1. GLOBAL OPTIMIZATION OF CLIMATE CONTROL PROBLEMS USING EVOLUTIONARY AND STOCHASTIC ALGORITHMS WSC7 2002, September 2002 1 Process 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 Carmen G. Moles 1, Adam S. Lieber 2, Julio R. Banga 1 and Klaus Keller 3

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

3. Introduction  Optimal reductions in CO 2 emissions reducing CO 2 emissions consumption increasing abatement costs reducing climate damages + -

4. Introduction: controlling CO 2 emissions involves economic tradeoffs world production climate impacts increase in global temperatures increase in atmospheric carbon dioxide emissions consumption capital stock economic damages investment into greenhouse gas abatement Total capital stock allocation via optimization of utility

5. Optimization of dynamic systems l 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 l 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

6. Global optimization methods l 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 l 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 â Classification of GO methods

7. Global Optimization methods Stochastic Deterministic Hybrids DIRECT approach and variants GCLSOLVE (Holmström, 99) MCS (Neumaier, 99) GLOBAL (Csendes, 88) 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)

8. Optimal climate control problem â 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 CO 2 level from Stocker and Schmittner (1997) Stabilizing CO 2 below critical CO 2 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 TCH changes the “optimal” policy Realistic thresholds can introduce local optima into the objective function and require global optimization algorithms

10. âObjective 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 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 The 94 decision variables represent the investment and CO 2 abatement over time (after discretization of the time horizon)

11. Results and discussions â Results DESRESMCS N. eval 3.5e63.5e571934 CPU time, min 110.8710.67 4.10 U* 26398.713326398.64126397.009 ICRSGCLSOLVELJ N. eval 3868606500020701 CPU time, min 10.00 103.78 0.97 U* 26383.716226377.064926375.8383 ICRS presented the most rapid convergence initially but was ultimately surpassed by DE and SRES. The best result is obtained by DE. SRES converged to almost the same value but about 10 times faster. CPU time,s 10 10 1 3 5 -6 10 -4 10 -2 10 0 CPU time,s Relative error Convergence curves ICRS SRES LJ DE GCLSOLVE

12. Results and discussions â Best profiles 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 2000205021002150 10 15 20 25 30 35 40 45 Abatement % years DE SRES Best solution 2000205021002150 16 17 18 19 20 years Investment % DE SRES Best solution (Keller et al.) Investment profile Abatement profile

13. Results and discussions â Multi-start procedure Histogram for the MS-SQP Objective function Frecuency 10000150002000025000 The best MS-SQP result was C=23854.71 SQP always converged to local solutions (even with multi-start N=100)

14. Conclusions âThe local algorithm (SQP), even with a multi-start procedure, converged to multiple local solutions â Evolutionary strategies (SRES method) presented the fastest convergence to the vicinity of the best known solution â Differential evolution (DE) arrived to the best solution, although at a rather large computational cost â Simple adaptative stochastic methods presented an interesting first period of fast convergence which suggest new hybrid approaches