[PPT] - Modeling Dynamic Network Modeling Dynamic Network Systems with PowerPoint Presentation

SLIDE 1

Modeling Dynamic Network Modeling Dynamic Network Systems with State-Contingent Penalty Functions Penalty Functions

Richard Howitt, Kristiana Hansen

University of California Davis & Universite de Louvain University of California, Davis & Universite de Louvain

CompSus09 Conference Cornell University Cornell University June 8-12, 2009

SLIDE 2

The Dynamic Network Problem The Dynamic Network Problem

Solved by restricted optimizing models
Two decision aspects

– The Network problem- allocation over a spatial network within a year – The Carryover problem- allocation of states between years with stochastic supplies

Dimensionality restrictions usually prevent

their simultaneous solution

Optimal spatial dynamic policy requires

p p y p y q joint solution

SLIDE 3

Current Solution Approaches

Standard Approach to the Network problem

– Solved by spatial Network Flow Program St h ti h d l t d b hi t i l – Stochastic hydrology represented by historical hydrologic sequences – Problem.. Spatial monthly allocation is nested within p y the annual stochastic state allocation problem

The annual dynamic allocation problem

– Solved by stochastic dynamic programming So ed by s oc as c dy a c p og a g – Synthetic hydrology – Problem.. The curse of dimensionality prevents a realistic spatial specification and dynamic risk and realistic spatial specification and dynamic risk and preferences are hard to specify.

SLIDE 4

A State-Contingent approach

Managers operate with limited foresight.

– They know the current stocks and states – They know the probability of future water year types.

State Contingent Calibration

State Contingent Calibration. – Calibrated to reproduce observed behavior for a set

f water year types.

Observed behavior reflects the effect of agency risk – Observed behavior reflects the effect of agency risk and intertemporal preferences – Having reproduced past water management, we can now optimize under alternative scenarios now optimize under alternative scenarios.

Two sets of nonlinear ( quadratic) calibration functions.

M thl f l t ti l lib ti d – Monthly for select spatial calibration nodes – Annual for storage carryover values

SLIDE 5

Modeling Approach

Characterize a small set of (3-5) years classified as a

given water year type. U t f b d i l t d fl d t

Use sets of observed or simulated flows and storage

with an objective function and calibration constraints for each year.

Solve each year and store the lagrangian values for

nodal and carryover calibration constraints. Obt i th lib ti l f ti b i

Obtain the calibration value functions by regressing on

the lagrange values for each set of years in each water year type. Impose curvature properties on the estimates. y yp p p p

Use the calibration values to simulate spatial dynamic

decisions by solving recursively linked annual

ptimization problems one year Bellman solution
ptimization problems- one year Bellman solution.

SLIDE 6

Case Study- The Northern California Water network network

124 nodes 211 arcs

124 nodes, 211 arcs

13 reservoirs, 9 groundwater basins

15 U b d d i t 9 i lt l

15 Urban demand points, 9 agricultural

demand points.

72 years simulated hydrology
Eight years used for calibration between

g y 1960-1980- normal, dry and drought years. y

SLIDE 7

State Contingent Value Functions- Shasta

1200 1000 1100 800 900 500 600 700 $ per 10 Ac Ft Critical Dry Normal 300 400 500 100 200 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250 4500 4750 5000

SLIDE 8

Sacramento Valley Water network

SR-NBB: New Bullards Bar Reservoir and Englebright Lake SR CFW SR PR SR-NHL New Hogan Lake CALAVERAS RIVER MOKELUMNE RIVER N, M, & S FORK YUBA RIVER, FRENCH DRY CREEK, DEER CREEK GREENHORN CREEK & BEAR RIVER COSUMNES RIVER N, M, and S ACCRETION CAMP FAR WEST TO WHEATLAND GAGE DA67 AG Urban Areas in DA70 but

utside CVPM7

Self-Calibrating Limited Foresight Netflow Model Schematic Adapted from CALVIN Schematic (Draper et al 2003) SR-6: Lake Oroville, Thermalito Fore- Aftebay SR-8 Folsom Lake, Lake Natomas, Nimbus Dam C31 D42 COTTONWOOD CREEK DA 58 LOCAL WATER SUPPLY INC. COW CREEK & BATTLE CREEK ANTELOPE, MILL,DRY,DEER & BIG CHICO CREEKS, DA 10 LOCAL WATER SUPPLY Accretion: American River Folsom to Fair Oaks DA70 local t Losses SR-CFW: Camp Far West Reservoir DA 15 D517 D98 SR-PR Pardee Reservoir / Camanche Reservoir DRY CREEK, LOCAL WATER Hogan Lake LAKE SHASTA FEATHER RIVER, KELLY RIDGE Stockton C39 GW-5 C32 C26 AD 5 AD 7 AD 8 C36 GW-7 C34 Redding DA 10 D37 GW-5 FORKS AMERICAN RIVER C43 C37 Sacramento GW-8 Depletion DA69 DA 15 Sac East Refuges C307 DA 14: BUTTE CREEK & LITTLE CHICO CREEK DA69 local water C311 UD 8 GW-1 UD 5 DA 59 C80 PAYNES AND SEVEN MILE CREEKS DA 70

utside CVPM7

Urban Demand Yuba Urban D5 SR-4 Shasta Lake D76b D77 D73 D66 SR-3 Clair Engle Lake / Whiskeytown Lake D30 D31 D43 D61 TRINITY RIVER, CLEAR CREEK, LEWISTON LAKE INFLOW water Sac West Refuges D517 D503 D507 D511 D515 D525 D523 D522 D521 D513 D550 Tracy Pumping Plant Harvey Banks Pumping Plant SR-CL-IVR Clear Lake/Indian Valley Reservoir SR-LB Lake Berryessa Monticello Dam D59 THE DELTA INFLOW D74 AD 2 AD 6 Contra Costa P i C4 GW-1 GW-6 AD 1 C3 GW-2 C6 C12 C69 C13 C15 GW-4 C14 AD 4 Glenn-Colusa Canal C17 C18 C68 C67 C42 C5 C8 SR BBL Napa-Solano Conties GW- 9 Trinity River Minimum Flows C306 C301 C11 C303 C1 C305 C302 DA12 local water C309 C2 AD 9 C313 C314 UD 2 UD 3 UD 9 UD 4 DA 59 C86 C87 DA 12 DA 65 DA65 local water DA55 local water UD 6 AD 3 CMWD Southern CA Demand C9 REGION 1 THOMES & ELDER CREEKS D528 D509 Contra Costa Required and Surplus Delta Outflow Pumping Plant EBMUD PUTAH CREEK CACHE CREEK SR- EBMU Pumping Plant C71 SR- LV C70 STONY CREEK GW-3 SR-BBL Black Butte Lake C201 C310 Mallard Slough Pumping Plant Walnut Creek Pumping Plant UD 9 DA 55 AD 3 Los Vaqueros Pumping Plant REGION 2

SLIDE 9

Shasta Storage (KAF)-1960-1965 In-sample calibration

5000 6000 4000 5000 3000 base Model 2000 1000 OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP 1960 1961 1962 1963 1964 1965

SLIDE 10

Shasta Storage 1980-1993 ( Out of Sample)

6000 4000 5000

LFN

3000 4000 K A F

LFN Actual

1000 2000

1 9 80 1 9 81 1 9 82 1 9 83 1 9 84 1 9 85 1 9 86 1 9 87 1 9 88 1 9 89 1 9 90 1 9 91 1 9 92 1 9 93 Normal Dry Normal Normal Normal Dry Normal Dry Critical Dry Critical Critical Critical Normal y y y y

SLIDE 11

T i it St 1980 1993 Trinity Storage 1980- 1993 (out of sample)

3000 2500 3000 1500 2000 K A F

LFN Actual

500 1000

Actual

500

1 9 8 0 1 9 8 1 1 9 8 2 1 9 8 3 1 9 8 4 1 9 8 5 1 9 8 6 1 9 8 7 1 9 8 8 1 9 8 9 1 9 9 0 1 9 9 1 1 9 9 2 1 9 9 3 Normal Dry Normal Normal Normal Dry Normal Dry Critical Dry Critical Critical Critical Normal Normal Dry Normal Normal Normal Dry Normal Dry Critical Dry Critical Critical Critical Normal

SLIDE 12

O ill St 1980 1993 Oroville Storage 1980-1993 (Out of Sample)

4500 3500 4000 500 2000 2500 3000 K A F

LFN Actual

1000 1500

ctua

500

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 Normal Dry Normal Normal Normal Dry Normal Dry Critical Dry Critical Critical Critical Normal

a

y

a
a
a

y

a

y C t ca y C t ca C t ca C t ca

a

SLIDE 13

Computation times Computation times

Calibration and Estimation time- 3 year types- 8

years in total– years in total Desktop Time 14.6 minutes

Simulation time desktop – 5.4 minutes/year

average 14 (1980 93) years 1 25 hours average– 14 (1980-93) years 1.25 hours . S l ti ti bl f t th t ti Solution times are comparable or faster than static linear programming network program solutions.

SLIDE 14

Spatial Dynamic Conclusions

The contingent calibrated functions are able to

model spatial dynamic problems using recursive ti i ti

ptimization.

The model reser oir and gro nd ater

The model reservoir and groundwater

management responds well to different year types, particularly drought years. yp , p y g y

Solution times make recursive optimization

p models a practical tool for dynamic network problems.

SLIDE 15

Salinity Projections 2004- 2030 Sa ty

ject o s

00 030

Sources--- Shoups & Hopmans 2005, Shoups(2004), Orlob(1991),

S J i V ll D i (1990) “R i b R ” San Joaquin Valley Drainage report(1990) “Rainbow Report”.

Average annual net salt increase 499,000 tons
Change in salt affected area- Shoups (2004)

Change in salt affected area Shoups (2004) 0.5% / year- Increase of 240,000 acres (13%) by 2030

Salinity levels and areas- DWR SJ Valley Drainage Monitoring

Program 2001- Plate 1. g

5 salt levels in shallow saline water. Current salt affected area 1.85

million acres

Deep aquifer salinity accumulation Shoups & Hopmans 2005 50%

percolation– net average aquifer salinity change 2004- 2030— 264mg/L – 343 mg/L.

SLIDE 16

Relative change in the shallow groundwater table (0.46 - 0.58% /pa-- Shoups 2004).

SLIDE 17

Saline Affected Areas (DWR 2001)

SLIDE 18

Field Level Crop Data (DWR)

SLIDE 19

Interaction of Salinity and cropping

SLIDE 20

Soil Capacity Class and Electrical Conductivity in Shallow Groundwater CVPM 19

SLIDE 21

Natural Neighbor Interpolation Natural Neighbor Interpolation

SLIDE 22

Marginal Effects of Salinity Ordered by Salt Tolerance

Evaluated Separately at Average and by Respective Salinity Zone

Marginal Effects Crop Salt Tolerance dS/m* CVPM 10 CVPM 14 CVPM 15 CVPM 19 CVPM 21 Grapes 1

0.20%**
1.06%**
8.67%**
0.94%
13.02%

Orchard 1.4

12.29%**
4.69%**
17.40%**
5.68%**
6.22%

Truck (Lettuce) 1.5

2.95%*
1.56%*

0.22%*

0.76%*
11.78%

Tomato 1.7 n/a

2.07%*

0.75%*

0.07%**

n/a Grain 4.5 0.60% 1.55%* 3.83%* 2.82%** 6.74% Sugar Beet 4.7 1.10%* 0.75%* 0.39%**

0.19%**

0.00% Field 5 2.21%**

0.45%**

0.69%

0.96%*

6.40% Cotton 5.1 6.30%* 4.57%* 9.30%* 5.80%** 7.80% Alfalfa 8 5.79%* 2.71%* 4.52%*

0.40%**

6.87% Fallow n/a

0.30%

0.21% 6.04%** 0.46%* 3.21% Obtained from http://www agric nsw gov au/reader/wm plants waterquality

Obtained from http://www.agric.nsw.gov.au/reader/wm-plants-waterquality
*Denotes significance at 5%
**Denotes significance at 1%

SLIDE 23

A Multinomial Logit Model of Farmer Salinity Response Salinity Response

'

12

Pr( )

i k

e Crop k

β

= =

x

'

12 1

( )

i l

l

p e

β =

∑

x

13 Crop groups
Salinity – Continuous measure of shallow

groundwater salinity by field

Soil – Integer 0-7 with decreasing soil quality

A C ti f l

Acres – Continuous measure of parcel area
Between 4,000 and 10,000 observations per CVPM

region, approximately 48,000 observations across all region, approximately 48,000 observations across all salinity affected CVPM regions

SLIDE 24

Micro-Modeling Region 19

Kern County California

C l Q i Gi h f

Central Question: Given that farmers

adjust crop rotations in response to li it h t i th ff t f li it salinity, what is the effect of salinity on crop yields in practice?

– Experimental vs. Behavioral

Focus on a single region

– 4,700 observations total, 2,400 over saline land

SLIDE 25

SLIDE 26

Experimental Yield Reduction Function

SLIDE 27

Behavioral Risk Model

Focus on 5 crop groups in Kern County, CA
Farmers as profit maximizing crop portfolio
Farmers as profit maximizing crop portfolio

managers

Model must be scaleable
Model must be scaleable
Estimate farmer risk aversion

M V framework – M-V framework – 1980-2005 time series of crop prices and yields

Given risk aversion estimate “behavioral rho”

Given risk aversion, estimate behavioral rho

– CVPM Region 19, 1998 observed crop proportions – Given risk aversion, what is the value of rho that , leads to observed crop proportions

SLIDE 28

Estimation of Behavioral Salinity Response Coefficients

Crop Group Behavioral Rho Experimental Rho* Orchard/Citrus 0.51** unavailable Grape 0 72** unavailable Grape 0.72** unavailable Truck 0.61** 2.86 Grain 1.68** 2.90 C tt 2 59** 3 00 Cotton 2.59** 3.00

*From VanGenuchten and Gupta 1993 **Robust to salinity bandwidth

Ordering by salt tolerance Ordering by salt tolerance

MAX

Yield Yi ld

Fundamental Equation:

1 *

MAX

Yield c scale

ρ

= ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠

50

c ⎜ ⎟ ⎝ ⎠

SLIDE 29

Example of Experimental and Behavioral Salinity Response

Grape Salinity Response

Experimental Behavioral 0 80 1.00 1.20 Max 0 20 0.40 0.60 0.80 eld/Yield 0.00 0.20 . . 2 3 . 4 5 . 6 8 . 9 1 . 1 3 1 . 3 6 1 . 5 8 1 . 8 1 2 . 3 2 . 2 6 2 . 4 9 2 . 7 1 2 . 9 4 3 . 1 6 3 . 3 9 Yi C/C50

2 55 ρ = 2.55 0.72

EXEPERIMENTAL BEHAVIORAL

ρ ρ = =

SLIDE 30

Example of Experimental and Behavioral Salinity Response

Grain Salinity Response

Behavioral Experimental 0 80 1.00 1.20 Max 0.20 0.40 0.60 0.80 Yield/Yield 0.00 0.00 0.15 0.31 0.46 0.61 0.77 0.92 1.07 1.23 1.38 1.53 1.69 1.84 1.99 2.15 2.30 Y C/C50

2.90

EXPERIMENTAL

ρ = 1.68

EXPERIMENTAL BEHAVIORAL

ρ ρ =

SLIDE 31

Salinity Modeling Conclusions Salinity Modeling Conclusions

Economic response to salinity can be modeled

Economic response to salinity can be modeled through deductive and inductive methods

Micro-modeling over salinity regions to

g y g determines behavioral salt response

Increased data availability continues to improve

y p results

Farmer salinity response functions can be used

to reduce economic impacts of salinity, and move toward sustainability.

SLIDE 32

Salinity Modeling Conclusions Salinity Modeling Conclusions

Economic response to salinity can be modeled

Economic response to salinity can be modeled through deductive and inductive methods

Micro-modeling over salinity regions to

g y g determines behavioral salt response

Increased data availability continues to improve

y p results

Farmer salinity response functions can be used