Optimierung und robuster Betrieb komplexer Systeme unter Unsicherheiten mit der stochastischen Programmierung
Final Report Abstract
The objective of the project was to develop efficient computational methods for the optimal, robust and reliable performance of large-scale and complex dynamic systems in the presence of uncertainties. In practical industrial processes, uncertainties can arise due to variations in supply of raw materials, random disturbances of feed flows, variations in feed qualities, changing market conditions, errors in reaction constants, etc. As a result product quality specification, safety conditions and process constraints may not hold as required. Such problems can be effectively addressed by using stochastic optimization models with chance constraints. Most practical processes are dynamic in nature and are described using a large number of nonlinear (differential algebraic) equations. In order to control such dynamic processes along reference trajectories, the chance constrained nonlinear model predictive control (NMPC) scheme is found to provide the efficient control mechanism. In this scheme, the dynamic model equations are discretized on time intervals and then nonlinear chance constrained optimization problems with large number of equality constraints are solved on a moving horizon. The major difficulty of this approach lies in the computation of the values and gradients of the chance constraints. Due to the nonlinearity of the model equations, it is difficult to directly determine a distribution function for the chance constrained output variables. To overcome this difficulty, chance constraints on output variables can be transformed into chance constraints using input variables (with known distribution). This back-projection of chance constraints is facilitated if there is a strict monotony relation between a chance constrained output variable and some uncertain input variable. Therefore, the major deliberations of this research have been: (i) investigation of a necessary and sufficient condition for the existence monotony relations to facilitate the back-projection of chance constraints (ii) to develop fast and efficient algorithms for the evaluation of multidimensional integrals related with values and gradient of chance constraints (iii) to test and verify the developed strategies on realistic process engineering applications. The required monotony relation has been studied based on a mathematical analysis of the nonlinear model equations by applying the global implicit function theorem. In addition to the chance constraints, the evaluation of the objective functions involves the computation of multidimensional probability integrals. This has been for long a bottleneck in the numerical solution of stochastic optimization problems causing the curse-of-dimensions. For this purpose, sparse-grid integration techniques have been introduced. Sparse-grid integration techniques facilitated the reduction of computational time decisively by avoiding computational redundancy. This enables the solution of realistic steady state and dynamic process engineering optimization problems with several uncertain input variables. This accomplished work can be taken as an important contribution to the field of nonlinear chance constrained optimization. In particular, exact representation of output chance constraints through back-projection using monotonicity and fast computation of the resulting chance constraints are very demanding in several fields of applied sciences, where optimality as well as reliability are critical. Nonlinear dynamic process can experience bifurcations and chaotic situations. The optimal and reliable control of a process under such instances is of at most importance. In general, global monotony relations may not always exist in order to facilitate back-projection of chance constraints from the space of output variables into the space of input variables. This issue requires a detailed investigation to come up with innovative ideas for the computation of chance constraints based on the analysis of the nonlinear dynamic process model equations. In addition, in this completed project, uncertain input variables are assumed to have Gaussian normal distributions. But, in reality, practical process engineering applications can have non-Gaussian distributions. Nowadays, process engineering applications use computer-aided control mechanisms. Automatic data collection, evaluation and statistical analysis on large set of data can be accomplished without major difficulties. Hence, the nature of process input variables and random disturbances can be characterized on the basis of historical or experimental data. Hence, scientific investigations have revealed that many process input data can have non-Gaussian distributions. Furthermore, in the immediate future renewable energy will play a more and more important role in energy generation and supply. However, a large part of renewable energy (e.g. power from wind and solar energy) is stochastic in nature and thus leads to disturbances to the operations of power supply networks. Due to the increasing amount of energy penetration from renewable energy these disturbances will be more and more significant. It is well-known that the power load (i.e. the power demand from consumers) also represents a considerable uncertain disturbance. In fact, recent scientific investigations reveal that distribution of uncertainties like the Weibull distribution, Beta-distribution, etc., are prevalent in wind power generation. In addition, the clearness index for the solar energy can be well described with an extended exponential distribution. These findings show that it is more accurate for these uncertain variables to be described with multimodal and multi-parametric distributions. Therefore, for the design and operation of power supply networks the uncertainty of these disturbances has to be taken into account. Future investigations will consider operations planning for such networks based on the theoretical results of chance constrained optimization. The objective will be to develop optimal as well as reliable short-term transportation strategies for power supply networks under these uncertainties so as to minimize the power loss and/or maximize the power transportation capacity. In general, the characteristics of random errors and disturbances varies with the type of process under consideration; therefore, they cannot be taken as being Gaussian in every process engineering application. This reality should be taken seriously in the upcoming research activities. The development of sparse-grid integration techniques for the computation of multidimensional integrals with non-Gaussian weight-functions is one of the central future issues. This requires a detailed theoretical analysis and the development of efficient computational algorithms for the determination of integration nodes and weights.
Publications
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2009. A moving horizon approach to a chance constrained nonlinear dynamic optimization problem. IFAC Workshop on Control and Applications of Optimization. 6 – 8 May 2009, Agora, Finland
Geletu, A.; Klöppel, M.; Li, P.; Hoffmann, A.
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2009. A Numerical Approach to a Chance Constrained Nonlinear Dynamic Optimization Problem, 4th German Polish Conference on Optimization Methods and Applications, March 14 – 18, 2009, Moritzburg, Saxony
Klöppel, M.; Geletu, A.; Li, P.; Hoffmann, A.
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2010. Einsatz von Sparse-Grid-Integrationstechniken in der wahrscheinlichkeitsrestringierten Optimierung, GOR Workshop: Vector Optimization of Complex Structures, March 17 – 19, 2010, Erlangen, Germany
Klöppel, M.; Geletu, A.; Hoffmann, A.; Li, P.
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2010. Efficient solution of chance-constrained nonlinear dynamic process optimization problems with non-Gaussian uncertainties, 2nd International Conference on Engineering Optimization, 6 - 9 September 2010, Lisboa, Portugal
Geletu, A.; Klöppel, M.; Li, P.; Hoffmann, A.
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2010. Optimierung von Energietransportsystemen unter Unsicherheiten, 126. GDNÄ-Versammlung, September 17 – 21, 2010, Dresden, Germany
Klöppel, M.; Geletu, A.; Hoffmann, A.; Li, P.
