1. Monte Carlo Methods CEE 6410 – Water Resources Systems Analysis Nov. 12, 2015

2. Learning Objectives 1.Monte Carlo simulate uncertain model parameters 2.Apply Monte Carlo simulations to Reservoir optimization problem (HW #7) Household water use in Amman, Jordan

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

4. Monte Carlo and Vegas!!!

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

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

7. Probability Distributions By Raphael Briand

8. Monte Carlo Sampling Inverse transform sampling with discrete variables 1. Find probability distribution 2. Find cumulative fraction (CDF) 3. Sample cumulative fraction value (uniform distribution [0 1]) 4. Find inverse value Sample Sampled CF value Hot water efficiency (%) 10.810.91 20.270.62 30.630.89

9. 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 (simulate) Step 5: Repeat Step #4 a large number of times!

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

11. Example 1. How do uncertain initial reservoir storage and inflows affect total net benefits (HW #7)? 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 FlowProb. (%)CF (%) 10.1 30.30.4 5 0.8 80.21.0 Table 1. Flow likelihood in Month 1 Flow in Time t+1 1358 Flow in Time t 10.20.50.20.1 30.20.3 0.2 50.10.4 0.1 80.20.30.40.1 Table 2. Transition probabilities

12. 12 Insufficient/unreliable public supplies Complex (tiered) rate structures Municipal water often considered unsafe to drink Expensive alternative sources Limited water conservation data Example 2. Modeling Water Use by Households in Jordan Right: Tanker truck refills storage tank Far Right: Rooftop water tanks Significant differences among users Red Sea SAUDI ARABIA EGYPT IRAN PAL.

13. 13 Modeling Approach 1.Identify options 2.Characterize options 3.Describe interdependencies 4.Identify availability events Above: Low-flow faucet, bidet, and toilet Right: Drinking water shop sells 20-liter jugs 5.Optimize 6.Repeat for a wide range of data values Top: Treatment barrels for grey-water reuse (CSBE)

14. 14 What decision variables are needed to model household options?

15. 15 Non-linear program with recourse Objective: Minimize expected annual costs [Eq 1] 1 st stage actions: Infrastructure investments (L hi ) Stochastic events: Public water availability (e) 2 nd stage actions: Public and alternative supply uses; behavior modifications (S hje ) Subject to: – Meet water requirements in each event Upper limits on actions – Mass balance – Storage capacity – Block pricing on network use 1 st Stage Event 2 nd Stage = Decision = State where stochastic information acquired

16. 16 Characterize action costs and effectiveness Above: Drip irrigation store

17. Run optimization many times (with different parameter values) to represent residential users in Amman, Jordan

18. 18 Calibrate to the distribution of piped water use 500 Monte- Carlo simulated households Adjust occupancy parameter (vacant residences)

19. 19 Distributions of water savings for conservation actions in Amman (error bars represent 10 th and 90 th percentiles)

20. 20 Major Findings 1.Modeling integrates source, availability, quality, storage, costs, conservation, and user behaviors. 2.Empirically estimate water use in Amman, Jordan. 3.Simultaneous output of:  Conservation technology adoption  Water use response  Household willingness-to-pay 4.Target conservation to select customers.. Above: Store selling rooftop water tanks Rosenberg et al, (2007) “Modeling Integrated Water-User Decisions in Intermittent Supply Systems." Water Resources Research. 43. W07425.Modeling Integrated Water-User Decisions in Intermittent Supply Systems

21. 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 analysi s