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

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

Monte Carlo and Vegas!!!

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?

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

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

Probability Distributions (Raphael Briand)

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

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

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!

2. Apply Monte Carlo Methods in Water Resources Optimization Problems

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

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

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

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 2000-03 88 4,036 753,076 New Single Family Homes 9 2005-09 305 3,885 648,719

Behavior and Demographic Technology Key Parameters Energy Factors Behavior and Demographic

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

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

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

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

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

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

Payback periods for actions

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

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

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