Performance Analysis of On-line Batch Optimization Systems

In this paper, the on-line optimization of batch reactors under parametric uncertainty is considered. A method is presented that estimates the likely economic performance of the on-line optimizer. The nmthod of t)rthogonal collocation is employed to convert the differential algebraic optimization problem (DAOP) of the dynamic optinfization into a nonlinear t)rogram (NLP) and determine the nominal optimum. Based on the resulting NLP, the optimization steps are approximated by neighbouring extremal problems and the average deviation from tile true process optimum is determined dependent on the measurement error and the parametric uncertainty. A back off from the active path and endpoint constraints is determined at each optimization step which ensures the feasible operation of the process. The method of the average deviation from optimum is developed for time optimal problems. The theory is demonstrated on an example. I N T R O D U C T I O N parameters using past and present process measureA wide variety of products in the chemical industries ments. The updated model is then optimized with are produced in batch mode. Due to disturbances respect to the manipulated variables and a new optiduring operation and uncertainties in process parammal input trajectory over the remaining time horizon eters, such as reaction kinetic parameters, there is a is determined. This sequence of an estimation and danger of producing unsatisfactory batches where the optimization step is referred to in the following as product or safety specifications are not met and path an Estimation-Optimization-Task, EOT (Ruppen et or endpoint constraints are violated. Therefore, it is at., 1997). The first part of the calculated input tradesirable to supervise and optimize the process durjectory is applied to the process until a new EOT is ing its operation in order to meet the product and carried out at some future point in time. safety specifications while maximizing an objective function, for example the yield of the desired product. O R T H O G O N A L C O L L O C A T I O N This can be achieved by acquiring on-line process inSince in batch processes the dynamic behaviour is formation which is then used to determine an imdominating and usually no steady state is reached, proved operation policy for the rest of the batch. This the objective function needs to be optimized with reresults in the following on-line optimization scheme spect to the dynamic model equations. Time optimal which consists of two steps, as depicted in Figure 1. problems, where the only objective is the minimizaIn a first step, the process model is identified or uption of the final batch time, have the following form: dated by estimating the state variables and/or a set of min t / (1) ~ uncertainty/disturbances u,tt I I s.t. x = f ( x , u , p , t ) , x(to) = xO (2) ~1 Process I ~ 9(X,U,p, t ) < O, (3) past and present past and present inputs measurements with x representing the states, u the control inputs _1 I_ -~ and p the uncertain process parameters. Since beEstimation r[ V ~timation. sides the dynamic model equations, f a set of alJ ~parameter/state estimates Optimizationgebraiv path/endpoint constraints, g is present, this future inputs ] Optimization ] Task, EOT problem is called a differential algebraic optimization problem, DAOP. One or more conditions, which are represented by a subset of the constraints g, need Figure 1: General structure of an on-line batch optito be reached in the minimum possible time. This mization system, subset of the constraints corresponds to the so-called