Rolling-Horizon Algorithm for Scheduling under Time-Dependent Utility Pricing and Availability

Transcription

1 Rolling-Horizon Algorithm for Scheduling under Time-Dependent Utility Pricing and Availability Pedro M. Castro Iiro Harjunkoski Ignacio E. Grossmann Lisbon, Portugal; Ladenburg, Germany; Pittsburgh, USA

2 Introduction Process operations are often subject to energy constraints Heating and cooling utilities, electrical power Availability Price Challenging aspect of plant scheduling Current practice heuristic rules for feasibility Due to complexity, choices are far from optimal No continuous-time formulation for time-dependent utility profiles Proposed approach general for continuous plants Focus on cement industry Grinding process major consumer of electricity June 8, 2010 Session: Integrated Management 2

3 Motivating problem Multiproduct, single stage plant Intensive use of electricity When and where to produce a certain grade? How much to keep in storage? Meet product demands (multiple due dates for each product) Minimize total energy cost Satisfy power availability constraints June 8, 2010 Session: Integrated Management 3

4 Electricity market Contracts between electricity supplier and plants Energy cost [ /kwh] Varies up to factor of 5 during the day Maximum power consumption [MW] Harsh cost penalties if levels are exceeded Optimal scheduling with large impact on electricity bill Goal is to produce in low-cost periods June 8, 2010 Session: Integrated Management 4

5 Process modeling From flowsheet to Resource-Task Network Convert problem data Shared storage units June 8, 2010 Session: Integrated Management 5

6 1.Discrete-time Slot1 Slot 2 Slot 3 Slot T-2 Slot T-1 δ T-2 T-1 1 T Elegant and compact formulation Discrete-events handled naturally Time intervals of 1 hour ( ) for 1 week horizon Minor limitations Can lead to slightly suboptimal solutions With too many changeovers June 8, 2010 Session: Integrated Management 6

7 2.Continuous-time Slot 1 Slot 2 Slot T-2 Slot T T-2 T-1 1 T General and accurate formulation Difficult to account for discrete events Location of event points unknown a priori Electricity pricing & availability Due dates Location of event points At demand points At some energy pricing/availability levels June 8, 2010 Session: Integrated Management 7

8 3. Aggregate model Slot1 Slot 2 Slot 3 Slot T-2 Slot T T-2 T-2 T-1 T-1 1 T Looks in between consecutive demand points Merges periods with same energy pricing/power level Low cost energy level Medium cost High cost Demand point Demand point Power availability (MW) Low cost Medium cost High cost Demand point Demand point Power availability (MW) Valid for single stage plants & instantaneous demands June 8, 2010 Session: Integrated Management 8

9 Important properties aggregate model It is a planning approach Not concerned with actual timing of events Continuous-time within a time interval without event points Different resource balances Equipments Slot duration processing times Utilities Energy balances instead of power balances Predicts # slots for continuous-time model It is a relaxation May underestimate total electricity cost 5 h@4 MW 4 h@5 MW but have same energy June 8, 2010 Session: Integrated Management 9

10 4. Rolling-horizon algorithm Combined aggregate/continuous-time model Time grid is part continuous and part discrete June 8, 2010 Session: Integrated Management 10

11 Computational statistics Case (P,M,S) Power Model T RMIP [ ] MIP [ ] CPUs Gap (%) EX5a (3,2,2) R DT ,351 31, AG 20 29,657 29, RH 17 41,124 41, CT 10 25,625 94, EX6 (3,2,3) U DT , AG 19 43, ,250 RH CT 9 35,517 Inf EX7 (3,3,4) U DT AG 18 68,282 68, RH CT 12 48,852 no sol EX8 (3,3,5) R DT , , AG , , RH , EX9 (4,3,4) U DT , AG 19 87, ,817 RH EX10 (5,3,4) U DT , ,505 AG ,550 0 RH 23 86, DT difficult to close optimality CT limited to very small problems AG very fast & accurate for unrestricted power RH generates full schedule in acceptable time June 8, 2010 Session: Integrated Management 11

12 Conclusions New aggregate model is very powerful Rigorous approach for unlimited power Vs. traditional discrete-time approach (DT) Lower degree of degeneracy 1/10 problem size Up to 4 orders magnitude reduction CPUs DT best approach under restricted power Finds very good solutions (<0.8 %) rapidly (5 min) New rolling-horizon algorithm tackles real-life problems to (near) optimality Considers whole remaining problem while simultaneously scheduling part of the time horizon Handles Complex energy & storage policies Multiple due dates Day T LB ( ) OBJ ( ) MO 5 65,300 65,300 TU 10 66, WE 15 66,274 66, TH 18 66,275 67, , FR 23 67,629 68, SA 29 68,302 69, SU 30 69,216 Inf ,216 June 8, 2010 Session: Integrated Management 12