1. CEE 6410 – Water Resources Systems Analysis
Monte Carlo Methods
CEE 6410 – Water Resources Systems Analysis
Thanks Dr. Bishop for this class. It helped me in my research a lot.
David E. Rosenberg

2. Learning Objectives Monte Carlo simulate uncertain model parameters
Apply Monte Carlo simulations to
Reservoir optimization problem (HW #7)
Household Water-Energy Use in SLC, Utah

3. Monte Carlo and Vegas!!!

4. 1. Monte Carlo Simulation
Quantify interactive effects of uncertainty, variation, and randomness on a final product e.g., What net benefits does a reservoir generate with uncertain initial storage, inflows, and in-stream flow requirements?

5. Stochastic!
What is the likelihood or probability of hitting within the center shade?
(Whitney King, 2013): Technical Archery

6. Deterministic VS Stochastic Approaches
chance
uncertain
Deterministic
certain
sure
1.6 gal/flush

7. Probability Distributions (Raphael Briand)

8. Hot water efficiency (%)
Monte Carlo Sampling
1. Describe probability distribution
2. Calc. cumulative fraction (CDF)
3. Sample CF value (uniform on [0 1])
4. Find associated hot water efficiency value
Number of observations
Sample
Sampled CF value
Hot water efficiency (%) 1
0.81
0.91 2
0.27
0.62 3
0.63
0.89
Skip this slide, but bring it up in case someone asks about the sampling

9. Example 1. Use 250 Initial Reservoir Storage values in HW7_mc_plots
Example 1. Use 250 Initial Reservoir Storage values in HW7_mc_plots.xls (sheet InitStorS) to…
Describe the probability distribution
Write the cumulative distribution function (CDF)
Sample initial storage values for the cumulative fractions 0.21, 0.48, 0.78

10. Stochastic Simulation Steps
Step 1: Define the model
Step 2: Identify the uncertain parameters and dependencies
Step 3: Specify a probability distribution for each uncertain parameter
Step 4: Sample values for each uncertain parameter and propagate uncertainties and dependencies
Step 5: Solve the optimization model using values in Step 4.
Step 6: Repeat Steps #4 and #5 a large number of times!

11. 2. Apply Monte Carlo Methods in Water Resources Optimization Problems

12. Example 2. How do uncertain initial reservoir storage and inflows affect total net benefits (HW #7)?
Table 1. Flow likelihood in Month 1
Assume:
Initial storage varies uniformly between 0.5 and 10 units
Inflow in month 1 varies according to observations (Table 1)
Inflows in subsequent months exhibit lag-1 correlation (Table 2)
Flow at A must be at least 1 unit or 20% of largest simulated flow
Use 250 samples
Results in HW7_mc_plots.xls (sheet NetBenS)
Flow
Prob. (%)
CF (%) 1
0.1 3
0.3
0.4 5
0.8 8
0.2
1.0
Table 2. Transition probabilities
Flow in Time t+1 1 3 5 8
Flow in Time t
0.2
0.5
0.1
0.3
0.4

13. Example 3. Synergistic Water and Energy Savings in SLC, Utah
What home-owner actions jointly conserve water and energy?
Which cost effective actions should cities synergistically promote?
How to target households to adopt actions?
1960
1977

14. Data-Driven Simulation-Optimization (Phd research of Adel Abdallah)
1. Collect high-freq. behavior data
2. Identify key parameters & distributions
3. Monte Carlo simulate
4. City-scale optimization
5. Mine results to target

15. High-Frequency Behavioral Data
Water (Aquacraft, 2005, 2009)
Energy (DOE, 2009)
Water heater market shipments (709 models)
Plumbing/heating contractor firms (343)
Average annual potable water temperatures (74 cities across the U.S.)
Dataset
Number of Cities
Data collection period
Number of houses
Monitoring days
Water use events
USEPA Retrofit 3 88
4,036
753,076
New Single Family Homes 9
305
3,885
648,719

16. Behavior and Demographic
Technology
Key Parameters
Energy Factors
Behavior and Demographic

17. Monte Carlo Simulation (1,000 households)
NEED TO UPDATE to include:
Water Uses (including outdoor)
Conservation actions (technical and behavioral)
Use and saved by conservation actions

18. Model Validation – Salt Lake City
Water use (gallons/household/year)

19. Simulated water and energy uses (largest 12% of users use 21% and 24% of water and energy)
The green circle in the middle represents the average value which is widely used to model water and energy linkages. But in our study we want to exploit this heterogeneity by targeting the high users first

20. Mixed Integer City-Scale Optimization
Decisions
Conservation actions implemented (binary) by:
Household (1,000)
End use/Appliance (8)
Method (4)
Objective function ($)
Minimize total cost to implement conservation actions
Subject to:
Meet city water reduction target
Meet city direct energy reduction target
Lower and upper bounds on number of actions
Mutually exclusive actions
Upper bound on payback period for actions
Action
Cost
Retrofit toilet
$342
Retrofit shower
$30
Retrofit faucet
$50
Retrofit clothes washer
$819
Reduce outdoor 10%
$200
Lower heater to 120oF

21. Monte-Carlo simulated household energy and water uses before and after conservation actions

22. Heterogeneity of household savings and payback periods
The Payback legend is sorted to match the Action legend. So on average, the shower action has the shorted payback period of 1 year while the Toilet action has the longest payback period of 4 years

23. Payback periods for actions

24. Costs to meet reduction targets
Mass-Applied
Targeted
Solving for Max water and energy savings s.t. to the upper bound of a city action (1000) I found that Max water % is about 10% and energy about 8%.
The red circle points out to an example target that I want to show its results. The target is 4% water and 5% energy
I want to pint out to the tradeoff between saving water and energy for the same cost line

25. Apply the results Target customers with large water and/or energy use
Encourage to implement one or more conservation actions
Shower and faucet actions conjunctively save water and energy
Reduce heater temperature to save energy
Outdoor conservation actions save water

26. Monte Carlo Wrap Up
Powerful tool to incorporate real world uncertainties
Also provides probabilistic outputs
Offers water system management and policy insights not available from deterministic analysis