Water Resources Planning and Management

Daene C. McKinney

Optimization

ReservoirsHoover Dam

158 m35 km3

2,074 MW

Grand Coulee Dam100 m

11.8 km36,809 MW

Toktogul Dam140 m

19.5 km31,200 MW

Dams

• Masonry dams– Arch dams

• Gravity dams

• Embankment dams• rock-fill and earth-fill dams

• Spillways

Reservoir

Qt Rt

St

K

Rt

K

St

Qt

Example• Allocate reservoir release Rt to 3 users and provide instream flow Qt

Operating Policy Allocation Policy

release Rt

inflow It

storage St

Optimization

• Benefit

• Decision variables

• Objective:

• Constraints:

Optimization model

Simulation

Operating Policy

Allocation Policy

Simulation vs Optimization• Simulation models: Predict response to given design• Optimization models: Identify optimal designs or policies

Modeling Process• Problem identification

– Important elements to be modeled – Relations and interactions between them– Degree of accuracy

• Conceptualization and development– Mathematical description– Type of model – Numerical method - computer code– Grid, boundary & initial conditions

• Calibration– Estimate model parameters– Model outputs compared with actual outputs– Parameters adjusted until the values agree

• Verification– Independent set of input data used – Results compared with measured outputs

Problem identificationand description

Model verification & sensitivity analysis

Model Documentation

Model application

Model calibration & parameter estimation

Model conceptualization

Model development

Data

Present results

Example – Water Users

Allocate release to users and provide instream flow

Obtain benefits from allocation of xi, i = 1,2,3

Bi(xi) = benefit to user i from using amount of water xi

3,2,1)( 2 ixbxaxB iiiiii

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10Allocation, x

Ben

efit

, B

B1

B2

B3

Example

• Decision variables:

)(

3

1

2

QRx

subject to

xbxa maximize

ii

3

1iiiii

x

Note: if sufficient water is available the allocations are independent and equal to

How?

8,33.2,3 *3

*2

*1 xxx

• Objective:

• Optimization model:

• Constraint:

3

1iiiii xbxamaximize )( 2

QRxxx 321

3,2,1, ixi

Optimization Problems

Objective function

Decision variables

Constraint set

Optimization Problems

while satisfying constraints

x

f(x)

x*

minimum

x*

x

f(x)

Xa b

X={x: a<x< b}Feasible region

Find the decision variables, x, that optimize (maximize or minimize) an objective function

Example

22221

2121 3),( Minimize xxxxxxxf

032

02

212

211

xxx

f

xxx

f

2

1*2

*1

x

x

01

23

0

1

2

3

-4

-2

0

2

4

6

8

10

F(x1,x2)

X2

X1

Existence of Solutions

• Weierstrass Theorem – Describes conditions on the objective function and the

constraint set so that we are guaranteed that solutions will always exist• Constraint set is compact (closed and bounded)• Objective function is continuous on the constraint set

x*

x

f(x)

Xa b

X={x: a<x< b}Feasible region

Convex Sets

convex

nonconvex

x

y

x

y

If x and y are in the set, then z is also in the set, i.e., don’t leave the set to get from x to y yxz )1( aa

Convex Functions

])1([ yx aaf

)(xf

x y

)( yf

)(xf

)()1()( yx faaf

yx )1( aa x

Line segment joining points on a convex function does not lie below the function

)()1()()]()1()([ yxyx faaffaaff

Linear functions are convex.

Existence of Global Solutions

• Local-Global Theorem (maximization)– Describes conditions for a local solution to be global

• Constraint set is compact and convex• Objective function is continuous on the constraint set and

concave• Then a local maximum is global

x

f(x)

x*

Global maximum

Concave function

X

Solutions – Global or Local?

x

Global Max Local Max

Feasible region: X

f(x)

Global Min

Local Min

Xff

X

xxx

x

)(*)(

and,*

*))(()(*)(

and,*

xxxxx

x

NXff

X

Global Max

Local Max

Solutions

Local - Global Theorem:  

1. If X is convex and f(x) is a convex function, then a local minimum is a global minimum

x

f(x)

x*

Global minimum

Convex function

X

x

f(x)

x*

Global maximum

Concave function

X

2. If X is convex and f(x) is a concave function, then a local maximum is a global maximum

Types of Optimization Problems

NonlinearProgram

Linear

Program

)(

)(

subject to

)( minimize

0xg

0xh

x

x

f

0x

bAx

x

subject to

minimize1

n

iii xc

Classic

Program

0xh

x

x

)(

subject to

)( minimize

f

No ConstraintsSingle Decision Variable

• First-order conditions for a local optimum

x

f(x)

x*

Global minimum

Convex function

X• Second-order conditions

for a local optimum

No constraints

Tangent is horizontal

Curvature is upward

Scalar

x

x

)( minimize f

No ConstraintsMultiple Decision Variables

• First-order conditions for a local optimum

• n - simultaneous equations

No constraintsVector

Classical Program

General Form Example

0xh

x

x

)(

subject to

)( minimize

f

02

subject to

21 Minimize

21

22

21

xx

xx

All equality constraints

Single ConstraintMultiple Decision Variables

One constraint

Vector

Single ConstraintMultiple Decision Variables

Lagrangian

First-order conditionsN+1 equations

Example

3tosubject

Minimize

321

313221

xxx

xxxxxx

03

0

0

0

321

213

312

321

xxxL

xxx

L

xxx

L

xxx

L

2

1

1

1

*

*3

*2

*1

x

x

x

]3[

)]([)(),(

321313221

xxxxxxxxx

hfL

xxxLagrangean

First – order conditionsNotice the signs

Example

• Decision variables:

)(

3

1

2

QRx

subject to

xbxa maximize

ii

3

1iiiii

x

Note: if sufficient water is available the allocations are independent and equal to

How?

8,33.2,3 *3

*2

*1 xxx

• Objective:

• Optimization model:

• Constraint:

3

1iiiii xbxamaximize )( 2

QRxxx 321

3,2,1, ixi

Example

Lagrangean

QRxxbxa,xL

ii

3

1iiiii )(

3

1

2

0

02

02

02

321

3333

2222

1111

QRxxxL

xbax

L

xbax

L

xbax

L

3,2,1

2*

i

b

ax

i

ii

3

1

3

1*

1

22

i i

i i

i

b

QRb

a

First – order conditions Notice the signs

Example

3,2,1)( 2 ixbxaxB iiiiii

5.0;5.1;0.1

8;7;6

321

321

bbb

aaa

Q x1 x2 x3 R

Total Downstream

Flow 5.00 0.18 0.45 2.36 5.64 2.00 2.00 8.00 1.00 1.00 4.00 4.00 2.00 2.00

10.00 1.55 1.36 5.09 2.91 2.00 2.00 15.00 2.91 2.27 7.82 0.18 2.00 2.00 16.00 3.00 2.33 8.00 0.00 2.00 2.67 20.00 3.00 2.33 8.00 0.00 2.00 6.67

i

i

x

B

Q

NB

Equal marginal benefits (slopes)

for all users

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10Allocation, x

Be

ne

fit,

B

B1

B2

B3

x 1*x 2* x 3*

Release Allocation RuleAllocation rule tells you the amount of released

water allocated to each use

Classical ProgrammingVector Case – Multiple Constraints

Lagrangian

First-order conditions N+M equations

M constraintsVector

0xh

x

x

)(

subject to

)( minimize

f

m

iii

)(hfL1

)(),( xxx

0

)...( 11

1

j

mm

jj

m

i j

ii

jj

x

h

x

h

x

f

x

h

x

f

x

L

0xhλ

)(L N equations

M equations

Example

0,edunrestrict

10

5

subject to

,

4 minimize

21

21

1

21

121

xx

xx

x

x x

xxxZ

)10()5(4

),(

2122311121

xxxxxxx

L

x

010

05

02

04

014

212

231

1

313

212

2121

xxL

xxL

xx

L

xx

L

xx

L

solution?validathisIs

5,5

:Suppose

21 xx

From Revelle, C. S., E. E. Whitlach, and J. R. Wright, Civil and Environmental Systems Engineering, Prentice Hall, Upper Saddle River, 1997

Nonlinear Program

General Form Example

)(

)(

subject to

)( minimize

0xg

0xh

x

x

f

0,0

32

subject to

1xln Maximize

21

21

21

xx

xx

x

Reservoir with Power Plant

Hoover Dam

earliest known dam - Jawa, Jordan - 9 m high x1 m wide x 50 m long, 3000 BC

Reservoir with Power Plant

Qt Rt

St

K

Et

Rt

K

St

EtHt

Qt

Reservoir with Power

Tt

RHkE

SHSHH

KS

SRQS

E

ttt

ttt

t

tttt

T

tt

,...,1

2

)()(

tosubject

Maximize

1

1

1

K

Qt Rt

St

Et

Qt Inflows (L3/time period)St Storage volume (L3)K Capacity (L3)Rt Release (L3 /period)Et Energy (kWh)Ht Head (L)k Coefficient (efficiency, units)

Maximize power production given capacity and inflows

Nonlinear