(454d) Optimal Scheduling Under Variable Electricity Cost and Power Availability

A challenge that needs to be addressed by scheduling approaches concerns the optimal policy of energy consumption in process operations. The large majority assumes a constant energy profile but in real-life this may not be the case. As an example, steam production may be shared by different sections of a plant, each with its own profile, leading to a variable availability for a given section. Disturbances may cause significant production delays and should be given full consideration when deriving the schedule.

A more substantial economical impact can be felt if one fails to consider variable utility pricing. In a liberalized market, electricity pricing is likely to change from week days to weekends and even within an operating day. The aim is to execute tasks during the low-cost hours and also to keep the power consumption below the contracted value to avoid incurring in additional costs. Since pricing can vary within a day up to a factor of five, scheduling tools that can cope with such energy constrains can lead to major savings in the energy bill.

We have recently proposed a new Resource-Task Network (RTN) continuous-time formulation for the short-term scheduling of continuous plants that effectively handles energy constraints related to pricing and availability and multiple intermediate due dates (New Continuous-Time Scheduling Formulation for Continuous Plants under Variable Electricity Cost, accepted for publication in Ind. Eng. Chem. Res.). However, despite its generality, the model can only solve very small problems to optimality when considering a 1-week horizon with end-of-the-day demands and frequent energy price changes.

Such discrete events are handled much more naturally with a discrete-time formulation. Indeed, that paper results show a superior performance over its continuous-time counterpart, with the RTN discrete-time model being able to tackle industrial problems of industrial relevance to near optimality (<1%) in a few minutes of computational time. There were however two important issues. First, slightly suboptimal solutions may result due to the duration of the task being approximated to a multiple of the specified interval length. Second, multiple instances of a task will be required to meet the demands leading to high solution degeneracy and the generation of unpractical schedules.

In this paper, we combine the complementary strengths of discrete and continuous-time formulations to propose a combined aggregate/continuous-time model. It uses a time grid featuring both continuous and discrete time points, the latter being spaced non-uniformly. The continuous part of the model generates the detailed schedule between two consecutive demand points. The aggregate part of the model can be viewed as a planning approach, somewhat related to the detailed discrete-time model. It finds the optimal production targets for the remaining time horizon without being concerned with the actual timing of events.

The combined model is incorporated into a rolling horizon algorithm that iteratively proceeds one demand period after the other. The aggregate model provides a lower bound, which is the exact optimum under unlimited power availability. Under such conditions, if the continuous-time model fails to reach the target, the cause is an insufficient number of event points, a well-known drawback of time grid based continuous-time models. The algorithm keeps increasing the number until the optimum is found thus effectively overcoming their most significant drawback.

Under limited power availability, the inaccurate prediction from the aggregate model may lead to suboptimal solutions. In particular, it may lead to overconsumption of power in cases where it is possible to respect such constraints. Overall, the rolling horizon algorithm is able to find very good practical solutions for industrial size problems in minutes of computational time.