Global Optimization for Scheduling Refinery Crude Oil Operations

Ramkumar Karuppiah1, Kevin C. Furman2 and Ignacio E. Grossmann1

(1) Department of Chemical EngineeringCarnegie Mellon University

(2) Corporate Strategic ResearchExxonmobil Research and Engineering

Enterprise-wide Optimization Project

2

MotivationMotivationScheduling and Planning of crude oil is key problem in petrochemical refineries

Large cost savings can be realized with an optimum schedule for the movement of crude oil

VesselsStorageTanks

Charging Tanks

Crude-Distillation Unit

How to coordinate discharge of vessels with loading to storage?How to synchronize charging tanks with crude-oil distillation?

Economic refinery operation

3

Problem StatementProblem StatementCrude Supply Streams Storage

TanksCharging Tanks

Crude Distillation Units

Given:(a) Maximum and minimum inventory levels for a tank(b) Initial total and component inventories in a tank(c) Upper and lower bounds on the fraction of key components in the crude inside a tank(d) Times of arrival of crude oil in the supply streams(e) Amount of crude arriving in the supply streams(f) Fractions of various components in the supply streams(g) Bounds on the flowrates of the streams in the network(h) Time horizon for scheduling

Determine:(i) Inventory levels in the tanks at various points of time(ii) Flow volumes from one unit to another in a certain time interval(iii) Start and end times of the flows in the network

Objective: Minimize Cost

4

Previous WorkPrevious Work

Lee, H.; Pinto, J. M.; Grossmann, I. E.; Park, S. (1996). Lee, H.; Pinto, J. M.; Grossmann, I. E.; Park, S. (1996). Mixed Integer Linear Mixed Integer Linear Programming Model for Refinery ShortProgramming Model for Refinery Short--Term Scheduling of Crude Oil Term Scheduling of Crude Oil Unloading with Inventory Management. Unloading with Inventory Management. I&EC Res.I&EC Res., 35, 1630 , 35, 1630 --1641.1641.

JiaJia, Z.; , Z.; IerapetritouIerapetritou, M. G. (2003). , M. G. (2003). Refinery ShortRefinery Short--Term Scheduling UsingTerm Scheduling UsingContinuous Time Formulation: Crude Oil Operations. Continuous Time Formulation: Crude Oil Operations. I&EC Res.I&EC Res., 42, 3085 , 42, 3085 --3097. 3097.

FurmanFurman, K. C.; , K. C.; JiaJia, Z.; , Z.; IerapetritouIerapetritou, M. G. (2005). , M. G. (2005). A Robust EventA Robust Event--BasedBasedContinuous Time Formulation for Tank Transfer Scheduling. Continuous Time Formulation for Tank Transfer Scheduling. Work in Progress.Work in Progress.

5

Scheduling ModelScheduling Model

Continuous time formulation by Furman et al. (2006)

Based on time events where inputs and outputs for a unit can take place in the same time event

Assumption: No simultaneous input and output for a tankTransfers from one tank to another are denoted by streams

Formulation reduces number of binary variables required in the scheduling model

Minimize cost objectives.t. Tank constraints

Distillation unit (CDU) constraintsSupply stream constraintsVariable bounds

OpOptimization model

(P)(P)

– Total inventory in tank b at end of time event t– Inventory of component j in tank b at end of time

event t– Total flow in stream s in time event t– Flow of component j in stream s in time event t– Start time of flow in stream s in time event t– End time of flow in stream s in time event t– Existence of flow in stream s in time event t

tottsV ,jtsV ,

1,tsT2,tsT

tsw ,

Variables in the modeltot

tbI ,

jtbI ,

Binary Variables

ss

6

Time representationTime representation

Tank ATank A

Tank BTank B1

,ab tT 2,ab tT 1

, 1ab tT +2

, 1ab tT +

-- Instead of using global times t, events t are usedInstead of using global times t, events t are used

-- Instead of timing of individual operations, timing of transfersInstead of timing of individual operations, timing of transfers is usedis used

Event tEvent t Event t+1Event t+1

7

Model ConstraintsModel Constraints

Crude inflow

Crude outflow

Tank constraints

Total Inventory balances

Individual component balancesNon-linear equations containing Bilinearities

Crude Tank

Duration constraintsTo bound the flow of a stream into/from a tank in a particular time event

Simple sequencing constraints

Bounds on component fractions inside a tank

8

Model Constraints (Model Constraints (ContdContd …… ))Distillation unit constraints

Continuous operation constraint

Allocation constraintsAt most one CDU can be charged by a charging tank at a timeAt most one charging tank can charge a CDU at any point of time

Crude-mix demand constraints

Overall mass balances

Component mass balances

Start and end timing constraints

Crude supply stream constraints

Feed

Products

..

.

9

NonNon--convex MINLPconvex MINLP

Minimize a cost objective similar to the one by Jia and Ierapetritou (2003)min total cost = waiting cost for supply streams

+ unloading cost of supply streams+ inventory cost for each tank over scheduling horizon+ setup cost for charging CDUs with different charging tanks

Overall model ≡ (P) Non-convex MINLP

Convex relaxation of (P)(obtained by linearizing non-linear equations in Tank constraints and introducing McCormick estimators (1976) for bilinear terms)

MILP≡ (R)

Objective function:

Scheduling problem modeled as a Mixed Integer Nonlinear Program (MINLP)Discrete variables used to determine which flows should exist and whenModel is non-linear and non-convex

10

Global Optimization of MINLPGlobal Optimization of MINLPLarge-scale non-convex MINLPs such as (P) are very difficult to solve

Commercial global optimization solvers fail to converge to solution in tractable computational times

Special Outer-Approximation algorithm proposed to solve problem to global optimality

Guaranteed to converge to global optimum given certain tolerance between lower and upper bounds

Upper Bound : Feasible solution of (P)

Lower Bound : Obtained by solving a MILP relaxation (R) of the non-convex MINLP model with Lagrangean Decomposition based cuts added to it

NLP fixed 0-1

Master MILP

Upper Bound

Lower Bound

11

Spatial Decomposition of the NetworkSpatial Decomposition of the Network

D1

D2

Crude Supply Streams Storage

TanksCharging Tanks

Crude Distillation Units

Network is split into two decoupled sub-structures D1 and D2Physically interpreted as cutting some pipelines (Here a, b and c)Set of split streams denoted by p {a , b, c }∈

ab

c

12

Decomposition of the modelDecomposition of the modelCreate two copies of the variables pertaining to the split streams and get two sets of duplicate variables :

These duplicate variables are related by equality constraints which are added to (R) to get model (RP):

and

The equations involving the split streams are re-written in terms of the newly created variables

Non-anticipativity constraints in (RP) are multiplied by Lagrange multipliers and transferred to objective function to bring model to a decomposable form which isdecomposed into sub-models (LD1) and (LD2)

Non-anticipativityconstraints

{ }tptptpjtp

tottp wTTVV ,

2,

1,,, ,,,,

{ }1,

1,2,

1,1,

1,,

1,, ,,,, tptptp

jtp

tottp wTTVV

{ }2,

2,2,

2,1,

2,,

2,, ,,,, tptptp

jtp

tottp wTTVV

tpVV tottp

tottp ,02,

,1,

, ∀=−

tpjVV jtp

jtp ,,02,

,1,

, ∀=−

tpTT tptp ,02,1,

1,1, ∀=−

tpTT tptp ,02,2,

1,2, ∀=−

tpww tptp ,02,

1, ∀=−

13

Decomposed SubDecomposed Sub--modelsmodelsmin z1 = waiting cost for supply streams + unloading

cost of supply streams + inventory cost for tanks in D1 over scheduling horizon + setup costs for charging CDUs in D1 with different charging tanks + Optimize to

get solution *1z

Sub-problem involves duplicate variables

{ }1,

1,2,

1,1,

1,,

1,, ,,,, tptptp

jtp

tottp wTTVV

(LD1)

∑∑∑∑∑∑∑∑∑∑∑ ++++p t

tpw

tpp t

tpT

tpp t

tpT

tpj p t

jtp

Vtpj

p t

tottp

Vtottp wTTVV 1

,,1,2

,2,

1,1,

1,

1,,,,

1,,, λλλλλ

s.t. Tank constraintsDistillation unit constraintsSupply stream constraints Variable bounds

min z2 = inventory cost for tanks in D2 over scheduling horizon + setup costs for charging CDUs in D2 with different charging tanks +

Optimize to

get solution *2z

{ }2,

2,2,

2,1,

2,,

2,, ,,,, tptptp

jtp

tottp wTTVV

(LD2)

∑∑∑∑∑∑∑∑∑∑∑ −−−−−p t

tpw

tpp t

tpT

tpp t

tpT

tpj p t

jtp

Vtpj

p t

tottp

Vtottp wTTVV 2

,,2,2

,2,

2,1,

1,

2,,,,

2,,, λλλλλ

s.t. Tank constraintsDistillation unit constraintsVariable bounds

Sub-problem involves duplicate variables

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Cut GenerationCut GenerationUsing solutions and we develop the following cuts :

Add above cuts to (R) to get (R′) which is solved to obtain a valid lower bound onglobal optimum of (P)

Remark: Update Lagrange multipliers and generate more cuts to add to (R)

*1z *

2z

waiting cost for supply streams + unloading cost of supply streams + inventory cost for tanks in D1 over scheduling horizon + setup costs for charging CDUs in D1 with different charging tanks +

≤*1z

∑∑∑∑∑∑∑∑∑∑∑ ++++p t

tpw

tpp t

tpT

tpp t

tpT

tpj p t

jtp

Vtpj

p t

tottp

Vtottp wTTVV ,,

2,

2,

1,

1,,,,,, λλλλλ

inventory cost for tanks in D2 over scheduling horizon + setup costs for charging CDUs in D2 with different charging tanks +

≤*2z

∑∑∑∑∑∑∑∑∑∑∑ −−−−−p t

tpw

tpp t

tpT

tpp t

tpT

tpj p t

jtp

Vtpj

p t

tottp

Vtottp wTTVV ,,

2,

2,

1,

1,,,,,, λλλλλ

Lagrange Multipliers

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Advantages of Cut GenerationAdvantages of Cut Generation

Lower bound obtained is at least as strong as one from conventional Lagrangeandecomposition or LP relaxation of (R)

Alternative decomposition schemes can be used to generate more cuts to add to relaxation (R)

D3

D4Crude Distillation Units

Crude Supply Streams

StorageTanks

Charging Tanks

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Proposed AlgorithmProposed Algorithm

Preprocessing

Bound Contraction optional

Solve Lower BoundingProblem

Solve Upper BoundingProblem

UB – LB <= tolerance ?STOP

Solution = UB

Add Integer Cuts

No Yes

Includes cutting planegeneration

Variant of Outer-Approximation ( Duran and Grossmann, 1986)

17

Illustrative ExampleIllustrative Example3 Supply streams – 6 Storage Tanks – 4 Charging Tanks – 3 Distillation units

Scheduling Horizon 15 hours

Number of supply streams 3

Arrival Time

Incoming Volume of

crude

Fraction of key component

IN1 1 60 0.03

IN2 6 60 0.05

IN3 11 60 0.065

0.075 (0.07 – 0.08)6010 – 90Tank6

0.075 (0.07 – 0.08)3010 – 90Tank5

0.065 (0.06 – 0.07)4010 – 110Tank4

0.05 (0.04 – 0.06)5010 – 110Tank3

0.03 (0.02 – 0.04)1010 – 110Tank2

0.031 (0.025 – 0.038)6010 – 90Tank1

Initial fraction of key component (min –

max)

Initial InventoryCapacity

6Number of Storage Tanks

0.075 (0.071 – 0.08)3080Tank4

0.0633 (0.06 – 0.065)3080Tank3

0.0483 (0.043 – 0.05)3080Tank2

0.0317 (0.03 – 0.035)580Tank1

Initial Fraction of key component (min –

max)

Initial InventoryCapacity

3Number of Charging Tanks

Bounds on flowrates in the streams: Lower Bound : 1, Upper Bound : 40

Number of CDUs : 3Waiting cost for supply streams (Csea): 5Unloading cost for supply streams (Cunload): 7Tank inventory costs (Cinv(b)): storage tanks – 0.05

charging tanks – 0.06Changeover cost for charged oil switch (Cset): 30Demand of mixed oils by CDUs : oil mix 1 60

oil mix 2 60 oil mix 3 60 oil mix 4 60

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Optimal Crude Flow ScheduleOptimal Crude Flow ScheduleGantt chart of optimal schedule

Start time of crude unloading

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

time (hrs) -->

CT3

CT2

CT4

CT1

ST6

ST5

ST4

ST3

ST2

ST150

10

20

30

30

5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

time (hrs) -->

ST1

ST2

ST3

ST4

ST5

ST6

IN1

IN3

IN2

IN1

60

60

555

Crude vessel to storage tanks

Crude transfers between storage and charging tanks

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Optimal Crude Flow ScheduleOptimal Crude Flow Schedule

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

time (hrs) -->

CDU1 being Charged CDU2 being Charged CDU3 being Charged

DU2

DU1

DU3CT4CT3CT4

CT2CT3

CT1CT2

303030

3030

6030

Gantt chart of optimal schedule

Charging schedule for distillation units

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Computational ResultsComputational ResultsOriginal MINLP model (P)

Example Number of Binary Variables

Number of Continuous Variables

Number of Constraints

1 48 300 946

2 42 330 994

3 57 381 1167

383.698928.60383.69383.693

361.636913.92.27359.48351.322

291.93827.70.37282.19281.141

Local optimum(using DICOPT)

Total time taken for

one iterationof algorithm*

(CPUsecs)

Relaxation gap (%)

Upper bound [on solving

(P-NLP) using

BARON ](zP-NLP)

Lower bound [obtained by

solving relaxation (RP) ]

(zRP)

Example

8025.91258100189.19383.6915874.83029600147.24383.693

5873.2310600133.80351.3214481.7931700113.35351.322

758.833430068.45281.141953.3940800-55.24281.141

Time taken to solve (RP)*

(CPUsecs)

No. of nodes

LP relaxation at root node

Solution (zRP)

Time taken to solve (R)* (CPUsecs)

No. of nodes

LP relaxation at root node

Solution(zR)

Solving MILP model (RP)(including proposed cuts)Solving MILP model (R)

Example

* Pentium IV, 2.8 GHz , 512 MB RAM

Solvers : MILP CPLEX 9.0, NLP BARON 7.2.5 (Sahinidis, 1996)

3 Supply streams 3 Supply streams –– 6 Storage tanks 6 Storage tanks –– 4 Charging tanks 4 Charging tanks –– 3 Distillation units3 Distillation units

3 Supply streams 3 Supply streams –– 3 Storage tanks 3 Storage tanks –– 3 Charging tanks 3 Charging tanks –– 2 Distillation units2 Distillation units

3 Supply streams 3 Supply streams –– 3 Storage tanks 3 Storage tanks –– 3 Charging tanks 3 Charging tanks –– 2 Distillation units2 Distillation units

BARON could not guarantee global optimality in more than 10 hours*

21

SummarySummary

New continuous time formulation used to represent the scheduling of crude oilat the front-end of a refinery

Scheduling model is a non-convex MINLP

Special Outer-Approximation algorithm proposed to solve problem to global optimality

Main idea : Generation of cutting planes for speeding up solution of MILP relaxation