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Transcript of Hanoi, January 28 th 2015 Rodolfo Soncini-Sessa DEI – Politecnico di Milano IMRR Project 8 –...
![Page 1: Hanoi, January 28 th 2015 Rodolfo Soncini-Sessa DEI – Politecnico di Milano IMRR Project 8 – Design algorithms INTEGRATED AND SUSTAINABLE WATER MANAGEMENT.](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e855503460f94b878d5/html5/thumbnails/1.jpg)
Hanoi, January 28th 2015
Rodolfo Soncini-SessaDEI – Politecnico di Milano
IMRR Project
8 – Design algorithms
INTEGRATED AND SUSTAINABLE WATER MANAGEMENT OF RED-THAI BINH RIVER SYSTEM
IN A CHANGING CLIMATE
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IMRR phases
econnaissance
odeling the system
ndicators identification
cenarios definition
lternative design
valuation
RMISAE
Soncini Sessa, 2007
omparison … C
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The Design Problem
(It)
scenario
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Design algorithm
SDPStochastic Dynamic Programming
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The Design Problem
(It)
scenario
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Assumptions
The objectives are separable
Compensation is acceptable
then
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The Design Problem for SDPIf et+1 is a white process
(It)
scenario
If et+1 is a white process
and
we do not consider exogenous information …
thenStochastic Dynamic Programming (SDP)
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SDP algorithm
xt+1= ft (xt,ut,et+1)
et+1 ~ Ft (• )
utUt (xt)
accordingly
p= {mt(•); t= 0,1,…,h}
step costOptimal expected
Cost-to-go
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SDP algorithm
Pros:1. It guarantees the best solution
(provided assumptions are satisfied)
Cons:2. Only one solution per run!
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J1
J2
Only one solution !
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Gestione delle Risorse Naturali, Politecnico di Milano
Stochastic Dynamic ProgrammingStochastic Dynamic Programming (SDP) suffers from a dual curse:
1) computational cost grows exponentially with state, control and disturbance dimension (curse of dimensionality [Bellman, 1967]);
Look-up tableH-function
unknown H-function
computations are numerically performed on a discretized variable domain
2) a dynamic model of any variable considered among the operating rule’s arguments has to be embedded in the algorithm (curse of modelling [Bertsekas and Tsitsiklis, 1996]).
timet t+1
models are use in a multiple one-step-ahead-simulation mode
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Number of iterations for 1 reservoir:
101 x 801 x 52 x (365) x 3 = 22 x 106
x 3
Time per evaluation: 9 x 10-6 sec.
Total time: 3 minutes
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Number of iterations for RTBR system:
104 x 804 x 55 x (365) x 3 = 1.4 x 1018
x 3
Time per evaluation: 3.7 x 10-5 sec.
Total time: 1,650,000 years!
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Gestione delle Risorse Naturali, Politecnico di Milano
Stochastic Dynamic ProgrammingStochastic Dynamic Programming (SDP) suffers from a dual curse:
1) computational cost grows exponentially with state, control and disturbance dimension (curse of dimensionality [Bellman, 1967]);
Look-up tableH-function
unknown H-function
computations are numerically performed on a discretized variable domain
2) a dynamic model of any variable considered among the operating rule’s arguments has to be embedded in the algorithm (curse of modelling [Bertsekas and Tsitsiklis, 1996]).
timet t+1
models are use in a multiple one-step-ahead-simulation mode
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Design algorithm
Genetic Algorithm
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4th December 2013
GA are search methods based on two principles inspired by nature:
WHAT ARE GENETIC ALGORITHMS?
Genetics = recombination of structuresNatural Selection = survival of the fittest
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The Design Problem
(It)
scenario
(It, θ)
(It, θ)
scenario
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4th December 2013Gestione delle Risorse Naturali, Politecnico di Milano
Universal function approximators
Artificial Neural Networks with some particular features can be used as universal function approximators, i.e. as policies.
Multi-layer Perceptron
u1,t
uq,t
θ = [γ11,1, …., γ1
m,n, … , βL1, …, βL
q]
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4th December 2013
SOLVING APPROACH: ANN to describe the control law ; GA to find the optimal ANN parameterization .
ALGORITHM:
Gestione delle Risorse Naturali, Politecnico di Milano
Run a system simulation for each individual
Selection, crossover and mutation
new population
initial population
time series of historical inflow
objectives
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J1
J2
Initial (random) population
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J1
J2
selection of the “best” solutions according to the Pareto dominance criterion
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J1
J2
survival of the fittest
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J1
J2
generation of a new population
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J1
J2
selection of the “best” solutions according to the Pareto dominance criterion
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J1
J2
survival of the fittest
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J1
J2
iterating….
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J1
J2
iterating….
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J1
J2
iterating….
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J1
J2
final approximation of the Pareto front
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GA algorithm
Pros:1. The whole Pareto boundary is generated in one run
Cons:2. It does not guarantees the best solution, neither an
asymptotic convergence
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Time per policy evaluation over 39 years for the RTBR system: 0.53 sec.
Dimθ = (2 x Ninput + Noutput) x Nneur Nneur ≥ Ninput + Noutput
Ninput= 4+2 Noutput = 4 Nneur = 10 Dimθ = 160 Num policies = 10160
4 reservoirs
Ninput= 3+2 Noutput = 3 Nneur = 9 Dimθ = 117 Num policies = 10117
3 reservoirs
Ninput= 1+2 Noutput = 1 Nneur = 5 Dimθ = 35 Num policies = 1035
1 reservoir
Too large!
Might be feasibleRunning time: 29 days
Numevaluations about 5.5 106
SDP 250 seconds = 470 policy evaluations
SDP is surely faster
How to reduce
the number of reservoirs
to 3 only?
We will see tomorrow.
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GA with extreme events
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Design scenario
20 normal years
10 extreme years
regular indicators
extreme indicators
JF , JS , JH ….. JeF , JeS
↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓
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Extreme events
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Pareto boundary (qualitative)
Flo
od
IHP1: Hydrop. Production
Extreme floods
Irrigation
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Extreme vs regular floods
Regular Flood
Ext
rem
e fl
oods
Irri
gati
on
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Trade off between extreme and standard floods
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Hoa Binh and Ha Noi flooding
r
t
r,a
A
A
It is feasible only when A <C
A flood of volume A is coming.How to minimize flooding in Ha Noi?
Catch.
C
r
a
r
inflowa
C Capacity
releaser
r flooding threshould
HN
HB
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Hoa Binh and Ha Noi flooding
t
r,a
C
A flood of volume A is coming.How to minimize flooding in Ha Noi?
C
r
a
r
inflowa
C Capacity
releaser
r flooding threshould
HN
HB
If A> C the spillway starts
acting
r Flooding!What can we do?
![Page 40: Hanoi, January 28 th 2015 Rodolfo Soncini-Sessa DEI – Politecnico di Milano IMRR Project 8 – Design algorithms INTEGRATED AND SUSTAINABLE WATER MANAGEMENT.](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e855503460f94b878d5/html5/thumbnails/40.jpg)
Hoa Binh and Ha Noi flooding
t
r,a
A flood of volume A is coming.How to minimize flooding in Ha Noi?
C
r
a
r
inflowa
C Capacity
releaser
r flooding threshould
HN
HB
*r
Intentionally produce a small flood!
What if the big
flood doesn’t
arrive?
r
C
We have flooded
for nothing!
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Thanks for your attention
XIN CẢM ƠN
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A51 A87 A20
H 22 22 20
I 330 316 124
F 90 88 106
extF 3973 2844 1490
F>13.4 814 230 0
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A51 A87 A20
H 22 22 20
I 330 316 124
F 90 88 106
extF 3973 2844 1490
F>13.4 814 230 0
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A51 A87 A20
H 22 22 20
I 330 316 124
F 90 88 106
extF 3973 2844 1490
F>13.4 814 230 0
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4th December 2013Gestione delle Risorse Naturali, Politecnico di Milano
The evaluation scheme
a(m3/s)
r (m3/s)
s (m3)
q_YB(m3/s)
q_HY (m3/s)
h_HN(m)
q_ST(m3/s)
g_hyd(kwh)
g_flo(cm)
Hydropowerplant
(conceptual)
Flow routing
(data-driven)
Flow routing
(data-driven)
flooding cost deficit cost
g_sup(m3/s)2 2
Reservoirs model
(conceptual)
hydropower cost
P(kwh)
u (m3/s)
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Gestione delle Risorse Naturali, Politecnico di Milano
Universal Approximation Theorem (Cybenko 1989, Funahashi 1989, Hornik et al. 1989)
Every continuous function defined on a closed and bounded set can be approximated arbitrarily closely by a Multi-Layer Perceptron, provided that the number n of neurons in the hidden layers is sufficiently high and that their activation function belongs to a restricted class of functions with particular properties. Precisely,
must be differentiable and monotonically increasing;
the input to the j-th neuron (denoted with ) must enjoy the following property:
Universal function approximators
Sigmoidal functions meet both the requirements.
e.g., the hyperbolic tangent is a sigmoidal function:
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Gestione delle Risorse Naturali, Politecnico di Milano
Universal Approximation Theorem (Cybenko 1989, Funahashi 1989, Hornik et al. 1989)
Every continuous function defined on a closed and bounded set can be approximated arbitrarily closely by a Multi-Layer Perceptron, provided that the number n of neurons in the hidden layers is sufficiently high and that their activation function belongs to a restricted class of functions with particular properties.
Universal function approximators
In practice, a 2-layer perceptron is enough
output
parameters