Smart Systems for Urban Water Demand Management

Smart Systems for Urban Water DemandManagement

Pantelis Sopasakisjoint work with A.K. Sampathirao, P. Patrinos & A. Bemporad.

IMT Institute for Advanced Studies LuccaMonte Verita, Switzerland, 22-25 Aug 2016

Today’s talk

We will learn how to:

I model water networks

I identify control objectives

I make decisions under uncertainty

I formulate MPC problems

I devise algorithms to solve them

I parallelise them on GPUs

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Introd

uctio

n

Con

trol

Alg

orith

ms

Simul

atio

ns

Wha

t to

mod

el

Lite

ratu

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verv

iew

Con

trol-o

rient

ed m

odel

s

Fore

cast

ing

The

MPC

con

cept

Wor

st-c

ase

MPC

Stoc

hast

ic M

PCSc

enar

io-b

ased

MPC

Prox

imal

ope

rato

r

Con

vex

conj

ugat

e

Dua

lity

APG

alg

orith

mD

WN

SM

PC p

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ems

Para

llelis

atio

n

KPIs

Sim

ulat

ion

resu

ltsCon

clus

ions

Modoller’s todos

Modelling and Control

of DWNs

PHYSICALPHYSICAL

mass balancespressure dropschlorine balances

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Modoller’s todos


of DWNs

PHYSICALPHYSICAL

TIME SERIESTIME SERIES


seasonal ARIMABATS modelsneural networksSVM and other ML

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Modoller’s todos


of DWNs

PHYSICALPHYSICAL


UNCERTAINTYUNCERTAINTY



parametric PDFsscenario trees

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Modoller’s todos


of DWNs

PHYSICALPHYSICAL



CONSTRAINTSCONSTRAINTS




tanks pumps pressure drops

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Modoller’s todos


of DWNs

PHYSICALPHYSICAL



CONSTRAINTSCONSTRAINTSOBJECTIVESOBJECTIVES




tanks pumps pressure drops

energy consumptionoperating costsmoothness of actuationsafety storage

`

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Control of water networks

ControllerDrinking Water

Network

Water demand

Electricity prices

Measurements

Actuation

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Taxonomy of control methodologies

Open Loop Closed Loop

CertaintyEquivalent Worst case Stochastic Risk-averse

Control of DWNs

● No feedback● Assumes perfect knowledge● No contingency plan● Requires human intervention Model Predictive

Control

Creasy 1998; Yu et al., 1994; Zessler and Shamir, 1989

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Control of DWNs

● Feedback compensates for modelling errors

Model Predictive Control


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Control of DWNs

● Suitable for MIMO● Control under uncertainty● Goal-driven● Requires simple model


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Control of DWNs

● Model assumed accurate● Constraints may be violated● Suboptimal (we can do better)


Sampathirao et al., 2014; Bakker et al., 2013; Leirens et al., 2010; Ocampo et al., 2009.

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Control of DWNs

● Pessimistic and conservative● Leads to small domain of attraction


Ocampo et al., 2009 and 2010.

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Control of DWNs

● Makes use of probabilistic info● Less conservative, more realistic


Sampathirao et al., 2014; Goryashko and Nemirovski, 2014; Tran and Brdys, 2009, Watkins and McKinney, 1997

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Control of DWNs

● Beyond the state of art● Uncertainty in uncertainty


Sampathirao et al., 2016

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Objectives

To MODEL (water demands, hydraulics, uncertainty, etc), pose astochastic predictive CONTROL problem (define objectives, constraints)

and devise algorithms to SOLVE it numerically.

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Intro

duct

ion

Con

trol

Alg

orith

ms

Simul

atio

ns

Wha

t to

mod

el

Lite

ratu

re o

verv

iew

Con

trol-o

rient

ed m

odel

s

Fore

cast

ing

The

MPC

con

cept

Wor

st-c

ase

MPC

Stoc

hast

ic M

PCSc

enar

io-b

ased

MPC

Prox

imal

ope

rato

r

Con

vex

conj

ugat

e

Dua

lity

APG

alg

orith

mD

WN

SM

PC p

robl

ems

Para

llelis

atio

n

KPIs

Sim

ulat

ion

resu

ltsCon

clus

ions

MPC

Control-oriented models

T1

T2

D1

T3

D2

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T1

T2

D1

T3

D2

Actuation

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T1

T2

D1

T3

D2

Uncertainty

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T1

T2

D1

T3

D2

States

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T1

T2

D1

T3

D2

Algebraic Conditions

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Our case study

DWN of Barcelona: 63 tanks, 114 pumping stations and valves, 88 demand nodes & 17 pipe intersection nodes.

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Simple mass balance equation (in discrete time)

xk+1 = Axk +Buk +Gddk,

0 = Euk + Eddk,

xk: tank volumes, uk: flows (controlled by pumping), dk: demands —along with the constraints

xmin ≤ xk ≤ xmax,

umin ≤ uk ≤ umax.

Sampathirao, Sopasakis et al., 2016, 2014; Wang et al., 2014; Ocampo et al., 2010; Ocampo et al., 2009.

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20 40 60 80 100 1202

3

4

5

6

7x 104 d100CFE

time (h)

m3

20 40 60 80 100 120

200

300

400

500

d114SCL

time (h)

m3

20 40 60 80 100 120

1000

2000

3000

4000

d115CAST

time (h)

m3

20 40 60 80 100 120

0.5

1

1.5

x 104 d130BAR

time (h)

m3

20 40 60 80 100 120500

1000

1500

2000

2500

3000

d132CMF

time (h)

m3

20 40 60 80 100 120

400

600

800

1000d135VIL

time (h)

m3

20 40 60 80 100 120200

400

600

800

1000

d176BARsud

time (h)

m3

20 40 60 80 100 120

5001000

1500

2000

25003000

d450BEG

time (h)

m3

20 40 60 80 100 120

1000

2000

3000

d80GAVi80CAS85

time (h)m

3

SimulatorReal dataUpper limitSafety level

Source: EFFINET, deliverable report D2.123 / 89

Demand forecasting

Demand prediction concept:

dk+j(εj) = dk+j|k + εj

where

1. dk+j : actual demand at time k + j

2. dk+j|k: prediction of dk+j using info up to time k

3. εj : j-step-ahead prediction error

and dk+j|k is a function of observable quantities up to time k.

Sampathirao, Sopasakis et al., 2016.

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Demand forecasting

Common approaches:

1. Neglect the error: (ε0, ε1, . . . , εN ) u (0, 0, . . . , 0)

2. Error bounds: (ε0, ε1, . . . , εN ) ∈ E , e.g., εj ∈ [εminj , εmax

j ]

3. Independent normal distributions: εj ∼ N (mj , σ2j )

4. (ε0, ε1, . . . , εN ) is random and admits finitely many values

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Demand forecasting

Common approaches:



j ]



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Demand forecasting

Common approaches:



j ]



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Demand forecasting

Common approaches:



j ]



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Error bounds

0 20 40 60 80 100 120 140 160 180 2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time [h]

Wa

ter

De

ma

nd

Flo

w [

m3/h

]

Forecasting of Water Demand

Future Past

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Continuous independent errors

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Predicted scenarios

Time [hr]

4370 4380 4390 4400 4410 4420 4430

Wa

ter

de

ma

nd

[m

3/s

]

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

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Control objectives

Stage costs:

1. Economic cost: `w(uk, k) = Wα(α1 + α2,k)′uk

2. Smooth operation cost: `∆(∆uk) = ∆u′kWu∆uk

3. Safe operation cost: `S(xk) = Wx‖[xs − xk]+‖4. Total cost: ` = `w + `∆ + `S .

We define ∆uk = uk − uk−1

Sampathirao et al., 2014; Cong Cong et al., 2014

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Control objectives

Stage costs:






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Control objectives

Stage costs:



3. Safe operation cost: `S(xk) = Wx‖[xs − xk]+‖

4. Total cost: ` = `w + `∆ + `S .



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Control objectives

Stage costs:






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Rawlings and Mayne, 2009

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Problem formulation

minimiseπ=({uk+j|k}j ,{xk+j|k}j)

V (π) :=

N−1∑j=0

`(xk+j|k, uk+j|k, uk+j−1|k, k),

subject to

xk+j+1|k = Axk+j|k +Buk+j|k +Gddk+j|k dynamics

Euk+j|k + Eddk+j|k = 0 algebraic

xmin ≤ xk+j|k ≤ xmax vol. constr.

umin ≤ uk+j|k ≤ umax flow constr.

xk|k = xk, uk−1|k = uk−1 initial cond.

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Worst-case MPC


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Worst-case MPCProblem formulation


maxdk+j|k

V (π),

subject to

dk+j = dk+j|k + εj predictions

εj ∈ Ej err. bounds

xk+j+1|k = Axk+j|k +Buk+j|k +Gddk+j dynamics

Euk+j|k + Eddk+j = 0 algebraic




Here uk+j|k is a function of εj – not a fixed value!

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Worst-case MPC

Attention! We are looking for control laws (functions) uk+j|k. We mayparametrise (why?) these functions as

uk+j|k = Kjej + bj ,

and solve for Kj and bj .

There exist other parametrisations as well.

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Worst-case MPC (tube-based)

Safe region


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Worst-case MPC (tube-based)

Problem formulation


V (π),

subject to

xk+j+1|k = Axk+j|k +Buk+j|k +Gddk+j|k dynamics

Euk+j|k + Eddk+j|k = 0 algebraic

xk+j|k ∈ X S volume constr.



* the constraint Euk+j|k + Eddk+j|k = 0 (certainty-equivalent) will not be satisfied for all εj

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

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Stochastic MPC

Problem formulation


EV (π),

subject to

xk+j+1|k = Axk+j|k +Buk+j|k +Gddk+j|k(εj) dynamics

Euk+j|k + Eddk+j(εj) = 0 alg. cond.




again, we’re looking for control laws uk+j|k(εj).

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Scenario trees

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Scenario trees

Demand forecasting:

dik+j = dk+j|k + εij

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Scenario trees

Input-disturbance coupling:

Euik+j|k + Eddik+j = 0

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Scenario trees

System dynamics:

xik+j+1|k=f(xanc(j+1,i)k+j|k , uik+j|k, d

ik+j|k)

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Scenario-based stochastic MPC

Problem formulation:

minimiseπ=({uk+ji|k}i,j ,{xk+ji|k}i,j)

EV (π)︷︸︸︷N−1∑j=0

µ(j)∑i=1

pij`ij ,

subject to


ik+j|k) dynamics

Euik+j|k + Eddik+j|k = 0 algebraic

xmin ≤ xik+j|k ≤ xmax volume constr.

umin ≤ uik+j|k ≤ umax flow constr.

x1k|k = xk, u

1k−1|k = uk−1 initial cond.

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Stochastic MPC (chance constraints)

Grosso et al., 2014.

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Intro

duct

ion

Con

trol

Alg

orith

ms

Simul

atio

ns

Wha

t to

mod

el

Lite

ratu

re o

verv

iew

Con

trol-o

rient

ed m

odel

s

Fore

cast

ing

The

MPC

con

cept

Wor

st-c

ase

MPC

Stoc

hast

ic M

PCSc

enar

io-b

ased

MPC

Prox

imal

ope

rato

r

Con

vex

conj

ugat

e

Dua

lity

APG

alg

orith

mD

WN

SM

PC p

robl

ems

Para

llelis

atio

n

KPIs

Sim

ulat

ion

resu

ltsCon

clus

ions

theory

The proximal operator

Let g : IRn → IR ∪ {+∞} be a proper, closed function and γ > 0. Define

proxγg(v) = arg minz

{g(z) + 1

2γ ‖z − v‖2}

If this is easy to compute we call g prox-friendly.

Parikh and Boyd, 2014; Combettes and Pesquet, 2010.

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Example 1. Take

g(x) = δ(x | C) =

{0, if x ∈ C+∞, otherwise

Then, proxγg(v) = proj(v | C).


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Example 2. Take

g(x) = d(x | C) = infy∈C‖y − x‖

Then,

proxλg(v) =

{v + proj(v|C)−v

d(v|C) , if d(v | C) > λ

proj(v | C), otherwise


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Key Property. Suppose g is given as

g(x) =

κ∑i=1

gi(xi),

then(proxλg(v))i = proxλgi(vi).


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Key Property. Suppose g is given as

g(x) =

κ∑i=1

gi(xi),

then(proxλg(v))i = proxλgi(vi).


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Convex Conjugate

Let f : IR→ IR ∪ {+∞} be convex, proper and closed. We define itsconvex conjugate as

f∗(y) = supx

{x′y − f(x)

}.

When f∗ is differentiable, then

∇f∗(y) = arg minz{x′y + f(z)}.

If f is prox-friendly, then

proxλf (v) + λproxλ−1f∗(λ−1v) = v.

If f is strongly convex, then f∗ is continuously diff/ble (Rockaffelar and Wets, 2009; Prop. 12.60).

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Convex Conjugate


f∗(y) = supx

{x′y − f(x)

}.






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Convex Conjugate


f∗(y) = supx

{x′y − f(x)

}.






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Forward-Backward Splitting

The forward-backward splitting is the representation of an optimisationproblem as follows

minimisez

f(z) + g(z),

where

I f, g: closed, convex

I f : diff/ble with Lipschitz gradient

I g: prox-friendly

FB Splittings are not unique.

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Example 1. `1-regularized least squares:

minimisez

12‖Az − b‖

2 + ‖z‖1.

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Example 2. Box-constrained QP

minimisez

12z′Qz + q′z + δ(z | C),

where C = {z : zmin ≤ z ≤ zmax}.

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Example 3. Constrained QP

minimisez

12z′Qz + q′z + δ(Hz | C).

but, g(z) := δ(Hz | C) is not prox-friendly!

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Dual optimisation problem

If g(z) is prox-friendly, but g(Hz) is not we may formulate the dualoptimisation problem

Under certain conditions these two problems have the same minimum and

z? = ∇f∗(−H ′y?).

Fenchel duality generalises Lagrangian duality (Rockafellar, 1972).

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Dual optimisation problem

If g(z) is prox-friendly, but g(Hz) is not we may formulate the dualoptimisation problem

Under certain conditions these two problems have the same minimum and

z? = ∇f∗(−H ′y?).

Fenchel duality generalises Lagrangian duality (Rockafellar, 1972).

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Proximal gradient algorithm

The proximal gradient method for solving

minimisez

f(z) + g(z)

where f is differentiable with L-Lipschitz gradient runs

zν+1 = proxγg(zν − γ∇f(zν)),

with γ ∈ (0, L−1).

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Dual proximal gradient algorithm

The proximal gradient method applied to the dual

minimisez

f∗(−H ′y) + g∗(y)

where f∗ is differentiable with L-Lipschitz gradient is

yν+1 = proxγg∗(yν + γH∇f∗(−H ′yν))

If f is L−1-strongly convex, then f∗ has L-Lipschitz gradient.

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The dual proximal gradient method

yν+1 = proxγg∗(yν + γH∇f∗(−H ′yν))

can be written as

zν = ∇f∗(−H ′yν)

tν = proxλ−1g(λ−1yν +Hzν)

yν+1 = yν + λ(Hzν − tν).

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Nesterov’s accelerated proximal gradient method converges as O(1/k2)instead of O(1/k):

wν = yν + θν(θ−1ν−1 − 1)(yν − yν−1)

zν = ∇f∗(−H ′wν)

tν = proxλ−1g(λ−1wν +Hzν)

yν+1 = wν + λ(Hzν − tν)

θν+1 = 12(√θ4ν + 4θ2

ν − θ2ν)

with θ0 = θ−1 = 1 and y0 = y−1 = 0 (Nesterov, 1983).

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The DWN control problem

P : minimiseπ=({uk+ji|k}i,j ,{xk+ji|k}i,j)

EV (π)︷︸︸︷N−1∑j=0

µ(j)∑i=1

pij`ij ,

subject to


ik+j|k) dynamics

Euik+j|k + Eddik+j|k = 0 algebraic

xmin ≤ xik+j|k ≤ xmax volume constr.

umin ≤ uik+j|k ≤ umax flow constr.

x1k|k = xk, u

1k−1|k = uk−1 initial cond.

We will write this problem as: minimisezf(z) + g(Hz).

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Let z = {xij , uij} and define

f(z) =

N−1∑j=0

µ(j)∑i=1

pij(`w(uij) + `∆(∆uij)) + δ(uij |Φ1(dij))

+ δ(xij+1, uij , x

anc(j+1,i)j |Φ2(dij)),

whereΦ1(d) = {u : Eu+ Edd = 0}

andΦ2(d) = {(x+, x, u) : x+ = Ax+Bu+Gdd}

Sampathirao et al., 2016.

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In other words:

f(z) = smooth cost + dynamics + alg equations

and

g(z) = all the rest

= nonsmooth cost + constraints

=

N−1∑j=0

µ(j)∑i=1

`S(xij) + δ(xij | X) + δ(uij | U),

but this g is not prox-friendly (it is not separable!).

Sampathirao et al., 2016.

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We create a copy of {xij}j,i which we denote by χij and introduce

t = ({xij}j,i, {χij}j,i, {uij}j,i)

with xij = χij . Then

g(t) =

N−1∑j=0

µ(j)∑i=1

`S(xij) + δ(χij | X) + δ(uij | U)

is prox-friendly.

It is t = Hz.

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Dual gradient computation

To compute the dual gradient we use

∇f∗(y) = arg minz{x′y + f(z)}

= arg minz:dynamics

Euij+Eddij=0

{x′y +∑j,i

quadratic(zij)}

This is an equality-constrained quadratic problem which can be solvedvery efficiently using dynamic programming.

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MV

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MV + Sum MV + Sum MV + Sum

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MV

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MV + Sum MV + Sum

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Intro

duct

ion

Con

trol

Algo

rithm

s

Simul

atio

ns

Wha

t to

mod

el

Lite

ratu

re o

verv

iew

Con

trol-o

rient

ed m

odel

s

Fore

cast

ing

The

MPC

con

cept

Wor

st-c

ase

MPC

Stoc

hast

ic M

PCSc

enar

io-b

ased

MPC

Prox

imal

ope

rato

r

Con

vex

conj

ugat

e

Dua

lity

APG

alg

orith

mD

WN

SM

PC p

robl

ems

Para

llelis

atio

n

KPIs

Sim

ulat

ion

resu

ltsCon

clus

ions

Control scheme

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KPIs

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KPIs

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KPIs

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KPIs

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it is fast

# scenarios

100 200 300 400 500

Ru

ntim

e (

s)

0

200

400

600

800

1000

1200

Gurobi

APG (500 iterations)

0 100 200 300 400 500

Ru

ntim

e s

lop

e

0

2

4

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it is efficient

Number of scenarios

50 100 150 200 250 300

KP

I E/1

00

0

1.55

1.6

1.65

1.7

1.75

1.8

KP

I S/1

000

1

2

3

4

5

6KPIE

KPIS

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Closed-loop simulations

20 40 60 80 100 120 140 160

Contr

ol action [%

]

0

0.2

0.4

0.6

0.8

Time [hr]

20 40 60 80 100 120 140 160

Wate

r C

ost [e

.u.]

60

70

80

90

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Intro

duct

ion

Con

trol

Algo

rithm

s

Sim

ulat

ions

Wha

t to

mod

el

Lite

ratu

re o

verv

iew

Con

trol-o

rient

ed m

odel

s

Fore

cast

ing

The

MPC

con

cept

Wor

st-c

ase

MPC

Stoc

hast

ic M

PCSc

enar

io-b

ased

MPC

Prox

imal

ope

rato

r

Con

vex

conj

ugat

e

Dua

lity

APG

alg

orith

mD

WN

SM

PC p

robl

ems

Para

llelis

atio

n

KPIs

Sim

ulat

ion

resu

ltsCon

clus

ions

Open problems

I Nonlinear

I Risk-averse

I Distributed

I Robust Economic MPC

I Faster Algorithms

Problem: introduce nonlinear pressure drop equations

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Open problems

I Nonlinear

I Risk-averse

I Distributed


I Faster Algorithms

Challenge: the distribution of future errors is not exactly known

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Open problems

I Nonlinear

I Risk-averse

I Distributed


I Faster Algorithms

Problems: spatial decomposition, communication constraints

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Open problems

I Nonlinear

I Risk-averse

I Distributed


I Faster Algorithms

Questions: performance guarantees, recursive feasibility

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Open problems

I Nonlinear

I Risk-averse

I Distributed


I Faster Algorithms

Ongoing work: quasi-Newtonian LBFGS-type algorithms

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Thank you for your attention!

References (i)

1. A. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, “Fast parallelizable scenario-based stochasticoptimization,” EUCCO 2016, Leuven, Belgium, 2016.

2. A. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, “GPU-accelerated stochastic predictive control of drinkingwater networks,” IEEE CST (submitted, provisionally accepted), 2016 (on arXiv).

3. A. Sampathirao, J. Grosso, P. Sopasakis, C. Ocampo-Martinez, A. Bemporad, and V. Puig, “Water demand forecastingfor the optimal operation of large-scale drinking water networks: The Barcelona case study,” 19th IFAC World Congress,pp. 10457–10462, 2014.

4. A. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, “Distributed solution of stochastic optimal controlproblems on GPUs, 54th IEEE Conf. Decision and Control, (Osaka, Japan), Dec 2015.

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