(215c) Optimal Scheduling with Energy Constraints

Although significant progress has occurred in the last few years, there are still some challenges that need to be addressed by scheduling formulations. One of them is related to the optimal policy of energy consumption in process operations. On the one hand, the price of electricity changes from week days to weekends and even within an operating day, and it is desired to execute tasks during the low-cost hours. On the other hand, in the planning stage, there is sometimes the need to make a commitment to certain energy levels. These come at a certain cost, which is not lowered if the actual operation consumes less energy. If, on the contrary, the levels are exceeded, the plant incurs on a large penalty that should be minimized. Efficient scheduling tools that can cope with such constraints can have an important impact on the energy bill.

Modeling energy costs/profiles that change during the day is relatively straightforward with a general RTN/STN discrete-time formulation. There are however some well-known issues related to discrete-time models. Large mathematical problems result, due to high number of time points that normally needs to be used to represent the problem data accurately. While this may not be that much of an issue for batch plants, for continuous plants, even a very fine time grid may be inappropriate. For these type of plants, using a continuous-time formulation is generally a better approach. One of the problems with continuous-time is that the number of event points of the single or multiple time grids that are used, has a profound effect on computational effort, and going beyond 10 may render the mathematical problem intractable. Thus, postulating a large number is simply not an option.

In order to model variable energy costs/profiles with a continuous-time formulation, we need to place a subset of the time points at certain real times. Although the effect of using fixed instead of variable time points has received little or no attention in the literature, a lower impact in computational effort is expected so hopefully, the number of event points can be increased. However, we do need to place the other event points so fixing some of them is actually leading to another problem. How many event points should be placed between two consecutive fixed points?

In this paper, we address a real scheduling problem from a continuous multiproduct plant. It is basically single stage with a few processing units in parallel sending material to some storage units. Although storage is shared, only one product can be stored at a particular unit at a given time. In order to model variable energy costs and profiles of energy availability, an algorithm is developed that relies on a fairly accurate discrete-time formulation to predict the number and location of the variable event points, and on a continuous time formulation to find an exact solution. A comparison to the stand alone approaches is also provided.