EWRI 2008 May 12, 2008

1. A New Algorithm for Water Distribution System Optimization: Discrete Dynamically Dimensioned Search (DDDS) EWRI 2008 May 12, 2008 Dr. Bryan Tolson1 Masoud A. Esfahani1 Dr. Holger Maier2 Aaron Zecchin2 Department of Civil & Environmental Engineering University of Waterloo, Canada School of Civil, Environmental and Mining Engineering, University of Adelaide

2. Research Goal • Develop a simple, parsimonious algorithm for constrained single objective Water Distribution System (WDS) design optimization • Algorithm design goals: • Eliminate need to fine tune algorithm parameters (regular algorithm + penalty function parameters) • Avoid poor solutions with a high reliability • Build off efficient and effective DDS algorithm for continuous optimization

3. Background: DDS Algorithm • Simple and fast approximate stochastic global optimization algorithm • For continuous optimization problems • Single-solution search (not population based) • Designed originally for computationally expensive automatic hydrologic model calibration: • Generate good* results in modeler’s time frame • Algorithm parameter tuning is unnecessary • Tolson & Shoemaker (2007), WRR

4. DDS Description • General DDS search strategy: 0. User inputs: - maximum function evaluations - decision variable ranges - perturbation size parameter (0.2*) • Initialize starting solution • Perturb current best solution to generate candidate solution • Compare candidate solution to best solution and update best solution if necessary • Repeat from step 2 until maximum objective function evaluations completed.

5. DDS Description • key to DDS is perturbation in step 2: • search globally at the start of the search by perturbing all decision variables (DVs) from their current best values • search locally at the end of the search by perturbing typically only 1 decision variable (DV) from its current best value • perturbed DVs are generated from a normal probability distribution centered on current best value • global to local search strategy scaled to user-specified maximum number of objective function evaluations • the only information used to direct candidate solution sampling is the current best solution

6. DDS to Discrete DDS (DDDS) • only modification is to discretize the DV perturbation distribution Discrete probability distribution of candidate solution option numbers for a single decision variable with 16 possible values and a current best solution of xbest=8. Default DDDS-v1 r-parameter of 0.2*

7. Start of SearchEnd of Search Best Current Solution (red) Pipe 1 Pipe 2 Pipe 3 Pipe 4 Global to Local Search • key to DDS and DDDS is to search globally at the start of the search and finish by searching locally • consider a WDS example with 4 decision variables: Example Candidate Solutions

8. Pipe 1 Pipe 2 Pipe 3 Pipe 4 Pipe 1 Pipe 2 Pipe 3 Pipe 4 Global to Local Search • key to DDS and DDDS is to search globally at the start of the search and finish by searching locally • consider a WDS example with 4 decision variables: Start of SearchEnd of Search Best Current Solution (red) Example Candidate Solutions

9. General Constrained WDS Optimization Formulation Given pipe layout, its connectivity & nodal demands choose pipe diameters (the decision variables) that: Minimize Total Pipe Costs Subject to: • meeting minimum nodal pressure requirements • selecting pipe diameters from a set of discrete alternatives Note that the hydraulic solver (e.g. EPANET2) determines a flow regime that automatically satisfies hydraulic constraints (conservation of mass, energy)

10. evaluated with all pipes at max diameter min required pressure actual pressure for solution DDDS for WDS Optimization • Add constraint handling technique to account for nodal pressure constraints • DDDS only explicitly handles DV bound constraints • DDDS compares two solutions based only on rank (which one is better) to update current best solution • therefore, objective function scaling is irrelevant • use a parameterless penalty function such that objective (Cost) is defined as: • Costs = total pipe costs for feasible solutions, or • for infeasible solutions • Same as Deb (2000) tournament selection-based method

11. DDDS for WDS Optimization second modification: 2. At end of the search, avoid wasting excessive function evaluations on candidate solutions with only one pipe perturbed from best solution • depending on # of DVs, this waste can be substantial (e.g. ~50 or fewer DVs) • try something more productive! • one pipe perturbations from a good solution will generally not improve solution since good solutions are typically ‘just’ feasible

12. Experimental Approach • Determine if DDDS extension to DDS for WDS optimization is competitive with high quality Ant Colony Optimization (ACO) results (HP & NYTP) • Assess improvements of multi-cycle DDDS approach over basic DDDS • Apply DDDS to large scale WDS optimization problem (hundreds of pipes to size) No algorithm parameter tuning in steps above

13. WDS Case Studies • EPANET2 used as hydraulic solver and library functions from EPANET Toolkit link to DDDS code in Matlab. • all previous results in literature for other algorithms utilize EPANET2 as hydraulic solver

14. Results in Proceedings Paper • evaluated very simple fix to excessive 1-pipe perturbations by DDDS (called DDDS-v1) showed • DDDS-v1 results for NYTP of comparable quality to various ACO algorithms in Zecchin et al. (2007) • DDDS-v1 results for HP that were better on average than the best ACO algorithm in Zecchin et al. (2007) • Our new approach shows good potential! • Remaining slides highlight some new results to appear in extension to conference paper … NFS

15. Basics of Multi-Cycle DDDS Specify maximum # of model evaluations, M • NOTE: • point at which DDDS search perturbs a single DV varies mainly with problem dimension and secondarily with M • with hundreds of DVs, multiple cycles unnecessary because this point is not reached until >95% of M completed (not wasting effort)

16. 2-NYTP Case Study • from Zecchin et al. (2007) • 6 ACO algorithms in Zecchin et al. use 500,000 function evaluations • optimal algorithm parameters determined for each algorithm using millions of evaluations • For multi-cycle DDDS, specify approx. maximum of 300,000 function evaluations • no algorithm parameter tuning • simply observe improvement achieved by each cycle • 20 optimization trials per algorithm

17. 2-NYTP Case Study – Cycle 1 performance Empirical CDF of best obj. func. values

18. 2-NYTP Case Study – impact of cycle 2 60,000 function evaluations not long enough for C2 (different result for NYTP)

19. 2-NYTP Case Study – impact of cycles 3 and 4

20. 2-NYTP Case Study – impact of 2P local search heuristic 2P change heuristic very effective polisher at end* of search

21. 2-NYTP Case Study – add best of 6 ACO algorithms (MMAS) from Zecchin et al

22. Constraint Handling Assessment for DDDS • Consider results for Hanoi network where many studies report algorithm difficulty in locating any feasible solution (Euseff & Lansey, 2003; Zecchin et al., 2005 and Zecchin et al., 2007)

23. Constraint Handling Assessment for DDDS: HP • Simple approach with no penalty parameters works very well best of 6 algorithms in Zecchin et al. 2007

24. Large Scale WDS: Balerma

25. Large Scale WDS: Balerma Algorithm response to smaller user-specified computational budget

26. Large Scale WDS: Balerma

27. Large Scale WDS: Balerma all studies use EPANET2

28. Conclusions • DDDS for WDS optimization is parsimonious: • no algorithm parameter-tuning • no penalty parameter-tuning • no parameter adjustment here for case studies with 21-454 pipe size decision variables • DDDS for WDS optimization is very effective: • 1-cycle and multi-cycle DDDS show improved results over alternative algorithm results • to the best of our knowledge DDDS (1-cycle and multi-cycle) found new best known solutions to two WDS design problems in the literature • Two-pipe change heuristic appears to be new

29. QUESTIONS ?

31. Keys to DDS • Algorithm scales to user-specified computational limits • Early in search  favours global search • Late in search  favours local search • STEP 1. Define DDS inputs for D dimensional problem: • neighborhood perturbation size parameter, r (0.2 is default) • maximum # of function evaluations, m • STEP 2. Evaluate objective function at initial solution • STEP 3. Randomly select a subset of the D decision variables for perturbation from the current best solution. • STEP 4. Perturb the decision variables selected in Step 3 from their current best solution (reflect at decision variable bounds if necessary) • STEP 5. Evaluate new solution and update current best solution if necessary • STEP 6. Update function evaluation counter, i=i+1, and check stopping criterion: • IF i = m STOP • ELSE repeat STEP 3 • Size of subset decreases as maximum function evaluation limit approached  normally distributed perturbations with adequate variance ensures global search

32. Robustness of DDS • DDS has been applied to a number of case studies, for example: • 6, 9, 10, 14, 20, 26, 30, 34 & 50 calibration parameters (= decision variables) • Anywhere from 100 to 100,000 model evaluations • Uncorrelated to very correlated decision variables • In each case, DDS was applied with the same algorithm parameter value & typically generated the best comparative results

33. Pipe 1 Pipe 2 Pipe 3 Pipe 4 Local Search Procedure for Polishing/Refining • Use two procedures: • One pipe change • Two pipe change • One pipe change procedure cycles through all possible one-increment pipe diameter reductions until none can improve solution

34. Two Pipe Change • an improved solution that differs in two pipes will have one pipe diameter reduced and another increased such that: • total WDS cost is reduced (*this does not require running EPANET*) • reduced pressures due to pipe diameter decrease are potentially mitigated by an increase in another pipe diameter

35. Two Pipe Change • How long does this take? • How long to confirm a solution is a locally optimal solution where no possible two pipe change will improve results? • the maximum number of combinations to be evaluated can be determined and is between: