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WSC7 2002, September 2002
GLOBAL OPTIMIZATION OF CLIMATE CONTROL
PROBLEMS USING EVOLUTIONARY AND
STOCHASTIC ALGORITHMS
Carmen G. Moles1, Adam S. Lieber2,
Julio R. Banga1 and Klaus Keller3
1Process
2
3
Engineering Group, IIM (CSIC), Vigo, Spain.
Mission Ventures, San Diego, CA, U.S.A
Department of Geosciences. The Pennylvania State University, U.S.A
Summary

Introduction

Optimization of dynamic systems


Global Optimization methods


Definition of the optimal control problem
Classification and brief description
Optimal climate control problem

Mathematical formulation

Results and discussion

Conclusions
Introduction
 Optimal reductions in CO2 emissions
reducing CO2
emissions
increasing
abatement
reducing
-
+
costs
climate
damages
consumption
Introduction: controlling CO2 emissions involves economic tradeoffs
Total capital stock allocation
via optimization of utility
consumption
world
capital stock
production
economic
damages
climate
impacts
investment into
greenhouse gas
abatement
carbon dioxide
emissions
increase in
atmospheric
carbon dioxide
increase in
global
temperatures
Optimization of dynamic systems

Objective of optimal control problems
 find a set of control variables (functions of time) in order to maximize (or minimize)
the performance of a given dynamic system, measured by some functional, and all
this subject to a set of path constraints
 dynamics usually described in terms of differential equations or in equations in
differences

Climate-economy system (case study)
 not smooth system with significant hysteresis responses which introduce
multimodality
 the traditional local optimization algorithms fail to obtain the global optimum, they
converge to local solutions
Global optimization methods
 Classification of GO methods

Deterministic methods: different approaches (Floudas,Grossmann, Pintér, etc.)

Guarantee global optimality for certain GO problems

Main drawbacks:
•significant computational effort even for small problems
•most of them not applicable to black-box models
•several differentiability conditions required

Stochastic methods: several approaches (Luus, Banga, Wang, etc.)

Aproximate solutions found in reasonable CPU times
 Arbitrary black-box DAEs can be considered (incl. discontinuities etc.)

Main drawback:
•Global optimality can not be guaranteed
Global Optimization methods
• Genetic algorithms (GAs) and variants
 DE (Storn & Price, 99)
Stochastic
• Adaptive stochastic methods
 ICRS (Banga and Casares, 87)
 LJ (Luus and Jaakola, 73)
• Evolution strategies (ES)
 SRES (Runarsson, 00)
Deterministic
• DIRECT approach and variants
 GCLSOLVE (Holmström, 99)
 MCS (Neumaier, 99)
Hybrids
GLOBAL
(Csendes, 88)
Optimal climate control problem
 Model formulation: important assumptions
 Based on the Dynamic Integrated model of Climate and the Economic (DICE),
economic model of Nordhaus (1994). It integrates
• economics
• carbon cycles
• climate science
• impacts
 Critical CO2 level from Stocker and Schmittner (1997)
 Stabilizing CO2 below critical CO2 level preserves the North Atlantic Thermohaline
circulation (THC) collapse, keller et al. (2002)
 THC collapse is the only abrupt climate change
 Future costs/benefits are discounted
Optimal climate control problem
 Preserving the TCH changes the “optimal”
policy
 Realistic thresholds can introduce
local optima into the objective function and
require global optimization algorithms
Optimal climate control problem
 Objective function formulation
 Radical simplification: At a given time, just one type of individuals
 At a given time, just the sum of individual utilities
 Over time, discount future people's utility
 The optimization problem maximizes the social welfare:
t*
U (t )
U*= 
t
(1


)
t=t0
• Agregate utility at a point in time : U(t)  L(t) ln c(t)
• Individual utility : ln c(t)
• Population : L
• Per capita consumption : c
• Pure rate of social time preference : ρ
 The 94 decision variables represent the investment and CO2 abatement
over time (after discretization of the time horizon)
Results and discussions
 Results
DE
 The best result is obtained by DE.
 SRES converged to almost the same
value but about 10 times faster.
10
SRES
MCS
3.5e6
3.5e5
71934
N. eval
10.67
4.10
CPU time,min 110.87
26398.7133
26398.641
26397.009
U*
ICRS
GCLSOLVE
LJ
386860
65000
20701
N. eval
103.78
0.97
CPU time,min 10.00
26383.7162 26377.0649 26375.8383
U*
Convergence curves
0
Relative error
 ICRS presented the most rapid
10
DE
-2
convergence initially but was
ICRS
GCLSOLVE
ultimately surpassed by DE
LJ time ,s
CPU
10
and SRES.
-4
SRES
-6
10
10
-1
10
1
CPU time ,s
10 3
10
5
Results and discussions
 Best profiles
Investment profile
DE
SRES
Best solution (Keller et al.)
Investment %
19
18
17
40
35
Abatement %
20
45
Abatement profile
DE
SRES
Best solution
30
25
20
15
16
2000
2050
years
2100
2150
10
2000
2050
2100
2150
years
 Significant differences in the optimal investment and abatement policies
even for very similar objective function values (frequent result in dynamic
optimization)
 It is due to low sensitivity of the cost function with respect to the decision
variables
Results and discussions
 Multi-start procedure
Histogram for the MS-SQP
The best MS-SQP result
was
C=23854.71
15
10
5
0
10000
15000
20000
25000
Objective function
SQP always converged to local solutions (even with multi-start N=100)
Conclusions
The local algorithm (SQP), even with a multi-start procedure,
converged to multiple local solutions
 Evolutionary strategies (SRES method) presented the fastest
convergence to the vicinity of the best known solution
 Differential evolution (DE) arrived to the best solution, although at a
rather large computational cost
 Simple adaptative stochastic methods presented an interesting first
period of fast convergence which suggest new hybrid approaches