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Optimization of a Time-discrete Nonlinear

Dynamical System From a Problem of Ecology.

Analytical and Numerical Approach.

Presented by Julie Pavlova

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Overview• Introduction

• The model

• Numerical results

• Controllability

• The problem of controllability

• An iterative solution

• An application of a gradient method

• Conclusion

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1. Introduction• 1997 - Kyoto Protocol was drawn by EU countries to solve most important ecological problem

• One of its mechanisms - “Joint-Implementation” intends to strengthen international cooperation (on reducing CO2)

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Joint-Implementation =JI• 1 step : a developed country gives a credit for a developing country to decrease its pollution level

• 2 step : the developing country uses these investments to realize a certain alternative energy project and then it will pay back the credit returning received quotas to the developed country

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Examples

• Netherlands signed projects of “JI” with Central and Eastern European countries:

a modernization project of a hydroelectric facility in Romania;

a landfill-gas project at eight different sites in Slovakia;

switch from coal to biomass at a power plant in Hungary.

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2. The Technology- Emission- Means Model (the TEM model)

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The TEM model:

i:=1,...,n -actors;

-emission of the i-th actor

-technology caused that emission

- financial means

actor “i” actor “j”

|||

the non-linear time-discrete dynamics of the TEM model

i

i

i

M

T

E

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Relationship between financial means and reduced emissions:

- reduced emissions of actor i in percent;

- financial means of actor i;

- describes the effect on the emissions of the i-th actor if the j-th actor invests money.

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λ

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- budget, upper bounds for the financial investigation i=1,...,n;

- the memory parameter which describes the effect of the preceeding investments;

- the growth parameter.

- implies that the actor have not reached yet the demanded value ( -normalized Kyoto level) => reduction of

in the 2nd equation.

- implies that the emissions are less than the requirements of the treaty => will increase in the 2nd equation.

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3. Numerical results

Data of the TEM model

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T),,( 111actor 1:... actor 2:_._._ actor 3:_ _ _

Influence on the emissions:

12actor 1:... actor 2:_._._._ actor 3:_ _ _ _

Influence on the financial means:

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14

Influence on the emissions:

actor 1:... actor 2:_._._ actor 3:_ _ _

T),,( 303030

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Influence on the financial means:

actor 1:... actor 2:_._._ actor 3:_ _ _

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4. Controllability

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Simplifying conditions:

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Consider our model as follows:

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Regarding the Jacobi-matrix of the right-hand side for the special case emij(t)=em*ij, where the economic relationship is const over a long period, we get:

The eigenvalues:

The fixed points under the simplifying conditions are not attractive.

njtEMλλλλ jj*j

*jn

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* ,...,1),(^

1,1,...,1 *1

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5. The problem of controllability

.)),(),...,(()(

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(5.1)

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(5.4)conditionsInitial

(5.3) andfor

(5.2) andfor

:and

andseemptynon

for

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(5.6) and

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Shortly: given -initial state of the dynamic system under consideration , find control functions (satisfying 5.2) and steer the system (under conditions 5.3) into the steady state of uncontrolled system:

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where

(5.7)

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6. An iterative solution

actor. th-ithe of function payoff-

(6.3)

consider For

.(6.2)

where

(6.1)

:definewe For

).values(5.4 initial-where

choose For

given. some For

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,...,n}{i

,...,n}{,i||||u||x)(u)(t||x(u):a

(t)),(t),...,x(xx(t)

,...,n},{(x(t),u),if(t)x):(u)(tx

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,...,n}{,ix)(xt

,...,n}{,iXtxt

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and for

(6.7)

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alsoare 5 section of 2) and 1) sAssumption

. for Put

for Put

(5.2). formthe of is (6.8)

(5.1) formthe of is (6.7)

(6.7) of sides hand-rightthe as

Define

for Put

and

(6.8) (

:function control a has nation everySo,

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,...,n}{i}MM|R),M{(EX

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fulfilled.are for

and

(6.6)(6.5),(6.3),(6.2),

conditionsthe that such

N 2.

1. find

given

:ility"controllab of problemThe " Now,

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1

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7. An application of the gradient method

}.,...,1{,0 niu

a

(u)a

i

ti

ti

:step-time

eachat yiterativelsolve to have We

(6.3). from

minimize to tries actor each

:casee cooperativ-non Consider

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Numerical results:

Column <Kyoto> means the emission targets mentioned in Kyoto Protocol.

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It shows that the insertion of the calculated parameters might be successful.

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8. ConclusionKyoto Protocol demands for reductions in greenhouse gas emissions by the industrialized countries. On the other hand, developing countries are expanding their energy consumptions that leads to increasing levels of greenhouse gas emissions.

The preparation of an optimal management tool requires the possibility to identify, assess and compare several technological options. For that reason the mathematical TEM model was elaborated. Control parameters have to be determined iteratively according to negotiation process.

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Thanks For

Your Attention!