Choosing among infinite alternatives NRMLec17 Andrea Castelletti Politecnico di Milano.

Choosing among infinite alternatives

NRMNRMLec17Lec17

Andrea CastellettiPolitecnico di Milano

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The pure planning problem

xt+1 = ft xt,u

p,wt( ) t=0,...,h-1

up ∈U p

w0h−1 given scenario

x0 given

any other constraint t=0,...,h-1

This is the more general formulation of the problem, accounting for both the transient period and the steady state conditions of the system.

Often the DM is only interested in the steady state conditions

J (u p*) =min

upi(x0 ,u

p,w0h−1)

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Transient vs. steady-state

A project has been proposed by Israel for exploiting the depression (-400 m a.s.l.) for hydropower production.

Let’s call up the capacity of the canal connecting the two seas.

Let’ assume that the flow through the canal is constant

Dead Sea

Mediterranean Sea

Jordan

El Araba

Jerusalem40 km

Mediterranean Sea

Dead Sea

Israel Jordan

Example: the Dead Sea Proj.

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Hp: deterministic and periodic inflow.

There are not outflow

Storage oscillations are peridodic with T


s1= f0 s0 ,0,w0( )

Ls0 = fT−1 sT−1,0,wT−1( )

What happens when the canal is constructed?The level of the lake rise until a new equilibrium (cycle) has been reached

Dead Sea

Mediterranean Sea

Jordan

El Araba

Jerusalem40 km

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Transient or steady-state?

The choice is up to the DM, but she will presumably be interested in the staedy state conditions.

- With the TDC formulation more importance is given to the transient.

TDC i = γτgτ (xτ ,u

p,wτ )τ=0

∞

∑- With the AEV formulation only the steady state conditions are

accounted for, while the transient ignored

otherwise

re-formulation of the planning problem.

AEV i =lim

h→ ∞

1h+1

gτ (xτ ,up,wτ )

τ=0

h

∑


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Pure planning problem

xt+1 = ft xt,u

p,wt( ) t=0,...,h-1

up ∈U p

w0h−1 given scenario

x0 given

any other constraint t=0,...,h-1

J (u p*) =min

upi(x0 ,u

p,w0h−1)

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The Long-term pure planning problem

x1= f0 x0 ,u

p,w0( ), L , x0 = fT−1 xT−1,up,wT−1( )

up ∈U p

w0T−1 given scenario

any other constraint t=0,...,T -1

J (u p*) =min

upi(up,w0

T−1)

With the AEV indicator this formulation produces the same optimal alternative as the previous one. However, for some particular forms of the objectives and the constraints the alternative can be singled out more easily.

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The disturbance scenarioVery often, when dealing with the long-term planning problem the average or the worst year is considered as the trajectory of the deterministic disturbances

However this relationship holds only for the linear case.

%w

t=

εt+kT

n+1k=0

n

∑ for t=1,...,T average year

w

t=max

k=0,...,nεt+kT for t=1,...,T worst year

Eεt{ }t=1,...,T

i x0 ,up,ε1

T( )⎡⎣

⎤⎦

maxεt{ }t=1,...,T

i x0 ,up,ε1

T( )⎡⎣

⎤⎦

based on the idea that the average perfomance can be computed by evaluating the system perfomance in the average conditions.

=i x0 ,up, %w0

T−1( )

=i x0 ,up,w0

T−1( )

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Solution algorithm

When U p includes few elements: the objective is evaluted for each alternative and that corresponding to the best value selected.

Finite horizon:

u p* =argmin

upi(x0 ,u

p,w0h−1)

∀ up ∈U p, given w0h−1 and x0 , the system is simulated over H with

x

t+1 = ft xt,up,wt( ) the trajectory x0

h is obtained

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Solution algorithm

Infinite horizon: a simulation based approach is not suitable as the simulation would last forever…

AEV

i =limh→ ∞

1h+1

gt(xt,up,wt)

t=0

h

∑

... future costs are discounted increasingly, the further away they are in time, so that the variations of the value of the indicator, sampled every T steps, become progressively smaller with the time t.

... if the system holds a ciclostationary condition in the long term, the variations of the value of the indicator become progressively smaller with the time t.

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Solution algorithm

Infinite horizon: a simulation based approach is not suitable as the simulation would last forever…

Operationally, the simulation can be stopped when the variations in the value of the indicator – sampled every T steps – are small enough.

Termination condition | i(k+1)T (g)−ikT (g) |<α

The bigger the value of α the shorter the simulation, but the more approximate the evaluation of the indicator will be.

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Solution algorithm

When U p includes a very high, or infinite, number of elements the exhaustive, simulation-based procedure is not feasible: one should settle for a sub-optimal solution.

Constrained evolutionary methods are used for this purpose:

• Via simulation the value of the indicator is computed for a

limited number of elements in Up;

• The algorithm evolves in Up searching for a solution as closes as possible to the optimal one;

• When the number of components of up is high, this approach quickly becomes computationally unfeasible.

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The Sinai Plan

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The Sinai Plan

GOAL: improving the quality of life by increasing the cultivated surface.Actions:To reclaim desertic lands in the 7 zones Zj (j=1-7) by exploiting the water from the 4 sources Ss (s=1-4) and by choosing for each one the more appropriate crop rotation Ri (i=1-5) and irrigation techniques Ih (h=1-3).

Evaluation criterion:Net benefit in steady state conditions.

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The Sinai Plan

j Reclaimable zone Zj Aj [feddan]

1 Costa Nord-Est del Sinai 515 000

2 El Tina Plains 135 000

3 Territory West of the Suez Canal 235 000

4 Territory North of the Salhia Canal 102 000

5 Territory between the Ismailia and Salhia Canals

120 000

6 Territory South of the Ismailia Canal 45 000

7 Territory South of Lake Manzala 29 000

max reclaimable area in zone Zj

4 212 m2

s Canal SS ws [109m3]

1 Ismailia 4.4

2 Bahr Hadous 2.4

3 Bahr El Baqar 1.4

4 Damietta-Salam 2.3

average annual volume derivable from SS

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The Sinai Plan

j Reclaimable zone Zj Aj [feddan]

1 Costa Nord-Est del Sinai 515 000

2 El Tina Plains 135 000

3 Territory West of the Suez Canal 235 000

4 Territory North of the Salhia Canal 102 000

5 Territory between the Ismailia and Salhia Canals

120 000

6 Territory South of the Ismailia Canal 45 000

7 Territory South of Lake Manzala 29 000

max reclaimable area in zone Zj

4 212 m2

s Canal SS ws [109m3]

1 Ismailia 4.4

2 Bahr Hadous 2.4

3 Bahr El Baqar 1.4

4 Damietta-Salam 2.3

avearge annual volume derivable from SS

i Crop Rotation Ri bi [EGP/feddan]

1 Clover-Rice 298

2 Clover-Cotton 374

3 Clover-Maize 152

4 Clover-Soya 158

5 Wheat-Maize 184

h Irrigation Techniques Ih Lh [EGP/feddan]

1 Flood 650

2 Sprinkler 1500

3 Drip 2500

unitary agricoltural benefit produced by

rotation Ri

unitary installation cost for irrigation technique Ih

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Final (political) decision

reasonable alternatives

2. Conceptualisation

3. Designing Alternatives

4. Estimating effectsS

takeh

old

ers

1. Reconnaissance

5. Evaluation

noMitigation,

and compensatio

n, Agreement

?

6. Comparison or negotiation

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The Sinai Plan: Phase 1 - Reconnaissance

Conditions imposed by the Ministry:

• the Plan must be prepared in a non-participatory way;

• the DM is the Ministry;

• the evaluation must be carried out using the Cost Benefit Analysis;

• the Plan must be evaluated on the long term (steady state).

Conditions identified by the Analyst:

• the Stakeholders are the farmers and the current water users (industries and hydropower) ;

• the quality of the water withdrawn plays a crucial role: salinity Qs [ppm] might strongly affect productivity.

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ools

yes






takeh

old

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1. Reconnaissance

5. Evaluation

noMitigation,

and compensatio

n, Agreement

?


Defining Actions

(measures)

Identifying the Model

Defining Criteria and

Indicators

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The Sinai Plan: Phase 2a – Defining actions

The actions proposed by the Ministry concern:• area to be reclaimed in each zone• crop rotation• irrigation technique

area [feddan] to be devoted to rotation Ri in zone Zj with the irrigation technique Ih

u

ijh1 :

annual water volume [m3] to be conveyed into the canal Ss in zone Zj

u

sj2 :

• supply volume• source for the supply

They can be formalized through these variables

However not all the combinations are feasible (e.g. drip irrigation is not allowed for rice).

Let’s call F the set of the feasible combinations of i,j,h,s

The vector up of the planning decisions is the following The vector up of the planning decisions is the following

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takeh

old

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1. Reconnaissance

5. Evaluation

noMitigation,

and compensatio

n, Agreement

?


Defining Actions

(measures)



Indicators

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The Sinai Plan: Phase 2b – Def. criteria and indicators

unitary agricoltural benefit

productivity coefficient [%]: the productivity decreases as the salinity Qj increases

i crop

j zone

h irrigation technique

The project indicator (objective) has to be formulated with the CBA

max (Social benefits- Social costs)

The Sinai Plan: Phase 2b – Def. criteria and indicators

The project indicator (objective) has to be formulated with the CBA

max (Social benefits- Social costs)

• installation cost of the irrigation techniques

C

irr= Lhuijh

1

( i, j ,h)∈F∑

Unitary installation costs of technique Ih

• transportation costs of the water to the reclaimed zone

• opportunity costs

C

tr= ctrlsjusj

2

(s, j )∈F∑

the specific cost [EGP /m3 km] constant over all the territory

distance [km] from the canal Ss and the zone Zj

C

opp= osusj

2

(s, j )∈F∑

opportunity costs [EGP /feddan] of the water in canal Ss

In conclusion the project indicator is the followingIn conclusion the project indicator is the following

i(u p ,w) =Bagr −Cirr −Ctr −Copp

vector of the volumes ws that can be withdrawn from the canals

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ools

yes






takeh

old

ers

1. Reconnaissance

5. Evaluation

noMitigation,

and compensatio

n, Agreement

?


Defining Actions

(measures)



Indicators

25

The Sinai Plan: Phase 2c – Identifying the model

The interest is in the long-term situation (according to the Ministry’s request) and the annual volumes ws are known:

• Design problem = long-term pure planning prolem;

• Time step = year;

The model is

x = f0 x,up,w( )

up ∈U p w given scenario any other constraint

the state is the salinity at the equilibrium

Qj=

[1−αlsj ]usj2Qs

s∈ s: (s, j )∈F{ }∑

[1−αlsj ]usj2

s∈ s: (s, j )∈F{ }∑

∀j

distance [km] between Ss and the zone Zj

unitary loss [m3/km]

Set of the feasible actions. Let’s see how it is defined.

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The Sinai Plan: the feasibility set

It is defined by 3 constraints

1. The total volume withdrawn from Ss must not be greater than the volume available ws

u

sj2 ≤ws

j∈ j: (s, j )∈F{ }∑ ∀s

U p

2. The total reclaimed area in zone Zj must not be greater than the maximum reclaimable area Aj

3. The components of vector up cannot assume negative meaningless values

(N.B. The alternative zero is the null vector.)

u

ijh1 ≤Aj

( i,h)∈ ( i,h): ( i, j ,h)∈F{ }∑ ∀j

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The interest is in the long-term situation (according to the Ministry’s request) and the annual volumes ws are known:

• Design problem = long-term pure planning prolem;

• Time step = year;

The model is:

x = f0 x,up,w( )

up ∈U p w given scenario any other constraint

volume ws that can be withdrawn from the canal

All the constraints not included above, precisely

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The unitary benefit bi has been estimated by assuming that the crop water demand is totally covered by the water supply. Therefore in order for the agricoltural benefit

to be estimated in the correct way, the total water volume supplied to a zone must be greater than the water demand of the crop rotation considered

(1−αlsj )usj

2

s∈ s: (s, j )∈F{ }∑ − Wihuijh

1

( i,h)∈ ( i,h): ( i, j ,h)∈F{ }∑ ≥0 ∀j

unitary water demand [m3/feddan] of rotation Ri irrigated with technique Ih

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ICT T

ools

yes






takeh

old

ers

1. Reconnaissance

5. Evaluation

noMitigation,

and compensatio

n, Agreement

?


30

The Sinai Plan: Phase 3 – Designing the alternatives

The design problem has the following form

J (u p*) = max

up ,Q1 ,...,Q7

biPi (Qj )uijh1

( i, j ,h)∈F∑ - Cirr up( )- Ctr up( )- Copp up( )

⎡

⎣⎢

⎤

⎦⎥

subject to Non-linear mathematical programming

55 decision variables

80 constraints

Qj=

[1−αlsj ]usj2Qs

s∈ s: (s, j )∈F{ }∑

[1−αlsj ]usj2

s∈ s: (s, j )∈F{ }∑

∀j

u

sj2 ≤ws

j∈ j: (s, j )∈F{ }∑ ∀s

u

ijh1 ≤Aj

( i,h)∈ ( i,h): ( i, j ,h)∈F{ }∑ ∀j

(1−αlsj )usj

2


1

( i,h)∈ ( i,h): ( i, j ,h)∈F{ }∑ ≥0 ∀j

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The Sinai Plan: Phase 3 – Designing the alternatives

The design problem has the following form

J (u p*) = max

up ,Q1 ,...,Q7

biPi (Qj )uijh1

( i, j ,h)∈F∑ - Cirr up( )- Ctr up( )- Copp up( )

⎡

⎣⎢

⎤

⎦⎥

subject to

Qj=

[1−αlsj ]usj2Qs

s∈ s: (s, j )∈F{ }∑

[1−αlsj ]usj2

s∈ s: (s, j )∈F{ }∑

∀j

u

sj2 ≤ws

j∈ j: (s, j )∈F{ }∑ ∀s

u

ijh1 ≤Aj

( i,h)∈ ( i,h): ( i, j ,h)∈F{ }∑ ∀j

(1−αlsj )usj

2


1

( i,h)∈ ( i,h): ( i, j ,h)∈F{ }∑ ≥0 ∀j

REMARKS:

if the value of Qj (j=1,...,7) were given the problem would be linear

REMARKS:

if the value of Qj (j=1,...,7) were given the problem would be linearidea for the

solution

Evolutionary algorithm in the space Qj (j=1,...,7) minimizing

J (Q

1,...,Q

7) =max

upi up,w( )

Subject to the same constraints of the original problem

simplex algorithm

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The Sinai Plan: the optimal alternative

The solution of the problem is the optimal Alternative(we assumed os = 10 [EGP/1000 m3])

Total reclaimed land = 448 000 feddan

Annual net benefit (objective) = 80 000 000 EGP

Zone Reclaimed area

[feddan]

Canals used

Crop Rotation

Irrigation techniques

1 0 * * *

2 135 000 1 e 3 5 2

3 235 000 2 e 3 2 1

4 0 * * *

5 0 * * *

6 45 000 3 5 2

7 29 000 2 e 4 2 1

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yes





Sta

keh

old

ers

1. Reconnaissance

4. Estimating effects

5. Evaluation

noMitigation,

and compensatio

n, Agreement

?


OPTIMAL ALTERNATIV

E

34

Reading

IPWRM.Theory Ch. 8, 10

Choosing among infinite alternatives NRMLec17 Andrea Castelletti Politecnico di Milano.

Documents

Transcript of Choosing among infinite alternatives NRMLec17 Andrea Castelletti Politecnico di Milano.