[2007]Effective MILP model for oil refinery-wide production

Effective MILP model for oil refinery-wide productionplanning and better energy utilization *B.J.Zhang *,B.HuaThe Key Laboratory of Enhanced Heat Transfer and Energy Conservation,Ministry of Education,South China University of Technology,Guangzhou 510640,ChinaReceived 21June 2005;accepted 18August 2005Available online 26October 2005AbstractGenerally,a refinery complex consists of process system and utility system.Process system not only produces liquefied petroleum gas,gas-oline,diesel and so on,but also some byproducts,such as fuel gas and residual fuel oil,which supply utility system as fuel.The utility system converts fuel gas and fuel oil to high pressure or medium pressure steam and electricity to meet the energy demand of the process system.A novel approach to the integration of the process system and utility system for better energy utilization is presented in this paper.A plant-wide multi-period planning mathematical model is proposed and three improvements are identified.First,the process unit energy-consumed model is refor-mulated,because energy consumed by a unit not only relates the throughput of the unit,but also varies with the operation modes of the unit and season.Second,The MILP (Mixed Integer linear Programming)model of utility system is embedded in the plant-wide model to gain the overall optimization and for better energy efficiency.Third,steam,fuel oil and fuel gas are balanced in the whole plant.Finally,the proposed approach was used in a real industrial example to determine the optimal mass and energy flow and some important results have been obtained and are presented.Ó2005Elsevier Ltd.All rights reserved.Keywords:Process optimization;Utility system;Oil refinery process optimization;Multi-period optimization;MILP1.IntroductionThe refinery industry is a vital component of the national industry.The effective integration of energy in refinery com-plex can create a great impact on our economy as well as cleaner production.In a refinery,all process plants rely on con-sistent steam and electricity supply that requires high capital and operating costs in utility system.Thus,the interactions be-tween utility plants and process plants have to be considered.Refinery production planning optimization involves many aspects ranging from quality analysis to selection of crude oils and products,from arrangement to selection of crude oils and products,from arrangement of processing modes to decisions on process operation modes,and so on.To focuson the integration of process system and utility system and as-sociated modeling issues,the refinery-wide production plan-ning optimization problem is defined as follows [1,2].For a given refinery system,the overall objective is to max-imize the overall refinery profit (or margin)by optimizing feedstock selection,process operation modes,flow direction between processes,finished product blending,stock level,and utility.The objective is defined as the total sales revenue minus the total operating costs.The total operating costs in-volve the purchase costs of all kinds of raw materials such as crude oil,fuel,power,catalysts,inventory cost,transporta-tion cost,other fixed cost.Quite a few research works have been reported which relate to these issues.Zhang et al.presented a method for overall re-finery optimization through integration of hydrogen network and steam network with the material systems,and an MINLP model has been formulated and linearization techniques are applied to covert the MINLP problem to a MILP problem.This approach is efficient particularly for a small or medium*Supported by Advanced Scientific Problems in High Efficient energy Sav-ing (G2000026308).*Corresponding author.Tel.:862087113614.E-mail address:bjzhang@ (B.J.Zhang).0959-6526/$-see front matter Ó2005Elsevier Ltd.All rights reserved.doi:10.1016/j.jclepro.2005.08.004Journal of Cleaner Production 15(2007)439e448/locate/jcleprochemical plant.Nevertheless,for large industry sites,its appli-cation is difficult because of the increasing complexities of the MINLP model,and it is limited in single periods [2].Kim et al.[3]proposed a plant-wide multi-period optimization model to determine the optimal energy supply to meet the varying energy demands and unit price changes.Wenkai et al.[4]proposed a material and energy integration model of a real world industrial petrochemical complex for support-ing investment decisions.Grossmann et al.presented a new model for heat integration that overcomes difficultiesNomenclature Index T Set of periods M Set of materialsM o Subset of M that involves the raw materials M p Subset of M that involves the products M i Subset of M that involves the intermediates M n Subset of M that involves power and electricity M w Subset of M that involves water and various lev-els of steamM x Subset of M that involves only non-inventoriable materialsM y Subset of M that involves only inventoriable materialsMF u ,e Subset of M that involves feeds of operation mode e of unit uMP u ,e Subset of M that involves products of operation mode e of unit u U Set of unitsU r Subset of U that involves processing units U u Subset of U that involves blend headers U t Subset of U that involves utility equipments E Set of operation modesE u Subset of E that operates on unit u ASet of properties for materialsParameters a m ,t Upper bounds for the amount of product sale orraw material m limited by government during time period tb m ,t Lower bounds for the amount of product sale or rawmaterial m available in market during time period tg m ,t Price of raw material m during time period t d m ,t Penalty cost for shortfall of product m duringtime period t3u Upper bounds for capacity of unit u z u Lower bounds for capacity of unit u y u ,e Upper bounds for capacity of operation mode eof unit uq u ,e Lower bounds for capacity of operation mode eof unit uk u ,e ,m Production rate for material m of operationmode e of unit uf u The amount of maintenance periods during plan-ning horizonp u ,e ,m ,t Coefficient for utility material m consumed byoperation mode e of unit u during time period tj u ,e ,t The constant term for utility material m con-sumed by operation mode e of unit u duringtime period tl u ,e ,m ,t Coefficient for utility material m generated byoperation mode e of unit u during time period tm u ,e ,t The constant for utility material m generated byoperation mode e of unit u during time period tx u ,t Maximal number of available operation modes of unit u during time period tO u ,e ,e #Changeover cost for changing from operation mode e to e #of unit ur m Upper bound for inventory amount of inventori-able material m2m Lower bound for inventory amount of inventori-able material ms m ,t Inventory cost for material m during time period t t m Transport cost for material m between market and planth u Efficiency of utility equipment u 4m ,a Property of material m H m Enthalpy of material m cFixed costVariablesPR m ,t Amount of raw material m purchased duringtime period tSP m ,t Amount of product p sold to market during timeperiod tFP m ,t Amount of shortfall of product p during time pe-riod tEF u ,e ,t Throughput of operation mode e of unit u duringtime period tUF u ,t Throughput of unit u during time period tEP u ,e ,m ,t Amount of material m produced from operationmode e of unit u during time period tEC u ,e ,m ,t Amount of material m consumed by operationmode e of unit u during time period tED u ,e ,m ,t Amount of utility material m consumed by oper-ation mode e of unit u during time period tEG u ,e ,m ,t Amount of utility material m generated by oper-ation mode e of unit u during time period tMI m ,t Amount of inventory of material m during timeperiod tUO u ,t 0e 1variables that denotes whether unit u is on EO u ,e ,t 0e 1variable that denotes whether unit u oper-ates with operation mode e during time period tX u ,e ,e #,t 0e 1variable that denotes whether unit u oper-ates with operation mode e during time period t and operates with operation mode e #during the next period t C 1Profit Objective function440 B.J.Zhang,B.Hua /Journal of Cleaner Production 15(2007)439e 448experienced either with direct integration approaches,or with the Duran and Grossmann model[5]when handling isother-mal streams.The proposed model used logic disjunctions to explicitly model the relative placement of stream for various potential pinch locations,and explicitly considers the non-isothermal and isothermal streams as separate cases[6]. Xuan and Yuan has built the overall refinery-wide model which combined all energy costs into one category,and the overall re-finery-wide model cannot cope with the recycle streams[7].2.Problem statementsGenerally,a refinery complex consists of process system and utility system as shown in Fig.1.Process system not only produces liquefied petroleum gas,gasoline,diesel and so on,but also some byproducts,such as fuel gas and residual fuel oil,which supply utility system as fuel.Utility system converts fuel gas and fuel oil to high pressure or medium pres-sure steam.A part of steam from utility system supplies pro-cess system as heat sources when the high pressure or medium pressure steam produced by process system cannot meet its demand.When the high pressure or medium pressure steam produced by the process system exceeds its self de-mand,the excess steam is transported to utility system,which combines with the steam from boilers,to pass turbines to extract low pressure steam and to generate electricity.Low-pressure steam supplies process system as heat sources and electricity supplies all users in the refinery.A trade contract between the refinery and the power company was signed to balance the electricity demand in the whole plant.The fuel oil and gas network is given as a set of fuel oil and gas sinks,defined as those processes consuming fuel oil and gas,and a set of fuel oil and gas sources,defined as those pro-cesses producing fuel oil and gas,together with the intercon-nections between them.The objective of fuel oil and gas network integration is to optimize fuel oil and gas use and gen-eration in terms offlow rate among all of fuel and gas network sinks and sources.The steam and power network in the whole refinery is given by the steam header conditions,the steam net-work and the hardware capacity of boilers,turbines,etc.[2]. The objective of energy system integration is to optimize elec-tricity purchased and produced and steam distribution among the whole refinery site.These two integration problems dis-cussed above are considered simultaneously with the overall objective defined in Section1.In other words,these two inte-gration methods are employed to maximize the overall profit (or margin)under the given market conditions and hardware capacity and process constraints for a given refinery system.The conventional approach to process system and utility system is a sequential hierarchical method.In other words, production planning is modeledfirst.Then,utility system is scheduled to meet the needs of process operations.Tradition-al sequential(or two-stage)method shown in Fig.2is used to integrate the utility system and the process system. Fig.2is adopted from Zhang et al.[2]and Kim et al.[3] with some modifications.In thefirst stage,the simple overall LP optimization takes into account the interactions among all systems.The unit models of these systems are simplified into linear ones.The objective of thefirst stage is to obtain a rough planning strategy by determining the selection of crude oils,products,and simplistic allocation of utilities.In the second stage,the streamflows from/to the process systemFig.2.Sequential procedure of refinery optimization.liquefiedpetroleum gasGasolineDieselKeroseneLube &Chemical ProductElectricity Fig.1.Simplified configuration of refinery complex.441B.J.Zhang,B.Hua/Journal of Cleaner Production15(2007)439e448and the utility system and others systems arefixed as bound-ary conditions.The utility system and other systems are then optimized separately by considering their detailed unit mod-els to determine their minimal operation costs.The scope of each second stage model is smaller compared with thefirst stage model.If the models fail to reach feasible solutions, the processing conditions of thefirst stage are then modified and the iteration continues until feasible solutions are found [8].The conventional approach separates the utility system from the process system,the interrelationship between them is simplified or even neglected and two of them are op-timized,respectively.A noticeable drawback of this method is that process system and utility system are optimized indi-vidually and therefore cannot guarantee the global optimality of the whole refinery complex.As a result steam often vents or letdowns,sometimes process system even shutdowns ur-gently because of unbalanced utility.3.FormulationBased on above insights,a new optimization strategy is proposed that tackles the problem of refinery optimization by considering proper integration of the different systems in-cluding process system and utility system.A plant-wide multi-period planning mathematic model is proposed and three improvements are identified.First,process unit energy-consumed model is reformulated,because energy consumed by a unit not only relates the throughput of the unit,but also varies with the operation modes of the unit and season.Sec-ond,the MILP(Mixed Integer linear Programming)model of utility system is embedded in the plant-wide model to gain the overall optimization and for better energy efficiency. Third,steam,fuel oil and fuel gas balance in the whole plant.Although optimization can be stated in many different ways,the common optimization to an industrial process is to maximize the profitability of the process.In this work,the whole refinery is considered to be one process.This process uses the given crude to produce various petroleum products to achieve economic objectives.The objective of optimization in hand is to reach the max-imal profitability given the crude and refinery facilities.No major hardware change in the current facilities is considered in the optimization.The optimization tries tofind the optimal operation modes of units and streamflow that maximize the overall profit of the whole refinery while observing all the pro-cess constraints.In the integration model,the objective function is given by:Max profit ZXm˛M p Xt˛Tg m;t SP m;tÿXm˛M oXt˛Tg m;t PR m;tÿXm˛M y Xt˛Ts m;t MI m;tÿXm˛M pXt˛Tt m SP m;tÿXm˛M oXt˛Tt m PR m;tÿXu˛U Xe˛E uXe0˛E uXt˛TO u;e;e0X u;e;e0;tÿXm˛M pXt˛Td m;t FP m;tÿcð1Þwhere each term accounts for income from sales,purchasecost,inventory cost,transport cost of products,transport costof raw materials,operation mode change over cost,shortfallcost andfixed cost,such as the capital cost and salaries,respectively.3.1.Market demand for raw materials and productsEq.(2)forces raw materials purchased from markets notonly to be greater than the minimum amount in the givenperiod,but also lower than the available amount in marketsduring the period.The minimum amount of raw materialsmay be limited by government or process operations.b m;t%PR m;t%a m;t m˛M oð2ÞIn general,the demand of products is assumed to beflex-ible in the sense that it is given by a range of values havinga hard upper bound and a soft lower bound as Birewar andGrossmann presented[9].The lower bounds are given byfixed orders booked by sales department.Production short-falls with respect to the lower bounds stand for loss ofpotential sales that is penalized in the objective function,Eq.(1).SP m;t%a m;t m˛M pð3ÞSP m;t R b m;t C FP m;tÿ1ÿFP m;t;m˛M pð4Þ3.2.Material balance and inventoryAll materials in the whole refinery divide into tow catego-ries that are non-inventoriable and inventoriable.Non-inven-toriable materials such as steam,power and electricity,which must be consumed in the current period,otherwisethey will disappear and wasted.Inventoriable materials canbe stored in tanks or in other facilities;they can help tomake operations stable and to meet the changing demand ofthe markets.Eq.(5)gives the upper and lower bounds for the inven-tory of inventoriable materials.Eq.(6)corresponds for theinventoriable material balance.For a given material andperiod,the stock volume at the end of the current periodmust equal the stock volume at the end of the previousperiod plus the production volume and the purchase volumeduring the given period and minus the consumption volumeand sales during the given period.Eq.(7)illustrates thatthe relationships of the amount of non-inventoriable materialm production and purchase must equal or be more than theamount of its consumption and sales.Non-inventoriablematerials cannot cross periods,and they must be consumedinstantly.2m;t%MI m;t%r m;t;m˛Mð5Þ442 B.J.Zhang,B.Hua/Journal of Cleaner Production15(2007)439e448MI m ;t Z MI m ;t ÿ1CX u ˛U r Xe ˛E uEP u ;e ;m ;tÿX u ˛U r Xe ˛E uEC u ;e ;m ;t C PR m 0;t ÿSP m ;t ;m ˛M yð6ÞX u ˛U r Xe ˛E uEP u ;e ;m ;t C PRm 0;tRX u ˛U r Xe ˛E uEC u ;e ;m ;t C SP m ;t ;m ˛M xð7Þ3.3.Process unitIn this paper,a refinery unit is expressed in terms of a set of operation modes that determine side streams with different property,such as freezing point,which is an im-portant quality control specification for diesel blending and applied in many refineries.Furthermore,energy con-sumed by units is also related to the selected operation mode.The operation of refinery considered in this paper varies with seasons.This is true for most of refineries.The operationalchanges are necessary to cope with the market demands and price changes with respect to seasons.The price structure is greatly affected by the market demand.For example,the gasoline has a higher price in the summer when a lot of gaso-line is consumed by travelers.The diesel has a higher price in the winter because the demand of heating oil is high and diesel can be used to make the heating oil.Eq.(8)specifies the limitation on unit production.Only when throughput of units amount up to allowed hardware ca-pacity unit can be on operation,otherwise,it is set to zero.And throughput of operation modes constrained by Eq.(9)is similar to the throughput of units.Eq.(10)indicates that a re-finery unit capacity is break into several operation modes.In this way energy consumed by units and property of side streams can be differentiated by operation mode.z u UO u ;t %UF u ;t %3u UO u ;t ;u ˛U r ð8ÞLPS Fig.3.A simplified flow sheet of the refinery complex.Fig.4.The simplified flow sheet of the process system.Fig.5.The simplified flow sheet of the utility system.443B.J.Zhang,B.Hua /Journal of Cleaner Production 15(2007)439e 448q u;e EO u;e;t%EF u;e;t%n u;e EO u;e;t;u˛U r;e˛E uð9ÞXe˛E uEF u;e;t Z UF u;t;u˛U rð10ÞEq.(11)indicates that the most number of operation modes of a unit can be allowed during planning horizon.Generally, only one operation mode is allowed in short term planning be-cause of operating stabilization and the cost incurred by oper-ation mode change.Xe˛E uq u;e Z1;u˛U rð11ÞEq.(12)considers that throughput of operation mode equals its consumed volume during the given period.Eq.(13)corresponds to the side streams of each operation mode. We assume that side streams of operation modes are expressed in terms of linear unit ratio to the throughput of the corre-sponding operation mode.Eq.(14)forces the sum of all pro-duction rates of an operation mode must equal1.In other words,the materials must balance.EF u;e;t ZXm˛MF u;eEC u;e;m;t;u˛U r;e˛E uð12ÞEP u;e;m;t Z k u;e;m EF u;e;t;u˛U r;e˛E u;m˛MP u;eð13Þ1ZXm˛MP u;e k u;e;m;u˛U r;e˛E uð14ÞA refinery complex contains variety of process units and util-ity equipments,and all of them require regular maintenance toenhance their reliability and to avoid emergency shutdowns.Nevertheless,plant maintenance is unfavorable on productionsuspension,huge capital and human resources involvement.Italso affects the material and utility balances between the processand utility plants.An effective maintenance planning shouldhence be able to balance all the above factors and provide a fea-sible and economical result[10].To optimize the maintenanceplanning Eq.(15)is considered.CARD(T)is the number ofperiods in the planning horizon.Blend headers are excludedfrom Eq.(15)only because they have stand-by headers.XtUO u;t%CARDðTÞÿf u;u˛U r W U tð15ÞIn general,when the unit changes operation modes,change-over costs are involved.To take this into account,a0e1vari-able,UX u,e,e#,t denoting the production changeovers,isintroduced.That is,UX u,e,e#,t should take the value of1ifand only if unit u operates with operation mode e duringtime horizon t(EO u,e,t Z1),and it operates with operationmode e#during the following time horizon t C1(EO u,e#,t Z1).This logical condition can be expressed bythe following proposition:EO u;e;t^EO u;e0;t5X u;e;e0;t;u˛U r;e;e0˛E uð16ÞWe can derive the following inequalities to represent theabove condition as Raman and Grossmann presented[11].Table1Prices of raw materials and products(RMB/T)T1T2T3T4T5T6T7T8T9T10T11T12 Crudeÿ1119ÿ1176ÿ1167ÿ1367ÿ1576ÿ1585ÿ1511ÿ1554ÿ1575ÿ1684ÿ1554ÿ1684 Methanolÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984ÿ2984 Electricityÿ0.5ÿ0.5ÿ0.6ÿ0.6ÿ0.6ÿ0.6ÿ0.7ÿ0.7ÿ0.7ÿ0.5ÿ0.5ÿ0.5 LPG154213651286161518492008202122492336226622492155 CLO129812981393161117881805175317551774193517551937 Coke000000000000 Propene254026922987364133583301335833763911383933763931 Propane230723072018187719852071213622112341239322112393 90#G162316231741201422352225219121942218241921942423 93#G172017201845213523692359232323262351256323262568 97#G181718171949225625042492245424582485271024582714 0#D161416141732197921972187215221562180237221562376 N20#D179217921923219624382427238924362420263324362638 N35#D185618561992227625262515247524802507272824802733 FG131113231420162318011793176517681787194517681948 Table2Optimal result of purchased raw materials(kT/period,kKW h/period)T1T2T3T4T5T6T7T8T9T10T11T12Sum Crude500.0123.9500.0500.0400.0500.0500.0500.0500.0500.0500.0500.05523.9 Methanol 1.90.0 1.9 1.9 1.9 3.80.0 1.9 1.9 3.80.0 1.920.6 Electricity42671747461640161148242933743355399237834657337840762 444 B.J.Zhang,B.Hua/Journal of Cleaner Production15(2007)439e4488<:EO u ;e ;t C EO u ;e 0;t C 1ÿ1%X u ;e ;e 0;tEO u ;e ;t R X u ;e ;e 0;t EO u ;e 0;t R X u ;e ;e 0;t;u ˛U r ;e ;e 0˛E u ð17ÞRefinery units are energy sinks,and also are energy sour-ces.Most units produce fuel oil and gas,even some high pres-sure steam,on the same time,they consume high,medium and low pressure steam,power and electricity.In traditional plan-ning model,energy consumption of a unit equals the product of the throughput and fixed specific energy consumption with-out considering the different operation mode [7].But in fact,energy consumption is not only related to the throughput,but also to the selected operation mode of unit and season.In this paper,a new method of calculating energy consumptionand generation for units is formulated as Eqs.(18)and (19),respectively.Energy consumed by units:ED u ;e ;m ;t Z p u ;e ;m ;t EF u ;e ;t C j u ;e ;t ;u ˛U r ;e ˛E u ;m ˛M w W M nð18ÞEnergy generated by units:EG u ;e ;m ;t Z l u ;e ;m ;t EF u ;e ;t C m u ;e ;t ;u ˛U r ;e ˛E u ;m ˛M w W M nð19Þwhere the first term indicates that the operation modes of a unit have different specific energy consumption.The second term is the influence of different seasons on energy consump-tion.We all know that the energy consumption in winters andTable 3Optimal results of sales (kT/period)T1T2T3T4T5T6T7T8T9T10T11T12Sum LPG 27.500017.8019.737.8105.121.133.927.3290.2CLO 07.6000.20.20000008Coke 23.2 5.822.522.515.922.522.522.522.52223.122.5247.5Propene 00031.7000033.110.3014.289.3Propane 2.10 3.5 2.6 2.60 5.300.5 2.6 1.5 2.122.8Loss 2.90.8 2.7 2.7 1.9 2.8 2.7 2.6 2.6 3.4 5.4 2.633.190#G 00097.548.7 1.400077.700225.393#G 48.310707*********.4707058.670741.397#G 94.81093.510.310.893.594.698.71196.999.198.5811.70#D 136.50208.5171.5168.6241.8225.3225.3154.2191.773.7248.22045.3-20#D 34.61026.526.524.200021.52936.432.2240.9-35#D 33.51021.515.5000030.622.825.533.5192.9FG 0000000000000FO 025.93027.822.654.828.128.528.327.927.728.2329.8Sum403.480.1478.7478.6383.3487468.2479.8479.4575.4384.9579.35278.1Table 4Throughput of units,fuel consumed by boilers and steam passing turbines (kT/period)T1T2T3T4T5T6T7T8T9T10T11T12CDU1300.00.0300.0300.0350.0350.0300.0300.0300.0300.0300.0300.0CDU2200.0123.9200.0200.00.0200.0200.0200.0200.0200.0200.0200.0Reformer 0.00.00.00.00.00.00.00.00.00.00.00.0FCC1100.082.9100.0100.00.0100.0100.0100.0100.0100.0100.0100.0FCC2180.00.0180.0180.0180.0180.0180.0180.0180.0180.0180.0180.0DH 45.80.032.340.634.032.927.627.640.635.953.827.8GFU10.00.09.30.00.00.00.00.00.00.00.00.0GFU223.70.030.030.030.030.030.0 5.80.030.017.423.7MRBE 15.00.015.015.015.015.015.015.015.015.015.015.0GUU017.80.07.87.87.87.87.87.87.87.87.87.8GUU027.80.07.87.87.87.87.87.57.87.87.87.8GUU030.00.00.00.00.00.00.00.00.00.00.00.0GUU04 1.00.50.8 1.00.0 3.50.30.00.10.7 1.20.2GUU05 3.9 3.9 3.9 3.90.0 3.9 3.9 3.9 3.9 3.9 3.9 3.9GUU06 3.90.0 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9TUU01100.00.0100.0100.00.0100.00.096.1100.0100.095.329.8TUU0270.70.072.072.072.00.00.072.00.00.072.00.0TUU030.00.0 5.6 1.6 6.355.072.00.00.00.00.072.0TUU040.00.00.00.00.045.031.90.045.0 2.40.00.0TUU050.027.80.00.072.00.072.00.019.072.00.072.0TUU060.00.00.00.00.00.00.00.00.00.00.00.0445B.J.Zhang,B.Hua /Journal of Cleaner Production 15(2007)439e 448summers is very different.p u ,e ,m ,t and l u ,e ,m ,t are regression pa-rameters from a large history data.3.4.Blending constraintsBlending is different from other process units of the refin-ery in the sense that it is only a mixing process.There is no chemical reaction or separation involved in blending.Blending is the process of blending several blending stocks that are pro-duced in upper stream units or purchased from the market to make several grades of gasoline or diesel according to the specifications.The objective of blending is to allocate the available blending components in such a way to meet product demands and specifications at least cost and to produce prod-ucts which maximize the overall profit.So blend header is not only material balance expressed by Eq.(20),but also property constraints.Eq.(21)forces the properties of blending products must meet the product specification.Xm ˛MF u ;eEC u ;e ;m ;t ZXm ˛MP u ;eEP u ;e ;m ;t ;u ˛U u ;e ˛E u ð20ÞXm ˛MF u ;ef ÿ4m ;a ÁEC u ;e ;m ;t %ðor R ÞXm ˛MP u ;ef ÿ4m ;a ÁEP u ;e ;m ;t ;u ˛U u ;e ˛E uð21Þwhere the function f is to make the property change of mixing process a linear relationship [12].3.5.Utility modelA boiler for steam generation can be modeled as follows:EP u ;e ;m 0;t Zh uðH m 0ÿH m $ÞX m ˛MF u ;eðH m EC u ;e ;m ;t Þ;e ˛E u ;u ˛U t ;m 0˛MP u ;e ;m $Z BFWð22ÞIt is assumed that effective enthalpy H m of fuel m ,enthalpy H m #of steam m #,enthalpy H m $of boiler feed water and effi-ciency h u of boiler are constant.Thus,the steam generation EP u ,e ,m #,t is a linear function of the fuel consumption.Fuel used in the boils,including residual fuel oil and fuel gas,is supplied from refinery units.Hence,the fuel consumption in the boilers is part of the overall fuel balance in the overall re-finery planning optimization model.Back pressure turbines and extraction turbines are consid-ered in this paper.For a turbine,the pass-out steam flow can be modeled as follows:XmEP u ;e ;m ;t Z EC u ;e ;m 0;t ;e ˛E u ;u ˛U t ;m ˛MP u ;e X M w ;M 0˛MF u ;e X M w ð23ÞEP u ;e ;m 0;t Z h uH m $EC u ;e ;m $;t ÿXm 0H m 0EP u ;e ;m ;t !;e ˛E u ;u ˛U t ;m 0˛MP u ;e X M n ;M ˛MP u ;e X M w ;m $˛MF u ;e X M wð24ÞEq.(23)indicates that the consumed steam of a turbine equals the sum of the extracting steam and condensingwater.Sum Purchased ProducedPeriod123456789101112160001200080004000E l e c t r i c i t y (K K W .H )Fig.6.Optimal electricity purchase and production.LPS MPSPeriodS t e a m (K T)122001501005003456789101112Fig.7.Medium and low pressure steam consumed by process units.446 B.J.Zhang,B.Hua /Journal of Cleaner Production 15(2007)439e 448。