Storage scheduling constraints

The constraints considered in the mathematical formulation are divided into two modules. The first deals with the mass balance constraints and the second with the sequencing and scheduling constraints. The mass balance constraints for the case where there is no central storage are slightly different to those for the case where there is. The mass balances for each are described in the mass balance module below. The sequencing and scheduling module will be described, for both cases, in a subsequent section. The nomenclature for all the sets, variables and parameters can be found in the nomenclature list. 

Sequencing and Scheduling Constraints Associated with Storage... 

Constraints (6.42), (6.43) and (6.44) deal with the scheduling aspects of two streams leaving the storage vessel. Constraints (6.42) ensures that streams leaving the storage vessel at later time points correspond to a later absolute time within the time horizon. Constraints (6.43) and (6.44) ensure that if two water streams are leaving the storage vessel at the same time point, both streams leave at the same time in the time horizon. 

The constraints that comprise the scheduling module of the model are divided into four groups, namely, task scheduling, direct recycle/reuse scheduling, storage scheduling and time horizon constraints. 

Scheduling constraints have to be derived to account for the timing of multiple streams leaving a storage vessel. Constraint (7.36) ensures that water leaving a storage vessel at a later time point does so at a later absolute time in the time horizon. Constraints (7.37) and (7.38) ensure that the time at which two streams leave a storage vessel at a time point corresponds to the same time for each. 

The scheduling constraints presented above hold for each storage vessel in the process. It must be noted that there is no interaction between the various storage vessels, since each storage vessel is independent of the other. 

Apart from direct reuse scheduling, constraints also have to be derived to ensure the correct scheduling of indirect reuse through inherent storage. 

In published mathematical formulations scheduling constraints for wastewater storage in processing units are not required, since this option is ignored. However, in this formulation the option to store wastewater in idle processing units is included, hence scheduling constraints that govern this occurrence are required. 

In the case where there is a central storage vessel, scheduling constraints also have to be derived to account for the timing of water entering and exiting the vessel relative to the overall operation. 

As with the multiple contaminant wastewater minimisation model, the mathematical model for multiple storage vessels comprises of two modules, namely, a mass balance module and a scheduling module. The constraints that comprise the mass balance module are described first. 

The first constraints that are dealt with in the zero effluent scheduling formulation are the unit mass balance constraints. These constraints account for the movement of mass between the various units and storage vessels. 

Constraints (8.38) - (8.40) are constraints that deal with the scheduling of streams to and from a storage vessel. If water leaves a storage vessel at a time point after the time point at which the water entered the vessel, then the time at which this happens must occur at a later absolute time in the time horizon. This is given in constraint (8.38). The time at which a stream leaves a storage vessel and the time at which water enters a storage vessel must coincide, provided the two streams enter at the same time point. This is ensured through constraints (8.39) and (8.40). 

The constraints that ensure the correct scheduling of the central storage vessel are similar to those presented by Majozi (2005) and will therefore not be discussed in great detail. 

Enhanced continuous-time unit-specific event-based formulation for short-term scheduling of multipurpose batch processes Resource constraints and mixed storage policies. Ind. Eng. Chem. Res., 43, 2516-2533. 

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