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EWRI 2008 May 12, 2008

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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

- 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

- 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

- 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.

- 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

- 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*

Start of SearchEnd of Search

Best Current Solution (red)

Pipe 1

Pipe 2

Pipe 3

Pipe 4

- 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

Pipe 1

Pipe 2

Pipe 3

Pipe 4

Pipe 1

Pipe 2

Pipe 3

Pipe 4

- 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

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)

evaluated with all pipes at max diameter

min required pressureactual pressure for solution

- 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

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

- 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

- 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

- 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

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)

- 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

Empirical CDF of best obj. func. values

60,000 function evaluations not long enough for C2

(different result for NYTP)

2P change heuristic very effective polisher at end* of search

- 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)

- Simple approach with no penalty parameters works very well

best of 6 algorithms in Zecchin et al. 2007

Algorithm response to smaller user-specified computational budget

all studies use EPANET2

- 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

- 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

- 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

Pipe 1

Pipe 2

Pipe 3

Pipe 4

- 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

- 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

- 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: