Process unit shutdowns within a multi-unit process commonly induce additional unit shutdowns and, furthermore, they invariably result in economic consequences. Possibilities of such consequences include the loss of production as well as the creation of off-specification product. Mitigation of these effects may be possible if sufficient buffer capacity is available to introduce a level of independence between the upstream and downstream processes that allow the remainder of the plant to continue operation until the affected units are back in operation. Maximizing the potential of buffer storage, however, is not trivial. A question that arises is what the optimal levels of the buffer storage units should be for use in normal operation, with insight as to what abnormal operation conditions may occur. As demonstrated by Balthazaar (2005), this is a function of the expected unit failure frequencies, failure lengths and recoveries, and process dynamics. The optimization goal is to maximize plant economics by accounting for profit from production as well as costs incurred by process unit shutdowns.

Research into the determination of optimal nominal buffer tank levels is relatively sparse. Related work is that of Tolio at al. (2002) in which a similar process is studied, but with the goal of determining average buffer levels, and without the possibility of control or corrective action to the processing units. In addition, this work focused on discrete part manufacturing, as opposed to continuous process operation. The work by Lee and Reklaitis (1989a,b) while related, focuses on batch systems, which results in a different problem formulation and consequently different results.

Multiple key failure scenarios are simultaneously considered through multi-period dynamic optimization in order to determine a single set of optimal nominal buffer levels common to all scenarios. These scenarios refer to the combination of failure characteristics including failure location, failure length, recovery time, and failure frequency. The use of mixed-integer linear programming (MILP) captures the continuous and discrete decisions that exist within this mathematical model. Discrete decisions are present due to highly constrained processing rates, the triggering of a shutdown when throughput falls beneath a specified minimum value, and the need to count shutdowns.

A case study of two processors in series separated by one intermediate storage unit is first examined. Initially, two failure scenarios (one for the upstream unit, one for the downstream unit) are considered and results studied in detail to gain insight into the influence of various model parameters on the optimization results. Next, a configuration of three processors in series separated by two buffer tanks is examined. Here, three failure scenarios are considered (one for each process unit), but with the added complication of the possibility of sending intermediate product to a purge stream in order to avoid an induced unit shutdown when economically logical. The two processor configuration was then re-examined, but now with ten failure scenarios - five each for the upstream and downstream unit. The purpose of including a large number of failure scenarios is to simulate one of two realistic conditions. The first condition is of several possible failure “modes” for each process unit. For example, different failure durations may be associated with a mechanical failure, electrical disruption, and off-specification product. Alternatively, a distribution in failure durations can simulate uncertainty in the failure duration. Finally, the problem formulation is applied to a typical pulp mill process that includes a recycle stream.

Solution non-uniqueness is a computational issue that arose during this work. A multi-tiered optimization combined with a grid-search approach was used to uncover all optimal solutions for the smaller test cases (two and three processors in series). A soft-constraint problem formulation approach, which avoided the use of integer variables, was also considered. The advantage of this approach is a decreased problem complexity due to the elimination of integer variables. However, accurate model representation is lessened due to the inability to account for costs associated with shutdowns (besides production loss).

Current extensions to this work include the consideration of discrete process unit flow rates, common in manufacturing processes, as well as production and maintenance scheduling.

References

Balthazaar, A. K. S. (2005). Dynamic Optimization of Multi-Unit Systems Under Failure Conditions. M.A.Sc. thesis, McMaster University, Hamilton, Ontario, Canada.

Lee, E.S. and Reklaitis, G.V. (1989a). Intermediate storage and operation of batch processes under batch failure. Comput. Chem. Eng., 13, 491–498.

Lee, E.S. and Reklaitis, G.V. (1989b). Intermediate storage and the operation of periodic processes under equipment failure. Comput. Chem. Eng., 13, 1235–1243.

Tolio, T., Matta, A., and Gershwin, S.B. (2002). Analysis of two-machine lines with multiple failure modes. IIE Transactions, 34, 51–62.