Schedulability Analysis of Short-Term Scheduling for Crude Oil Operations in Refinery With Oil...

190 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 8, NO. 1, JANUARY 2011

Schedulability Analysis of Short-Term Scheduling forCrude Oil Operations in Refinery With Oil ResidencyTime and Charging-Tank-Switch-Overlap Constraints

NaiQi Wu, Senior Member, IEEE, Chengbin Chu, Feng Chu, and MengChu Zhou, Fellow, IEEE

Abstract—For short-term scheduling of crude oil operations inrefinery, often crude oil residency time constraint and charging-tank-switch-overlap constraint are ignored by mathematical pro-gramming models to make the problem solvable. Thus, a scheduleobtained by such mathematical programming models is infeasibleand cannot be deployed. To solve this problem, this work studiesthe short-term scheduling problem of crude oil operations in a con-trol theory perspective. The system is modeled by a hybrid Petri netand a short-term schedule is seen as a series of control commands.With this model, schedulability analysis is carried out and schedu-lability conditions are presented. These conditions can be used asconstraints for finding a realizable and optimal refining schedule.Moreover, based on the proposed approach, a detailed schedule canbe easily obtained given a realizable refining schedule. In this way,the complexity for the short-term scheduling problem of crude oiloperations in refinery is greatly reduced and effective techniquesand tools for practical applications can be obtained.

Note to Practitioners—Because of the complexity of short-termscheduling for oil refinery processes, it lacks effective techniquesand software tools for practical applications. This paper presentsa unique methodology for the short-term scheduling problemof crude oil operations. By studying this problem in a controltheory perspective, schedulability conditions are developed byconsidering all constraints, some of which are ignored by a mathe-matical programming method. With the schedulability conditions,a detailed short-term schedule can be easily found. Thus, it can beapplied to solve industrial-size problems.

Index Terms—Discrete-event systems, hybrid systems, oil re-finery, Petri net (PN), scheduling.

Manuscript received January 02, 2010; revised March 08, 2010, May 01,2010, and June 19, 2010; accepted June 28, 2010. Date of publication September13, 2010; date of current version January 07, 2011. This paper was recom-mended for publication by Associate Editor L. Shi and Editor Y. Narahari uponevaluation of the reviewers’ comments. This work was supported in part by theNational Science Foundation (NSF) of China under Grant 60474061, in partby NSF of Guangdong Province under Grant 6104659, in part by le ConseilRegional de la Champagne-Ardenne, France, in part by the National Basic Re-search Program of China under Contract 10CB328100, and in part by the ChangJiang Scholars Program, PRC Ministry of Education.

N. Q. Wu is with the Department of Industrial Engineering, School of Mecha-tronics Engineering, Guangdong University of Technology, Guangzhou 510006,China (e-mail: [email protected]).

C. Chu is with the Laboratoire Génie Industriel, Ecole Centrale Paris,Châtenay-Malabry Cedex 92295, France (e-mail: [email protected]).

F. Chu is with the Laboratoire IBISC EA 4526, Université d’Evry Vald’Essonne, CE 1455 Courcouronnes 91020 Evry Cédex, France (e-mail:[email protected]).

M. C. Zhou is with The Key Laboratory of Embedded System and ServiceComputing, Ministry of Education, Tongji University , Shanghai 200092, China,and also with the Department of Electrical and Computer Engineering, NewJersey Institute of Technology, Newark, NJ 07102-1982 USA ([email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TASE.2010.2059015

I. INTRODUCTION

A S A TYPE of process industry, oil refinery takes an im-portant part in manufacturing. When an oil refinery plant

is well operated it may increase profit by $10 per ton of productor more [13]. Thus, attention has been paid to the developmentof effective techniques for the operation of refinery. With theinstallation of advanced control systems for process control andthe planning technique well-developed by using linear program-ming-based commercial software [1], [16], there is a gap be-tween the process control and planning levels. To effectivelyoperate an oil refinery plant, it is necessary to integrate processcontrol and production planning by short-term scheduling. Ashort-term schedule should provide all the activities in every de-tail for the whole scheduling horizon. It is the detail that makesthe short-term scheduling problem so difficult as pointed out byHonkomp et al. [7]. Besides, it is subject to various constraints,including physical and process oness. If one of them is violated,a short-term schedule becomes infeasible.

However, because of the extreme complexity of a short-termscheduling problem for oil refinery, it lacks effective techniquesand software tools. In fact, this job is still done manually bya planner in oil refinery plants. Thus, it is very meaningful tosearch for effective techniques and tools for short-term sched-uling, and effort has been made to do so in the recent years. Themathematical programming models that are applied to studythe scheduling problem of batch processes [12], [15] are modi-fied for short-term scheduling of oil refinery processes and aresolved by an exact way. In such models, a crucial and difficultproblem is how the time variable should be described. Thereare mainly two categories: discrete-time and continuous-timemodels. In the former, the scheduling horizon is divided into anumber of time intervals with uniform time durations. An event,such as the beginning and ending of an operation, should happenat the boundary of a time interval. The work in [6], [8], [11],[14], [17], [18] belongs to the discrete-time representation do-main. The main drawback with such models is that the uniformtime interval must be small enough so as to obtain acceptableaccuracy. This leads to a large number of binary variables andmakes the problem very difficult to solve.

To overcome the drawback in the discrete-time represen-tation, continuous-time representation is adopted by someresearchers [4], [8]–[10]. Although, with these models, thenumber of discrete variables is reduced significantly, there arenonlinear constraints in it, which makes the problem difficultto solve. Because of the complexity for the requirement of pro-viding detailed activity, to make the problem solvable, in both

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WU et al.: SCHEDULABILITY ANALYSIS OF SHORT-TERM SCHEDULING FOR CRUDE OIL OPERATIONS 191

discrete-time and continuous-time models, often some con-straints, such as the oil residency time (RT) constraint for bothstorage and charging tanks and charging-tank-switch-overlapconstraint in feeding distillers, are ignored. This leads theirsolutions to infeasible ones. In fact, due to the discrete decisionsinvolved, a short-term scheduling problem is inherently com-binatorial in nature, and is known to belong to NP-completeproblems [5]. Thus, although there is significant progress intheory for the short-term scheduling problem of oil refinery,there is a serious gap between theory and applications [24].It is very crucial to search for effective techniques that canovercome the problem of time representation, computationalcomplexity, and solution feasibility so that they can be usablein practice.

Short-term scheduling for crude oil operations is one of themost difficult scheduling problems in operating an oil refineryplant. To reduce the computational complexity, the problem ofshort-term scheduling can be decomposed into two subproblemsin a hierarchical way [26], [27]. At the upper level, it finds therefining schedule to optimize objectives, such as crude oil inven-tory and production rate. For each distiller, a refining scheduledetermines just the production rate, crude oil types, crude oilamount for each type to be processed, and the process sequencewithout involving the detailed schedule. Thus, it is much easierthan the original short-term scheduling problem. At the lowerlevel, it provides the detailed schedule to realize a given re-fining schedule. In doing so, it gives rise to another problem: 1)how can we guarantee that the refining schedule found is realiz-able, because, at the upper level, it does not decide the detailedschedule and 2) how can we guarantee that a detailed schedulecan be found when a refining schedule is realizable. If schedu-lability conditions are known, the upper level subproblem canbe solved by taking them as constraints. In this way, the real-izability of the obtained refining schedule is guaranteed. Thus,if given a refining schedule, a detailed one can be efficientlyobtained with the schedulabilty conditions, the issue of compu-tational complexity can be resolved.

In viewing each operation decision in a short-term scheduleas a control command to the system, Wu et al. [24], [25] suggestto study the schedulability problem of short-term scheduling forcrude oil operations in a control theory perspective. The execu-tion of an operation decision transfers the system from one stateto another. Thus, if a system is schedulable, the initial state mustbe safe and an operation decision should transfer the systemfrom a safe state to another one. Then, a newly reached statecan be seen as an initial state at the time point. This impliesthat all the reached states should be safe. Hence, safeness of thesystem is equivalent to schedulability. With safeness conditionsknown and the system state given, it must not be difficult to findan operation decision such that it transfers the system from thegiven safe state to another one, and the detailed schedule for agiven realization refining schedule can be obtained by creatingoperation decisions one by one in an easy way.

In scheduling crude oil operations, there are various con-straints, including resource and process ones to be explainedin Section II. Most of the existing methods consider only theresource constraints and parts of the process constraints. Theyoften ignore the oil RT and charging-tank-switch-overlap con-straint because of the complexity resulting from them. Espe-

cially, the latter is never considered in the existing methods.Thus, feasibility of the resulting schedule cannot be guaranteed.

Following the suggestion in [24] and [25], short-termscheduling analysis is done and schedulability conditionsare presented in [27] and [29] with resource constraints andparts of process constraints considered. By considering all theconstraints, this work studies the schedulability problem ofshort-term scheduling for crude oil operations. The system ismodeled by a kind of Petri nets (PN), such that the state of thesystem under control can be described. With the PN model,schedulability conditions are presented.

II. SHORT-TERM SCHEDULING PROBLEM

A. Crude Oil Operations in Refinery

For a general refinery process, crude oil is carried to the portby crude oil tankers, where it is unloaded into storage tanks bythe port. It is then transported to charging tanks in the refineryplant through a pipeline. It is next fed into distillers for distil-lation. The middle products from the distillers are then sent toother production units for fractionation and reaction. The prod-ucts after fractionation and reaction are blended to produce thefinal products that are ready for delivery. This paper addressesonly short-term scheduling problems for crude oil operationsfrom a tanker to distillers.

In a refinery, various types of crude oil are processed. Thecomponents are different with different types of crude oil. Crudeoil can be unloaded into only an empty storage tank unless thesame type of crude oil is in it. After filling a storage or chargingtank, it must stay in it for a certain amount of time to separatethe brine, and then can be transported to charging tanks via thepipeline. We call this time delay oil residency time (RT). Whenit is transported through a pipeline to charging tanks in a re-finery plant, different types of crude oil may be mixed to obtainsuitable components for distillation. However, we do not con-sider the mixing here for a mixture can be treated just as a newtype of crude oil. Usually, a pipeline takes tens of kilometerslong with capacity of tens of thousand cubic meters. It is full ofcrude oil all the time and cannot be emptied. Crude oil in thepipeline should be taken as inventory and cannot be neglected.Notice that all the types of oil are transported from storage tanksto charging tanks via the pipeline. To do so, it needs to switchfrom one type of oil to another from time to time. There may bea number of crude oil segments in the pipeline with each seg-ment having different types of crude oil. When it is transported,the pipeline can feed one charging tank at a time.

Each charging tank can hold one type of crude oil at a timeas well. Besides, a tank cannot receive and send oil simultane-ously for both storage and charging tanks. Crude oil can alsobe mixed when it is charged into distillers. Notice that when acharging tank feeds a volume of crude oil into a distiller in timeinterval , this tank must be dedicated to the distiller andcannot be charged in . Thus, if crude oil is mixed in crudeoil feeding, two or more charging tanks should be used to serveone distiller at the same time. In this way, a large number ofcharging tanks are required for distiller feeding. Hence, crude oilis not mixed when feeding distillers due to the limited numberof charging tanks. Thus, only one tank is needed to feed onedistiller at a time unless it switches from one charging tank to


another tank for feeding a distiller. In refining, all distillers workconcurrently and no distiller can be interrupted unless mainte-nance is required.

Besides, in feeding a distiller, there is a charging-tank-switch-overlap constraint explained as follows. Assume thatcharging tank is feeding distiller and the oil in

will be used up at time if the distiller is fed byjust . However, the refining process requires that thesuccessor tank ready to feed should start workingat time that is before . Thus, in time interval with

, both and are feeding distiller. During this time, the remaining oil in is used up

and also a certain amount of crude oil in is fed into. This is called a charging-tank-switch-overlap constraint.

In summary, scheduling crude oil operations includes the fol-lowing resource and process constraints. The former include: 1)the limited number of storage and charging tanks, and the ca-pacity of each tank; 2) the limited flow rate of oil unloading andpipeline; and 3) the volume of various crude oil types availablein storage and charging tanks, and in coming tankers. The latterinclude: 1) a distiller should be kept in working all the time unin-terruptedly; 2) at least one charging tank should be dedicated toa distiller at any time for feeding it; 3) a tank cannot be chargedand discharged at the same time; 4) oil RT constraint; and 5)charging-tank-switch-overlap constraint.

B. Short-Term Scheduling

A crude oil operation process is composed of a series of oper-ations. The questions are when an operation should take place,what should be done, and how it should be done. For each oper-ation to take place, a decision should be made to answer thesequestions. To describe a short-term schedule, we first presentthe definition of an operation decision.

Definition 2.1: An operation decision is defined aswhere crude oil type;

volume of crude oil to be unloaded from a tanker to astorage tank, or transported from a storage tank to a chargingtank, or fed from a charging tank, to a distiller; the sourceplace from which the crude oil is to be delivered; the des-tination to which the crude oil is to be delivered and is atime interval in which and are the start and end time pointsof the operation.

The flow rate in delivering crude oil in can be variable.However, to ease the operations, in the reality, the flow rate for asingle operation is kept as a constant. Thus, given volume andtime interval in an , the flow rate is deter-mined and used. Each is a control command that transfersthe system from a state to another.

There are three types of : crude oil unloading, trans-portation, and feeding, denoted by , , and ,respectively, and their time interval is denoted as , ,and , respectively. For , is a tanker and is astorage tank. For , is a storage tank, is a charging tank,and the transportation must be conducted through a pipeline. For

, is a charging tank and is a distiller. We useto denote the th for feeding distiller during the schedulehorizon. Let be the schedule horizon that often lastsfor a week or ten days and , , and

denote flow rates for a tanker unloading, pipeline

transportation, and distiller feeding decided by . Let bethe index set of the distillers. Given the system state at , i.e.,the inventory of crude oil and state of all the devices, and infor-mation of tanker arrival, the short-term scheduling problem isto find a series of described as follows:

(2.1)

Subject to the resource and process constraints given above,where ,consists of all the for feeding distiller .

C. Safeness and Schedulability

A schedule for crude oil operations is infeasible if one of theconstraints is violated. It follows from the short-term schedulegiven in (2.1) that an operation decision in a schedule is,in fact, a control, in other words, a short-term schedule is com-posed of a series of control commands. These control commandsact on the system sequentially. Surely, the action of antransfers the system from one state to another. Thus, the actionof a short-term schedule will result in a series of state transitions.

In [27], safe state and safeness of the system are de-fined. By safeness, if a short-term schedule

is obtained such that transferthe system from state to state , from toand from to , and all the satesare safe, the schedule is feasible. On the other hand, at any time,if the system is in a safe state, there must exist a short-termschedule so that the system is always in safe states.

Therefore, to make the short-term scheduling problem ofcrude oil operations tractable, the key is the safeness of thesystem just as it does in discrete event systems. For example,when is known and is safe, we can create easily.Then, with one-step look ahead, we can find . If is alsosafe, must be feasible. By doing so, a short-term schedule

, which is feasible, can befound by a computationally efficient algorithm. Thus, it is verymeaningful to make effort in studying the safeness problemof the system to present conditions for the identification of asafe state set. The most difficult part in short-term schedulingfor crude oil operations is to find the and . Thispaper focuses on analyzing the safeness of the states that areaffected by and , i.e., operation decisions fortransportation and feeding.

III. PETRI NET (PN) MODELING

If there exists a short-term schedule such that the system isalways in safe states, the system is schedulable. To check thesafeness of the system, a model that can exactly describe thesystem’s dynamic behavior is necessary, just like that a differen-tial equation is needed to model a continuous system for analysisof dynamic properties. In this section, a hybrid PN is presentedfor the system studied in this paper.

From the short-term schedule defined in the last section, anactivity in a schedule contains its start time, source and desti-nation. These variables are discrete event variables. However,the volume to be delivered in an activity is a continuous vari-able. Thus, crude oil operations belong to a hybrid process, and


Fig. 1. Icons in the model.

a hybrid model suitable for short-term scheduling is needed. Al-though there are PN models for a hybrid process, such as thatin [2], [3], [19], they cannot be applied in oil refining processdirectly because of the special constraints and requirements dis-cussed above. For example, in these studies, a pipeline is treatedjust as a tank, but it is not suitable to do so for our purpose.Here, the segments of different oil types in pipeline should beidentified. Besides, the operation of an oil refinery process isgoverned by the decisions given in the short-term schedule. Inother words, the process is under control. Thus, a model for thisprocess should describe these decisions (commands). By sim-plifying the PN models in [22] and [23], we present a PN modelhere for the purpose of schedulibility analysis. We first presentthe models for devices, e.g., tanks and pipeline. Then, based onthe models for devices, the model for the whole system is de-veloped. A reader is referred as to [20], [21], [30]–[32] for thebasic knowledge of .

The PN model used here is a kind of colored-timeddefined as

, where and are sets of discreteand continuous places; , , and are sets of discrete,timed, and continuous transitions; and are input and outputfunctions; and represent the color sets of the placesin and transitions in ; and is the initial marking. Theicons are shown in Fig. 1.

A. Device Modeling

We model the devices in the system as modules, and basedon them we can develop the PN model for the overall system.The devices are tanks (storage and charging tanks) and pipelineonly. A storage tank is modeled by a PN shown in Fig. 2(a). Atoken in a discrete place is a discrete token and acts just as thatin regular PN. A token in a continuous place represents a type ofmaterials with a real number of volume, called token volume forshort. A discrete transition acts just as that in regular PN. Whena timed transition fires, it delivers a token from a place to anotherafter a constant time delay. When a continuous transition fires,some material of a type represented by a token is delivered froma place to another with a known flow rate determined by an .Whether the token is completely removed from its input placeis determined by its firing duration.

In Fig. 2 (a), continuous places and that can hold atmost one token at a time model the state of a tank. A token inthem represents that there is crude oil in the tank. A token inrepresents that the oil in the tank is not ready for discharging.Oil in a tank is ready for discharging only if has a tokenbut has none. Continuous transitions and model the pro-cesses of filling to and discharging from a tank. The time delayby timed transition models the oil residency-time constraint.

Fig. 2. The PN models for tanks.

Thus, when charging oil to a tank stops, a token representing atype of crude oil must stay in for some time. After the timedelay specified by , a token with a real number of volumemoves from to , and is ready to be discharged from a tank.Because there is only one token in discrete place , only oneof transitions , , and can fire at a time. This model guaran-tees that a tank cannot be filled and discharged at the same time.The self-loop between and together with the inhibitor arc

guarantees the oil residency-time requirement of the oilin the tank before it can be discharged. The volume associatedwith the token in continuous place models the capacity of thetank available at the current marking. Thus, the behavior of thecharging and discharging of a storage tank is exactly described.Thereafter, by , , and (or in short)we mean places , , and the model for tank .

Beside the oil RT constraint, for charging tanks, there is also acharging-tank-switch-overlap constraint. To model it, we shouldfind a way in which a schedule can be found so that it seems nosuch a time overlap to the schedule, but when the schedule is ex-ecuted, the charging-tank-switch-overlap can be implemented.Consider that charging tank is feeding distillerand the successor charging tank is . Assume that thedistiller will be fed only by till time and then by

. By the charging-tank-switch-overlap process, it meansthat the crude oil in to be fed into the distiller must beready at a suitable time and is available forrecharging only at suitable time . Thus, if we prop-erly schedule the oil ready time for and the avail-able recharging time for without considering the timeoverlap, the charging-tank-switch-overlap can be implementedwhen the schedule is executed. The key is the determination of

and . For this purpose, we have the following proposition.Proposition 3.1: Assume that at time tank with

oil is feeding a distiller at rate , and during the charging-tank-switch-overlap interval the oil volume ratio to be fedto the distiller from and is : . Let

be the time point when the oil in is used up ifonly feeds the distiller. Then, and for time interval

can be determined asand , where isa known constant.

The overlap time duration is specified by processrequirement while is calculated. By setting andaccording to Proposition 3.1, if we schedule the system suchthat a distiller is fed by and sequentially without


Fig. 3. The PN model for the pipeline.

overlap, but the oil in is ready for feeding at leasttime units ago and is recharging at

least time units late, then when the scheduleis executed, the charging-tank-switch-overlap constraint can besatisfied. Let be the oil RT, and

. We have the following proposition.Proposition 3.2: Assume that in scheduling the system: 1)

there is no charging-tank-switch-overlap constraint; 2) a distilleris fed first by , then followed by ; and 3) the systemis scheduled such that the RT for is greater than or equalto and is recharged at least time units later after it isemptied. Then, the charging-tank-switch-overlap constraint canbe satisfied when the schedule is executed.

Based on Propositions 3.1 and 3.2, the behavior of a chargingtank is modeled by the PN shown in Fig. 2(b). Compared withthe PN model for storage tank shown in Fig. 2(a), it adds a con-tinuous place , timed transition , inhibitor arc , andself loop between and . Because of the self-loop between

and , cannot fire before the firing of ends. Transitionspecifies the time delay for recharging a just emptied chargingtank. Consider that when a charging tank is charged, it may becharged not to its capacity, so there is a token in . Inhibitorarc disables the firing of when there is a token ineven there is a token in . In this way, it guarantees that a justemptied charging tank can be recharged only after a time delayspecified by . It should also be pointed out that the time asso-ciated with is .

In practice, when the system is scheduled by a planner man-ually, it is the residency-time and charging-tank-switch-overlapconstraints that impose a great burden on the planner. The PNmodel in Fig. 2(b) makes them easy to be checked.

The PN model for the pipeline that can hold three differenttypes of crude oil is shown in Fig. 3, as discussed in [27]. Let

and . Because wedo not consider crude oil mixing here in transportation via thepipeline, only one transition in and one in can fire ata time. Since a transition in and a transition in shouldfire with the same flow rate, we model the pipeline by a macrotransition . Then, place in can be denoted as . When

fires, it implies that one transition in and another in firewith the same rate simultaneously, or crude oil is delivered froma storage tank to the pipeline (a place in ) by a transition in ,and at the same time crude oil in is being delivered into acharging tank by a transition in .

B. PN Model for the Whole System

The PN in Fig. 4 models the whole system with two storageand two charging tanks. For simplicity, we omit the discreteplace and its associated arcs, and the inhibitor arc in a tankPN model. Crude oil in a storage tank is discharged through a

pipeline, transition modeling the pipeline is the dischargingtransition for every storage tank. It is also the charging transitionfor both charging tanks. In Fig. 4,and model the two storage tanks,respectively, and and

the two charging tanks,respectively. Place models a tanker. The number inrepresents that a tanker carries types of crude oil when itarrives. A token can be moved into from only if isempty. Places and at the right end represent two distillers.It should be pointed out that the PN model shown in Fig. 4just describes the structure of the system, but not the detaileddynamics of crude oil flow. From the definition of a short-termschedule, we know that firing of a continuous transition in thePN model must be triggered by an . Thus, the dynamics ofcrude oil flow in the PN model should be governed by the flowrate given in the . This can be modeled by the transitionenabling and firing rules that are defined below.

We use to denote the set of input places of transition(input transitions of place ) and the set of output placesof (output transitions of ). Let denote the volumeof material in at marking .

Definition 3.1: A discrete transition (including discrete onesin ) is said to be enabled at marking if ,and , . When fires, is changed into with

, and , .By Definition 3.1, when some part of oil in in Fig. 3(b)

is discharged, there is still a token in and cannot fire, so atoken in cannot go to . Only if all the oil in is dischargedsuch that is empty, then can fire and a token in can go to

. This guarantees that each continuous place in Fig. 3 can holdone type of crude oil. A timed transition is used only in the PNfor a tank. Its enabling and firing rules are defined as follows.

Definition 3.2: A timed transition t with delay is said tobe enabled at marking if , . When fires attime , is changed into such that: 1) at time ,

, if and , and ifand and 2) at time ,and , and , and ,

, .Notice that when a timed transition fires, there may be a

token in its output place . When the token in the input placemoves into these two tokens merge into one with the volumebeing the sum of their respective values. This definition is usedto describe such a fact that when a tank is neither full nor empty,this tank can still be charged. When it is charged, the volume ofoil should be added. Meanwhile, the time delay associated withthe timed transition guarantees the oil RT constraint, or timedelay to recharge a charging tank. In this way, the behavior of atank is precisely modeled by the PN.

Because there are multiple types of crude oil, it is necessaryto distinguish them. To do so, colors are introduced into the PNmodel. We use to denote the color of crude oil and say thata token in place representing crude oil of type has colorand the number of tokens in p with color at marking isdenoted by , and the volume for this token at marking

is denoted by . When , itimplies that the number of tokens with color in is zero.


Fig. 4. The PN model for the whole system.

If a continuous transition is firing to move crude oil typefrom a place to another, we say that is firing with color . Acontinuous transition must fire with a color. As discussed above,the volume of a token in in Fig. 2(a) or in Fig. 2(b) modelsthe capacity of a tank available. Let denote the color for sucha token. In other words, or in each tank’s PN model hasthe same color as the token in and/or .

Definition 3.3: A continuous transition (including transitionsin and in ) is said to be enabled with color atmarking if: a) or , for anyand b) for some , then or

, or musthold.

or say that the oilin has color that is same as that in . This impliesthat if there is crude oil in a tank, only the same type of crudeoil can be filled into it. However, if a tank is empty, any typeof crude oil can be filled into it. When a continuous transitionis enabled and triggered by an , it can then fire. This firingmust be associated with a flow rate given in the , so the flowof crude oil is governed by the flow rate. The transition firingrules below describe this dynamics. We assume that the firingwith color begins at time , ends at time , ,and the flow rate for firing triggered by an is , then themarking changes as follows.

At , if and

(3.1)

At , if and

(3.2)

At , if , and there is a token with color

(3.3)

If , and there is a token with color

(3.4)

If , and there is a token with color

(3.5)

Because any firing of a continuous transition leads to a crudeoil operation determined by an , expressions (3.1)–(3.5) de-scribe the dynamics of crude oil flow in the PN model for the

system. Up to now, we have completed the PN modeling of thesystem. It should be pointed out that only one of the output tran-sitions of in Fig. 4 and one of input transitions of a place rep-resenting a distiller can fire at a time. Transition can fire withonly one input and one output transition. Further, the startingtime and duration of a transition firing should be determined bya control policy. Thus, to run the above PN model, a controlpolicy is necessary and we will discuss that in Section IV.

IV. SCHEDULABILITY AND SAFENESS ANALYSIS

When a continuous system is described by a differential equa-tion and a control is applied, one can check if the system isstable. Also, with the differential equation model, stability canbe analyzed. Similarly, when an is executed, the system ofcrude oil operations is transferred from one state to another. Bythe PN model developed in the last section, it is easy to check ifthe resulting state is feasible. Just like stability in a continuoussystem, the problem here is if there exists a schedule such thatthe system is always in a safe state. It will be shown that the PNmodel can be used similarly to a differential equation in a con-tinuous system in some sense.

Because the most difficult part of short-term scheduling is todetermine and , here, it is assumed that there isalways enough crude oil in the storage tanks to be processed.Hence, the schedulability is independent of the number ofstorage tanks and their capacity. In fact, the availability ofcrude oil can be easily dealt with when a refining scheduleproblem is solved.

Definition 4.1: A system of crude oil operations with initialstate is said to be schedulable if there exists a feasible short-term schedule for a horizon .

Notice that the key here is the schedule horizon ,because the operation of distillers cannot be interrupted. The ex-istence of a feasible short-term schedule for schedulehorizon does not guarantee that a feasible short-termschedule for schedule horizon , with ,can be found.

Therefore, with Definitions 4.1, to analyze the schedulabilityof crude oil operations is to analyze the safeness of the systemunder Definition 4.1. We use , , , and to denotethe feeding rate to distiller , the maximal oil transportationrate of the pipeline, the oil RT for charging tanks, and the timedelay for a charging tank recharging after being emptied, respec-tively. Further, places , , , and are used for chargingtank in the PN model. The schedulability problem for


system with one distiller is studied in [28]. Here, we focus onthe schedulability analysis for systems with multiple distillers.

A. Systems With Two Distillers

In general, it is desirable to schedule a system such that itsproduction rate is near its maximal one, or ,where is the feeding rate to distiller . The discussionbelow is based on the assumption that the system is operatedunder the maximal production rate. Often, different distiller pro-cesses different crude oil type at the same time and the feedingrate of one distiller is different from that of another.

Property 4.1: Assume that: 1) and; 2) there are four charging tanks – with capacity

, , , and , and andare for , and for ; 3)

; 4) initially, the volume of oil type 1 in andis and , and the volume of oil type 2 inand is and , the oil in andis ready for feeding, and is ready for charging. Then, thesystem is not schedulable. The PN model for this situation isshown in Fig. 12.

Proof: See the Appendix.Property 4.1 shows that a system with four charging tanks for

two distillers is not schedulable. Notice that, in this case, it canbe shown that if there is no charging-tank-switch-overlap con-straint the system is schedulable. It is this constraint that makesthe system not be schedulable. In Property 4.1, we assume that

, a special case. If , the situation is muchmore complicated and is more difficult to schedule. In that case,it is easy to show that the system is not schedulable. It is clearthat if the number of charging tanks is less than four, the systemmust not be schedulable. Now, we consider the situation thatthere are five charging tanks for two distillers. The PN modelfor this situation is shown in Fig. 13.

Property 4.2: Assume that: 1) , ,and ; 2) there are five charging tanks –with capacity , , , , and

; 3) ; 4) initially, the volumeof oil type 1 in and is , and thevolume of oil type 2 in is , andare empty, the oil in and is ready for feeding, and

is ready for charging; and 5). Then, the system is schedulable.Proof: See the Appendix.

To make the system schedulable, the key is to schedule itsuch that is emptied before by some time andthe difference between and is not too large. It is easyto verify that when the capacity of the charging tanks is greaterthan that given in Property 4.2 and initially there is more oil inthe charging tanks, then it is easier to schedule the system. Gen-erally, lasts for 6–7 h and for 2–3 h. Thus, if ,from , we require .This is not difficult to be satisfied in practice. By

, the system is operated under the maximal produc-tion rate. It follows from the proof of the property that when

it is easier to make the oil RT constraintsatisfied than the situation . Thus, when

, if the system is schedulable, it must beso when .

From the proof of Property 4.2, we also know that only twocharging tanks can be used for distiller , so if it needs toswitch from processing one type of oil to another and there is asmall amount of oil for the first type, then to make the systemschedulable, such a small amount of oil should be processed inmixing with other type of oil. Now, we discuss the situation thatthere are six charging tanks. The PN model for such a system isshown in Fig. 14.

Property 4.3: Assume that: 1) , ,and ; 2) there are six charging tanks –with capacity , , , ,

, and ; 3) ; 4) initially,the volume of oil type 1 in and is ,

, and the volume of oil type 2 in andis , , and are empty, the oilin and is ready for feeding, and is readyfor charging; and 5) . Then, thesystem is schedulable.

Proof: See the Appendix.It is clear that if, initially, the volume of oil in the charging

tanks , , , and is , , ,and with , respectively, the system is schedulable.In this case, the state of the system is safe, but is away fromthe safeness boundary given in Property 4.3. Thus, in this case,it is easier to obtain a feasible schedule, and one needs onlyto charge a tank with a right type of oil when it is ready forcharging. With , if , itrequires that . When charging tanks ,

, and are charged with volume , it requiresthat . Often, the capacity of a chargingtank is a number of times of . Thus, when ,

is easy to satisfy in practice. In fact, when a tankis charged, it is charged to its capacity, and thus, often .

Next, we discuss the schedulability problem when a distillerneeds to switch from processing one type of crude oil to anothertype and there is a small amount of oil for the former type. Wehave the following proposition.

Property 4.4: Assume that: 1) , ,and ; 2) there are six charging tanks –with capacity , , , ,

, and ; 3) ; 4) ini-tially, the volume of oil type 1 in and is

, and the volume of oil type 2 in is ,, , and are empty, the oil in andis ready for feeding, and and are ready

for charging; 5) the volume of crude oil of type 2 remaining instorage tanks is , and after processing this amount ofoil type 2, distiller should switch to process oil type 3; and6) . Then, the system is schedu-lable and the volume of oil type 2 can be used up withoutbeing mixed with another oil type.

Proof: See the Appendix.In comparison with the result of [28, Theorem 4.3], after

switching from processing one type of oil to another, the amountof oil in the charging tanks is decreased when there are threecharging tanks for one distiller. However, here, the amount of


Fig. 5. PN model for a system with six charging tanks and three distillers.

oil in the charging tanks is not decreased, although there arethree charging tanks for each distiller.

B. Systems With More Than Two Distillers

After the discussion for systems with two distillers, we candiscuss the safeness conditions for systems with more than twodistillers so as to obtain the results for general situations.

Consider systems with six charging tanks and three distillersand the following state: 1) and

; 2) there are six charging tanks – withcapacity , , and and arefor , and for , and for

; 3) ; 4) initially, the volume of oil type 1in and is and , the volume ofoil type 2 in and is and , andthe volume of oil type 3 in and is and

, respectively. The oil in , , and isready for feeding, and is ready for charging.

Observing the PN model for this system shown in Fig. 5, atthe initial marking (time ), we can fire and withcolor in and to feed ,respectively, and fire with color and with color tofeed and , respectively. Meanwhile, we fire with color

to charge . At time , marking is reachedsuch that , ,

, and . At thismarking, we fire with color , with , and with tofeed , , and , respectively. At the same time, we fire with

to charge . At time , marking is reachedsuch that , ,

, , and. At this marking, we continue

firing with color , with , and with tofeed , , and , respectively, fire with to charge

. Then, at time , marking is reachedsuch that , ,

, , and. At this marking, no charging

tank is ready for charging, cannot fire until time unitslater. In this way, the amount of crude oil in the charging tankswill be decreased step by step, and finally, the system willreach an infeasible state. Notice that, here, we assume that

that is a special and easy case. This

shows that it is not schedulable if there are six charging tanksfor three distillers. However, we have the following result.

Theorem 4.1: Assume that: 1) ,, , and ;

2) there are charging tanks with ca-pacity ,

and ,; 3) ;

4) initially, the volume of oil type 1 in andis , the volume of oil type 2 inis the volume of oil type in is

the volume of oil type in is, the other tanks are empty, and the

oil in is readyfor feeding, and is ready for charging; and 5)

. Then, the system isschedulable.

Proof: See the Appendix.Theorem 4.1 extends Property 4.2 to a general situation. No-

tice that when , Theorem 4.1 does not hold, for in thiscase there is not enough oil in the charging tanks to make thestate safe. Although if there are charging tanks for dis-tillers with , the systemis not schedulable, the system becomes schedulable with onemore charging tank added. Of course, the more oil is there inthe charging tanks, the easier is the system to be scheduled.When there is more oil in the charging tanks, constraint

is eased to some extent. For the sit-uation that a distiller needs to switch from processing one typeof oil to another, we have the following theorem.

Theorem 4.2: Assume that: 1) ,, and

; 2) there are charging tankswith capacity ,

and, , ; 3)

; 4) initially, the volume of oil type 1 inand is , the volume of oil

type 2 in is the volume of oil typein is the volume of oil type in

is , the other tanks are empty, the oilin and is ready forfeeding, and and are ready for charging;5) the volume of crude oil of type remaining in storage tanksis , and after processing this amount of oil type , dis-tiller should switch to process oil type ; and 6)

. Then, the system is schedulableand the volume of oil type can be used up without beingmixed with another oil type.

Proof: See Appendix.In Theorems 4.1 and 4.2, the amount of oil in the charging

tanks for a safe state depends on , the number of distillers.However, there are no more than four distillers for most of re-fineries and the capacity of a charging tank is greater than ,in general.

Theorem 4.3: Assume that: 1) ,, and


TABLE ITHE INITIAL STATE FOR THE STORAGE TANKS

TABLE IITHE INITIAL STATE FOR THE CHARGING TANKS

; 2) there are charging tanks with capacity

; 3); 4) initially, the volume of oil type 1 in and

is and the volume of oil typein and is and ,

, the other tanks are empty, the oil in ,, , is ready for feeding, and is

ready for charging; and 5) .Then, the system is schedulable.

Proof: See the Appendix.It should be pointed that the result in Theorem 4.3 is in-

dependent of , the number of distillers. Besides, for a safestate, it requires less oil in the charging tanks and the amountof oil in the charging tanks is independent of too. Thus,when each time a charging tank is charged, it is charged to ca-pacity, all the constraints can be easily satisfied. For

, if and there are four dis-

tillers, then, it requires . It alwaysholds, for is the largest one. Furthermore, if each time acharging tank for distiller is charged to , it requires

, which must hold. Notice that,in the proof, we first charge , not . If we first charge

Fig. 6. The refining schedule for the case study.

Fig. 7. The detailed charging tank schedule for feeding distillers.

, and assume that: 1) there are four distillers, or ;2) ; and 3) each time a tank for is charged to ,it requires , which can be easilysatisfied. This implies that the charging tanks can be rechargedin any order. Hence, when there are charging tanks fordistillers and each time a charging tank is charged as much aspossible, the state must be safe and a detailed schedule can beeasily obtained by using the approach presented here.

We assign three charging tanks for each distiller. Unlike theresults obtained in Theorems 4.1 and 4.2, with three chargingtanks for each distiller, a safe state is independent of the numberof distillers. Thus, it is much easier to obtain a detailed schedule.If there are more charging tanks, there is no problem for ob-taining a detailed schedule. Hence, in general, it requires morethan charging tanks for a refinery with distillers. Fortu-nately, for most refineries, that is the case.

With the safeness conditions given above, at any safe state fora system, one can create an operation decision such that when itis executed, the system is transferred into another safe one. Thisimplies that if the initial state is known and a refining schedule isgiven, we can check if the schedule is realizable. Furthermore,if so, a detailed schedule can be found by creating the operationdecisions one by one by following the proofs of the theorems.In this way, the starting time, amount of oil and its type to be de-livered, and flow rate for an operation decision can be easily de-termined, and the time can be any continuous time point. Thus,it is very computationally efficient to find a detailed schedule.

V. INDUSTRIAL CASE STUDY

This section presents a case study to show the power of theapproach proposed in this paper. This case problem arises froma practical application scenario in China. The refinery has threedistillers , , and with 12 storage tanks and 12charging tanks. Three times each month, a short-term schedule


Fig. 8. The detailed tanker unloading and storage tank filling schedule.

Fig. 9. The detailed storage tank discharging schedule.

Fig. 10. The detailed charging tank filling schedule.

should be created for the next 10 days. The initial state for the 12storage tanks and 12 charging tanks for the case study is shownin Tables I and II. There is 12,000 tons of crude oil type #7. Atanker with 65,000 tons of crude oil type #5 will arrive at time30 h and another tanker with 55,000 tons of crude oil type #2,60,000 tons of crude oil type #8, and 135,000 tons of crude oiltype #3 will arrive at time 65 h. The crude oil unloading ratesfor both tankers are 2950 tons/h and 3400 tons/h, respectively.The maximal flow rate of the pipeline is tons/h.A refining schedule is given as shown in Fig. 6 for the next tendays with refining rates for distillers , , and being333.3, 291.7, and 625 tons/h, respectively, where #1 denotescrude oil type #1. Both oil RT for charging tanks and chargingtank switch overlap time are 4 h with : being 1: 1. Thus,we have and h, respectively.

Observing the refining schedule, we know that pro-cesses 38,000 tons of crude oil type #1 first and then 42,000tons of crude oil type #5. 38,000 tons of crude oil type #1 isalready in the charging tanks, and 16,000 tons of crude oil type#5 is in the charging tanks, another 16,000 tons of crude oiltype #5 is in the storage tanks. It takes more than 200 h for

to process 70,000 tons of oil and crude oil type #5 in thefirst arriving tanker is available for use after time 80 h. Thus,the oil to be processed by is available. For and ,the crude oil to be processed is already in the charging tanks,pipeline, and storage tanks. Thus, the basic schedulabilitycondition is satisfied.

Notice that charging tanks C121 and C128 hold crude oil type#2, but that crude oil in these charging tanks is not to be re-fined. This implies that these two charging tanks are just used asstorage tanks, but not as charging tanks for the next 10 days. Ex-amining the refining schedule and the initial sate of the charging

tanks, we find that: 1) processes 38,000 tons of oil type #1held in charging tanks C122 and C129 first and then switchesto process crude oil type #5. This implies that charging tanksC122, C129, and C182 are usable; 2) first processes thecrude oil held in tanks C124 and C181, then switches to processcrude oil type #7, or charging tanks C124 and C181 are usable;3) process the oil held in tank C115 first and then switchesto process oil type #4, or charging tanks C115 and C127 are us-able; 4) charging tanks C116, C125, and C180 are empty. Theyare free to use; 5) the total production rate is equal to the max-imal transportation rate of the pipeline. Thus, there are totallyten charging tanks are usable for three distillers. Initially, C125is ready for charging and it takes more than 10 h to be chargedto capacity, or Condition 5) in Theorem 4.3 is satisfied. Thus,the conditions for the number of charging tanks given in The-orem 4.3 are satisfied. Furthermore, the volume of crude oil foreach type is more than that required by the conditions given inTheorem 4.3. Therefore, the system is schedulable.

With the above observation and following the scheduling ideapresented in this paper, we find the detailed schedule for thiscase study, as shown in Figs. 7–10, where #1 stands for crudeoil type #1 and 4.0 means tons of crudeoil. In fact, to obtain the detailed schedule, we need only to filla charging tank with an appropriate crude oil type and the tankis filled to its capacity. Meanwhile, we can do the tanker un-loading, storage tank filling, and storage tank discharging sim-ilarly. This is consistent with the analysis given in the last sec-tion. The detailed schedule is created in a recursive way byfinding the one by one such that the system is always in asafe state.

With the detailed schedule obtained, we can find the imple-mentation such that both oil RT constraint and charging-tank-


Fig. 11. The implementation of the schedule.

switch-overlap constraint are satisfied. This implementation isshown in Fig. 11.

From the case problem, when the conditions are satisfied forcreating a detailed schedule, the key is to fill the charging tankswith the proper amount and type of crude oil for each distillersuch that the system is kept in a safe state. To do so, everytime when an is completed one needs to select a dis-tiller for which there is the least crude oil remaining in its cor-responding charging tanks. Then, one creates another tofill a charging tank for this distiller with appropriate type andamount of crude oil. This can be easily done and its computa-tional complexity is independent of the constraints considered,the number of charging tanks, the number of distillers, and thelength of the scheduling horizon.

If a mathematical programming model is used, the size ofthe problem is greatly dependent on the constraints consideredand the number of charging tanks and distillers. In [17], it ispointed out that, to guarantee the solution accuracy, a time slotof 15 min should be used when a mixed integer programmingmodel with discrete time representation is used. For the dis-cussed case problem, it has 960 time slots, 24 tanks, six types ofcrude oil, and at least three constraints for each tank. Thus, bydoing so, if the discussed case problem is formulated as a mixedinteger programming problem, it has more than

binary variables! It is difficult to be solved in anexact way by a mixed integer programming model. However,by using the proposed approach, a set of good initial solutionscan be obtained, which might lead to much faster global conver-gence using mathematical programming techniques. The issuesneed to be further studied.

With the schedulability conditions as constraints, a realizablerefining one can be found to optimize the operations. Such arefining scheduling problem can be solved by using linear pro-gramming models. Notice that the objectives in scheduling thecrude oil operations are associated with the refining scheduleonly, but not the detailed schedule. Thus, by doing the schedu-lability analysis, it makes the short-term scheduling problemtractable without undermining the optimality.

VI. CONCLUSION

It is a great challenge to search for efficient tools for short-term scheduling for oil refinery processes. One of the most diffi-cult subproblems is the short-term scheduling problem for crudeoil operations. Because of its great complexity, it lacks efficienttechniques and software tools and this job is still done manually

by planners. Thus, in recent years, research has been conductedin searching for such techniques and tools. In most studies inthis field, the problem is modeled by mathematical program-ming and solved in an exact way. Although, some advancementhas been made, the existing methods are not applicable in prac-tice due to the computational complexity and the ignorance ofsome constraints to ease its solution.

To solve this problem, our previous work [25] suggestsstudying this problem in control theory perspective. Followingthis idea, this paper conducts the schedulability analysis intaking account of all the detailed constraints including oil RTand charging-tank-switch-overlap that are not considered in theexisting studies. It is shown that the problem can be equivalentto another problem that has no charging-tank-switch-overlapconstraint. Thereby the analysis can be simplified. Based on theequivalence, a hybrid PN is developed to model the short-termscheduling processes. With it, schedulability conditions arepresented. By using the presented conditions, it can check if agiven refining schedule is realizable. If so, a detailed short-termschedule can be found easily. Furthermore, the proposedconditions can be used as constraints for finding a realizablerefining schedule. In this way, the problem can be solved in ahierarchical way. At the upper level, it finds a realizable refiningschedule, and then, at the lower level, the detailed scheduleis obtained. In this way, the problem is greatly simplified andat the same time the feasibility is guaranteed. Therefore, it isapplicable in practice.

It is sure that schedulability conditions can be used as con-straints to find a realizable and optimal refining schedule. How-ever, this paper does not answer how to do that, and this is ourfuture work. Up to now, in studying the short-term schedulingproblem of refinery, the objective is to minimize the costs causedby tanker waiting, tanker unloading delay, crude oil inventory,and so on. Essentially, such costs are related only to a refiningschedule, but independent of a detailed schedule. However, incrude oil operations, a great deal of cost is from frequent oiltransportation switch from one type of oil to another, frequentswitch from one charging tank to another in feeding a distiller,non-proper oil mixing, and so on as pointed out in [24]. Suchcost is related to the detailed schedule. It is shown that when thesystem is not at the safeness boundary, we have more freedom(more choices), or such cost can also be minimized. It is our fu-ture work to do so.

APPENDIX

The Proof of Property 4.1: Consider the PN model forthe system, as shown in Fig. 12. At the initial marking(time ), we have

, ,and . At this marking, we can fire only

with color and with color to feed and ,respectively, and, at the same time, fire with color tocharge . At time , marking is reached suchthat ,

, , and. At this marking, we can fire with

color to feed , and with color to feed ,and continues its firing to charge , respectively. At


Fig. 12. PN model for a system with four charging tanks and two distillers.

Fig. 13. PN model for a system with five charging tanks and two distillers.

time , marking is reached such that,

, , and. At this marking, and con-

tinue their firing to feed and , respectively, and atthe same time fire with color to charge . Then,at time , marking is reached such that

,, , and

. At this marking, can continueits firing to feed and can fire with color to feed .Thus, it is a feasible marking. However, if continues its firingto charge , then time units later, will be empty andthere is no transition enabled for feeding . At the same time,

, and cannot fire to charge . Thus, the crudeoil in the charging tanks will be decreased. In this way, thecrude oil in the charging tanks will become less and less, untilthere is no crude oil to feed the distillers. In other words, thesystem is not schedulable.

The Proof of Property 4.2: With the PN model shownin Fig. 13, the system can be scheduled as follows. Ini-tially, at marking (time ), we have

,, , and other

places are empty. At this marking, we fire and withcolors and to feed and , respectively. Noticethat and

Fig. 14. PN model for a system with six charging tanks and two distillers.

. Thus, we can firewith color during time interval

to charge and with color during time intervalto charge , for ,

and at time , musthold. At time , marking is reached such that

,, ,

, , and. At this marking, after

time units, must be empty and the token in movesinto . Hence, we fire with color to feed , firewith color to charge , and fire with color in timeinterval and fire with color in time interval

to feed . Then, at time , markingis reached such that

, ,and . This marking is equivalent to theinitial marking . Thus, the above process can be repeated ina cyclic way, or it is schedulable.

The Proof of Property 4.3: With the PN modelshown in Fig. 14, we can schedule the system as fol-lows. At the initial marking (time ), we have

, , ,, and . Because

,. Meanwhile, we have

. This implies thatafter is charged with volume of oil, is ready forcharging. Hence, at marking , we can fire with colorand with color to feed and , respectively. At the sametime, with in mind, we fire with color

in time interval to charge , andthen with color in time interval tocharge . At time , marking is reached suchthat

, ,, ,

and . At this marking, we can firewith color and with color to feed and ,

respectively, and continue firing with color to charge .


Then, at time , marking is reached such that

, , ,, and . This

marking is equivalent to the initial marking . The opera-tion can be continued in a cycle way. Hence, the system isschedulable.

The Proof of Property 4.4: With PN model shownin Fig. 14, the system can be scheduled as follows. Atthe initial marking (time ), we have

, , and. Notice that

and imply that .Hence, at this marking, we fire with color and withcolor to feed and , respectively. Meanwhile, we firewith color in time interval to charge ,with color in time intervalto charge , and to charge

. At time , marking is reached suchthat

, ,, ,

, , and. At this marking, we fire

with color to feed , fire with color in timeinterval to feed , fire with color in timeinterval to feed , fire withcolor in time interval to feed

. Meanwhile, we fire with color to charge . Then,at time , marking is reached such that

, ,and . This marking is equivalent to theinitial marking, or the system is schedulable.

The Proof of Theorem 4.1: Let a charging tankbe modeled by and distiller

be modeled by in the PN. Then, this system canbe scheduled as follows. Initially, at marking (time

), we have , ,, ,

, ,, ,

and the other places are empty. It follows fromthat and

.

This implies that .

With , we have

. This implies that it needs at least timeunits to charge with volume and when ischarged, is ready for charging. Thus, at this marking,we fire to feed , and to feed , ,respectively. At the same time, we fire in the following order:with color to charge with volume withcolor to charge with volumewith color to charge with volume

with color to charge with volume with colorto charge with volume . Thus, at

time , marking is reached suchthat , ,

,and , , and

. At this marking, we fire tofeed , and to feed , , respectively,and to feed , meanwhile fire with color to charge

. Then, at time , marking is reached such that, ,

, ,, , and

. This marking isequivalent to marking . Hence, the previous process canbe repeated in a cycle way and a feasible schedule can beobtained.

The Proof of Theorem 4.2: Initially, at marking(time ), we have ,

, ,, ,

, ,, and the other places

are empty. Similar to the proof of Theorem 4.1, at this marking,we fire to feed , and to feed , ,respectively. At the same time, we fire in the followingorder: with color to charge with volume ,and then with color to charge with volume

with color to charge withvolume with color to chargewith volume with color to charge withvolume with color to charge with volume

. Thus, at time ,marking is reached such that ,

, ,and ,

, , and. At this marking,

we fire with color to feed , and with colorto feed , , respectively, withcolor and volume to feed and with color

and volume to feed . Meanwhile we firewith color to charge . Then, at time ,

marking is reached such that, , ,

,, , ,

and . This marking isequivalent to marking in Theorem 4.1. It follows fromTheorem 4.1 that the system is schedulable.

The Proof of Theorem 4.3: Let a charging tankbe modeled by and

distiller be modeled by in the PN, then, thissystem can be scheduled as follows. Initially, at marking

(time ), we have ,, ,

, , ,,


, and other places are empty. At this marking,in time interval , we fire with color to feed ,

with color to feed , , respectively; and intime interval , we fire with color to feed

, with color to feed , , respectively. Atthe same time, we fire with the following order: with colorto charge with volume with color to charge

with volume with color to chargewith volume . Notice that by assumption 5), we

have

.This implies that when is charged with volume

, places , , are ready for charging.Thus, the above transition firing is feasible. Then, attime , marking is reached such that

,, , ,

, ,, . This

marking is equivalent to the initial marking . Hence, theprevious process can be repeated in a cyclic way and a feasibleschedule can be obtained.

REFERENCES

[1] Bechtel, PIMS (Process Industry Modeling System) User’s Manual.Houston, TX: Bechtel Corp, 1993.

[2] H. Chen and H.-M. Hanisch, “Analysis of hybrid system based on hy-brid net condition/event system model,” Discrete Event Dynamic Sys-tems: Theory and Applications, vol. 11, pp. 163–185, 2001.

[3] R. David and H. Alla, “On hybrid Petri nets,” Discrete Event DynamicSystems: Theory and Applications, vol. 11, pp. 9–40, 2001.

[4] C. C. Diego and J. Cerda, “Dynamic scheduling of multiproductpipelines with multiple delivery due dates,” Comput. Chem. Eng., vol.32, pp. 728–753, 2008.

[5] C. A. Floudas and X. Lin, “Continuous-time versus discrete-time ap-proaches for scheduling of chemical processes: A review,” Comput.Chem. Eng., vol. 28, pp. 2109–2129, 2004.

[6] K. Glismann and G. Gruhn, “Short-term scheduling and recipe op-timization of blending processes,” Comput. Chem. Eng., vol. 25, pp.627–634, 2001.

[7] S. J. Honkomp, S. Lombardo, O. Rosen, and J. F. Pekny, “The curse ofreality—Why process scheduling optimization problems are difficultin practice,” Comput. Chem. Eng., vol. 24, pp. 323–328, 2000.

[8] Z. Jia, M. Ierapetritou, and J. D. Kelly, “Refinery short-term sched-uling using continuous time formation: Crude oil operations,” Ind. Eng.Chem. Res., vol. 42, pp. 3085–3097, 2003.

[9] Z. Jia and M. Ierapetritou, “Efficient short-term scheduling of refineryoperations based on a continuous time formulation,” Comput. Chem.Eng., vol. 28, pp. 1001–1019, 2004.

[10] R. Karuppiah, K. C. Furmanb, and I. E. Grossmann, “Global optimiza-tion for scheduling refinery crude oil operations,” Comput. Chem. Eng.,vol. 32, pp. 2745–2766, 2008.

[11] H. Lee, J. M. Pinto, I. E. Grossmann, and S. Park, “Mixed integer linearprogramming model for refinery short-term scheduling of crude oil un-loading with inventory management,” Ind. Eng. Chem. Res., vol. 35, pp.1630–1641, 1996.

[12] C. A. Mendez, J. Cerda, I. E. Grossmann, I. Harjunkoski, and M. Fahl,“State-of-the-art review of optimization methods for short-term sched-uling of batch processes,” Comput. Chem. Eng., vol. 30, pp. 913–946,2006.

[13] L. F. L. Moro, “Process technology in the petroleum refining in-dustry—Current situation and future trends,” Comput. Chem. Eng.,vol. 27, pp. 1303–1305, 2003.

[14] L. F. L. Moro and J. M. Pinto, “A mixed-integer model for short-termcrude oil scheduling,” in AIChE Annual Meeting, Miami Beach, FL,1998, paper 241c.

[15] S. Mouret, I. E. Grossmann, and P. Pestiaux, “Tightening the linearrelaxation of a mixed integer nonlinear program using constraint pro-gramming,” in Integration of AI and OR Techniques in Constraint Pro-gramming for Combinatorial Optimization Problems, W. van Hoeveand J. N. Hooker, Eds. New York: Springer, 2009, pp. 208–222.

[16] R. Pelham and C. Pharris, “Refinery operation and control: A futurevision,” Hydrocarbon Process., vol. 75, no. 7, pp. 89–94, 1996.

[17] J. M. Pinto, M. Joly, and L. F. L. Moro, “Planning and schedulingmodels for refinery operations,” Comput. Chem. Eng., vol. 24, pp.2259–2276, 2000.

[18] N. Shah, “Mathematical programming techniques for crude oil sched-uling,” Comput. Chem. Eng., vol. 20, pp. S1227–S1232, 1996.

[19] M. Silva and L. Recalde, “Petri nets and integrality relaxations: A viewof continuous Petri net models,” IEEE Trans. Syst., Man, Cybern., PartC, vol. 32, no. 4, pp. 317–327, 2002.

[20] J. Wang, Time Petri Nets: Theory and Application. Boston, MT:Kluwer Academic Publishers, 1998.

[21] J. Wang, “Charging information collection modeling and analysis ofGPRS networks,” IEEE Trans. Syst., Man, Cybern., Part C, vol. 37,no. 4, pp. 473–481, Jul. 2007.

[22] N. Q. Wu, L. P. Bai, and C. B. Chu, “Hybrid Petri net modeling forrefinery process,” in Proc. 2004 IEEE Conf. Syst., Man, Cybern., 2004,pp. 1734–1739.

[23] N. Q. Wu, L. P. Bai, and C. B. Chu, “Modeling and conflict detection ofcrude-oil operations for refinery process based on controlled-colored-timed Petri net,” IEEE Trans. Syst., Man, Cybern., Part C, vol. 37, no.4, pp. 461–472, Jul. 2007.

[24] N. Q. Wu, M. C. Zhou, and F. Chu, “Short-term scheduling for refineryprocess: Bridging the gap between theory and applications,” Int. J. In-tell. Control Syst., vol. 10, no. 2, pp. 162–174, Jun. 2005.

[25] N. Q. Wu, M. C. Zhou, F. Chu, and Y. M. Qian, “Issues on short-term scheduling of oil refinery,” in Proc. 2006 IEEE Conf. Syst., Man,Cybern., Taiwan, Oct. 2006, pp. 2920–2925.

[26] N. Q. Wu, M. C. Zhou, and F. Chu, “A Petri net based heuristic algo-rithm for realizability of target refining schedule for oil refinery,” IEEETrans. Autom. Sci. Eng., vol. 5, no. 4, pp. 661–676, Oct. 2008.

[27] N. Q. Wu, F. Chu, C. B. Chu, and M. C. Zhou, “Short-term schedu-lability analysis of crude oil operations in refinery with oil residencytime constraint using Petri net,” IEEE Trans. Syst., Man, Cybern., PartC, vol. 38, no. 6, pp. 765–778, Nov. 2008.

[28] N. Q. Wu, C. B. Chu, F. Chu, and M. C. Zhou, “Short-term schedula-bility analysis of crude oil operations in refinery with hybrid Petri net,”in Proc. 2008 IEEE Conf. Syst., Man, Cybern., Singapore, Oct. 12–15,2008, pp. 1916–1921.

[29] N. Q. Wu, F. Chu, C. B. Chu, and M. C. Zhou, “Short-term schedula-bility analysis of multiple distiller crude oil operations in refinery withoil residency time constraint,” IEEE Trans. Syst., Man, Cybern., PartC, vol. 39, no. 1, pp. 1–16, Jan. 2009.

[30] N. Q. Wu and M. C. Zhou, System Modeling and Control With Re-source-Oriented Petri Nets. New York: CRC Press, Taylor & FrancisGroup, 2009.

[31] M. C. Zhou and K. Venkatesh, Modeling, Smulation and Control ofFlexible Manufacturing Systems: A Petri Net Approach. Singapore:World Scientific, 1998.

[32] B. Hruz and M. C. Zhou, Modeling and Control of Discrete Event Dy-namic Systems. London, U.K.: Springer, 2007.

NaiQi Wu (M’04–SM’05) received the M.S. andPh.D. degrees in systems engineering both fromXi’an Jiaotong University, Xi’an, China, in 1985 and1988, respectively.

From 1988 to 1995, he was with the ChineseAcademy of Sciences, Shenyang Institute of Au-tomation, Shenyang, China, and from 1995 to1998, with Shantou University, Shantou, China.From 1991 to 1992, he was a Visiting Scholarat the School of Industrial Engineering, PurdueUniversity, West Lafayette, IN. In 1999, 2004, and

2007–2009, he was a Visiting Professor with the Department of IndustrialEngineering, Arizona State University, Tempe, the Department of Electricaland Computer Engineering, New Jersey Institute of Technology, Newark,and the Industrial Systems Engineering Department, Industrial Systems Op-timization Laboratory, University of Technology of Troyes, Troyes, France,respectively. He is currently a Professor of Industrial and Systems Engineeringwith the Department of Industrial Engineering, School of MechatronicsEngineering, Guangdong University of Technology, Guangzhou, China. He isthe author or coauthor of many papers published in the International Journalof Production Research, the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND


CYBERNETICS, the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION,the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING,IEEE/ASME TRANSACTIONS ON MECHATRONICS, the IEEE TRANSACTIONS ON

SEMICONDUCTOR MANUFACTURING, the Journal of Intelligent Manufacturing,Production Planning and Control, Computers & Chemical Engineering, andRobotics and Computer Integrated Manufacturing. His research interestsinclude production planning and scheduling, manufacturing system modelingand control, discrete-event systems, Petri net theory and applications, logisticsand transportation, and information assurance.

Dr. Wu is an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS,MAN, AND CYBERNETICS, PART C and the IEEE TRANSACTIONS ON

AUTOMATION SCIENCE AND ENGINEERING, and Editor-in-Chief of the In-dustrial Engineering Journal. He was a Program Committee Member of the2003, 2004, 2005, 2006, 2007, 2008, 2009, and 2010 IEEE International Con-ference on Systems, Man, and Cybernetics, a Program Committee Member ofthe 2005, 2006, 2007, 2008, 2009, and 2010 IEEE International Conference onNetworking, Sensing and Control, a Program Committee Member of the 2006,2008, and 2009 IEEE International Conference on Automation Science andEngineering, a Program Committee Member of the 2006 IEEE InternationalConference on service systems and service management, a Program CommitteeMember of the 2007 International Conference on Engineering and SystemsManagement, and reviewer for many international journals.

Chengbin Chu received the B.Sc. degree inelectrical engineering from Hefei University ofTechnology, Hefei, China, in 1985 and the Ph.D.degree in computer science from Metz University,Metz, France, in 1990.

He was with the National Research Institute inComputer Science and Automation (INRIA), France,from 1987 to 1996. He was a Professor with theUniversity of Technology of Troyes, France, from1996 to 2008, where he was also Founding Directorof the Industrial Systems Optimization Laboratory.

He is currently a Senior Professor of the Chair Supply Chain at Ecole CentraleParis. He is interested in research areas related to operations research and mod-eling, analysis, and optimization of supply chain and production systems. Heis author or coauthor of three books and more than 100 articles in internationaljournals such as Operations Research, SIAM Journal of Computing, the IEEETRANSACTIONS ON ROBOTICS AND AUTOMATION, the IEEE TRANSACTIONS

ON AUTOMATION SCIENCE AND ENGINEERING, the International Journal ofProduction Research, Naval Research Logistics, and so on. He also publishedmany papers in conference proceedings.

Dr. Chu received the First Prize of Robert Faure Award in 1996 for his re-search and application activities. He also received the 1998 Best TransactionsPaper Award from the IEEE Robotics and Automation Society. He was nomi-nated Chang Jiang Scholars Program Chair Professor by the Chinese Ministryof Education in 2005. He is an Overseas Visiting Professor and Overseas Di-rector of the Department of Industrial Engineering at Xi’an Jiaotong Univer-sity. He served as Associate Editor of the IEEE TRANSACTIONS ON ROBOTICS

AND AUTOMATION from 2001 to 2004. He is currently an Associate Editor ofthe IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING and theIEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS.

Feng Chu received the B.S. and M.S. degrees inelectrical engineering from Hefei University ofTechnology, Hefei, China, in 1986 and InstitutNational Polytechnique de Lorraine (France) in1991, respectively, and the Ph.D. degree in computerscience from University of Metz. Metz, France, in1995.

She was with the Jiangsu University of Tech-nology (China), for two years and at the NationalResearch Institute in Computer Science and Au-tomation (INRIA), France, for four years. She was

an Associate Professor at the University of Technology of Troyes, France, from1999 to 2009. She joined the Laboratory of IBISC in 2009. She is currently aFull Professor at the Université d’Evry Val d’Essonne. She has published morethan 40 papers in international journals such as the European Journal of Op-erational Research, Computers & Operations Research, Applied Mathematicsand Computation, Computers & Industrial Engineering, the InternationalJournal of Computer Integrated Manufacturing, the International Journal of

Production Research, Transportation Research, the IEEE TRANSACTIONS ON

SYSTEMS, MAN, AND CYBERNETICS, the IEEE TRANSACTIONS ON ROBOTICS

AND AUTOMATION, the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND

ENGINEERING, etc., and over 50 papers in academic conferences. She is mainlyinterested in modeling, analysis and optimization of complex systems includingintelligent transportation systems, logistics and production systems based onPetri nets and operations research.

MengChu Zhou (S’88–M’90–M’93–F’03) receivedthe B.S. degree in electrical engineering from Nan-jing University of Science and Technology, Nanjing,China, in 1983, the M.S. degree in automatic con-trol from the Beijing Institute of Technology, Beijing,China, in 1986, and the Ph.D. degree in computer andsystems engineering from the Rensselaer PolytechnicInstitute, Troy, NY, in 1990.

He joined the New Jersey Institute of Technology(NJIT), Newark, in 1990, and is a Professor of Elec-trical and Computer Engineering and the Director

of Discrete-Event Systems Laboratory. He is presently a Professor at TongiUniversity, Shanghai, China. He has over 350 publications including 10 books,�� journal papers (majority in the IEEE TRANSACTIONS), and 17 bookchapters. He coauthored with F. DiCesare Petri Net Synthesis for Discrete EventControl of Manufacturing Systems, Kluwer Academic, Boston, MA, 1993,edited Petri Nets in Flexible and Agile Automation, Kluwer Academic, 1995,coauthored with K. Venkatesh Modeling, Simulation, and Control of FlexibleManufacturing Systems: A Petri Net Approach, World Scientific, 1998, coeditedwith M. P. Fanti, Deadlock Resolution in Computer-Integrated Systems, MarcelDekker, 2005, coauthored with H. Zhu, Object-Oriented Programming in C++:A Project-based Approach, Tsinghua University Press, 2005, coauthored withB. Hruz, Modeling and Control of Discrete Event Dynamic Systems, Springer,London, 2007, and coauthored with Z. Li, Deadlock Resolution in AutomatedManufacturing Systems: A Novel Petri Net Approach, Springer, New York,2009.

He was invited to lecture in Australia, Canada, China, France, Germany,Hong Kong, Italy, Japan, Korea, Mexico, Singapore, Taiwan, and the U.S.and served as a plenary speaker for several conferences. He served as anAssociate Editor of the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION

from 1997 to 2000, and the IEEE TRANSACTIONS ON AUTOMATION SCIENCE

AND ENGINEERING from 2004 to 2007, and currently is an Editor of the IEEETRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, and AssociateEditor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS:PART A, and the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS. He servedas Guest Editor for many journals including the IEEE TRANSACTIONS ON

INDUSTRIAL ELECTRONICS and the IEEE TRANSACTIONS ON SEMICONDUCTOR

MANUFACTURING. His research interests are in intelligent automation, lifecycleengineering and sustainability evaluation, Petri nets, wireless ad hoc and sensornetworks, system security, semiconductor manufacturing, and energy systems.

Dr. Zhou was the recipient of the NSF’s Research Initiation Award, the CIMUniversity-LEAD Award by the Society of Manufacturing Engineers, the PerlisResearch Award by NJIT, the Humboldt Research Award for U.S. Senior Scien-tists, the Leadership Award and Academic Achievement Award by the ChineseAssociation for Science and Technology-U.S., the Asian American Achieve-ment Award by the Asian American Heritage Council of New Jersey, and theDistinguished Lecturership of the IEEE SMC Society. He was General Chair ofthe IEEE Conference on Automation Science and Engineering, Washington DC,August 23–26, 2008, General Co-Chair of the 2003 IEEE International Confer-ence on System, Man and Cybernetics (SMC), Washington DC, October 5–8,2003, Founding General Co-Chair of the 2004 IEEE International Conferenceon Networking, Sensing and Control, Taipei, March 21–23, 2004, and GeneralChair of the 2006 IEEE International Conference on Networking, Sensing andControl, Ft. Lauderdale, FL, April 23–25, 2006. He was Program Chair of the1998 and 2001 IEEE International Conference on SMC and the 1997 IEEE In-ternational Conference on Emerging Technologies and Factory Automation. Heorganized and chaired over 80 technical sessions and served on program com-mittees for many conferences. He is Program Committee Chair of the 2010 IEEEInternational Conference on Mechatronics and Automation, August 4–7, 2010,Xi’an, China. He has led or participated in 40 research and education projectswith total budget over $10M, funded by National Science Foundation, Depart-ment of Defense, Engineering Foundation, New Jersey Science and TechnologyCommission, and industry. He was the founding Chair of Discrete-Event Sys-tems Technical Committee and founding Co-Chair of the Enterprise InformationSystems Technical Committee of the IEEE SMC Society, and Chair (founding)of the Semiconductor Manufacturing Automation Technical Committee of theIEEE Robotics and Automation Society. He is a life member of the Chinese As-sociation for Science and Technology-U.S. and served as its President in 1999.

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