The University of Adelaide, AustraliaSchool of Civil, Environmental & Mining Engineering

Water Systems and Infrastructure Modelling & Management Group

(WaterSIMM)

Using Genetic Algorithms to Optimise Network Design and System Operation Including Consideration of Sustainability

Professor Angus Simpson

Victorian Modelling Group24 March 2010

Outline

1. Simulation of water distribution systems

2. Various formulations of equations

3. Todini and Pilati solution method

4. Genetic algorithm optimisation of water distribution system networks

5. Genetic algorithms for optimising operations of pumping systems

6. Optimising for sustainability

My research interests

1. Optimisation of the design and operation of water distribution systems using genetic algorithms [including sustainability considerations (GHGs)]

2. Monitoring health and assessing condition of pipes non-invasively in water distribution systems using small controlled water hammer events

My research interests

3. Steady state computer simulation analysis of water distribution systems - improving solvers and modelling of PRVs and FCVs

4. Water hammer modelling in pipelines

The research team

• Total of 14• 9 academics (local and overseas)• 1 Research Post Doc.• 4 PhDs

The research team - academics

• Prof. Angus Simpson• Prof. Martin Lambert (condition assessment and

genetic algorithms)• Prof. Holger Maier (genetic algorithms and

sustainability)• Dr. Sylvan Elhay – School of Computer Science,

The University of Adelaide (steady state solution of pipe networks)

• Prof. Lang White – School of Electrical and Electronic Engineering, The University of Adelaide (condition assessment)

International collaboration

• Professor Caren Tischendorf – Head of Dept. of Mathematics and Computer Science, University of Cologne, Germany (steady state solution of pipe networks)

• Dr. Jochen Deuerlein – 3S Consult, Germany (steady state solution of pipe networks, correct modelling of PRVs, FCVs in water networks) – ex-University of Karlsruhe

• Dr. Arris Tijsseling – University of Eindhoven, The Netherlands (condition assessment)

• Prof. Wil Schilders – University of Eindhoven, The Netherlands (speeding up solution of nonlinear steady state pipe network equations)

Components in a Water Distribution System

Simulation of water distribution systems

• Solution of a set on non-linear equations for – flow (Q) and – pressure (head or hydraulic grade line – H)

• EPANET uses Todini and Pilati (1987) method – very fast

• There are many sophisticated commercially available software packages

Assumptions

• Fixed demands – assumed to be not pressure dependent, for example - 100 houses aggregated to a node

• Various water demand loadings cases– Peak hour (hottest day in summer)– Peak day (extended period simulation usually

over 24 hours to check tanks do not run empty)– Fire demand loading cases– Pipe breakage cases

Decisions to be made

• Diameters and locations of pipes• Location of pumps• Locations and setting of valves (PRVs, FCVs)• Locations and elevations of tanks• Operation of pumps – off-peak electricity rates• Minimising carbon footprint• Reliability considerations

Design objectives

• For the economic cost component minimise sum of – Capital cost of the water distribution system

(say for a 100 year life)– Present value of pump replacement/refurbishment

(every 20 years)– Present value of pump operating costs (100 years)

)()()(),(11

pumpingNPVpcLdcPDC k

NPUMP

kkkk

NPIPE

kk

Accounting for Time

• Present value analysis (PVA)

• Usually the discount rate i is selected to be cost of capital 6 to 8%

• For social projects, such as WDSs, a social discount rate should be used, for example, i = 1.4% (intergenerational equity)

tti

CPV

1

C: Payment/cost on a given future date

t = Number of time periods

i = Discount rate

Design objectives

• Satisfy design criteria – for example–Minimum/maximum allowable pressures–Maximum allowable velocities–Tanks must not empty

NDEMANDSjNJitHtHH jijiji ,...,1;,...,1,,)( max,,

min,

NDEMANDSjNPitVtV jiji ,...,1;,...,1,,)( max,,

Trial and error approach

• User has to choose - diameters of pipes, locations of tanks, pumps, and PRVs

• Engineering judgment and experience• Run simulation model for demand load cases• Check design criteria and compute cost• Adjust sizes of elements in response to

pressures• Rerun simulation model

New York Tunnels problem

New York City Water Supply Tunnels

Brooklyn

Hillview ReservoirEL 340 ft

16

9 21

8

20

19

11

7

6

5

4

17 18

13

14

3

2City Tunnel No. 1

City Tunnel No. 2

Bronx

Queens

Richmond

Manhattan

10

12

15

A very very large search space size

• Any one of the 21 existing pipes could be duplicated

• Choose from 16 allowable pipe sizes to meet demands

• Search space size = 1.43 x 1021

• Eliminate 99.99% of possible solutions by engineering experience (leaves 1.43 x 1017)

A very very large search space size

• At 10,000 evaluations per second – can compute 3.15 x 1011 per year

• It will take 454,630 years to fully enumerate 0.01% of the total search space (only for 21 decision variables)

A typical simple water distribution system

Simulation – Solving for The Unknowns

q Q1 Q2 QNP T=

• Only consider systems with pipes and reservoirs • The unknown flow vector (10 pipes in examples)

• The unknown head vector (7 nodes in example) (gives pressures)

h H1 H2 H NJ T=

A total of 17 unknowns Qs and Hs

Continuity Equation of Flow at a Junction

• Flow In = Flow Out + Demand (or Withdrawal Discharge)

• where Qj = flow in pipe j (m3/s or ft3/s) NPJi = number of pipes attached to node i DMi = demand at the node i (m3/s or ft3/s) NJ = total number of nodes in the water distribution system

(excluding fixed grade nodes such as reservoirs)

Q j

j 1=

NPJi DMi+ 0=

Pipe Head Loss Equations in Terms of Nodal Heads

H i Hk– r jQj Qjn 1–

=

• where• = nodal head at node i in the water distribution system (m or ft)

• rj = resistance coefficient for the pipe j depending on the head loss relationship (for example, Darcy–Weisbach or Hazen–Williams)

• Qj = flow in pipe j (m3/s or ft3/s)

• n = exponent of the flow in the head loss equation (Darcy–Weisbach n = 2 or Hazen–Williams n = 1.852)

Hi

Four different non-linear formulations

• #1 Q-Equations• #2 H-Equations• #3 LF- Equations (Loop Flow Equations)• #4 Todini and Pilati H-Q Equations

#1 - The Q-equations formulation (10 unknowns)

The Q-equations (10 unknowns)

q Q1 Q2 QNP T=

The Q-equations (Newton iterative solution technique)

#4 -Todini and Pilati Q-H formulation

• Define topology matrices• Develop block form of equations• Use an analytic inverse of block matrices to

reduce matrix size from 17 unknowns to 7 unknowns (same as unknown heads H)

• Fast algorithm

Todini and Pilati Q-H formulation

• Unknowns

h H1 H2 H NJ T=

q Q1 Q2 QNP T=

Todini and Pilati Q-H formulation - Define topology matrices

Todini and Pilati Q-H formulation - Define topology matrices

Todini and Pilati Q-H formulation - continuity at nodes

Todini and Pilati Q-H formulation – head loss equations for pipes

Note that later on theinverse of this matrix will give problems for zero flows

Todini and Pilati Q-H formulation – head loss equations for pipes

Todini and Pilati Q-H formulation–two sets of equations

The Todini and Pilati equations

Research Issues

• Improving solution speed• Making solution algorithms more robust – zero

flows cause Todini and Pilati method to fail – a regularization method has been developed to control the condition number

• An improved convergence criterion for stopping has been developed

Research Issues

• Decomposing networks into trees, blocks and bridges to speed up analysis

• Growing typical networks that have correct mix of loops, links and junctions

GENETIC ALGORITHMS FOR OPTIMISATION OF WATER DISTRIBUTION SYSTEMS

Types of Evolutionary Algorithms

• Genetic algorithms (Holland 1976; Goldberg 1989)

• Ant Colony Optimisation (ACO)• Tabu search• Simulated annealing• Particle swarm optimisation (PSO)• Evolutionary strategy (Germany)

History of genetic algorithms applied to water distribution systems

• Pioneered at the University of Adelaide by Laurie Murphy under my supervision in an honours project in 1990 and a PhD starting in 1991

• Initial focus was on the optimisation of the design of water distribution systems

• A spinoff company of Optimatics Pty Ltd formed by University of Adelaide in 1996 – operates in Australia, NZ, USA and UK (employs 20 people)

• Research focus is now on optimising operations and accounting for multiple objectives (sustainability, reliability)

Genetic algorithm optimisation

• Population orientated technique (select a population size of say 500)

• Based on mechanisms of natural selection and genetics

• Selection, crossover and mutation operators produce new generations of designs

• Fitness of strings drives process• Uses EPANET type simulation model to

assess performance of all trial water distribution networks in each generation

Creating a string from sub-strings - an example

BINARY CODING

Chromosome of decision variables

• Choice tables are required for each decision variable

Chromosome

Existing pipe [3] (binary)00 = no change, e = 2.5 mm01 = clean/line, e = 0.3 mm10 = duplicate 306 mm11 = close the pipe

Model Operation

Optimisation-Simulation Model Link

GA OPTIMISATION MODEL

HYDRAULIC SIMULATION MODEL

Configuration of water distribution and performance passes back and forth

Simulate hydraulics of water distribution system

• Decode each string using the choice tables• Run a computer simulation model• Simulate demand loading cases

consecutively – peak hour, fire, extended period simulation

• Record any violation of constraints (e.g. pressures too low, velocities too high)

Choices for the decision variables

A decoding lookup table Nominal Diameter

(mm)

Actual or Internal

Diameter (mm)

Binary Coding

Integer Coding

Roughness height (mm) for Darcy-Weisbach friction

factor

Unit Cost ($/m)

150 151 000 1 0.25 49 200 199 001 2 0.25 63 250 252 010 3 0.25 95 300 305 011 4 0.25 133 375 384 100 5 0.25 171 450 464 101 6 0.25 220 500 516 110 7 0.25 270 600 615 111 8 0.25 330

Total cost and corresponding fitness of the string

• Total cost is the: example1. Real cost of the water distribution system

design

PLUS 2. A penalty pseudo-cost (or costs) if the

constraint(s) are not met– For example

=K*Maximum pressure deficitwhere K=$50,000 per metre

• Fitness is often taken as the inverse of the total cost

Steps in a genetic algorithm optimisation

• Select a population size (e.g. N=100 or N=500)• Select a reproduction or selection operator• Select a probability of crossover (Pc)• Select a probability of mutation (Pm)

A Simple Genetic Algorithm

ThisGeneration

N=500

Selection Crossover &

Mutation

The NextGeneration

Mating Pool

N=500

Tournament selection

Randomly select pairs of chromosomes

versus

Evaluation of the fittest

Forming the Mating Pool

1 1 1 1 1

1 11 11 11

1 11 1

1 1

1

1111

1 1 1

11 1

1 11

1

1

11 1 1

1 11 11

1 1

11 1 1 1 1 1

11 1 1

1

1

11

1

00 0 0

0

0

0 0

00

00 0 0 0

00

0 00

0000 0

00000

00000

00000

0

00

0

0 0

0

0 0

0

1 11 1 11 1 000

65

112

94

83

98

143

87

130

Fittest strings win

Two sets of tournament selection are required

versus

versus

versus

FITNESS

Crossover (one-point)

Chromosome A

Chromosome B Parents

Chromosome A`

Chromosome B`Offspring

1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1

1 1 11 1 1

1 1 1 1

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

Randomly select a crossover point

Interchange the tails

The mutation operator

• Mutation occurs with a very small probability• One bit switches to a new value

Produce many generations

• Continue to create a series of new generations (say 100,000 different networks)

• Repeat selection, crossover and mutation

• Increasingly fit solutions are generated

• The 10 or so lowest cost solutions must be remembered along the way

Designs improve in each generationB

est C

ost

in E

ach

Gen

erat

ion

($ m

illio

n)

30

40

50

60

70

80

90

100

0 50,000 100,000 150,000 200,000

Number of Solution Evaluations

New York Tunnels Problem

Optimisation of pumping plant operations

Murray Bridge – operations optimisation

• United Utilities – Riverland water treatment plant project at Murray Bridge, South Australia

• GA optimisation minimises operations electricity costs by – maximizing off-peak pumping and – minimizing static pump head

Background

Pumps

Elevated Storage

Water Treatment Plant Clear Water

Storage

Case study – Murray Bridge system layout

White HillStorage Tank

(WHS)Off

Takes

Murray Bridge

Onkaparinga

Pipeline

Murray Bridge Water Treatment

Plant

Clear WaterStorage Tank (CWS)

3 Parallel Fixed Speed Pumps

Traditional approach to control

• Trigger levels in storage tanks OR• Pump scheduling

Controls based on trigger levels

Lower Trigger Level (Minimum Allowable Level)

Upper Trigger Level (Maximum Allowable Level)More Peak

Pumping than Necessary

Time

Tank

Lev

el

Peak Tariff Period

Tank not Full for Next

Peak Tariff Period

Tank not Full at Start of Peak Tariff Period

Off-Peak Tariff Period

7am 7am9pm

Optimisation-Simulation Model Link

GA OPTIMISATION MODEL

HYDRAULIC SIMULATION MODEL

Operating policies for pumping system - trigger levels, schedules for

pumps turning on and turning off

Model operation

Decision variables - operations optimisation at Murray Bridge

• Used a combination of real-value and integer value representation of the decision variables

• Four decision variable for final formulation:–Pump start time to fill tank–Pump stop time to drain tank to minimum level–Reduced upper trigger level for tank– Initial level in CWS tank

A system controlled with original trigger levels and schedules

Lower Trigger Level (Minimum Allowable Level)

Upper Trigger Level (Maximum Allowable Level)

Time

Peak Tariff Period

Tank Full at Start of Peak Tariff Period

Off-Peak Tariff Period

7am 7am9pm

Start Primary Pump

Stop Primary Pump

Tank at Minimum Level at End of Peak Tariff Period

Tan

k L

evel

A system controlled with both schedules and a reduced upper trigger level (1)

Time

Lower Trigger Level (Minimum Allowable Level)

Upper Trigger Level (Maximum Allowable Level)

Peak Tariff Period

Tank Full at Start of Peak Tariff Period

Off-Peak Tariff Period

7am 7am9pm

Start PumpReduced Upper Trigger Level

Tank at Minimum Level at End of Peak Tariff Period

Tan

k L

evel

Tan

k L

evel

Lower Trigger Level (Minimum Allowable Level)

Upper Trigger Level (Maximum Allowable Level)

Time

Peak Tariff Period

Tank Full at Start of Peak Tariff Period

Off-Peak Tariff Period

7am 7am9pm

Start Pump

Reduced Upper Trigger Level Extended into Off-Peak Tariff Period

Tank at Minimum Levelat End of Peak Tariff Period

Switch Time

A system controlled with both schedules and a reduced upper trigger level (2)

Modelled System - Original Trigger Levels

Lev

el (

m)

4

5

6

7

Daily electricity cost (averaged over 28 days) = $313.65

Time (hrs)

Daily peak pumping (averaged over 28 days): $286.05

Initial Level in CWS = 3.4m

Lower trigger level = 5.49m

Upper trigger level = 6.98m

0

1

2

3

8

11 am9 am7 am 7 am5 am3 am1 am11 pm9 pm7 pm5 pm3 pm1 pm

Daily off-peak pumping(averaged over 28 days): $27.60

WHS (m)

CWS (m)

WHS and CWS tank levels for first 24 hours of a 28-day simulation under fixed trigger level control for a 7.17 ML/D flow

Optimised system - new approach

Initial Level in CWS = 3.675 m

Lower trigger level = 5.49 m

Reduced trigger level = 5.76 m

Upper trigger level = 6.98m

WHS and CWS tank levels after optimisation for a 7.17 ML/D flow

0

1

2

3

4

5

6

7

8

Time (hrs)

Lev

el (

m)

WHS

CWS

Pump on at 1:54:13 am

Peak Pumping: $189.51 Off-peak Pumping: $67.63

Total electricity cost = $257.14. Hence a $56.51 or 18.0%

11 am9 am7 am 7 am5 am3 am1 am11 pm9 pm7 pm5 pm3 pm1 pm

Switch time 2:15 am

Results - savings from new approach

18.284.34378.76463.110

13.646.3293.02339.328

18.056.48257.16313.647.17

22.459.72207.13266.866

22.936.17121.57157.744

54.950.6141.6592.262

(%)($)Improved Controls

Current Controls

SavingsPumping Cost ($/Day) Daily

Demand (ML/Day)

Accounting for Sustainability in the Design and Operation of Water Distribution Pumping Systems

Research Objectives

• To construct a sustainability integrated multi-objective genetic algorithm optimisation model for the planning, design and evaluation of WDSs

• To explore the impacts different sustainability criteria (Greenhouse Gas emissions) will have on the results of WDS optimisation

Aspects of Sustainability-

Social

Environmental

Economic Technical

1: Total cost of the system

2: GHG emissions

3:System reliability 4: Robustness of

Pareto-optimal Front

Two of the Main Conflicting Objectives

Minimisation of thetotal life cycle system costs

Minimisation of thetotal life cycle system GHG emissions

Higher CostLower GHG

Lower Cost

Higher GHG

Big pumpSmall pipe

Small pumpBig pipe

Evaluating multi-objective optimisation results using a Pareto tradeoff curve

Cost

GHG

TradeoffCurve

Determining life cycle economic costs

Determining life cycle GHG emissions (no discounting i = 0% IPCC)

Optimisation Framework

Generate Options

MO

GA

Simulation

Evaluation

Comparison & Selection

WDS Optimisation

Obje

ctives

Minimisation of total cost

Minimisation ofGHG emissions

Maximisation ofsystem reliability

Maximisation of robustness

of Pareto-optimal solutions

Multi-objective optimisation using genetic algorithms• MOGA: NSGA-II Generate initial

population

Objective evaluation

Simulation models

Ranking

Generate global population

Non-dominated sorting

Crowding distance

Comparison & selection

Constraint handling

Generate child population

Crossover

Mutation

Stopping criteria met?

Stop

Yes

No

Case Study• The network consists

of a lower reservoir (water source), one pump, one rising main and an upper reservoir

1,500,00095

1,500120100

Average peakday flow (L/s)Design life (years)

Design conditions of case 1Annual demand (m3)

Static head Pipe length

Case Study• The aim

– to select the best combination of the pump size and pipe size

– deliver the minimum average peak-day flow

– minimise both the total cost and GHG emissions of the network during its design life

1,500,00095

1,500120100

Average peakday flow (L/s)Design life (years)

Design conditions of case 1Annual demand (m3)

Static head Pipe length

Pareto optimal tradeoff curve – multi-objective optimisation i = 6%

6% discount rate for costs

A

B

G H

C D EF

87.0

91.0

95.0

99.0

103.0

4.3 4.7 5.1 5.5

System cost (M$)

GH

G

(k-t

onne)

$21.8/tonneCO2-e

$134/tonneCO2-e $371/tonne CO2-e

(300)

(450)(375)

6% discount rate for costs

A

B

G H

C D EF

87.0

91.0

95.0

99.0

103.0

4.3 4.7 5.1 5.5

System cost (M$)

GH

G

(k-t

onne)

$21.8/tonneCO2-e

$134/tonneCO2-e $371/tonne CO2-e

(300)

(450)(375)

Diameter for lowest cost solution

Tradeoff for i = 1.4%

1.4% discount rate for costs

B

C DE

H

FG

88.5

89.5

90.5

91.5

92.5

10.1 10.3 10.5 10.7 10.9 11.1

System cost (M$)

GH

G

(k-t

onne

) $72.1/tonneCO2-e

$376/tonneCO2-e $1,468/tonne CO2-e

(375)

(450)

1.4% discount rate for costs

B

C DE

H

FG

88.5

89.5

90.5

91.5

92.5

10.1 10.3 10.5 10.7 10.9 11.1

System cost (M$)

GH

G

(k-t

onne

) $72.1/tonneCO2-e

$376/tonneCO2-e $1,468/tonne CO2-e

(375)

(450)

Lowest cost

Lowest GHGs

Conclusions

• Evolutionary algorithm optimisation has application in design and operation of water distribution systems

• It is relatively easy to tack on an evolutionary algorithm optimisation onto existing simulation models

• Capital costs and operating costs can be reduced significantly

• Sustainability can be optimised