Smart Systems for Urban Water Demand Management

Smart Systems for Urban Water Demand
Management
Pantelis Sopasakis
joint work with A.K. Sampathirao, P. Patrinos & A. Bemporad.
IMT Institute for Advanced Studies Lucca
Monte Verit´a, Switzerland, 22-25 Aug 2016

Today’s talk
We will learn how to:
model water networks
identify control objectives
make decisions under uncertainty
formulate MPC problems
devise algorithms to solve them
parallelise them on GPUs
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Introduction
Control
Algorithm
s
Sim
ulations
W
hattomodel
Literatureoverview
Control-orientedmodels
ForecastingTheMPC
concept
W
orst-caseMPC
StochasticMPCScenario-basedMPCProximaloperator
Convexconjugate
Duality
APG
algorithm
DW
N
SMPC
problems
Parallelisation
KPIs
Simulationresults
Conclusions

Modoller’s todos
Modelling
and Control
of DWNs
PHYSICALPHYSICAL
mass balances
pressure drops
chlorine balances
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Modoller’s todos
Modelling
and Control
of DWNs
PHYSICALPHYSICAL
TIME SERIESTIME SERIES
mass balances
pressure drops
chlorine balances
seasonal ARIMA
BATS models
neural networks
SVM and other ML
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Modoller’s todos
Modelling
and Control
of DWNs
PHYSICALPHYSICAL
UNCERTAINTYUNCERTAINTY
mass balances
pressure drops
chlorine balances
seasonal ARIMA
BATS models
neural networks
SVM and other ML
parametric PDFs
scenario trees
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Modoller’s todos
Modelling
and Control
of DWNs
PHYSICALPHYSICAL
CONSTRAINTSCONSTRAINTS
mass balances
pressure drops
chlorine balances
seasonal ARIMA
BATS models
neural networks
SVM and other ML
parametric PDFs
scenario trees
tanks
pumps
pressure drops
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Modoller’s todos
Modelling
and Control
of DWNs
PHYSICALPHYSICAL
CONSTRAINTSCONSTRAINTSOBJECTIVESOBJECTIVES
mass balances
pressure drops
chlorine balances
seasonal ARIMA
BATS models
neural networks
SVM and other ML
parametric PDFs
scenario trees
tanks
pumps
pressure drops
energy consumption
operating cost
smoothness of actuation
safety storage
`
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Control of water networks
Controller
Drinking Water
Network
Water demand
Electricity prices
Measurements
Actuation
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Taxonomy of control methodologies
Open Loop Closed Loop
Certainty
Equivalent
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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Certainty
Equivalent
Control of DWNs
● Feedback compensates
for modelling errors
Model Predictive
Control
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Model Predictive
Control
Certainty
Equivalent
Control of DWNs
● Suitable for MIMO
● Control under uncertainty
● Goal-driven
● Requires simple model
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Certainty
Equivalent
Control of DWNs
● Model assumed accurate
● Constraints may be violated
● Suboptimal (we can do better)
Model Predictive
Control
Sampathirao et al., 2014; Bakker et al., 2013; Leirens et al., 2010; Ocampo et al., 2009.
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Certainty
Equivalent
Control of DWNs
● Pessimistic and conservative
● Leads to small domain of attraction
Model Predictive
Control
Ocampo et al., 2009 and 2010.
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Certainty
Equivalent
Control of DWNs
● Makes use of probabilistic info
● Less conservative, more realistic
Model Predictive
Control
Sampathirao et al., 2014; Goryashko and Nemirovski, 2014; Tran and Brdys, 2009, Watkins and McKinney, 1997
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Certainty
Equivalent
Control of DWNs
● Beyond the state of art
● Uncertainty in uncertainty
Model Predictive
Control
Sampathirao et al., 2016
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Objectives
To MODEL (water demands, hydraulics, uncertainty, etc), pose a
stochastic predictive CONTROL problem (deﬁne objectives, constraints)
and devise algorithms to SOLVE it numerically.
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Introduction
Control
Algorithm
s
Sim
ulations
W
hattomodel
Literatureoverview
ForecastingTheMPC
concept
W
orst-caseMPC
Convexconjugate
Duality
APG
algorithm
DW
N
SMPC
problems
Parallelisation
KPIs
Simulationresults
Conclusions
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 80 100 120
2
3
4
5
6
7
x 10
4 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 10
4 d130BAR
time (h)
m3
20 40 60 80 100 120
500
1000
1500
2000
2500
3000
d132CMF
time (h)
m3
20 40 60 80 100 120
400
600
800
1000
d135VIL
time (h)
m3
20 40 60 80 100 120
200
400
600
800
1000
d176BARsud
time (h)
m3
20 40 60 80 100 120
500
1000
1500
2000
2500
3000
d450BEG
time (h)
m3
20 40 60 80 100 120
1000
2000
3000
d80GAVi80CAS85
time (h)
m3
Simulator
Real data
Upper limit
Safety level
Source: EFFINET, deliverable report D2.1
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Demand forecasting
Demand prediction concept:
dk+j( j) = ˆdk+j|k + j
where
1. dk+j: actual demand at time k + j
2. ˆdk+j|k: prediction of dk+j using info up to time k
3. j: j-step-ahead prediction error
and ˆdk+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: ( 0, 1, . . . , N ) (0, 0, . . . , 0)
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Demand forecasting
Common approaches:
2. Error bounds: ( 0, 1, . . . , N ) ∈ E, e.g., j ∈ [ min
j , max
j ]
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Demand forecasting
Common approaches:
j , max
j ]
3. Independent normal distributions: j ∼ N(mj, σ2
j )
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Demand forecasting
Common approaches:
j , max
j ]
3. Independent normal distributions: j ∼ N(mj, σ2
j )
4. ( 0, 1, . . . , N ) is random and admits ﬁnitely many values
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Error bounds
0 20 40 60 80 100 120 140 160 180 200
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Time [h]
WaterDemandFlow[m
3
/h]
Forecasting of Water Demand
FuturePast
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Continuous independent errors
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Predicted scenarios
Time [hr]
4370 4380 4390 4400 4410 4420 4430
Waterdemand[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
We deﬁne ∆uk = uk − uk−1
Sampathirao et al., 2014; Cong Cong et al., 2014
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Control objectives
Stage costs:
2. Smooth operation cost: ∆(∆uk) = ∆ukWu∆uk
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Control objectives
Stage costs:
3. Safe operation cost: S(xk) = Wx [xs − xk]+
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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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Model Predictive Control
Rawlings and Mayne, 2009
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Problem formulation
minimise
π=({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 + Gd
ˆdk+j|k dynamics
Euk+j|k + Ed
ˆdk+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 MPC
Attention! We are looking for control laws (functions) uk+j|k. We may
parametrise (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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Stochastic MPC
Problem formulation
minimise
EV (π),
subject to
ˆdk+j|k( j) dynamics
Euk+j|k + Eddk+j( j) = 0 alg. cond.
again, we’re looking for control laws uk+j|k( j ).
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Scenario trees
Demand forecasting:
di
k+j = ˆdk+j|k + i
j
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Scenario trees
Input-disturbance coupling:
Eui
k+j|k + Eddi
k+j = 0
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Scenario trees
System dynamics:
xi
k+j+1|k=f(x
anc(j+1,i)
k+j|k , ui
k+j|k, di
k+j|k)
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Scenario-based stochastic MPC
Problem formulation:
minimise
π=({uk+ji|k}i,j,{xk+ji|k}i,j)
EV (π)
N−1
j=0
µ(j)
i=1
pi
j
i
j,
subject to
xi
k+j+1|k=f(x
anc(j+1,i)
k+j|k , ui
k+j|k, di
k+j|k) dynamics
Eui
k+j|k + Ed
ˆdi
k+j|k = 0 algebraic
xmin ≤ xi
k+j|k ≤ xmax volume constr.
umin ≤ ui
k+j|k ≤ umax ﬂow constr.
x1
k|k = xk, u1
k−1|k = uk−1 initial cond.
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Stochastic MPC (chance constraints)
Grosso et al., 2014.
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Introduction
Control
Algorithm
s
Sim
ulations
W
hattomodel
Literatureoverview
ForecastingTheMPC
concept
W
orst-caseMPC
Convexconjugate
Duality
APG
algorithm
DW
N
SMPC
problems
Parallelisation
KPIs
Simulationresults
Conclusions
theory

The proximal operator
Let g : IRn
→ IR ∪ {+∞} be a proper, closed function and γ > 0. Deﬁne
proxγg(v) = arg min
z
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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Key Property. Suppose g is given as
g(x) =
κ
i=1
gi(xi),
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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 its
convex conjugate as
f∗
(y) = sup
x
x y − f(x) .
If f is strongly convex, then f∗
is continuously diff/ble (Rockaffelar and Wets, 2009; Prop. 12.60).
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Convex Conjugate
convex conjugate as
f∗
(y) = sup
x
x y − f(x) .
When f∗ is diﬀerentiable, then
f∗
(y) = arg min
z
{x y + f(z)}.
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Convex Conjugate
convex conjugate as
f∗
(y) = sup
x
x y − f(x) .
When f∗ is diﬀerentiable, then
f∗
(y) = arg min
z
{x y + f(z)}.
If f is prox-friendly, then
proxλf (v) + λproxλ−1f∗ (λ−1
v) = v.
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Forward-Backward Splitting
The forward-backward splitting is the representation of an optimisation
problem as follows
minimise
z
f(z) + g(z),
where
f, g: closed, convex
f: diﬀ/ble with Lipschitz gradient
g: prox-friendly
FB Splittings are not unique.
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Example 1. 1-regularized least squares:
minimise
z
1
2 Az − b 2
+ z 1.
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Example 2. Box-constrained QP
minimise
z
1
2z Qz + q z + δ(z | C),
where C = {z : zmin ≤ z ≤ zmax}.
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Example 3. Constrained QP
minimise
z
1
2z 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 dual
optimisation problem
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 dual
optimisation 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
minimise
z
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
minimise
z
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−1
yν
+ 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−1
ν−1 − 1)(yν
− yν−1
)
zν
= f∗
(−H wν
)
tν
= proxλ−1g(λ−1
wν
+ Hzν
)
yν+1
= wν
+ λ(Hzν
− tν
)
θν+1 = 1
2( θ4
ν + 4θ2
ν − θ2
ν)
with θ0 = θ−1 = 1 and y0 = y−1 = 0 (Nesterov, 1983).
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The DWN control problem
P : minimise
π=({uk+ji|k}i,j,{xk+ji|k}i,j)
EV (π)
N−1
j=0
µ(j)
i=1
pi
j
i
j,
subject to
xi
k+j+1|k=f(x
anc(j+1,i)
k+j|k , ui
k+j|k, di
k+j|k) dynamics
Eui
k+j|k + Ed
ˆdi
k+j|k = 0 algebraic
xmin ≤ xi
k+j|k ≤ xmax volume constr.
umin ≤ ui
k+j|k ≤ umax ﬂow constr.
x1
k|k = xk, u1
k−1|k = uk−1 initial cond.
We will write this problem as: minimisezf(z) + g(Hz).
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Let z = {xi
j, ui
j} and deﬁne
f(z) =
N−1
j=0
µ(j)
i=1
pi
j( w
(ui
j) + ∆
(∆ui
j)) + δ(ui
j|Φ1(di
j))
+ δ(xi
j+1, ui
j, x
anc(j+1,i)
j |Φ2(di
j)),
where
Φ1(d) = {u : Eu + Edd = 0}
and
Φ2(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
(xi
j) + δ(xi
j | X) + δ(ui
j | 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 {xi
j}j,i which we denote by χi
j and introduce
t = ({xi
j}j,i, {χi
j}j,i, {ui
j}j,i)
with xi
j = χi
j. Then
g(t) =
N−1
j=0
µ(j)
i=1
S
(xi
j) + δ(χi
j | X) + δ(ui
j | 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 min
z
{x y + f(z)}
= arg min
z:dynamics
Eui
j+Eddi
j=0
{x y +
j,i
quadratic(zi
j)}
This is an equality-constrained quadratic problem which can be solved
very 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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Introduction
Control
Algorithms
Sim
ulations
W
hattomodel
Literatureoverview
ForecastingTheMPC
concept
W
orst-caseMPC
Convexconjugate
Duality
APG
algorithm
DW
N
SMPC
problems
Parallelisation
KPIs
Simulationresults
Conclusions

it is fast
# scenarios
100 200 300 400 500
Runtime(s)
0
200
400
600
800
1000
1200
Gurobi
APG (500 iterations)
0 100 200 300 400 500
Runtimeslope
0
2
4
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it is eﬃcient
Number of scenarios
50 100 150 200 250 300
KPIE
/1000
1.55
1.6
1.65
1.7
1.75
1.8
KPIS
/1000
1
2
3
4
5
6KPI
E
KPIS
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Closed-loop simulations
20 40 60 80 100 120 140 160
Controlaction[%]
0
0.2
0.4
0.6
0.8
Time [hr]
20 40 60 80 100 120 140 160
WaterCost[e.u.]
60
70
80
90
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Introduction
Control
Algorithms
Simulations
W
hattomodel
Literatureoverview
ForecastingTheMPC
concept
W
orst-caseMPC
Convexconjugate
Duality
APG
algorithm
DW
N
SMPC
problems
Parallelisation
KPIs
Simulationresults
Conclusions

Open problems
Nonlinear
Problem: introduce nonlinear pressure drop equations
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Open problems
Nonlinear
Risk-averse
Challenge: the distribution of future errors is not exactly known
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Open problems
Nonlinear
Risk-averse
Distributed
Problems: spatial decomposition, communication constraints
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Open problems
Nonlinear
Risk-averse
Distributed
Robust Economic MPC
Questions: performance guarantees, recursive feasibility
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Open problems
Nonlinear
Risk-averse
Distributed
Robust Economic MPC
Faster Algorithms
Ongoing work: quasi-Newtonian LBFGS-type algorithms
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References (i)
1. A. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, “Fast parallelizable scenario-based stochastic
optimization,” EUCCO 2016, Leuven, Belgium, 2016.
2. A. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, “GPU-accelerated stochastic predictive control of drinking
water 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 forecasting
for 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 control
problems on GPUs, 54th IEEE Conf. Decision and Control, (Osaka, Japan), Dec 2015.
5. A. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, “Proximal Quasi-Newton Methods for Scenario-based
Stochastic Optimal Control,” IFAC 2017 (submitted).
6. M. Bakker, J. H. G. Vreeburg, L. J. Palmen, V. Sperber, G. Bakker, and L. C. Rietveld, “Better water quality and higher
energy efficiency by using model predictive flow control at water supply systems,” J Wat Supply: Research &
Technology – Aqua, 62(1), pp. 1–13, 2013.
7. S. Leirens, C. Zamora, R. Negenborn, and B. De Schutter, “Coordination in urban water supply networks using
distributed model predictive control,” ACC 2010, (Baltimore, USA), pp. 3957–3962, 2010.
8. C. Ocampo-Martinez, V. Puig, G. Cembrano, R. Creus, and M. Minoves, “Improving water management efficiency by
using optimization-based control strategies: the barcelona case study,” Water Science and Technology: Water Supply,
9(5), pp. 565–575, 2009.
9. C. Ocampo-Martinez, V. Fambrini, D. Barcelli, and V. Puig, “Model predictive control of drinking water networks: A
hierarchical and decentralized approach,” ACC 2010, (Baltimore, USA), pp. 3951–3956, 2010.
10. A. Goryashko and A. Nemirovski, “Robust energy cost optimization of water distribution system with uncertain
demand,” Automation and Remote Control 75(10), pp. 1754–1769, 2014.
11. U. Zessler and U. Shamir, “Optimal operation of water distribution systems,” J Wat Resour Plan & Mngmt, 115(6), pp.
735–752, 1989.
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References (ii)
12. G. Yu, R. Powell, and M. Sterling, “Optimized pump scheduling in water distribution systems,” J Optim Theory & Appl,
83(3), pp. 463–488, 1994.
13. V. Tran and M. Brdys, “Optimizing control by robustly feasible model predictive control and application to drinking
water distribution systems,” Artiﬁcial Neural Networks – ICANN 2009, vol. 5769 of Lecture Notes in Computer Science,
pp. 823–834, Springer, 2009.
14. J. Watkins, D. and D. McKinney, “Finding robust solutions to water resources problems,” J Wat Res Plan & Mngmt
123(1), pp. 49–58, 1997.
15. S. Cong Cong, S. Puig, and G. Cembrano, “Combining CSP and MPC for the operational control of water networks:
Application to the Richmond case study,” 19th IFAC World Congress, (Cape Town), pp. 6246–6251, 2014.
16. J. Grosso, C. Ocampo-Mart nez, V. Puig, and B. Joseph, “Chance-constrained model predictive control for drinking
water networks,” Journal of Process Control 24(5), pp. 504–516, 2014
17. P.L. Combettes and J.-C. Pesquet. “Proximal splitting methods in signal processing,” Technical report, 2010. URL
http://arxiv.org/abs/0912.3522v4.
18. N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim 1, pp. 127–239, 2014.
19. R. Rockafellar and J. Wets, “Variational analysis,” Berlin: Springer-Verlag, 3rd ed., 2009.
20. R. Rockafellar, “Convex analysis,” Princeton university press, 1972.
21. Yu. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2
),” Soviet
Mathematics Doklady 72(2), pp. 372–376, 1983.
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