Crude oil scheduling optimization including specific operational aspects of a Brazilian refinery.

Transcription

1 Crude oil scheduling opimizaion including specific operaional aspecs of a Brazilian refinery. Raiana R. Seixas, Valéria V. Muraa, Sérgio M. S. Neiro Federal Universiy of Uberlândia - Uberlândia MG Brazil raianars@homail.com, valeria@ufu.br, srgneiro@ufu.br 1. Absrac Opimizing he use of resources in a refinery is sraegic for mainaining he viabiliy and compeiiveness of one company. The objecive of his work is o develop an opimizaion model of he crude oil scheduling problem, which would be able o represen a large size refinery resriced o specific operaing rules of he Brazilian oil indusry. I aims a maximizing he conribuing margin by defining he mos suiable allocaion of resources, sequencing and iming of asks and operaing parameers. The sae-of-he-ar coninuous-ime formulaion MOS was used as a basis whereof, he following aspecs were added: ank heel, seling ime, spli of crude oil parcels and alignmen of muliple anks o he same disillaion uni. Moreover, a smoohening ransiion sraegy called TIMT (emporary injecion of muliple base anks) was imposed when swiching from one base ank o anoher. The resuling model was implemened in he GAMS sysem and wo soluion sraegies were employed o solve he nonconvex MINLP problem: a mmilp- NLP decomposiion and a linear-based approach. A performance analysis was carried ou for comparing he wo soluion sraegies. In general, TIMT was able o sofen he abrup aleraion suffered by CDU in erms of is inpu composiion, reducing he impac of disurbance generaed by loading ransiion, however, i caused a significan increase in CPU ime and a sharp reducion of he number of feasible soluions found by he model. Noneheless, robusness can be achieved by adding slack variables o seleced complicaing consrains. 2. Keywords: scheduling, crude oil supply and blending, emporary injecion of muliple anks (TIMT). 3. Inroducion Oil is one of he mos imporan energy sources in he world. However, one oil company, such as a commodiy producer, is individually unable o conrol he price of is producs in he inernaional marke, which depends on supply and demand, as well as global macroeconomic and geopoliical facors. In his conex, opimizing he use of resources in a refinery is sraegic for mainaining viabiliy and compeiiveness of he company. The use of opimizaion ools by he programmer can provide many economic and operaional benefis: beer use of oil ank capaciy, assessmen of he bes way o execue he monhly planning, mainenance of opimal operaion despie unexpeced changes, comparison beween he curren programming wih he planning, beer visibiliy and conrol hroughou he supply chain [1]. The objecive of his work is o develop an opimizaion model of he crude oil scheduling problem, which would be able o represen a large size refinery resriced o specific operaing rules of he Brazilian oil indusry. I aims a maximizing he conribuing margin by defining he mos suiable allocaion of resources, sequencing and iming of asks and operaing parameers. The sae-of-he-ar coninuous-ime formulaion MOS (muli-operaions sequence), proposed by Moure e al. (2011) was used as a basis, since i allows efficien symmery-breaking and requires fewer prioriy-slos when compared wih oher classic models, hus leading o smaller model sizes [3]. The following aspecs were added o he MOS model: ank heel, seling ime, spli of crude oil parcels and alignmen of muliple anks o he same disillaion uni, as well as a ank feeding muliple disillaion unis. Moreover, a smoohening ransiion sraegy called TIMT (emporary injecion of muliple base anks) was imposed when swiching from one base ank o anoher. The TIMT is he overlapping of he ime windows of he base ank leaving and he base ank aking over he CDU load. A se of rules was imposed for he ransiion operaion. The purpose of he TIMT sraegy is o sofen he composiion variaion during ank swiches, hus minimizing he CDU perurbaion. 4. Problem descripion The crude oil shor-erm scheduling problem addressed in his work was based on a real-world refinery locaed in Brazil. The esing scenarios sough o represen real siuaions ha usually are faced by programmers on heir rouine asks. The refinery under sudy encompasses nine charging anks, hree crude disillaion unis (CDU) and up o hiry-six crude oil ypes, which from now on will be referred o as crude oil componens. A se of differen crude parcels composed of a crude mix is scheduled o arrive a he refinery sie by pipeline a differen poins in ime along he scheduling horizon, which are pre-defined by planning. Crude oil parcels are unloaded direcly o charging anks. The se of charging anks is subdivided in wo disjoin subses, namely, he subse of base anks, which are used o compose he major porion of he CDU load, and he subjec injecing anks, which are used o only complemen he CDU load (from 5 o 30 percen of he oal load). Each disillaion uni can be fed by up o wo charging anks (one base ank and one injecing ank) and each charging ank can feed up o wo CDUs. The crude mix is no specified a he charging ank level bu a he CDU level, which means ha charging anks conaining crude mix ou of specificaion can be aligned o any of he disillaions unis as long as he final mix in he CDU load complies wih he imposed qualiy consrains. In order o conrol qualiy specificaions such as densiy, sulfur conen, and acidiy in he CDU load, virual mixers were inroduced in he model. While one ank is charging a CDU i canno be loaded, and afer one ank receive a crude oil mix, i has o sele for 24 hours before charging crude disillaion unis. All disillaion unis mus operae coninuously. Transfer operaions mus lie wihin a specified flow range, whereas he amoun of crude mix held in charging anks mus be wihin he ank heel and is maximum holding capaciy. Figure 1 shows he scheme for he crude oil supply and blending problem under sudy. On he lef mos side, green cylinders prc1-3 represen he differen parcels scheduled o arrive a he refinery along he scheduling horizon. The number, amoun and qualiy of each

2 parcel can vary configuring differen scenarios of he same problem. Each one of he charging anks, depiced in he picure by he orange polygons, are idenified by codes from AA o LL, which are he shor for he ank ags TQ101AA o TQ101LL. The iniial crude oil slae and oal amoun in each charging ank complees he definiion of a scenario ogeher wih he deailed informaion on he scheduled parcels o arrive a he refinery. On he righ mos side of he picure UC, UN and UV represen each one of he hree crude disillaion unis. I should be noed ha here is limied conneciviy beween anks and disillaion unis, whereas full connecion beween parcel and anks is allowed. For sake of faciliaing calculaion, virual mixers are placed beween anks and unis so ha he final crude mix flowrae and qualiies are deermined prior o being fed o he unis. There is one mixer assigned o each uni so ha hey can be used univocally. Each connecing line in Figure 1 represen a possible ransfer operaion beween wo resources. The problem under sudy feaures a large size problem composed of 40 differen possible ransfer operaions. The bigger he number of possible operaions, he bigger he problem size, which is consequence of he involved number of resources. Anoher imporan facor ha affecs he problem size is he diversiy of crude oil componens under consideraion, which in his case accouns for a oal of up o 36 crude oil componens. Figure 1. Scheme for he crude oil scheduling problem under sudy represening a real-world refinery 4.1. Descripion of esing scenarios Six differen scenarios were used o es he model performance, wo of which are illusraed in his paper. Each scenario is characerized by presening an iniial invenory and crude oil profile; number, volume and arrival ime of crude parcels including composiion; scheduling horizon lengh; and resources availabiliy. Table 1 shows some of he differences beween Scenario A and Scenario B in erms of daa inpu. I can be seen from he daa ha Scenario A has a smaller amoun of oil available o feed he disillaion unis and 2 of 8 he charging anks are ou of specificaion a he beginning of he ime horizon. These facors can cause some difficulies in mainaining he coninuous disillaion unis operaion. On he oher hand, Scenario B encompasses more resources and has a greaer variey of crude oil componens, which leads o a larger problem. Table 1. Daa inpu of Scenario A and Scenario B Time Horizon Charging anks available Charging anks ou of specificaion (=0) Types of crude oil Useful volume available (=0) Parcels arriving Toal oil volume arriving Scenario A 168h ,893.0 m ,000.0 m3 Scenario B 144h ,041.8 m ,000.0 m3 5. Mahemaical model 5.1. Objecive funcion: The objecive funcion seeks o maximize he conribuing margin. I penalizes he number of exchanges on he feeding anks and he use of slack variables, which allows he violaion of some key consrains. Max G (1) c i T r R d i T V ivc 1000 Y ivv v I r Spli of crude oil parcels unloaded by pipeline: FQ ivk i T v Wd v I r W TI v I r W TI r Unloading sar ime of one parcel is bigger or equal han he arrival ime of his parcel: 1000 FF iv 1000 FI iv i T i T v Wd

3 S iv Sar r ; i T,. v W u (2) Unloading end ime of one parcel is less or equal han he arrival ime of he parcel plus he max unloading duraion of he parcel: E iv Sar r + L ir FR v ; i T,. r R p, v O r A unique conneciviy is allowed beween he pipeline and anks a a given prioriy-slo: Z iv 1 ; i, r R p (3) (4) v O r In he case of parcel spli, when a porion of one parcel is unloaded o one ank, his consrain prevens he subsequen repeiive unloading of anoher porion of he parcel ino he same ank: Z iv 1 ; v I r, r R (5) i Unloading of parcels mus occur in crescen prioriy-slos: E i1 v + D iv + D ir S i2 v + H (1 Z i2 v) ; i 1 < i 2, r R v I r i 1 <i<i 2 v I r i 1 <i<i 2 v I r v I r (6) 5.3. Seling ime: + S i2v Minimum ime inerval beween he end of he loading operaion and he sar of he unloading operaion for a ank: E i1 v + D i1 v TR vv Z i1v S i2 v + (H + TR vv ) (1 Z i2 v) ; v W i T v W i 1 <i<i 2 v W v W v W v W v W v v v W i 1, i 2 T, i 1 < i 2, W CLIQUE TR (G NO ) No overlapping operaions mus be assigned o disjoin ime slos E i1 v + D i1 v + H (1 Z i2 v) ; v W i T v W v W v W i 1 <i<i 2 i 1, i 2 T, i 1 < i 2, W CLIQUE(G NO ) (7) (8) 5.4. Alignmen of up o wo anks o he same CDU (Mixer and injecing ank modeling): Time synchronizaion in he inpu and oupu of he mixer: S iv S iv H (1 Z iv ) ; i, r R m, v I r, v O r (9) S iv S iv + H (1 Z iv ) ; i, r R m, v I r, v O r (10) E iv E iv H (1 Z iv ) ; i, r R m, v I r, v O r (11) E iv E iv H (1 Z iv ) ; i, r R m, v I r, v O r (12) Minimum operaion duraion applied o unloading operaions: D iv D v. Z iv ; i T, v W u (13) Global mass balance a he mixer (No accumulaion erm): V iv = V iv ; i T, r R M (14) v I r v O r Mass balance by ype of oil a he mixer (No accumulaion erm): V ivc = V ivc ; i T, r R M, c C (15) v I r v O r In ordinary condiions, up o one base ank can be aligned o he same CDU. Up o wo anks can be aligned during TIMT: v (I r W TI ) Z iv (1 + X ir ) ; i, r R m (16) No injecing ank can compose he CDU load before one base ank has been seleced. Up o one injecing ank is allowed:

4 Z iv Z iv v I r W TI v (I r W TI ) X ir ; i, r R m (17) Qualiy consrains are specified in he disillaion column inpu, no in each ank. (Volume base consrain): x k. V iv x ck V ivc x. k V iv ; i, v W d, k K v Qualiy consrains are specified in he disillaion column inpu, no in each ank. (Mass base consrain): x k V ivc xk c x ck V ivc xk c x k V ivc xk c ; i, v W d, k K m (18) (19) The minimum conribuion required of a ank in he CDU load is 5%: V iv 0.05 V iv V v (1 Z iv ) ; i, r R m, v I r, v O (20) r The maximum conribuion of an injecing ank in he CDU load is 30%: V iv 0.30 V iv + V v (1 Z iv ) ; i, r R m, v I r W TI, v O (21) r 5.5. Parallel aciviy in anks: Each ank can be aligned wih up o wo CDU: Z iv 2 ; i, r R i O r (22) The oupu operaions of a ank, allocaed o he same prioriy-slo, mus be synchronized: S ir S iv + H(1 Z iv ) ; i, v O r, r R (23) S ir S iv H(1 Z iv ) ; i, v O r, r R (24) E ir E iv + H(1 Z iv ) ; i, v O r, r R (25) E iv H(1 Z iv ) ; i, v O r, r R (26) E ir Duraion calculus of one ank unloading operaion: E ir = S ir + D ir ; i, r R (27) If here is no movemen on he ank oule, he ank ime variables mus be zero. Maximum and minimum flow rae a he oupu of each ank. FR r D ir E ir H Z iv ; i, r R v O r V iv FR r D ir ; i, r R (28) (29) v O r 5.6. Temporary Injecion of Muliple Tanks (TIMT): Two TIMT may no occur in consecuive prioriy-slos: X ir + X i+1 r 1 ; 1 < i < I, r R m (30) Transiion idenificaion beween operaion v and v: Y iv v Z i 1v + Z iv Z i 1v 1 ; 1 < i < I, r R m, v, v (I r W TI ) (31) TIMT was conceived o occur in hree prioriy-slos: in he firs one, only operaion v' is aking place; in he hird one, here is only he presence of operaion v; in he second prioriy-slo he wo operaions v' and v overlap each oher. If a ransiion occurs, Y ivv mus assume he value of 1: 2 Y iv v Z iv + Z i+1v ; 1 < i < I, r R m, v, v I r W TI (32) The binary variable X ir is used o idenify wheher a ransiion is aking place involving he resource r (which in his case is one of he crude disillaion unis). The relaion beween X ir and Y iv v is described below: X ir Y iv v ; 1 < i < I, r R m, v, v I r W TI (33) Volume proporion beween he base ank aking over and he base ank leaving is given by he equaions below. The esablishmen of a volume raio beween he overlapping anks akes place in order o ensure, during TIMT, an inermediae poin of heir properies in erms of composiion and qualiy.

5 V iv 0.50 V iv V iv 0.30 V iv + V v V (1 v Yiv v ) ; 1 < i < I, r R m, v, v I r W TI (34) (1 Yiv v ) ; 1 < i < I, r R m, v, v I r W TI (35) Minimum TIMT duraion: D iv D pol + (H D pol )(1 X ir ) ; 1 < i < I, r R m, v O r (36) I defines he minimum duraion of operaions (relaxaion during TIMT): D iv D v Z iv (D v D pol ) X ir ; i, r R m, v O r (37) 5.7. Slack Variables: The inroducion TIMT makes he model harder o be solved. In order o avoid infeasible soluions, i will be allowed he infringemen of some consrains by inroducing slack variables, which are as follows: I allows a violaion in he consrain ha conrols he acidiy in he CDU inpu: Acidiy slack variable maximum limi: x ck V ivc x c k FQ ivk x k V ivc xk c ; i, v W d FQ ivk x k V ivc xk c ; i, v W d (39) I allows a violaion in he consrain ha conrols he minimum flow rae in he CDU load: V iv + FF iv FR r D iv ; i, v W d (40) Minimum flow rae slack variable maximum limi: FF iv FR r D iv ; i, v W d (41) I allows a violaion in he consrain ha conrols he upper limi of an injecing ank proporion: V iv 0.30 V iv + FI iv ; i, v W d (42) (38) Injecing ank proporion slack variable maximum limi: FI iv 0.10 V iv ; i, v W d (43) 6. Soluion mehods Two soluion sraegies were employed in order o solve he nonconvex MINLP problem resuled from his work: a mmilp-nlp decomposiion proposed by Chen e al. (2012) and a linear-based approach proposed by Reddy e al. (2004). All formulaions were coded using he GAMS 24.4 sysem and solved on an Inel(R) Core(TM) i7, CPU3.5 GHz and 16.0GB RAM. MILP problems were solved using CPlex 12 and he NLP problems were solved using CONOPT. For MILP problems, wo sop crierion were employed: relaive gap of 2% and a maximum soluion ime of 7200 CPU seconds, whaever is aained firs. All soluions repored were solved using 5 prioriy-slos, which was he minimum number necessary o ge feasible soluions for he se of scenarios esed mmilp-nlp decomposiion: This sraegy is derived from he MILP-NLP decomposiion proposed by Moure e al (2009), which solves he MINLP problem in wo seps: firs an MILP problem is solved, discarding he nonlinear consrains; nex, he ineger soluion obained from he MILP problem is fixed in he complee MINLP problem resuling in a NLP problem. The NLP subproblem is hen solved using he soluion of he MILP as a saring poin [2]. The MIP-NLP procedure was developed in order o reduce he CPU effor in solving MINLP problems. However, i migh lead o subopimal or infeasible oucomes. In order o increase he probabiliy of obaining feasible and beer soluions wih such sraegy, he mmilp-nlp decomposiion was devised, whereby muliple soluions of he MILP problems are generaed, and hen he relaxed NLP problems corresponding o hese MIP soluions are solved. This can be achieved by using he soluion pool seings of GAMS/CPLEX [4]. Parameers of he soluion pool seings employed in his work were as follows: solnpoolinensiy = 4, solnpoolpop = 2, solnpoolcapaciy = 100, solnpoolgap = 0.5 and populaelim = Linear-based approach: I is an ieraive algorihm o eliminae he crude composiion discrepancy wihou using nonlinear consrains by following hese seps: a. In he beginning of he ime horizon, he composiion of each crude oil componen (c) in each ank (r) is known and deermined by he equaion: Composiionrc = L 0rc /L 0r i, c, r R. b. This calculaed composiion is added as a parameer o he model in order o eliminae he composiion discrepancy in he firs prioriy-slo hrough he equaion:v ivc = Composiion rc V iv i = 1, c, v W u. For all prioriy slos wih i > 1 (or i > curren ieraion), he composiion discrepancy is allowed. The resuling MILP problem is hen solved wih he decisions relaed o he firs (curren ieraion) prioriy-slo being accuraely made. c. The crude oil profile is hen updaed wih respec o he anks ha were loaded in he firs (curren ieraion) prioriy-slo.

6 d. All decisions for he firs (curren ieraion) ime slo are fixed. This includes iming and volume decisions. Then, he consrain of sep b is updaed o be valid for he nex prioriy-slo, and he MILP problem is solved for he second prioriy-slo. The algorihm is repeaed unil he las prioriy-slo. 7. Resuls Table 2 shows resuls for he wo scenarios esed. In boh cases, he soluion sraegy mmilp-nlp decomposiion, when compared wih he linear based approach, presened higher values for he objecive funcion, wha is desirable for a maximizaion problem. Toal CPU ime was much smaller when using he linear-based approach for Scenario A and almos he same in boh soluions sraegies for Scenario B. Comparing he wo scenarios, CPU ime was almos he same when using he decomposiion procedure, bu was considerably smaller for Scenario A, when using he linear-based sraegy. For Scenario A, he model only found feasible soluions if violaions in some consrains were allowed, which is relaed (bu is no he only cause) wih he difficul siuaion presened in Scenario A in erms of qualiy specificaions and availabiliy of oil o feed he crude disillaions unis. Table 3 shows resuls for he model wihou he consrains relaed o he emporary injecion of muliple base anks (TIMT), which was called Simplified model. For his version of he model, i was no necessary adding slack variables, since in all esed cases, a leas one feasible soluion was found. In anoher hand, in previous ess of he model wih he TIMT modeling and wihou slack variables, from 6 cases esed, only 2 had feasible soluions. Comparing he resuls of he model wih and wihou he TIMT modeling, one can realize ha he inroducion of TIMT, besides decreasing he number of feasible soluions found by he model, increased abruply he CPU ime (in mos cases he resoluion ime was abou 5 imes higher when using TIMT). In anoher hand, he comparison of Figures 2 and 3, shows ha TIMT was able o sofen he abrup aleraion suffered by CDU in erms of is inpu composiion during load ransiion, reducing he impac of disurbance generaed by loading ransiion. In he case wihou TIMT, i may happens ha one column is receiving, in an insan, a kind of mixure of a ank wih cerain properies, and suddenly, in he nex insan, i may receive a mix from anoher ank wih compleely differen properies. Therefore, as he difficuly of finding feasible soluions was solved by he conrolled permission of violaion in some key consrains, and for he Complee model boh esing scenarios were solved in less han 15 minues, i may worh keeping he TIMT consrains in he model in order o ge beer soluions. Table 2. Compuaional resuls for Scenario A and Scenario B (Complee model - wih slack variables) Scenario A Scenario B Soluion sraegy mmilp-nlp decomposiion Linear-based approach mmilp-nlp decomposiion Linear-based approach MIP Soluion 36, NA 38, NA Bes MINLP Soluion 35, , ,66 37, MIP Time (seconds) NLP Time (seconds) NA 53,85 NA Toal Time (seconds) Number of feasible soluions 100 NA 100 NA Final GAP Binary variables number Toal variables number 7,816 7,816 13,110 13,110 Toal equaions number 11,999 11,999 19,092 19,092 Use of slack variable FQ ivk No Yes No No Use of slack variable FF iv No No No No Use of slack variable FI iv Yes No No No Table 3. Compuaional resuls for Scenario A and Scenario B (Simplified model - wihou slack variables) Scenario A Scenario B Soluion sraegy mmilp-nlp Linear-based mmilp-nlp Linear-based decomposiion approach decomposiion approach Toal Time (seconds) Number of feasible soluions 44 NA 100 NA Binary variables number Toal variables number 7,641 7,641 12,871 12,871 Toal equaions number 11,367 11,367 18,172 18,172

7 Figure 2. Profile properies in CDU during he ime horizon for Scenario A (complee model - wih TIMT) Figure 3. Profile properies in CDU during he ime horizon for Scenario A (Simplified model - wihou TIMT) Figures 4 o 6 presen Gan char for Scenario A. Gan char was divided in 3 pars in order o make he visualizaion easier. The volume ransferred in each operaion as well as he flow raes are placed inside he bars (As he esing scenarios have big volumes, all quaniies were divided by 1000). TIMI can be idenified in he Gan char by a recangle. Figure 4. Gan char for Case- sudy A (Par 1)

8 Figure 5. Gan char for Case- sudy A (Par 2) Figure 6. Gan char for Case- sudy A (Par 3)

9 Figures 7, 8, 9 and 10 show he invenory evoluion in each ank during ime horizon. Figure 6. Invenory evoluion for he anks AA and BB (Scenario A) Figure 7. Invenory evoluion for he anks CC and DD (Scenario A) Figure 8. Invenory evoluion for he anks FF and HH (Scenario A) Figure 9. Invenory evoluion for he anks II and LL (Scenario A)

10 8. Conclusion The objecive of his work was o generae an oil scheduling opimizaion model ha would be able o represen a real large size refinery considering specific operaing rules of he Brazilian oil indusry. The model developed has achieved his goal. TIMT was able o sofen he abrup aleraion suffered by CDU in erms of is inpu composiion, reducing he impac of disurbance generaed by loading ransiion, however, i caused a significan increase in CPU ime and i abruply reduced he number of feasible soluions found by he model. The addiion of slack variables in key consrains proved o be effecive in increasing he robusness of he model, making i possible o find feasible soluions for all esing scenarios. For mos cases esed, he mmilp-nlp procedure was more effecive in erms of maximizing he conribuion margin and finding feasible soluions, while he linear based approach was able o find soluions more quickly. 9. Acknowledgemens The auhors would like o acknowledge he financial suppor from CAPES and he echnical suppor from CENPES. 10. Nomenclaure Ses 11. References Posiive Variables T Se of all prioriy-slos. L ir Toal accumulaed level of crude in ank r before he operaion W Se of all operaions assigned o prioriy-slo i Wu Unloading operaions Siv Sar ime of operaion v in he prioriy-slo i Wm Mixer inle operaions Div Duraion of operaion v in he prioriy-slo i WTI Operaions of an injecing ank Eiv End ime of operaion v in he prioriy-slo i Wd CDU inle operaions V iv Toal volume ransferred by operaion v in he prioriy-slo i R Se of all resources Vivc Volume of crude c ransferred by operaion v in he prioriyslo i Rm Se of Mixers Vivk Value of key-propery k for operaion v in he prioriy-slo i Rp Se of Parcels Y iv v I appoins if he TIMT from operaion v o operaion v is assigned o he prioriy-slo i Rd Se of Disillaion unis (CDU) FQivk Slack variable of qualiy consrains (k=acidiy) R Se of Tanks FFiv Slack variable of minimum flow rae in he CDU load consrain RTI Se of Injecing anks FIiv Slack variable in he upper limi of an injecing ank proporion Ir Inle operaions of resource r Parameers Or Oule operaions of resource r Gc Conribuing margin of crude c c Type of crudes H Time horizon K Produc properies V v, V v Limis of oal volume ransferred during operaion v D v Minimum duraion of operaion v Km Produc properies in mass base FR r, FR v Flow rae limiaions for ransfer operaion v Kv Produc properies in volume base x. k, x k. Limis of propery k of he blended producs D pol, D pol Limis of TIMT duraion Binary Variables x ck Value of he propery k of crude c Zi,v X ir I assumes he value of 1 if operaion v is acive in he prioriy-slo i ; 0 oherwise x c k Densiy of crude c I assumes he value of 1 if here is one TIMT occurring in one CDU (r), in he prioriy-slo i; 0 oherwise TRvv Sar r k Minimum ime he ank remains saionary if he operaion v (load) precede he operaion v (unloading) Arrival ime of he parcels Acidiy 1. P. Chandra Prakash Reddy, I.A. Karimi, R. Srinivasan, A new coninuous-ime formulaion for scheduling crude oil operaions, Chemical Engineering Science, 2004, 59, S. Moure,, I.E. Grossmann, P. Pesiaux, A Novel Prioriy-Slo Based Coninuous-Time Formulaion for Crude-Oil Scheduling Problems, Ind. Eng. Chem. Res. 2009, 48, S. Moure,, I.E. Grossmann, P. Pesiaux, Time represenaions and mahemaical models for process scheduling problems, Compuers and Chemical Engineering, 2011, 35, X. Chen, I. Grossmann, L. Zheng, A comparaive sudy of coninuous-ime models for scheduling of crude oil operaions in inland refineries. Compuers & Chemical Engineering, v. 44, p , 2012.