rolling-horizon algorithm for scheduling under time-dependent utility … · 2018. 10. 20. ·...
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Rolling-Horizon Algorithmfor Scheduling under Time-Dependent Utility Pricing and Availability
Pedro M. CastroIiro HarjunkoskiIgnacio E. Grossmann
Lisbon, Portugal; Ladenburg, Germany; Pittsburgh, USA
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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 2Session: Integrated Management
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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
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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
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Process modeling• From flowsheet to Resource-Task Network
• Convert problem data
June 8, 2010 Session: Integrated Management 5
Shared storage units
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1.Discrete-time• 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
1 T2 4 T-2 T-1
Slot1
3δ
Slot 2 Slot 3 Slot T-2 Slot T-1
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2.Continuous-time• 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
1 T2 3 T-2 T-1
Slot 1 Slot 2 Slot T-2 Slot T-1
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3. Aggregate model• Looks in between consecutive demand points• Merges periods with same energy pricing/power level
• Valid for single stage plants & instantaneous demands
June 8, 2010 Session: Integrated Management 8
1 T2 4 T-2 T-1
Slot1
31
Slot 2 Slot 3 Slot T-2 Slot T-1
2 3 T-2 T-1
Demand pointDemand point
Low cost energy level Medium cost High cost
Power availability (MW)
Demand point Demand point
Low cost Medium cost High cost
Power availability (MW)
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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
June 8, 2010 Session: Integrated Management 9
• It is a relaxation– May underestimate total
electricity cost• 5 h@4 MW ≠ 4 h@5 MW
but have same energy
€21575 €18977
€21575 €21575
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4. Rolling-horizon algorithm• Combined aggregate/continuous-time model
– Time grid is part continuous and part discrete
June 8, 2010 Session: Integrated Management 10
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Computational statisticsCase (P,M,S) Power Model |T| RMIP [€] MIP [€] CPUs Gap (%)EX5a (3,2,2) R DT 169 31,351 31,798 7200 0.02
AG 20 29,657 29,657 0.240RH 17 41,124 41,124 7.06
CT 10 25,625 94,901 9829EX6 (3,2,3) U DT 169
43,25043,259 7200 0.02
AG 1943,250
0.370
RH 21 5.57CT 9 35,517 Inf. 2811 -
EX7 (3,3,4) U DT 16968,282 68,282
19.90AG 18 0.7
RH 12 3.12CT 12 48,852 no sol. 7200 -
EX8 (3,3,5) R DT 169 101,139 104,622 7200 0.22AG 19 104,375 104,375 2.05 0RH 31 - 151,257 17330 0.16
EX9 (4,3,4) U DT 16987,817
87,868 7200 0.06AG 19 87,817 0.71 0RH 25 917
EX10 (5,3,4) U DT 169 86,505 86,582 7200 0.09AG 19 86,550 3.57 0RH 23 86,550 1508
June 8, 2010 Session: Integrated Management 11
• 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
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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 (