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A New Algorithm for Water Distribution System Optimization: Discrete Dynamically Dimensioned Search (DDDS). EWRI 2008 May 12, 2008. Dr. Bryan Tolson 1 Masoud A. Esfahani 1 Dr. Holger Maier 2 Aaron Zecchin 2 Department of Civil & Environmental Engineering University of Waterloo, Canada

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EWRI 2008 May 12, 2008

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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

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

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

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.

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

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*

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

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

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)

evaluated with all pipes at max diameter

min required pressureactual 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

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

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

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

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 …


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)

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

2-NYTP Case Study – Cycle 1 performance

Empirical CDF of best obj. func. values

2-NYTP Case Study – impact of cycle 2

60,000 function evaluations not long enough for C2

(different result for NYTP)

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

2-NYTP Case Study – impact of 2P local search heuristic

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

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

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)

Constraint Handling Assessment for DDDS: HP

  • Simple approach with no penalty parameters works very well

best of 6 algorithms in Zecchin et al. 2007

Large Scale WDS: Balerma

Large Scale WDS: Balerma

Algorithm response to smaller user-specified computational budget

Large Scale WDS: Balerma

Large Scale WDS: Balerma

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


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

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

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

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

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:

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