1. Tingju Zhu1, Guilherme F. Marques2, Jay R. Lund3
Economic Optimization of Integrated Water Management
and Transfers under Stochastic Surface Water Supply
1International Food Policy Research Institute, Washington, DC
2Universidade Federal do Rio Grande do Sul, Brazil
3Department of Civil and Environmental Engineering, University of California, Davis, CA
World Environmental & Water Resources Congress
2013, Cincinnati, OH

2. Sectors, Choices, and Decision-making Structure
Perennial
o Citrus
o Grapes
o Fruit nuts
Annual
 Cotton
 Field crop
 Truck crop
 Alfalfa
 Misc grain
Long-term
o Toilet
o Dishwasher
o Washing
machine
o Leakage
o Xeriscaping
Short-term
 Toilet dam
 Dry lawn
 Dry shrub

3. Quantifying Seasonal Flow Forecasting Skill















r
KK
r
K
r
K
r
K
rr
r
K
rr
ppp
ppp
ppp
,...,,
...
,...,,
,...,,
21
22221
11211
r
P
Seasonal
Forecasting
Model
Historical
weather data
Seasonal
forecasts /
hindcast
Observed
flow data
 Tf
K
ff
ppp ,...,, 21f
P
 T
K
h
ppp ,...,, 21P

4. Formulation of Three-stage Stochastic Programming
Problem
max Z = -
v=1
V
å c1i × X1iv
i=1
I
å -cpc × IPC - IRv
v=1
V
å
- pj
f
pjk
r
v=1
V
å cR1i XL1ijkv
i=1
I
å +
v=1
V
å cR2l × XL2ljkv
l=1
L
å
æ
è
ç
ö
ø
÷
k=1
K
å
æ
è
çç
ö
ø
÷÷
j=1
K
å
+ pj
f
pjk
r
v=1
V
å b1i XH1ijkv
r
- a1iv +0.5g1iv XH1ijkv
r
( )XH1ijkv
r
( )i=1
I
å +
v=1
V
å b2l X2ljkv
r
- a2lv +0.5g2lv X2ljkv
r
( )X2ljkv
r
( )l=1
L
å
æ
è
ç
ö
ø
÷
k=1
K
å
æ
è
çç
ö
ø
÷÷
j=1
K
å
- cu1mY1m - pj pjk
r
cu2nY2njk
n=1
N
å
k=1
K
å
æ
è
ç
ö
ø
÷
j=1
K
å
m=1
M
å
- pj
f
pjk
r
CWTjk +cr XRjk +cpWPjk( )
k=1
K
å
æ
è
ç
ö
ø
÷
j=1
K
å
Subject to a set of constraints
D1: 1st stage decisions, being made only
once at the beginning of the entire planning
period, and are independent of any
particular hydrological year types.
D2: 2nd stage decisions, being made
when seasonal forecasting becomes
available.
D3: 3rd stage decisions, being made when
actual year type is known.

5.  

















J
j
jTjGj
J
j
j
J
j
j
TCGCP
gPgfPfZ
1
1
2j1211
1
2j1211
)()(
),()(),()(max YYYXXX
jTGqW jjj  ,),( 2j1 XX
jTqSD jjj  ,)1(),( 2j1 YY
  jWGP
J
j
jj 
,0),(
1
2j1 XX
   
   




















J
j
jj
J
j
jjjj
J
j
jjjj
J
j
jTjGj
J
j
j
J
j
j
WGPTqSD
TGqWTCGCP
gPgfPfL
1
2j1
1
2j1
1
2j1
1
1
2j1211
1
2j1211
),()1(),(
),()()(
),()(),()(
XXYY
XX
YYYXXX


Lagrangian
Analytical Analysis of Two-Stage Water Transfer Problem -
Formulation
Constraints
Objective

6. i
W
P
Wf
P
f J
j
j
J
j
j
J
j
j 











 
,
X
),(
X
),(
X
),(
X
)(
1 1i
2j1
1 1i
2j1
1 1i
2j12
1i
11
XXXXXXX

Equimarginal principle (1): Marginal benefit of growing permanent crop X1i equals MV of
irrigation water increase minus the MV of overdrafting the portion of increased irrigation
water use that percolates into the aquifer
kj
WGCW
P
f jG
j
j
,,
X
),(
G
)(
X
),(
)1(
X
),(
2jk
2j1
j2jk
2j1
2jk
2










 XXXXXX 2j1



Equimarginal principle (2): Marginal benefit of growing annual crop X2jk in year type j equals
the value of the portion of marginal applied water that does not percolate into aquifer plus
the cost to pump the rest of marginal applied water that percolates into the aquifer, in year j.
Analytical Analysis of Two-Stage Water Transfer Problem -
Equimarginal principle

7. i
Y
S
Y
g
P
Y
g J
j
j
J
j
j 








 
,
),(),()(
1 1i
2j1
1 1i
2j12
1i
11
YYYYY

   
kj
S
PY
g
j
j
,,
Y
,,
2jk2jk
2





 2j12j1 YYYY 
Marginal cost of implementing a long-term conservation measure equals the value of
water use reductions resulting from implementing the measure
Marginal cost of implementing a short-term conservation measure k in year type j equals
the value of water conserved from the marginal implementation
j
G
GC
P j
jG
j
j



 ,
)(


Marginal value of irrigation water in year type j equal the marginal cost of groundwater
pumping or recharge in year type j plus the expected marginal value of groundwater
overdraft
21
1
,,
)()(
2
2
2
2
1
11
jj
G
GC
PG
GC
P j
jG
j
j
j
jG
j
j








For any two different year types j1 and j2, the marginal value of irrigation water minus
marginal cost of groundwater pumping or recharging in year type j1 should equal that in j2
Analytical Analysis of Two-Stage Water Transfer Problem -
Equimarginal principle

8. j
T
TC
P j
jT
j
jj





,
)(
Under economically optimal situation the difference between urban water shadow
value and irrigation water shadow value in year type j should equal the marginal cost of
water transfer.
Analytical Analysis of Two-Stage Water Transfer Problem -
Equimarginal principle

9.   
   

 
 
   

 























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
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





















J
j
I
i
ijiRPjRjPjjTj
M
m
J
j
J
j
N
n
njn
M
m
mmjrjj
N
n
njnjmm
K
k
kjkjkkkjk
J
j
I
i
ijijiiijijiiINI
XLcXRcWPcWTUAWTAUcp
YeYeDcpYcpYc
XXXv
XXXvpXIcZMax
1 1
1,
1 1 1 1
22
1
11
1
2211
1
222222
1 1
1111111,
2
1
2
1


jWTUAWTAUXRcapWPqXwXw jjjRjj
K
k
kjk
I
i
iji   
,
1
22
1
11 
jWTUAWTAUqYeYeD jjj
N
n
njn
M
m
mmj   
,)1(
1
22
1
11 
jXwXwXRcapWPp
h
j
L
l
kjk
I
i
iijRjj 

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







  
,0
1 1
22
1
11
Stochastic mass conservation of groundwater aquifer
Capacity constraints: Land, water, infrastructure
Water balance in urban sector
Water balance in ag sector
Two-stage Programming Model
Objective function

10. Discrete probabilities of surface water availability for the
normal, dry, and wet scenarios

11. Base –
Conjunctive
use plus
water
transfers
noWT –
Conjunctive
use without
water
transfers
NoCU –
Water
transfers
without
conjunctive
use
NWNC – No
conjunctive
use plus no
water
transfers
Inflow
Scenario -
Normal
X X X X
Inflow
Scenario - Dry
X X X X
Inflow
Scenario -
Wet
X X X X
Hydrologic and Water Management Scenarios

12. -200
-150
-100
-50
0
50
100
150
200
250
300
0.044
0.264
1.242
4.022
9.652
18.688
30.564
44.252
57.934
69.916
79.862
86.932
91.898
95.196
97.282
98.488
99.228
99.600
99.792
99.892
99.930
99.964
99.982
99.992
99.998
Agriculturalwatersupply,allocationanduse
(106m3/yr)
Non-exceedence frequency of hydrologic year type (%)
U-A transfer
A-U transfer
Artificial recharge
GW pumping
Surface water supply
Ag Use
Agricultural water supply and optimal pumping, recharge,
transfers and use decisions in various year types

13. 0
2000
4000
6000
8000
10000
12000
14000
16000
Base noCU noWT NWNC
Perennialcroparea(ha)
Normal Dry Wet
Perennial crop areas under various surface water availability
and management scenarios

14. 60
70
80
90
100
110
120
130
140
150
80 130 180 230 280 330Irrigationwateruse(106m3)
Surface water availability (106 m3)
Base NoCU
NoWT NWNC
0
500
1000
1500
2000
2500
80 130 180 230 280 330
Annualcroparea(ha)
Surface water availability (106 m3)
Base NoCU
NoWT NWNC
Annual crop areas and water uses under normal surface
water availability scenario

15. -40
-20
0
20
40
60
80
100
0.044
0.264
1.242
4.022
9.652
18.688
30.564
44.252
57.934
69.916
79.862
86.932
91.898
95.196
97.282
98.488
99.228
99.600
99.792
99.892
99.930
99.964
99.982
99.992
99.998
Urbanwatersupply,conservation,transfers
anduse(106m3/yr)
Non-exceedence frequency of hydrologic year type (%)
Dry lawn
Leakage control
Toilet upgrade
A-U transfer
U-A transfer
Surface water supply
Urban water use
Urban water management decisions under normal surface
water availability scenario and base case water mgt

16. 0
10
20
30
40
50
60
70
80 130 180 230 280 330
Urbanwateruse(106m3)
Surface water availability (106 m3)
Base NoCU
NoWT NWNC
Urban water uses under normal surface water availability
scenario

17. -30
-20
-10
0
10
20
30
88.0
97.7
107.1
116.3
125.5
134.8
144.1
153.5
162.9
172.2
181.6
191.1
200.6
210.1
219.4
228.8
238.1
247.7
256.8
266.8
276.4
285.4
294.1
304.0
314.7
WaterTransfer(06m3)
Surface water availability (106 m3)
(a) Normal
-40
-30
-20
-10
0
10
20
30
79.5
87.5
96.3
105.1
114.0
122.9
131.8
140.8
149.8
158.8
167.8
176.8
185.8
194.8
203.8
212.9
222.4
231.2
240.0
248.6
257.5
266.9
276.0
288.6
294.6
WaterTransfer(06m3)
Surface water availability (106 m3)
(b) Dry
-30
-20
-10
0
10
20
30
115.0
124.4
133.7
142.9
152.1
161.5
170.9
180.3
189.8
199.2
208.6
218.1
227.5
237.1
246.6
256.0
265.7
274.9
285.1
294.0
303.2
312.6
322.9
333.0
344.1
WaterTransfer(06m3)
Surface water availability (106 m3)
(c) Wet
Water transfer from agricultural sector to urban sector (A-U)
and visa verse (U-A)

18. -100
-50
0
50
100
150
200
88
98
107
116
126
135
144
154
163
172
182
191
201
210
219
229
238
248
257
267
276
285
294
304
315
Waterquantity(106m3)
Surface water availability (106 m3)
-100
-50
0
50
100
150
200
88
98
107
116
126
135
144
154
163
172
182
191
201
210
219
229
238
248
257
267
276
285
294
304
315
Waterquantity(106m3)
Surface water availability (106 m3)
Groundwater management in the base (a) and NoWT (b)
water management cases, under normal surface water
scenario
(a) Base
(b) NoWT

19. 0
200
400
600
800
1000
1200
1400
1600
80 100 120 140 160 180 200 220 240 260 280 300 320 340
Marginalexpectedvalue(000$/106m3)
Surface water availability (106 m3)
Base
NoCU
NoWT
NWNC
0
200
400
600
800
1000
1200
1400
1600
80 100 120 140 160 180 200 220 240 260 280 300 320 340
Marginalexpectedvalue(000$/106m3)
Surface water availability (106 m3)
Base
NoCU
NoWT
NWNC
Marginal expected value of water in the agricultural district
and urban area for the four management cases under
normal surface water availability scenario
(a) Agricultural (b) Urban

20. 0
200
400
600
800
1000
1200
1400
80 100 120 140 160 180 200 220 240 260 280 300 320 340
Marginalexpectedvalue(000$/106m3)
Surface water availability (106 m3)
(a) Agriculture
Normal
Dry
Wet
0
200
400
600
800
1000
1200
1400
1600
80 100 120 140 160 180 200 220 240 260 280 300 320 340
Marginalexpectedvalue(000$/106m3)
Surface water availability (106 m3)
(b) Urban
Normal
Dry
Wet
Marginal expected value of water in (a) the agricultural area
and (b) urban center under normal, dry and wet surface
water availability scenario, base case management

21. Inflow Management
Agricultural benefit Urban cost System net benefit
Perennial
crops Total
Permanent
conservation Total Value
Change from base
(%)
Normal Base 142.9 142.4 -2.0 -23.4 116.7 0.0
NoCU 118.4 119.8 -2.4 -24.6 93.1 -20.2
NoWT 143.0 144.0 -2.4 -45.1 98.9 -15.2
NWNC 109.9 111.6 -2.4 -45.1 66.5 -43.0
Dry Base 137.0 136.4 -2.4 -23.3 110.3 0.0
NoCU 110.1 111.3 -2.4 -25.1 83.6 -24.2
NoWT 141.4 142.1 -3.1 -50.7 91.4 -17.1
NWNC 102.9 104.6 -3.1 -50.7 53.8 -51.2
Wet Base 144.2 145.2 -0.4 -22.1 121.6 0.0
NoCU 137.2 138.7 -2.4 -24.0 113.6 -6.5
NoWT 144.2 145.6 -2.4 -34.4 111.2 -8.5
NWNC 127.9 129.6 -2.4 -34.4 95.2 -21.7
Benefit and Cost – Three Inflow Scenarios & Three
management Scenarios

22. Conclusions
 Urban and agricultural water users have significant
ability to adjust to imperfect water supply reliability
through various water conservation and crop
production decisions
 Water transfers provide local incentives to facilitate
coordinated urban and agricultural water
conservation and water transfers
 Conjunctive use and water transfer operations
complement each other and increase flexibility in
local water management facing uncertain surface
water supply