GLOBAL OPTIMIZATION OF CLIMATE CONTROL PROBLEMS USING EVOLUTIONARY AND STOCHASTIC ALGORITHMS

1. 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 Engineering Group, IIM (CSIC), Vigo, Spain. 2 Mission Ventures, San Diego, CA, U.S.A 3 Department of Geosciences. The Pennylvania State University, U.S.A

2. Summary • Introduction • Optimization of dynamic systems • Definition of the optimal control problem • Global Optimization methods • Classification and brief description • Optimal climate control problem • Mathematical formulation • Results and discussion • Conclusions

3. Introduction reducing CO2 emissions increasing abatement costs - reducing climate damages + consumption • Optimal reductions in CO2 emissions

4. Introduction: controlling CO2 emissions involves economic tradeoffs Total capital stock allocation via optimization of utility consumption capital stock world production economic damages investment into greenhouse gas abatement climate impacts increase in global temperatures increase in atmospheric carbon dioxide carbon dioxide emissions

5. 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 algorithmsfail to obtain the global optimum, they converge to local solutions

6. 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

7. Global Optimization methods • Genetic algorithms (GAs) and variants • DE (Storn & Price, 99) • Adaptive stochastic methods • ICRS (Banga and Casares, 87) • LJ (Luus and Jaakola, 73) • Evolution strategies (ES) • SRES (Runarsson, 00) Stochastic Deterministic • DIRECT approach and variants • GCLSOLVE (Holmström, 99) • MCS (Neumaier, 99) Hybrids GLOBAL (Csendes, 88)

8. 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

9. Optimal climate control problem • Preserving the TCHchanges the “optimal” policy • Realistic thresholds can introduce • local optima into the objective function and require global optimization algorithms

10. Optimal climate control problem • The optimization problem maximizes the social welfare: • 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 • 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 94 decision variables represent the investment and CO2 abatement over time (after discretization of the time horizon)

11. Results and discussions DE SRES MCS N. eval 3.5e6 3.5e5 71934 0 10 CPU time,min 110.87 10.67 4.10 U* 26398.7133 26398.641 26397.009 ICRS GCLSOLVE LJ -2 10 N. eval 386860 65000 20701 CPU time,min 10.00 103.78 0.97 U* 26383.7162 26377.0649 26375.8383 -4 10 CPU time ,s Convergence curves -6 10 -1 1 3 5 10 10 10 10 Relative error CPU time ,s • Results • Thebest resultis obtained byDE. • SRESconverged to almost the same value but about10 times faster. • ICRSpresented themost rapid convergence initiallybutwas ultimately surpassedbyDE andSRES. DE ICRS GCLSOLVE LJ SRES

12. Results and discussions Abatement profile Investment profile DE SRES Best solution DE SRES Best solution (Keller et al.) 20 45 19 40 Abatement % Investment % 35 18 30 25 17 20 years years 15 16 2000 2050 2100 2150 10 2000 2050 2100 2150 • Best profiles • Significant differencesin theoptimal investment and abatement policies even for very similar objective function values (frequent result in dynamic optimization) • It is due to low sensitivity of thecost function withrespect to the decision variables

13. Results and discussions Histogram for the MS-SQP The best MS-SQP result was C=23854.71 Objective function Frecuency 10000 15000 20000 25000 • Multi-start procedure • SQP always converged to local solutions (even with multi-start N=100)

14. Conclusions • The local algorithm(SQP), even with a multi-start procedure,convergedtomultiple local solutions • Evolutionary strategies (SRES method) presented thefastest convergenceto the vicinity of the best known solution • Differential evolution (DE) arrived to thebest solution, although at aratherlarge computational cost • Simple adaptative stochastic methods presented aninteresting firstperiod offast convergencewhich suggestnew hybrid approaches