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Вычислительные технологии
Том 6, № 1, 2001
OPTIMIZATION OF A TIME-DISCRETE NONLINEAR DYNAMICAL SYSTEM FROM A PROBLEM OF ECOLOGY. AN ANALYTICAL AND NUMERICAL APPROACH
St. Pickl
Department of Mathematics, University of Cologne, Germany
G.-W. Weber
Department of Mathematics, Darmstadt University of Technology, Germany
e-mail: pickl@zpr.uni-koeln.de
Работа посвящена математическому анализу нелинейной дискретной по времени модели промышленных выбросов (TEM) для решения проблем защиты окружающей среды. Введение управляющих параметров в динамику модели TEM дает возможность получить новые результаты в области дискретных по времени систем управления. С другой стороны, эти результаты позволяют усовершенствовать важный экономический инструмент управления состоянием окружающей среды. Устойчивые состояния модели TEM можно рассматривать как предельные значения, указанные в Киотском протоколе. Приводятся результаты расчетов.
1. Introduction
The conferences of Rio de Janeiro 1992 and Kyoto 1997 demand for new economic instruments which focus on environmental protection in both macro and micro economy. An important economic tool in that area is Joint-Implementation (JI) which is mentioned explicitly in Kyoto Protocol [1]. It is an international program which intends to strengthen international cooperations between enterprises on reducing CO2-emissions. A sustainable development can only be guaranteed if the instrument is embedded in an optimal energy management. In this context, optimal energy management according to JI means that it must work on the micro level with minimal costs and it should be protected against misuse on the macro level.
For that reason, the TEM model (Technology-Emissions-Means model) was developed, giving the possibility to simulate such an economic behaviour. The realization of JI is determined subject to technical and financial constraints. In a JI program emissions reduced by technical cooperations are registrated at the clearing house whose establishment is also a demand of Kyoto Protocol. The associated cost reductions should then be allocated in an optimal way. This approach is as well integrated in the TEM-Model as the possibility to regard the influence of several cost allocations on the feasible set of control parameters. Furthermore, in the played cost-game a special solution, called the т—value is examined in [2]. This value stands for
© St. Pickl, G.-W. Weber, 2001.
a rational allocation process. Now, the question arises in which situations the t—value is equivalent to the necessary control-parameters to reach the regions, mentioned in the treaty of Kyoto. This question is answered by the Equivalence Theorem [2]. Numerical results are based on a qualitative analysis of the TEM model in order to simulate such a JI program. The results may lead to new insights in JI and improve that important management tool.
2. The model
The presented TEM model describes the economic interaction between several actors (players) which intend to minimize their emissions Ei caused by technologies Ti using expenditures of money Mi or financial means, respectively. The index i G {1, ... , n} stands for the i-th player. The players are linked by technical cooperations and the market, which expresses itself in the nonlinear time-discrete dynamics of the Technology-Emissions-Means model, in short: TEM model. The TEM model is based on a general model which was introduced in [3] and refined in [4] and [2]:
n
AEi(t) = J] emij (t)Mj (t), j=i
AMi(t) = —AiMi(t)(M* — Mi(t))(Ei(t) + ^iAEi(t)). (2. 1)
The first equation describes the time-dependent behaviour of the emissions reduced so far by each player. These levels are influenced by financial investigations which are determined by the second equation. The emij-parameters determine the effect on the emissions of the i-th actor, if the j-th actor invests money. We can say that it expresses how effective technology cooperations are, which is the kernel of a JI program. Besides that, the integration into the TEM model of the memory parameter ^ and the growth parameter A guarantees a realistic economic market behaviour while the Mi*(t) (i G {1, ... , n} t G N) are upper bounds for the financial investigations.
3. Numerical results
The TEM model can be regarded as a mathematical model which supports the development of a management tool in the area of the international climate change convention, namely in the creation of a JI program.
In the following we regard the TEM model as a routine with which we are able to compare several scenarios. Let us begin with the following example to get a first insight into the dynamics.
Table 1
Data of the TEM model
Player i Ei(0) Mi(0) M A 1/60 * em-matrix
1 -0.1 30 60 1/60 1 -0.525 -0.475
2 -0.1 20 60 1/60 -0.475 1 0.525
3 -0.1 10 60 1/60 -0.1 -0.1 0.2
Figure 1. Influence of the memory parameter on the emissions reduced (a) and on the financial
means (b), data of table 1, p = (1,1,1)T.
Figure 2. Influence of the memory parameter on the emissions reduced (a) and on the financial
means (b), data of table 2, (p = (30, 30, 30)T.
We regard the behaviour between three actors who have not yet reached the limiting value of Kyoto at the beginning of the time period, indicated by a normalized value of (-0.1, -0.1, -0.1)
for each actor (actor 1: •, actor 2: —• and player 3:--, in the figures 1, 2). The em -matrix
has positive and negative entries which means that we deal with cooperative and competitive behaviour. The memory parameter might be 1, the growth parameter 1/60 for each actor. Every actor starts with financial means of 30, 20 and 10 financial units, respectively. Each of the actors has a budget of 60 financial means at his disposal. We observe the strong oscillation of the curves. For example, with his emissions reduced the second actor has reached the value fixed in the treaty of Kyoto. After a reduction of the financial means the curve will even fall under the baseline.
The extraordinary case with a growth parameter vector (30, 30, 30) stands for a scenario where every actor reaches the demanded value.
Nevertheless, because of the nonlinear structure of the dynamics we can not guarantee that such a value exists in general.
Table 2
Data of the TEM model
Player i Ei (0) Mi(0) m; A em-matrix
1 0 0.3 i 30 0.01 1 -0.5 -0.5
2 0 0.5 i 30 0.01 -0.5 1 -0.5
3 0 0.2 i 30 0.01 -0.5 -0.5 1
With the numerical results, we can get an intuition for the situation in which developing countries are expanding their energy consumption.
4. A short remark about controllability
Under the simplifying conditions
^emij (t)Mj (t) = 0, Mi(t)[M* - Mi(i)] = 0, i e{l,...,n] j=i
we are able to determine the fixed points E,M of the dynamical system given by (2. 1). The fixed points are steady states and have no time-dependence.
Additionally, regarding the Jacobi-matrix of the right-hand side for the special case emij (t) = emj, where the economic relationship is constant over a long period, we get
/1 0 0 1
00 00 00
0
01
0 0
0 0
emu em.2i
emn i 1 - XiM^Ei 0
em 12 em22
emn2 0
em1n
emn
\
1 - A2M2*E.
1 - AnM*EnJ
\0 ...... c
and the eigenvalues: A^ = ... that the fixed points under the simplifying conditions mentioned above are not attractive. For further studies to this field see the fundamental contributions of [5] and [6].
An
1, A,
n+j
1 - A*MjEj(t) for j e {1, ... , n}. We see
0
5. The problem of controllability
In order to formulate the abstract controllability problem, we prefer the following notation of a system of difference equations:
Xi (t + l) = xi (t) + fi(x(t),u(t)) (5. 1)
for i G {l, ... , n} and t G N0 = N U {0}. Here xi : N0 ^ Rli and ui : N0 ^ Rmi (i G {l, ... , n}) are state and control vector functions, respectively, which are the components of
vector functions
x(t) = (xi(t), ... , xn(i)), u(t) = (ui(t), ...,u„(t)), t G N0.
Furthermore,
n n
fi : JJ R j x Yl Rm ^ R j j=i j=i
are given vector functions for i G {1,... , n}. Additionally, we assume for every i G {1, ... , n} non-empty sets Xi C Rli and Ui C Rmi to be given, and require control conditions
ui(t) G Ui for all i G {1, ... , n} and t G N0 (5.2)
as well as state constraints
xi(t) G Xi for all i G {1, ... , n} and t G N0. (5.3)
Finally, we require initial conditions of the form
Xi(0) = xoi for i G{1, ...,n} (5.4)
where x0i G Xi (i G {1, ... , n}) are given.
If one chooses n control functions ui : N0 ^ Ui (i G {1, ... , n}), then, by (5. 1) and (5. 4), n state vector functions xi : N0 ^ Rli are uniquely determined, i.e., we are able to compare several scenarios. Now we make the following assumptions:
1. For every i G {1, ..., n} the zero vector Qm. of Rmi belongs to Ui.
2. The nonlinear system
fi (xi, ...,in, Bmi,..., 0m„) = 0ii for i g{1, ...,n} (5.5)
has at least one solution
n
x = (xi, ..., in) G JJXj.
j=i
Under these assumptions we formulate the problem of controllability as follows:
n
Let i = (ii, ..., in) G n Xj be a solution of (5. 5). We are looking for control functions
j=i
ui : N0 ^ Rmi (i G {1, ... , n}) and some N G N0 such that the conditions (5.1)-(5.4) are satisfied as well as
ui(t) = 8m. and xi(t) = ii, i G{1, ...,n}, t G N, t > N. (5.6)
n
In other words: Given an initial state x0 G Xj of the dynamical system under consideration,
j=i
find control functions ui : N0 ^ Rmi (i G {1, ... , n}) which satisfy (5. 2) and steer the system, under the conditions (5. 3), into a steady state of the uncontrolled system whose dynamics is described by
xi(t + 1) = xi(t) + fi(x(t), 6), i G{1, ...,n}, t G N0, (5.7)
where 6 = (6mi, ... , 6m„).
6. An iterative solution
For some t G No we assume vectors
Xi(t) G Xi, i G {1, ... , n}
to be given. For t = 0 we choose
Xi(0) = xoi, i G {1, ... , n} with x0i G Xi being the initial values in (5. 4).
n
For every vector u G Rmj we define
j=i
Xi(u)(t +1) := Xi(t) + fi(x(t),u), i G {1, ... where x(t) = (xi(t), ... , xn(t)) , and
a*(u) := ||xi(u)(t + 1) - £j||2 + ||ui|2, i G {1, .
n},
, n}.
(6.1)
(6. 2)
Here, || • ||2 denotes the Euclidean norm. For every i G {1, ... , n} we consider the function
JJr"
j=i
j ^ R
+
(6. 3)
as payoff function of the i-th actor. This actor is regarded as the i-th player of a game in which he has the set Ui at his disposal as the set of strategies by which he tries to control the game. However, the players are linked by the set
Zt := {u G fi Uj | Xi(u)(t +1) G Xi for all i G(1,...,n}}
(6. 4)
j=i
of feasible controls. Every player intends to minimize the value a*(w) of his own payoff function. This value depends on all controls «i, ... , un and therefore cannot be determined by the i-th player alone. Let us comprise the individual payoff functions by putting
^t(u) := at(u), u G Zt
(6. 5)
i=1
Then, the players have to solve the following Problem: Find a global minimizer U G Zt of i.e.,
< ^t(u) for all u G Zt.
(6. 6)
We have seen that the fixed points are not attractive. Thus the steady states (E, On) G R2n are not even attractive. Therefore, we assume the dynamics (2. 1) to be controlled in the following way
Ei(t +1) = Ei(t) + J] emij (Mj (t) + Uj (t)),
j=i
Mi(t + 1) = Mi(t) + Ui (t) - Ai (Mi(t) + Ui(t))(M* - Mi(t) - Ui(t))Ei(t)
(6. 7)
t
a
for i £ {1, ... , n} and t £ No- This means that the i-th nation has at its disposal a control function ui : N0 ^ R which we assume to satisfy
Ui(t) £ Ui, i £ {1, ..., n}, t £ N0, (6.8)
where every Ui is some subset of R with 0 £ Ui.
Puting ni = 2, mi = 1, xi = (Ei, Mi) for i £ {1, ..., n} and defining functions fi : R2n x Rn ^ R2 by the right-hand sides of (6. 7), then (6. 7) is of the form (5. 1). Furthermore (6. 8) is of the form (5. 2). We put
Xi = {(Ei, Mi) £ R2 | 0 < Mi < M*}
for i £ {1, ..., n}, then (2. 1) can be rewritten in the form (5. 1). Finally, we put x0i = (E0i, M0i) for i £ {1, ... , n}. The assumptions 1) and 2) of Section 5 are also satisfied. Now, the problem of controllability reads as follows:
Given (E, 6n) £ R2n, find control functions u* : N0 ^ R (i £ {1, ... , n}) and some N £ N0 such that the conditions (6.2), (6.3), (6.5), (6.6) and
u*(t) = 0, (Ei(t),Mi(t)) = (Ei, 0),
for all i £ {1, ... , n} and t > N, are fulfilled.
Several algorithms to solve this problem are investigated in [2].
7. An application of the gradient method
Let us consider the noncooperative case where each actor tries to minimize the value a*(w) from (6. 3). Then we have to solve iteratively at each time step the following problem
da* dui
0, i £ {!,..., n}.
This can be done by an algorithm which was implemented by the first author. A detailed description is presented in [2]. For further numerical techniques, e.g., of second order, see [7] and [8].
Here, we only want to present the numerical results (see Fig. 3) which show that the insertion of the calculated control parameters might be successful. The column entitled by Kyoto indicates the emission targets mentioned in Kyoto Protocol.
Table 3
Data for applying the gradient method
Player i Ei(0) Mi(0) M? A em-matrix Kyoto
1 -1 20 40 0 0.01 1 -0.8 0.1 1
2 -1 10 40 0 0.02 0.2 1 -0.8 1
3 -1 30 40 0 0.08 -0.1 -0.5 1 1
Figure 3. Application of the gradient method.
8. Conclusion
The Framework Convention on Climate Change (FCCC) of Kyoto Protocol demands for reductions in greenhouse gas emissions by the industrialized countries. On the other hand, developing countries are expanding their energy consumption, leading to increased levels of greenhouse gas emissions. The preparation of an optimal management tool in that field requires the possibility to identify, assess and compare several technological options. For that reason, the mathematical TEM model presented in this paper was elaborated. According to the FCCC (Article 4, paragraph 2(a)), control parameters were incorporated which have to be determined iteratively, according to a negotiation process.
9. Acknowledgements
The authors want to acknowledge Werner Krabs and Yurii Shokin for their critical remarks and encouraged motivations.
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Received for publication August 15, 2000
