Pedro M. Castro Iiro Harjunkoski Ignacio E. Grossmann

Lisbon, Portugal Ladenburg, Germany Pittsburgh, USA

1

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

August 17, 2009 2

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

Respect power availability constraints

3August 17, 2009

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 can have 20% impact on electricity bill Produce in low‐cost periods

August 17, 2009 4

From flowsheet to Resource‐Task Network

Convert problem data

August 17, 2009 5

6

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

August 17, 2009 7

1 T2 4 T-2 T-1

Slot1

3δ

Slot 2 Slot 3 Slot T-2 Slot T-1

General and accurate formulation Difficult to account for discrete events

Location of event points unknown a priori Electricity pricing & availability Product due dates

Location of event points At demand points

At some energy pricing/availability levels

August 17, 2009 8

1 T2 3 T-2 T-1

Slot 1 Slot 2 Slot T-2 Slot T-1

Looks in between consecutive demand points Merges periods with same energy pricing/power level

Only valid for single stage plants

August 17, 2009 9

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)

Can be view as a planning approach Not concerned with actual timing of events

Continuous‐time within a time interval without event points Resource balances for equipments and utilities replaced by two sets of constraints Time interval duration greater than sum of processing times

Energy balances (soft constraints due to slacks Sr,t)

August 17, 2009 10

||,,max,, TtTtRr TC

Ii i

tiirt

c

||,,)( max,,

,, TtTtRrS UT

Ii i

tiirtr

intrt

c

Calls combined aggregate/continuous‐time model

August 17, 2009 11

12

August 17, 2009 13

Case (P,M,S) Power Model |T| DV SV EQ RMIP MIP [€] CPUs Gap (%)EX5 (3,2,2) U DT 169 2016 10953 6739 26738 26780 7200 0.04

AG 19 216 1146 736 26757 26758 0.33 0RH 13 50 214 152 26758 26758 2.26 0CT 11 660 1364 696 25625 26911 4241 0

EX5a (3,2,2) R DT 169 2016 10953 6739 31351 31798 7200 0.02AG 20 228 1208 776 29657 29657 0.24 0RH 17 50 214 142 41124 41124 7.06 0CT 10 603 1240 629 25625 94901 9829 0

EX6 (3,2,3) U DT 169 2520 14155 8423 43250 43259 7200 0.02AG 19 270 1498 920 43250 43250 0.37 0RH 21 56 262 166 43250 43250 5.57 0CT 9 552 1260 646 35517 Inf. 2811 -

EX7 (3,3,4) U DT 169 3528 18534 10780 68282 68282 19.9 0AG 18 357 1852 1112 68282 68282 0.7 0RH 12 44 203 121 68282 68282 3.12 0CT 12 880 2134 1156 48852 no sol. 7200 -

EX8 (3,3,5) R DT 169 4032 21736 12464 101139 104622 7200 0.22AG 19 432 2310 1360 104375 104375 2.05 0RH 31 336 1271 777 - 151257 17330 0.16

EX9 (4,3,4) U DT 169 4074 24092 13810 87817 87868 7200 0.06AG 19 504 2566 1506 87817 87817 0.71 0RH 25 53 258 151 87817 87817 917 0

EX10 (5,3,4) U DT 169 5880 29650 16840 86505 86582 7200 0.09AG 19 630 3174 1836 86505 86550 3.57 0RH 23 66 317 181 86550 86550 1508 0

6.25

6.25

6.25

6.25

6.25

6.25

0.75

0.75

0.75

0.75

0.75

0.75

0

2

0

4

4

4

0

8

4.25

13

13

13

13

2.571

2.571

2.571

2.571

2.571

2.571

2.571

4.429

4.429

4.429

4.429

4.429

0

1.929

0.679

4

4

4

4

8

8

10

10

10

10

2.915

3

3

3

3

7

7

7

11

11

11

5.5

3

4.25

13

13

13

7

7

7

7

7

7

4

4

4

5.5

5.5

5.857

13

13

13

7

7

7

11

11

11

5.5

4.25

5.5

13

13

13

9

9

8.75

13

13

13

2.429

3.5

3.5

3.5

3.5

3.5

3.5

2.5

2.5

2.143

5.5

5.5

5.5

5.5

2

2

2

2

2

24

24

3.143

24

24

24

24

0 24 48 72 96 120 144 168

M1

M2

M3

S1

S2

S3

S4

Time (h)

P1

P2

P3

P4

P5

0

200

400

600

800

1000

1200

0 24 48 72 96 120 144 168

(ton

)

Time (h)

S1

P1

P2

P3

P4

P5

0

200

400

600

800

0 24 48 72 96 120 144 168

(ton

)

Time (h)

S2

P1

P2

P3

P4

P5

0

200

400

600

800

1000

0 24 48 72 96 120 144 168

(ton

)

Time (h)

S3

P1

P2

P3

P4

P5

0

200

400

600

800

1000

1200

0 24 48 72 96 120 144 168

(ton

)Time (h)

S4

P1

P2

P3

P4

P5

DT difficult to close optimality CT limited to small problems RH generates full schedule relatively fast

Problems with unlimited power availability Aggregate model is rigorous and very powerful Lower degree of degeneracy 1/10 problem size 4 orders magnitude reduction CPUs vs. DT

Rolling‐horizon algorithm finds practical, global optimal solutionsefficiently Considers the whole remaining problem simultaneously

August 17, 2009 14

EX10

Day |T| LB (k€) OBJ (k€)

MO 4 86.55 86.674

5 86.55 86.55

TU 9 86.55 86.573

10 86.55 86.55

WE 12 86.55 86.55

TH 16 86.55 86.55

FR 18 86.55 86.55

SA 21 86.55 86.601

22 86.55 86.55

SU 23 86.55 86.55

Problems with restricted power Discrete‐time formulation is the best

Finds very good solutions (<0.8 %) rapidly (5 min) Aggregate model is a relaxation (underestimates cost)

Rolling horizon may generate suboptimal solutions

August 17, 2009 15

€21575

€18977

Discrete and continuous‐time models IECR 2009, 48, 6701‐6714

Aggregate model and rolling horizon approach To be submitted to CACE Special Issue PSE09

Acknowledgments Financial support from CAPD Center at Carnegie Mellon University, Fundação para a Ciência e Tecnologia, Fundação Calouste Gulbenkian

August 17, 2009 16

Upgrade from Castro et al. (2004) Binary variables

Ni,t task i executed at interval t

Yt,tp,e time period tp of level eactive during interval t

Yt,tdout event point t is demand

point td

New constraints Tasks starting time greater

than energy level lower bound

Must end before time interval

and energy level upper bound

Absolute time of a demand point equal to a due date

August 17, 2009 17

TtRrTsT TC

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tiirtt

c

,max

,,1

tYlbTsEe TPtp

etpttpete

,,,

etrNHYubTscee

ce Ii

tiirTPtp

etpttpeIi i

tiirt ,,)1( ,,,,,max

,,

tYHYtfxTYtfxTDtd

outtdt

TDtd

outtdttdt

TDtd

outtdttd

)1( ,,,Yt+1,tdout=1

tfxtd