∗Stochastic Optimization for Operating Chemical Processes under Uncertainty

Mathematical optimization techniques are on their way to becoming a standard tool in chemical process engineering. While such approaches are usually based on deterministic models, uncertainties such as external disturbances play a significant role in many real-life applications. The present article gives an introduction to practical issues of process operation and to basic mathematical concepts required for the explicit treatment of uncertainties by stochastic optimization. 1 OPERATING CHEMICAL PROCESSES Chemical industry plays an essential role in the daily life of our society. The purpose of a chemical process is to transfer some (cheap) materials into other (desired) materials. Those materials include any sorts of solids, liquids and gas and can be single components or multicomponent mixtures. Common examples of chemical processes are reaction, separation and crystallization processes usually composed of operation units like reactors, distillation columns, heat exchangers and so on. Based on market demands, those processes are designed, set up and put into operation. From the design, the process is expected to be run at a predefined operating point, i.e., with a certain flow rate, temperature, pressure and composition [22]. Distillation is one of the most common separation processes which consumes the largest part of energy in chemical industry. Figure 1 shows an industrial distillation process to separate a mixture of methanol and water to high purity products (methanol composition in the distillate and the bottom should be xD ≥ 99.5mol% and xB ≤ 0.5mol%, respectively). The feed flow F to the column is from outflows of different upstream plants. These streams are first accumulated in a tank (a middle buffer) and then fed to the column. The column is operated at atmospheric pressure. From the design, the diameter of the column, the number of trays, the reboiler duty Q and the reflux flow L will be defined for the given product specifications. For an existing chemical process, it is important to develop flexible operating policies to improve its profitability and reducing its effect of pollution. The everchanging market conditions demand a high flexibility for chemical processes under different product specifications and different feedstocks. On the other hand, the increasingly stringent limitations to process emissions (e.g., xB ≤ 0.5mol% in the above example) require suitable new operating conditions satisfying these constraints. Moreover, the properties of processes themselves change during process 458 R. Henrion et al. Figure 1. An industrial distillation column with a feed tank operation, e.g., tray efficiencies and fouling of the equipment, which leads to reduction of product quality if the operating point remains unchanged. Therefore, keeping a constant operating point given by the process design is nowadays an out-dated concept. That is to say, optimal and robust operating policies should be searched for and implemented online, corresponding to the real-time process situations. In the past, heuristic rules were used for improving process operation in chemical industry. However, since most chemical processes behave nonlinear, time-dependent and possess a large number of variables, it was impossible to find the optimal solutions or even feasible solutions by heuristic rules. Therefore, systematic methods including modeling, simulation and optimization have been developed in the last two decades for process operation. These methods are model-based deterministic approaches and have been more and more used in chemical industry [10]. 1.1 Process Modeling Conservation laws are used for modeling chemical processes. A balance space is first chosen, for which model equations will be established by balancing mass, momentum and energy input into and output from the space [3]. Thus variables of a space can be classified into independent and dependent variables. Independent variables are input variables including manipulated variables and disturbance variables. For instance, the reflux flow and the reboiler duty are usually manipulated variables for a distillation column, while the feed flow and composition are disturbance variables. Dependent variables are output variables (usually called state variables) which depend on the input variables. The compositions and temperatures on the trays inside the column are dependent variables. Besides conservation laws, corStochastic Optimization for Operating Chemical Processes 459 relation equations based on physical and chemical principles are used to describe relations between state variables. These principles include vapor-liquid equilibrium if two phases exist in the space, reaction kinetics if a reaction takes place and fluid dynamics for describing the hydraulics influenced by the structure of the equipment. Let us consider modeling a general tray of a distillation column, as shown in Figure 2, where i and j are the indexes of components (i = 1,NK) and trays (from the condenser to the reboiler), respectively. The dependent variables on each tray are the vapor and liquid compositions yi,j, xi,j, vapor and liquid flow rate Vj, Lj, liquid molar holdup Mj, temperature Tj and pressure Pj. The independent variables are the feed flow rate and composition Fj, zFi,j, heat flow Qj and the flows and compositions from the upper as well as lower tray. The model equations include component and energy balances, vapor-liquid equilibrium equations, a liquid holdup equation as well as a pressure drop equation (hydraulics) for each tray of the column: – Component balance: