Global Optimization of Sizing Problem in Pipe Networks

Global Optimization of Sizing Problem in Pipe Networks Arvind U. Raghunathan System Dynamics & Optimization United Technologies Research Center East Hartford, CT, USA

The International Conference on Continuous Optimization, 2010

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Introduction

Outline 1

Introduction Motivation Problem Statement Previous Work

2

Convex MINLP Formulation Hydraulic Calculation as Convex Program Convex MINLP Formulation

3

Linearizations-based MINLP Algorithm Deriving Linearizations

4

Results Water Distribution Networks

5

Summary

6

Appendix A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Introduction

Motivation

Motivation Networks with Nonlinear Resistances Resistances associated with transmission lines Cost and resistance are inversely correlated Minimize cost of distribution network

(a) Water Distribution A. U. Raghunathan (UTRC)

(b) Gas Pipeline Global Opt. of Sizing in Pipe Networks

(c) Electrical Transmission ICCOPT 2010

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Introduction

Problem Statement

Problem Statement: Minimum Cost Network

0

2

1

3

4

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Introduction

Problem Statement

Problem Statement: Minimum Cost Network Pump nodes (N pump ) Fixed pressure

pump (Pi )

Pressure Drop Function φd (Q)

Elevation (Hi )

0 Edges (E) Length (Lij ) Diameter choices (D := {D1 , . . . , Dnd }) max Flow bound (Qij,d ∀d ∈ D)

2

1

3 Q Cost Function cd * *

4 Demand nodes (N \ N pump )

*

Bounds on pressure (Pimin , Pimax ) Elevation (Hi ) Demand (Qidem ) A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Introduction

Problem Statement

Pressure Drop Constraint Pressure Drop Function: φd : R+ → R+ 1.85

φd (Q) ∝ Q d 4.87 2 φd (Q) ∝ Q d5

Hazen-Williams Darcy-Weisbach

j i

j i

Flow from i to j

Flow from j to i

(Hi +Pi )−(Hj +Pj ) = φd (Qij )Lij

(Hi + Pi ) − (Hj + Pj ) = −φd (−Qij )Lij

Non-smooth & Non-convex (Hi + Pi ) − (Hj + Pj ) = sgn(Qij )φd (|Qij |)Lij A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Introduction

Problem Statement

Non-Smooth, Non-Convex MINLP Formulation Objective Function

nd P P

cd Lij xij,d

d=1 (i,j)∈E

Choice of one diameter

nd P

xij,d = 1∀(i, j) ∈ E

d=1

xij,d ∈ {0, 1} Flows on edges

−Qijmax xij,d ≤ Qij,d ≤ Qijmax xij,d ∀(i, j) ∈ E, d = 1, . . . , n

Flow conservation

nd P P

Qij,d −

d=1 (j,i)∈E

∀i ∈ N \ N Pressure drop constraints

nd P P

Qij,d = Qidem

d=1 (i,j)∈E pump

(Hi + Pi ) − (Hj + Pj ) =

nd P

sgn(Qij )φd (|Qij,d |)Lij

d=1

∀(i, j) ∈ E Pimin ≤ Pi ≤ Pimax ∀i ∈ N A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Introduction

Previous Work

Previous Work Split-pipe Formulation: Shamir & co-workers [2, 4] Allow pipes of different diameters xij,d ∈ {0, 1} → xij,d ∈ [0, 1] Qij,d = Qij

Two-stage iterative algorithm Fix flows, Qij to satisy flow demands Solve resulting Linear Program (LP) Modify flows based on sub-gradients

MILP Formulation: Artina & Walker Piecewise linear inner approximation of pressure drop constraint Posed as SOS2 constraint

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Introduction

Previous Work

Previous Work (contd.)

MINLP Formulation: Bragalli & others [6] Smooth pressure drop constraint to obtain MINLP Fit cost as smooth function of diameter Use cost-fit for NLP relaxation of MINLP Non-convex MINLP solved by Bonmin’s Branch & Bound

Convergence to Global Optimum not guaranteed

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Global Opt. of Sizing in Pipe Networks

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Convex MINLP Formulation

Outline 1

Introduction Motivation Problem Statement Previous Work

2

Convex MINLP Formulation Hydraulic Calculation as Convex Program Convex MINLP Formulation

3

Linearizations-based MINLP Algorithm Deriving Linearizations

4

Results Water Distribution Networks

5

Summary

6

Appendix A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Convex MINLP Formulation

Hydraulic Calculation as Convex Program

Conventional Formulation

Nonlinear Equations (NLE) (Pi + Hi ) − (Pj + Hj ) = sgn(Qij )φdij (Qij )Lij Pi = Pipump P P Qji − Qij = Qidem j:(j,i)∈E

∀(i, j) ∈ E ∀i ∈ N pump ∀i ∈ N \ N pump

j:(i,j)∈E

max ≤ Q ≤ Q max −Qij,d ij ij,dij ij

∀(i, j) ∈ E

Assumption 1

φd (·) is strictly monotonically increasing

2

Pressure drop inversely proportional to diameter φd (Q) < φd 0 (Q) ∀d > d 0 , Q > 0

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Global Opt. of Sizing in Pipe Networks

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Convex MINLP Formulation

Hydraulic Calculation as Convex Program

Convex Formulation Definition Φd (Q) =

RQ

φd (Q 0 )dQ 0

0

Variational Formulation (VF) [3] min

P (i,j)∈E

(Φdij (Qij+ ) + Φdij (Qij− ))Lij

−

P i∈N pump ,(i,j)∈E

s.t. πij − λ+ ij , λij − Λ+ ij , Λij

P

(Hipump + Pipump )(Qij+ − Qij− )

(Qji+ − Qji− ) −

j:(j,i)∈E Qij+ , Qij− Qij+ , Qij−

P j:(i,j)∈E

(Qij+ − Qij− ) = Qidem ∀i ∈ N \ N pump

≥ 0∀(i, j) ∈ E max ∀(i, j) ∈ E ≤ Qij,d ij

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Convex MINLP Formulation

Hydraulic Calculation as Convex Program

Sketch of the Proof Theorem (Q, P) solves (NLE) iff (Q + , Q − ,π, λ,Λ) with Λ = 0. Proof. Only If Qij+ = max(0, Qij ); Qij− = min(0, Qij ); πi = Hi + Pi ; λ+ ij = max(0, (H + P = max(0, (Hj + Pj ) − (Hi + Pi )); λ− i i ) − (Hj + Pj )) ij Φd (·) is strictly convex =⇒ Solves (VF) If Solves (VF) =⇒ Qij+ > 0 or Qij− > 0 πj − πi +φdij (Qij+ )Lij − λ+ ij = 0 πi − πj +φdij (Qij− )Lij − λ− ij = 0 Choose, Pi = πi −Hi ; Qij = Qij+ − Qij− A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Convex MINLP Formulation

Hydraulic Calculation as Convex Program

Sketch of the Proof Theorem (Q, P) solves (NLE) iff (Q + , Q − ,π, λ,Λ) with Λ = 0. Proof. Only If Qij+ = max(0, Qij ); Qij− = min(0, Qij ); πi = Hi + Pi ; λ+ ij = max(0, (H + P = max(0, (Hj + Pj ) − (Hi + Pi )); λ− i i ) − (Hj + Pj )) ij Φd (·) is strictly convex =⇒ Solves (VF) If Solves (VF) =⇒ Qij+ > 0 or Qij− > 0 πj − πi +φdij (Qij+ )Lij − λ+ ij = 0 πi − πj +φdij (Qij− )Lij − λ− ij = 0 Choose, Pi = πi −Hi ; Qij = Qij+ − Qij− A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Convex MINLP Formulation

Hydraulic Calculation as Convex Program

Bound Multiplier as Flow Limiting Valve max (NLE) is ill-posed if Qij > Qij,d ij

(Hi + Pi ) − (Hj + Pj ) = φdij (Qij )Lij

Λ+ ij > 0 - Pressure loss in valve πi − πj = φdij (Qij+ )Lij + Λ+ ij Smooth model for flow limiters

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Convex MINLP Formulation

Convex MINLP Formulation

Convex MINLP Formulation Additional Variables

Bounds on Flows

Direction variables: xijdir ∈ + − Flow variables: Qij,d , Qij,d

{0, 1}

+ max x dir Qij,d ≤ Qij,d ij − max (1 − x dir ) Qij,d ≤ Qij,d ij

Convexify the Pressure Drop Constraint (Hi + Pi ) − (Hj + Pj ) ≥ (Hj + Pj ) − (Hi + Pi ) ≥

nd P d=1 nd P d=1

A. U. Raghunathan (UTRC)

+ φd (Qij,d )Lij − ∆P max (1 − xijdir ) − φd (Qij,d )Lij − ∆P max xijdir

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Linearizations-based MINLP Algorithm

Outline 1

Introduction Motivation Problem Statement Previous Work

2

Convex MINLP Formulation Hydraulic Calculation as Convex Program Convex MINLP Formulation

3

Linearizations-based MINLP Algorithm Deriving Linearizations

4

Results Water Distribution Networks

5

Summary

6

Appendix A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Linearizations-based MINLP Algorithm

Sketch of MINLP Algorithm Root Node

Converges to Global Optimum Integer Infeasible Check for violation of constraints Add linearization of violated constraint

Integer Feasible

Integer Feasible

Solve (VF) for given diameters

Solve (VF) for given diameters

Feasible w.r.t. pressure requirement

Infeasible w.r.t. pressure requirement

Add linearization of violated constraint

Fathom node(s)

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Global Opt. of Sizing in Pipe Networks

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Linearizations-based MINLP Algorithm

Deriving Linearizations

Linearizations at Integer Infeasible Nodes Local: Relaxation of Pressure Drop (RelPD) Let (x , x dir , P, Q + , Q − ) be solution to node LP. For each edge (i, j) ∈ E, Pick direction based on xijdir If xijdir > 0.5, evaluate pressure drop constraint from i → j cij = −(Hi + Pi ) + (Hj + Pj ) +

nd P d=1

+ φd (Qij,d )Lij + ∆P max (1 − xijdir )

If xijdir ≤ 0.5, evaluate pressure drop constraint from j → i cij = (Hi + Pi ) − (Hj + Pj ) +

nd P d=1

− φd (Qij,d )Lij + ∆P max xijdir

If cij > , add linearization

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Linearizations-based MINLP Algorithm

Deriving Linearizations

Linearizations at Integer Infeasible Nodes (contd.)

0

2

1

Global: Directed Cycles (DirCyc)

3

Let (x , x dir , P, Q + , Q − ) be solution to node LP. For each edge (i, j) ∈ E, If xijdir > 0.5, then flow from i → j If xijdir ≤ 0.5, then flow from j → i

4

A. U. Raghunathan (UTRC)

Check for directed cycle. If directed cycle exists, add constraint to invalidate the choice of edge directions.

Global Opt. of Sizing in Pipe Networks

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Linearizations-based MINLP Algorithm

Deriving Linearizations

Linearizations at Integer Infeasible Nodes (contd.) Maximum Pressure Drop Infeasibility + (Pi + Hi ) − (Pj + Hj ) ≥ max φd (Qij,d )Lij − ∆P max (1 − xijdir ) d=1,...,nd

Linearization provides info on a single diameter only ˆ+ ) − (Hi + Pi ) − (Hj + Pj ) ≥ (φd (Q ij,d ˆ + )Q ˆ + )Lij + ∇φd (Q ˆ + )Lij Q + ∇φd (Q ij,d ij,d ij,d ij,d −∆P max (1 − xijdir )

φd (Q)

ˆ + ) − ∇φd (Q ˆ + )Q + = φd (Q ij,d ij,d ij,d eq eq eq φd 0 (Qij,d 0 ) − ∇φd 0 (Qij,d 0 )Qij,d 0 eq ˆ+ Q Qij,d 0Q ij,d

A. U. Raghunathan (UTRC)

Global: Max. Pressure Drop (MaxPD) ˆ+ ) − (Hi + Pi ) − (Hj + Pj ) ≥ (φd (Q ij,d ˆ + )Q ˆ + )Lij + ∇φd (Q ˆ + )Lij Q + ∇φd (Q ij,d ij,d ij,d ij,d eq + max +∇φd 0 (Qij,d (1 − xijdir ) 0 )Lij Qij,d 0 −∆P

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Linearizations-based MINLP Algorithm

Deriving Linearizations

Linearizations at Integer Infeasible Nodes (contd.) Maximum Pressure Drop Infeasibility + (Pi + Hi ) − (Pj + Hj ) ≥ max φd (Qij,d )Lij − ∆P max (1 − xijdir ) d=1,...,nd

Linearization provides info on a single diameter only ˆ+ ) − (Hi + Pi ) − (Hj + Pj ) ≥ (φd (Q ij,d ˆ + )Q ˆ + )Lij + ∇φd (Q ˆ + )Lij Q + ∇φd (Q ij,d ij,d ij,d ij,d −∆P max (1 − xijdir )

φd (Q)

ˆ + ) − ∇φd (Q ˆ + )Q + = φd (Q ij,d ij,d ij,d eq eq eq φd 0 (Qij,d 0 ) − ∇φd 0 (Qij,d 0 )Qij,d 0 eq ˆ+ Q Qij,d 0Q ij,d

A. U. Raghunathan (UTRC)

Global: Max. Pressure Drop (MaxPD) ˆ+ ) − (Hi + Pi ) − (Hj + Pj ) ≥ (φd (Q ij,d ˆ + )Q ˆ + )Lij + ∇φd (Q ˆ + )Lij Q + ∇φd (Q ij,d ij,d ij,d ij,d eq + max +∇φd 0 (Qij,d (1 − xijdir ) 0 )Lij Qij,d 0 −∆P

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Linearizations-based MINLP Algorithm

Deriving Linearizations

Linearizations at Integer Infeasible Nodes (contd.) Maximum Pressure Drop Infeasibility + (Pi + Hi ) − (Pj + Hj ) ≥ max φd (Qij,d )Lij − ∆P max (1 − xijdir ) d=1,...,nd

Linearization provides info on a single diameter only ˆ+ ) − (Hi + Pi ) − (Hj + Pj ) ≥ (φd (Q ij,d ˆ + )Q ˆ + )Lij + ∇φd (Q ˆ + )Lij Q + ∇φd (Q ij,d ij,d ij,d ij,d −∆P max (1 − xijdir )

φd (Q)

ˆ + ) − ∇φd (Q ˆ + )Q + = φd (Q ij,d ij,d ij,d eq eq eq φd 0 (Qij,d 0 ) − ∇φd 0 (Qij,d 0 )Qij,d 0 eq ˆ+ Q Qij,d 0Q ij,d

A. U. Raghunathan (UTRC)

Global: Max. Pressure Drop (MaxPD) ˆ+ ) − (Hi + Pi ) − (Hj + Pj ) ≥ (φd (Q ij,d ˆ + )Q ˆ + )Lij + ∇φd (Q ˆ + )Lij Q + ∇φd (Q ij,d ij,d ij,d ij,d eq + max +∇φd 0 (Qij,d (1 − xijdir ) 0 )Lij Qij,d 0 −∆P

Global Opt. of Sizing in Pipe Networks

ICCOPT 2010

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Linearizations-based MINLP Algorithm

Deriving Linearizations

Cuts at Integer Feasible Nodes 0

Global: Infeasible Configurations (InfCfg) Solve (VF) to check for pressure satisfaction.

2

1

4

3

If not satisfied Add cut to invalidate this configuration. Invalidate configurations with same flow direction and smaller diamaters If satisfied Add cut to invalidate other flow directions in this configuration.

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Linearizations-based MINLP Algorithm

Deriving Linearizations

MINLP Algorithm Data: nrml = 10−2 ; slwp = 10−6 ; N RelPD = 151; N MaxPD = 40 begin Start Branch-and-Bound algorithm. while Gapintegral > 0 do  = (Gapintegral reduction over 100 nodes > 10−5 ) ? nrml : slwp for each node in tree do if Directed-Cycle then Add global cut (DirCyc) to invalidate choice of directions on edges Continue if integer infeasible and mod(N node , N MaxPD ) == 0 then Add global cut (MaxPD) if MaxDP-Infeas(i, j) >  ∀(i, j) ∈ E if integer infeasible and mod(N node , N RelPD ) == 0 then Add local cut (RelPD) if RelDP-Infeas(i, j) > ∀(i, j) ∈ E if integer feasible then Solve the (VF) for given configuration. Add global cut to invalidate flow direction choices for given configuration if PressDrop-Infeas or MaxFlow-Active then Add global to invalidate current configuration if PressDrop-Infeas then Add global cut (InfCfg) to invalidate other configurations with same flow direction and smaller diameter sizes

A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Results

Outline 1

Introduction Motivation Problem Statement Previous Work

2

Convex MINLP Formulation Hydraulic Calculation as Convex Program Convex MINLP Formulation

3

Linearizations-based MINLP Algorithm Deriving Linearizations

4

Results Water Distribution Networks

5

Summary

6

Appendix A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Results

Water Distribution Networks

Example Problems (Bragalli & others [6])

Problem name shamir hanoi new york blacksburg foss_poly_0 foss_iron foss_poly_1 pescara modena

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# nodes 8 33 21 32 38 38 38 74 276

# edges 8 34 21 25 58 58 58 99 317

Global Opt. of Sizing in Pipe Networks

# diameters 14 6 12 11 7 13 22 13 13

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Results

Water Distribution Networks

Computational Results - Moderate Instances Implementation CPLEX using CPXcutcallback IPOPT - NLP solver Pentium Duo Core - 2.63 GHz, 3 GB RAM, 1 thread Automatic variable priority Integrality Gap =0.0 % Problem name shamir hanoi new york blacksburg

Optimal Objective 419, 000 6, 109, 620.09 3, 9307, 799.72 118, 251.2

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# NLPs 34 135 53 252

# cuts 203 2072 1783 810

Global Opt. of Sizing in Pipe Networks

# CPU time (sec) 3.7 822.4 258.3 113

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Results

Water Distribution Networks

Computational Results - Large Instances

Difficulty in solving Unable to close gap after 2 hrs of computation. Unable to find feasible solution. Problem name foss_iron foss_poly_0 foss_poly_1 pescara modena

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Lower Bound 175, 423 6.74902 · 107 25, 799.4 1.63103 · 106 2.1108 · 106

Best Objective 221, 155 8.91541 · 107 −−− −−− −−−

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Results

Water Distribution Networks

Extension to Layout & Sizing Optimization Choice of edge & diameter

nd P

xij,d ≤ 1

∀(i, j) ∈ E

d=1

Afshar and Jabbari [5] Solved using Genetic Algorithm Problem name # Nodes # Edges # Diameters Reported Optimal Obj. Optimal Obj. CPU time (sec) Gap(%)

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Network I 9 12 13 39400 38600 4140 4.15

Global Opt. of Sizing in Pipe Networks

Network II 20 37 13 1783086 — 7200 —

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Summary

Outline 1

Introduction Motivation Problem Statement Previous Work

2

Convex MINLP Formulation Hydraulic Calculation as Convex Program Convex MINLP Formulation

3

Linearizations-based MINLP Algorithm Deriving Linearizations

4

Results Water Distribution Networks

5

Summary

6

Appendix A. U. Raghunathan (UTRC)

Global Opt. of Sizing in Pipe Networks

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Summary

Summary

Hydraulic calculation can be cast as Convex Program Sizing optimization problem is Convex MINLP Tailored linearizations-based algorithm (a la FILMINT) Computationally efficient on moderate-size problems Outlook Large-scale problems Dynamic optimization problems Extension to electrical grid

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Appendix

Outline 1

Introduction Motivation Problem Statement Previous Work

2

Convex MINLP Formulation Hydraulic Calculation as Convex Program Convex MINLP Formulation

3

Linearizations-based MINLP Algorithm Deriving Linearizations

4

Results Water Distribution Networks

5

Summary

6

Appendix A. U. Raghunathan (UTRC)

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Appendix

References I E. C. Cherry. Some general theorems for non-linear systems possessing reactance. Philosophical Magazine, (Ser. 7)42:1161–1177, 1951. E. Alperovits and U.Shamir. Design of optimal water distribution systems. Water Resources Research 13(6):885–900, 1977. M. Collins, L. Cooper, R. Legason, J. Kennington and L. LeBlanc. Solving the pipe network analysis problem using optimization techniques. Management Science, 24(7):747–760, 1978. G. Eiger, U. Shamir and A. Ben-Tal. Optimal design of water distribution networks. Water Resources Research, 30(9):2637–2646, 1994. A. U. Raghunathan (UTRC)

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Appendix

References II

M. H. Afshar and E. Jabbari. Simultaneous layout and pipe size optimization of pipe networks using genetic algorithm. The Arabian Journal for Science and Engineering, 33(2B):391–409, 2008. C. Bragalli, C. D’Ambrosio, J. Lee, A. Lodi and P. Toth. Water Network Design by MINLP, IBM Research Report RC24495, 2008.

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Appendix

Computational Results - Large Instances II Λ+ ij > 0 - Pressure loss in valve πi − πj = φdij (Qij+ )Lij + Λ+ ij Assume flow bounds are imposed through flow limiters

Problem name foss_iron foss_poly_0 foss_poly_1

Best Objective 175, 922 6.95878 · 107 28, 569.4

Gap (%) 0.0 4.15 10.7

CPU time (sec) 166 10800 10800

# Flow Infeas. 4 5 15

Modeling of flow bounds

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