Rolling-Horizon Algorithmfor Scheduling under Time-Dependent Utility Pricing and Availability

Pedro M. CastroIiro HarjunkoskiIgnacio E. Grossmann

Lisbon, Portugal; Ladenburg, Germany; Pittsburgh, USA

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

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

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

Process modeling• From flowsheet to Resource-Task Network

• Convert problem data

June 8, 2010 Session: Integrated Management 5

Shared storage units

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

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

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)

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

4. Rolling-horizon algorithm• Combined aggregate/continuous-time model

– Time grid is part continuous and part discrete

June 8, 2010 Session: Integrated Management 10

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

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 (