Optimization of a Time-discrete Nonlinear Dynamical System From a Problem of Ecology. Analytical and Numerical Approach.

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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 i1,...,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
technical cooperation
market
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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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- budget, upper bounds for the financial
investigation i1,...,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) gt reduction
of in the 2nd equation. - implies that the
emissions are less than the requirements of the
treaty gt will increase in the 2nd equation.
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3. Numerical results
Data of the TEM model
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Influence on the emissions
actor 1... actor 2_._._ actor 3_ _ _
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Influence on the financial means
actor 1... actor 2_._._._ actor 3_ _ _ _
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Influence on the emissions
actor 1... actor 2_._._ actor 3_ _ _
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Influence on the financial means
actor 1... actor 2_._._ actor 3_ _ _
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4. Controllability
Consider our model as follows
Simplifying conditions
The fixed points of dynamic system (steady
states, have no time-dependence)
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Regarding the Jacobi-matrix of the right-hand
side for the special case emij(t)emij, where
the economic relationship is const over a long
period, we get
The eigenvalues The fixed points under the
simplifying conditions are not attractive.
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5. The problem of controllability
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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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6. An iterative solution
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7. An application of the gradient method
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Numerical results
Column ltKyotogt 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. Conclusion
Kyoto 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!