Z2
F
10
C
-10
0
A 0
10
20
30
40
B
Z = 10
-10
Z1
Z=5
Z = 20
30
20
w=∞
E
D
Z2
F
w=0
10
C
-10
0
A 0
10
20
30
40
B
-10
Z1
Tradeoffs
• Tradeoff = Amount of one
objective sacrificed to gain an
increase in another objective,
i.e., to move from one
noninferior solution to another

Z i (x)

Z j (x)
• 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
Z2
E
D
C
A
Z1
B
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
 
Mazimize Z ( x )  {Z1 ( x ), Z 2 ( x ),, Z p ( x )}
subject to

g1 ( x )  b1


g m ( x )  bm

x  ( x1 , x2 ,..., xn )  0




Mazimize Z (x)  w1Z1 (x)  w2 Z 2 (x)  ...  w p Z p (x)
subject to

g1 (x)  b1


g m (x)  bm

x  ( x1, x2 ,..., xn )  0
•
Vary the weights over
reasonable ranges to generate
a wide range of alternative
solutions reflecting different
priorities.
Constraint Method
 
Mazimize Z ( x )  {Z1 ( x ), Z 2 ( x ),, Z p ( x )}
subject to

g1 ( x )  b1


g m ( x )  bm

x  ( x1 , x2 ,..., xn )  0

Mazimize Z k ( x )
subject to

Z i ( x )  Li

g1 ( x )  b1
i  1,..., p, i  k


g m ( x )  bm

x  ( x1 , x2 ,..., xn )  0
– 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