Cyclic Scheduling of Pulp Digesters with Integrated Heating Tasks

Abstract

This paper addresses a multistage batch plant scheduling problem under energy constraints. These reflect the limited availability of a thermal heating utility that is shared among parallel digesters of different capacities for the production of pulp. Depending on the processing sequence, more or less steam will be available for a given digester, which will affect the duration of its heating stage and the overall cycle time. Such integrated heating tasks resemble direct heat integration, which has been addressed through models based on generic frameworks for process representation (e.g., State-Task Network, ResourceTask Network, State-Sequence Network) and relying on a single time grid, either discrete or continuous. A new multiple time grid continuous-time model is now proposed where the complex energy constraints are derived from the higher level modeling framework that is Generalized Disjunctive Programming. The results show a considerable better performance compared to RTN discrete and continuous-time formulations, due to a substantially lower integrality gap and model size.

Optimal Scheduling of Multiproduct Pipelines in Networks with Reversible Flow

“…Equation features embedded disjunctions that can be transformed into simple disjunctions and logic propositions using the guidelines in Vecchietti and Grossmann (examples of real-world problems tackled by GDP models relying on embedded disjunctions can be found in refs − ). The logic proposition of the outer disjunction can be reformulated into eq , after eliminating variables Y s , t S,idle . …”

Continuous‐time scheduling formulation for straight pipelines

“…Constraints are essentially divided into three types, involving: (1) just continuous variables, which mostly involve volumetric balances; (2) continuous and binary variables, which are first formulated as disjunctions and then converted into mixed‐integer linear format using convex hull reformulations; and (3) solely binary variables, which are first stated as logic propositions. One purpose of the derivation starting from a GDP formulation is to highlight the differences between inclusive and exclusive disjunctions. To keep the nomenclature simple, we rely on the same letter for a particular type of parameter (lower case) or variable (upper case) and change the superscripts and subscripts to distinguish between the source.…”

Optimal scheduling of single stage batch plants with direct heat integration

“…However, employing a general scheduling formulation for a simpler plant topology, typically leads to a less efficient approach (Harjunkoski et al, 2014). In a recent article dealing with a pulp plant consisting of four parallel batch digesters competing for the same heating utility (steam), Castro et al (2014b) have shown that orders of magnitude reduction in computational time can be obtained by using a multiple time grid formulation rather than single grid discrete and continuous-time formulations. Part of the success resulted from relying on Generalized Disjunctive Programming (GDP) (Raman and Grossmann, 1994) for deriving the complex timing constraints between interacting digesters.…”