Article pubs.acs.org/IECR
Effective Solution Approach for Integrated Optimization Models of Refinery Production and Utility System Hao Zhao, Gang Rong,* and Yiping Feng State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, P.R. China S Supporting Information *
ABSTRACT: A typical refinery consists of the production system and the utility system that are routinely optimized in a hierarchical procedure. To meet the demand for higher profit and energy utilization, it is imperative to integrate the two systems for refinery-wide simultaneous optimization, although the integrated model results in a mixed-integer nonlinear programming (MINLP) problem in which the bilinear terms for the correlation of the blending process and gas emission directly result in inconsistency between solution quality and time. The main aim of this work is to propose a solution strategy based on heuristics to decompose the integrated model of the two systems into a mixed-integer linear programming (MILP) model and a nonlinear programming (NLP) model which are then solved iteratively through variables transferring to further reduce the solution time. The solution of the traditional sequential method is incorporated to generate better initial estimates for the decomposed model to gain better solution quality and efficiency. The proposed solution approach is compared with the basic sequential method and the standard MINLP solvers. The results obtained in two scenarios of a case study from a real refinery demonstrate the effectiveness of the proposed decomposition strategy. a flexible utility system.13 Improved models of steam and gas turbines were incorporated into an overall model for a utility cogeneration system.14 Environmental costs15 and equipment failures16 were considered, respectively, in operational optimization of the cogeneration system. The analysis of steam power plant performance and retrofit management were also incorporated to improve the refinery site operation.17 Moreover, the scheduling of a fuel gas system was also investigated to improve energy efficiency in a refinery.18 Upon the above research, the traditional approach to the refinery-wide planning optimization of the production system and the utility system follows a hierarchical approach. The production system is optimized first to obtain the optimal allocation of products and process flows to gain efficient use of raw material. Then, based on the production planning results, the total energy demands of the processing plants are calculated. Attaining the utility demands, the utility system is optimized to operate the utility equipment to minimize the total energy cost. Thus, the relationship of the production systems and the utility system is separated in the traditional method and can be described as rather “master and slave” than united equally.19 The interaction between the two systems has been neglected in the traditional optimization method. On one hand, the utility system is superstructured, mainly consisting of the steam network under different grades, and the utilities are often shared by multicomplexes, which make it difficult to be integrated into the whole system optimization for one complex. Also, the interaction between the two systems is inconvenient
1. INTRODUCTION Oil refining industries account for an important part of global energy consumption, and the main part of the energy consumption of oil refining is utility generation. A typical refinery consists of the production system aiming at producing refining products to meet the market demand and the utility system carrying out tasks in providing energy to the production system.1 During the past decades, planning optimization on the refinery production system and the utility system planning has been investigated intensively in plenty of research, respectively. The typical refinery production planning model was presented,2,3 aiming at optimizing unit production and product inventories. Nonlinear empirical models for the crude distillation unit (CDU) and the fluid catalytic cracking unit (FCC)4,5 were developed, and a more mechanical CDU model of fractionation index unit6 was proposed in formulating optimization models of refinery-wide planning. Multiple time periods7 and decision influence of energy and emission management8 were also considered in a refinery planning site. The improvement was achieved in refinery planning and scheduling optimization, considering the more precise model formulation of the distillation units based on the conventional swing-cut modeling for choosing the best solution.9 A novel technique was proposed using monotonic interpolation to optimize both the recipes of the blended material and its blending component distillation for petroleum fuels.10 Meanwhile, studies on the utility system have focused mainly on the operational optimization, assuming the energy demands are given or can be predicted. The classical operational optimization model of the utility system of the refinery has been presented11 and proposed later,12 focusing on optimizing the equipment operation. The method of thermodynamic analysis and mathematical analysis was introduced in designing © 2015 American Chemical Society
Received: Revised: Accepted: Published: 9238
February 20, 2015 July 29, 2015 August 31, 2015 August 31, 2015 DOI: 10.1021/acs.iecr.5b00713 Ind. Eng. Chem. Res. 2015, 54, 9238−9250
Article
Industrial & Engineering Chemistry Research
Figure 1. Interaction of material and energy in a typical refinery flow diagram.
utility balance, and the utility system is optimized under given demands. Actually, in an oil refinery, the refinery production system not only produces gasoline and diesel by consuming energy from the utility system but also produces some byproducts. Byproducts provide the utility system as fuel resources that can be in the forms of fuel oil and fuel gas. Meanwhile, the utility
to define in mathematical formulation and then neglected in the traditional optimization method. On the other hand, due to the low cost for energy in earlier ages, the economic benefit of energy system optimization is not that important as the production system. This has led to an optimization situation where the production system is optimized without considering 9239
DOI: 10.1021/acs.iecr.5b00713 Ind. Eng. Chem. Res. 2015, 54, 9238−9250
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Industrial & Engineering Chemistry Research
flow and energy flow interaction and gives a definition of the problem which specifies the nonlinearity of MINLP model between the blending process of production system and fuel consumption of boilers of the utility system. Section 3 presents a mathematical formulation of the correlation model from which the bilinear problem originates. Section 4 outlines the proposed solution approach to the integrated model involving the decomposition strategy for MINLP model and parameters transferring strategy in solution iteration process. Finally, the proposed solution strategy is implemented in two scenarios, and the solution results are discussed compared with several achieved solution methods.
system needs to provide consistent steam and electricity to all production plants. Moreover, gas emission from the steam generation process of boilers is also associated with the fuel property that is closely related to the mixing process of fuel oil and gas in the production system. Without taking those interactions between the two systems into consideration, the traditional hierarchical approach may result in a suboptimal decision in a production system site or even infeasible results in a utility system site. In the current global market, the pressure for the improvement of energy utilization efficiency and more strict environmental restrictions have united to force the enterprise to enhance the existing optimization approach for the refinerywide planning. Due to significant interactions between the two systems, it is imperative to consider the simultaneous optimization of the two systems for the global optimal solution. However, this simultaneous optimization problem leads to a large-scale model and makes it an arduous task to solve when a multiperiod planning horizon is taken into account. Some pioneer research has been done on the aspect of incorporating both systems on planning and operational scheduling recently by developing integrated models. A specific frame to integrate heat and power is proposed to get more optimization space in batch and semicontinuous machining processes.20 An MILP model was proposed to optimize the material flow of a refinery along with the steam power system with the linear yield models of production and cogeneration units.21 The resource task network (RTN) representation was used in characterizing the relationship between manufacturing system and utility system, and an integrated MILP model was presented and solved.22 Though focusing on the plant site, Zhang et al. developed a new approach to integrating the operation condition of distillation and heat recovery effect in a crude oil distillation unit.23 An integration scheme for process plants and the utility system on the site-scale steam integration was proposed to attain energy utilization efficiency.24 Several previous research has been focusing on the bilinear problem caused by the integration of process network with multicomponent streams25 and the crude oil scheduling problem for a particular MILNP model. 26,27 Also, a mathematical decomposition approach was addressed28 and proposed29 for integrating planning and scheduling optimization for the multisite and multiproduct plants. A two-step MILP-NLP procedure has been incorporated to solve the nonconvex MINLP model that is used for the crude-oil operations problem.30 A nonconvex nonlinear mixed-integer problem for process synthesis of refinery production system was decomposed into a two-stage stochastic programming model and solved iteratively.31 In the former research, an integrated model of the production system and the utility system model was presented to achieve higher profitable product interest and save more energy costs.32 Due to the model complexity, a preprocessing solution strategy is proposed to attain better initial estimates for the integrated model. Nevertheless, the nonconvexity and the bilinear relation in the model still cause difficulties in the solving process and make the solution time suboptimal. To tackle this problem, a model decomposition strategy and an iteration solution algorithm are proposed and implemented in this work based on the preceding progress. The rest of this work is outlined as below. Section 2 provides the process description of both the production system and the utility system of a typical refinery considering both material
2. PROBLEM STATEMENT For a given refinery, different material flow, multiutility flow, and the complex interaction between them can be shown in Figure 1. For the production system, whose flowchart is mainly outlined in the upper section of Figure 1, it not only produces gasoline, diesel, and so on by consuming energy from the utility system, but it also produces some byproducts. The crude oil is divided into some straight-run fractions including light straightrun naphtha (LSR), heavy straight-run naphtha (HSR), raw kerosene (RKER), light gas oil (LGO), atmospheric gas oil (AGO), vacuum gas oil (VGO), and residues (RED) in the crude distillation unit (CDU). The crude oil distillation unit consists of the atmospheric distillation unit (ADU) for separation of lighter components and the vacuum distillation unit (VDU) for heavier components. Portions of AGO from ADU is fed to VDU to produce VGO and RED. RED is processed through the delayed coking unit (DCU) and then fed to the fluid catalytic cracker (FCC) with VGO and portions of AGO to produce the cracked gasoline (CG), the crack gas oil (CGO), and the FCC fuel oil (FOF). HSR is transferred to the catalytic reforming unit (CRU) to produce reformer gasoline (RG). RKER is sent to the hydro-desulfurization unit (HDS) to produce final product kerosene (KER) within quality specification. The part of CGO is consumed by the hydrotreating unit (HT) to produce the diesel (DIE) and the HT fuel oil (FOH). LSR and RG are then blended with MTBE for 90# gasoline production, whereas portions of HSR and CG are blended with MTBE for 93# gasoline production in the gasoline blender. Respectively, portions of LGO, AGO and the part of DIE are blended as −10# diesel product, and the part of CGO and the part of DIE are blended as 0# diesel product in the diesel blender. The fuel gas (FG) and fuel oil (FO) produced by different units as byproducts are collected separately and can be used as the fuel resource of the utility system. Fuel gas produced by the processing unit DCU, FCC, HT, and CRU as a byproduct often contains a high content of sulfur and cannot be directly used as a fuel resource. Portions of fuel gas with high sulfur content (SFG) is required to be processed in the gas desulfurization unit (DS) before it is transferred to the gas tank as fuel gas with low-sulfur content (LSFG). Some byproducts as fuel gas and fuel oil can provide the utility system as fuel resources. The final products include two kinds of gasoline (90# gasoline and 93# gasoline), KER, and two kinds of diesel (−10# diesel and 0# diesel). The gasoline blending process requires the octane number of 90# and 93# gasoline products to be higher than 90 and 93, respectively. On the other hand, the diesel blending for diesel blending require the pour point of −10# and 0# diesel to be lower than −10 and 0 °C, respectively. 9240
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system and the utility system in a typical refinery is introduced briefly as below, and more elaboration is put on the correlation of two systems from which the nonconvexity of the integrated model originated. 3.1. Production Planning Model. In accordance with the process description in section 2 regarding the process from the crude oil purchase to final product production, the mathematical model is formulated as below.
Respectively, the utility system needs to provide consistent steam and electricity causing operation cost and fuel material costs to all production plants. It consists of five boilers (Bl) and six turbines (Tb). The boilers are supposed to consume fuel oil, fuel gas, or natural gas (NG). Portions of the high-pressure steam (HS) and the medium-pressure steam (MS) are consumed by the turbines to generated electricity (EL), which can be delivered to the power grid and consumed by the production process and to produce the low-pressure steam (LS) to meet the utility demand of production system. The part of the steam is consumed by the production units depending on the different requirement for steam pressure and quantity. The steam valve is used to transfer the higher pressure steam to lower pressure steam. In the case of the condition that the steam demand exceeds the steam generation capacity, highpressure steam is allowed to be purchased from the outside steam network. The remaining unused steam is condensed by cooling water, then recycled to the deaerator, and finally reused as boiler feedwater once the steam supply exceeds the processing demands. The system formulation of the nonlinear problem in this research is illustrated in Figure 2. The intermediate products as
DPcup, t ≥ SCc , t ≥ DPclo, t
∀ c ∈ CP , t
(1)
MIclo, t ≤ MIc , t ≤ MIcup, t
∀ c ∈ CV , t
(2)
MIc , t = MIc , t − 1 +
∑ ∑
∑ ∑
FPu , m , c , t −
u ∈ UP m ∈ MU
FCu , m , c , t
u ∈ UP m ∈ MU
(3)
∀ c ∈ CV , t
MIc , t = MIc , t − 1 +
∑ ∑
FPu , m , c , t − SCc , t
∀ c ∈ CP , t
u ∈ UBL m ∈ MU
(4)
∑ c ∈ CIu , m
FCu , m , c , t =
∑
FPu , m , c , t
∀ u ∈ UBL , m ∈ MU , t
c ∈ COu , m
(5)
∑
PIc , pFCu , m , c , t ≥
c ∈ CIu , m
∑
FPu , m , c , tPLc , p
c ∈ COu , m
∀ u ∈ UBL , m ∈ MU , p ∈ P , t
∑ c ∈ CIu , m
PIc , pFCu , m , c , t ≤
∑
(6)
FPu , m , c , tPUc , p
c ∈ COu , m
∀ u ∈ UBL , m ∈ MU , p ∈ P , t
(7)
Eq 1 presents the demand constraints where DPc,t indicates the market demand of product c of period t and SCc,t defines the amount of commodities sold (final product). Material inventory balance for final products including fuel gas and fuel oil is illustrated in eqs 2−4 where FPu,m,c,t and FCu,m,c,t denote the amount of commodities c produced and consumed, respectively. MIc,t indicates the material inventory for commodity c of period t. Eq 3 presents the inventory balance for intermediate products that can be not only produced by certain processing units but also consumed by other processing units. Inventory balance for final products is presented in eq 4 where the imported inventory from the production by the blender and the exported inventory for sale are defined. eq 5 illustrates the material balance of the blending process for two gasoline blenders (90# gasoline and 93# gasoline), kerosene tank, two diesel blenders (−10# diesel and 0# diesel), and fuel oil blender. Eqs 6 and 7 define the inequality constraints for blending process, and several properties (p) (e.g., octane number, pour point, sulfur content, and carbon content) of the intermediate product are introduced. PLc,p and PUc,p represent the lower and upper bound, respectively, for the blending property requirement of the gasoline and diesel products. For example, 90 and 93, for the octane number (p) of 90# gasoline (c) and 93# gasoline (c) as PLc,p, while −10 and 0, for the pour point (p) of −10# diesel (c) and 0# diesel (c) as PUc,p. PIc,p represents the property p of intermediate product c.
Figure 2. Correlation problem of blending process with fuel consumption.
the fuel oil and the fuel gas are produced by the processing units as byproducts and then delivered from tanks to blenders. The mixing process involves not only the material balance of unit consumption and production but also the property balance of certain content. The fuel mix, either from the oil blender (for fuel oil blending) or from the gas tank (for fuel gas mixing), is then fed into boilers in the utility systems as fuel resources to generate steam. During the process of steam generation, the amount of polluted gas emitted by the boiler is determined by the certain property (e.g., sulfur content and carbon content) of mixing fuel. The number of boilers is not limited to 2 (5 for the example in this research). Thus, the calculation of fuel mixing correlated with gas emission generates nonconvex bilinear constraints of the integrated model.
3. MATHEMATICAL FORMULATION OF INTEGRATED MODEL In previous work, an integrated planning model of refinery production system and utility system has been formulated including a production planning model and an operational planning model of utility system model respectively as well as the correlation model between the two systems.32 Due to space restriction, the integrated planning model for the production
xu , m , t FUulo, t ≤ FFu , m , t ≤ xu , m , t FUuup, t ∀ u ∈ UP , m ∈ MU , t 9241
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∑
xu , m , t = yu , t
(9)
m ∈ MU
FUulo, t ≤ FUu , t ≤ FUuup, t
∑
FFu , m , t = FUu , t
∀ u ∈ UP , m ∈ MU , t
∑
(10)
∀ u ∈ UP , t (11)
m ∈ MU
FFu , m , t =
calculated as piecewise linear functions of the maximum amount of fuel c consumed by boiler u in piece q (XFBq,u,c,t) and the maximum amount of steam generated (XSBq,u,c,t), 0−1 variable of piecewise segments (Aq,u,c,t), and the continuous 0−1 variable of boiler u (βq,u,c,t) in eqs 14 and 15. The last two are optimization variables in the function. Only one piece region can be valid and only one fuel resource can be consumed by each boiler in each period as expressed in eq 16. The value of continuous variables βq,u,c,t is restricted to vary from 0 to 1 in eq 17. Eq 19 indicates the energy balance constraints for boiler where Hstm and Hwat suggest the enthalpy values of saturated steam and water, respectively. Hc infers the enthalpy values of fuel c, and ηu,c means the efficiency of boiler u consuming fuel c. The nonlinear relationship between the steam generation and fuel consumption originates from the varying boiler efficiency ηu,c under different unit load. Figure 3 presents the relationship
∀ u ∈ UP , t
FCu , m , c , t
∀ u ∈ UP , m ∈ MU , t (12)
c ∈ CIu , m
FPu , m , c , t = αu , m , cFFu , m , t ∀ u ∈ UP , m ∈ MU , c ∈ COu , m , t
(13)
Production models for processing units are defined in eqs 8−13 where FUu,t denotes the unit flow rate, FUlou,t and FUup u,t infer the lower and upper bound of unit capacity, and FFu,m,t indicates the flow rate of unit u with operation mode m. xu,m,t is a 0−1 variable that implies whether processing unit u is on with operation m. yu,t is a 0−1 variable that not only suggests whether processing unit u is on of period t but also limits xu,m,t, which implies only one operation m of the processing unit allowed. The flow rate of the processing unit on operation m is limited by eq 8. Eq 9 restricts the operation condition for the processing unit that implies only one operation m of the unit u is allowed in each period t. eq 11 defines the throughput of each processing unit as the sum of the throughputs under all the operation modes. The throughput of the processing unit on each operation mode is equal to the sum of material consumption accordingly as expressed in eq 12. The product yield αu,m,c is given as fixed parameters in eq 13 where the material balance of unit output quantity FPu,m,c,t and unit processing quantity FFu,m,t is expressed under a particular operation mode. 3.2. Utility System Model. The operational planning model of the utility system mainly consists of process models of boilers and turbines and the balance constraints of steam demand and supply, as shown in the following equations. FBu , c , t =
∑ (Aq , u , c , t XFBq− 1, u , c , t
Figure 3. Piecewise approximation for nonlinear relationship of fuel consumption and steam generation of boilers.
between the boiler steam ratio of the generation capacity and the fuel ratio of the consumption capacity using the piecewise linearization method where ηu,c is treated as constant in several segments. It can be shown that boilers show a higher efficiency of fuel utilization in a superior region of the unit load. The approximation method is also applied in quantifying steam consumption with the electricity generation of turbines in eqs 20−25 where STCu,c,t and ETu,t represent the amount of steam consumed and electricity generated by the turbine u in period t, respectively.
+ βq , u , c , t (XFBq , u , c , t − XFBq − 1, u , c , t ))
q∈Q
(14)
∀ c ∈ CF , u ∈ UB , t
SBu , c , t =
∑ (Aq , u , c , t XSBq− 1, u , c , t
+ βq , u , c , t (XSBq , u , c , t − XSBq − 1, u , c , t ))
q∈Q
(15)
∀ c ∈ CS , u ∈ UB , t
∑ ∑
Aq , u , c , t ≤ 1
q∈Q
∀ u ∈ UB , t (16)
q ∈ Q c ∈ CF
0 ≤ βq , u , c , t ≤ Aq , u , c , t
∑ ∑
∀ c ∈ CF , u ∈ UB , q ∈ Q , t
Aq , u , c , t
ETu , t =
(21)
∀ u ∈ UT , t
∑ Bq,u ,t
(18)
≤1
∀ u ∈ UT , t (22)
q∈Q
FBu , c , t Hcηu , c = (Hstm − Hwat )SBu , c , t ∀ c ∈ CF , u ∈ UB , t
∑ (Bq , u , t XETq − 1, u , t + γq , u , t(XETq , u , t − XETq − 1, u , t )) q∈Q
∀ u ∈ UB , t
q ∈ Q c ∈ CF
(20)
∀ u ∈ UT , c ∈ CS , t
(17)
yu , t =
∑ (Bq , u , t XSTq − 1, u , t + γq , u , t(XSTq , u , t − XSTq − 1,u ,t ))
STCu , c , t =
STCu , c , t = FUu , t
(19)
0 ≤ γq , u , t ≤ Bq , u , t
The piecewise linear approximation method is introduced to express the nonlinear relationship between fuel consumption and steam generation of boilers in eqs 14−18. The amount of fuel consumption FBu,c,t and steam generation SBu,c,t is
yu , t =
∑ Bq,u ,t q∈Q
9242
∀ u ∈ UT , t
(23)
∀ u ∈ UT , c ∈ CS , q ∈ Q , t
(24)
∀ u ∈ UT , t (25) DOI: 10.1021/acs.iecr.5b00713 Ind. Eng. Chem. Res. 2015, 54, 9238−9250
Article
Industrial & Engineering Chemistry Research The steam consumed STCu,c,t and the electricity generated ETu,t are calculated as piecewise linear functions of the maximum amount of steam c consumed by turbine u in piece q XSTq,u,t and the maximum amount of electricity generated XETq,u,t, 0−1 variable of piecewise segment Bq,c,t, and continuous 0−1 variable of boiler u γq,u,t in eqs 20 and 21. Eqs 18 and 25 restrict the operation status of the utility equipment as boilers and turbines, which implies only one fuel in one region of piecewise segment is allowed to be fed to the boiler u and only one region of piecewise segments is valid for the turbine u. yu,t represents whether the boiler or turbine u is on in the period t. STGu,c,t denotes the amount of steam extracted by the turbine u of period t which is assumed equal to the amount of steam consumption of the turbine in this research.
∑
∑
SBu , c , t +
u ∈ UB
STGu , c , t +
u ∈ UT
∑ ∑ EGu ,m,c ,t − ∑ u ∈ UP m
+ LSIc , t − LSOc , t ≥
∑ ∑ EDu ,m,c ,t
∑
∑
ETu , t ≥
STCu , c , t + EPc , t
u ∈ UT
∀ c ∈ CS , m ∈ MU , t
∑ ∑ EDu ,m,c ,t
∀ c ∈ CE , t
u ∈ UP m
(27)
EPc , t ≤ EPUc , t
∀ c ∈ CU , t
FPu , m , c , tPc , p (32)
COc = ωCO2f (Pc , p)
∀ c ∈ CF , p ∈ CP
(33)
SOc = ωSO2f (Pc , p)
∀ c ∈ CF , p ∈ CP
(34)
XCu , c ,t = ωCO2f (Pc , p)FBu , c , t
∀ c ∈ CF , u ∈ UB , t
(35)
XSu , c , t = ωSO2f (Pc , p)FBu , c , t
∀ c ∈ CF , u ∈ UB , t
(36)
As provided in eqs 29 and 30, the amount of energy consumption and generation of the production unit is defined in terms of linear ratio to the unit throughput, the property of input material, and the corresponding operation mode. λu,m,c and μu,m,c are the coefficient of utility consumed and generated respectively by unit u on operation mode m. EDu,m,c,t and EGu,m,c,t denote the amount of utility c consumed and generated by the processing unit u, respectively. ρu,c,p and σu,c,p indicate the coefficient of feed property p on the demand and generation respectively of utility c of unit u. Eq 31 presents the material balance of byproduct as fuel oil/gas produced by connecting the fuel inventory from the preceding period MIc,t−1, the total quantities consumed by boilers FBu,c,t and sold to the market SCc,t with the fuel inventory during the current period MIc,t, and the quantities produced by the production system FPu,m,c,t and purchased from the market PCc,t. The property balance of blending fuel products as fuel oil and fuel gas is expressed as eq 32. The blending product property is also associated with the conversion coefficient ωCO2 and ωSO2 of the fuel resources in terms of the linear relation expressed as eqs 33 and 34 where COc and SOc are fuel emission factor of CO2 and SO2 respectively. f() is the linear function to calculate the molar weight of the gas (e.g., SO2) from the corresponding fuel property (e.g., sulfur content). The amount of CO2 emission XCu,c,t and SO2 emission XSu,c,t of the boiler is defined in eqs 35 and 36, respectively, by associating fuel consumed FBu,c,t of the boiler with fuel property. f() is the same linear function for Pc,p in eqs 33 and 34. The integrated objective is given by eq 37, aiming at maximizing the total profit considering product sales revenue, material purchase, inventory cost, production unit operation cost, fuel purchase, utility purchase, maintenance cost of energy equipment, operation cost of steam valve, and emission cost for CO2 and SO2.
(26)
u ∈ UT
∑ c ∈ COu , m
∀ u ∈ UBL , m ∈ MU , p ∈ P , t
u ∈ UP m
EPc , t +
FCu , m , c , tPIc , p =
c ∈ CIu , m
(28)
Eqs 26 and 27 infer the steam demand constraints and the electricity demand constraints, respectively where EPc,t denotes the amount of utility purchased of period t. LSIc,t and LSOc,t represent the flow rate of steam consumed and extracted, respectively, by the letdown valve, and EGu,m,c,t infers the amount of utility c produced of period t of unit u on operation m. Eq 26 indicates that the steam generated by boilers SBu,c,t, turbines STGu,c,t and the processing units EGu,m,c,t as well as purchased externally EPc,t subtracting the steam consumption by the turbines and the difference between the steam imported by steam valve LSIc,t from the higher grade and exported by the valve LSOc,t to lower grade, should satisfy the sum of the steam demand of processing units EDu,m,c,t. Eq 27 restricts that the sum of the total electricity generated by turbines ETu,t and purchased externally EPc,t should meet the total demand of processing units. The utility purchase limit is presented in eq 28. 3.3. Correlation between Two Systems. As described above, the production system and the utility system of refinery are not only interacted in the material flow as fuel gas/oil but also in the energy aspect of utility consumption by processing units and generation from boilers and turbines as well as gas emission linking blending process to fuel burning in boilers. The correlation model is formulated as follows. EDu , m , c , t = λu , m , cFFu , m , t + ρu , c , p Pc , p + CDu , m , c ∀ c ∈ CU , p ∈ P , m ∈ MU , u ∈ UP , t
(29)
4. SOLUTION STRATEGY The full-scale integrated model formulated above assembling eqs 1−37 is indeed a large-scale MINLP problem, and the nonconvexity comes from the bilinear relation in calculating the mixing concentration associating the blending process with gas emission in eq 32 and eqs 35 and 36. Thus, the use of the decomposition method and solution strategy is required to solve the problem effectively. 4.1. Model Decomposition. The bilinear constraints cannot be solved in an MILP or NLP model. The reformulation
EGu , m , c , t = μu , m , c FFu , m , t + σu , c , pPc , p + CGu , m , c ∀ c ∈ CU , p ∈ P , m ∈ MU , u ∈ UP , t MIc , t +
∑
(30)
FBu , c , t + SCc , t = MIc , t − 1
u ∈ UU
+
∑ ∑ u ∈ UBL m ∈ MU
FPu , m , c , t + PCc , t
∀ c ∈ CF , t (31) 9243
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Figure 4. Model decomposition strategy.
Figure 5. Proposed solution strategy for integrated model.
of the bilinear equations is introduced by some researchers26 to get a relaxed LP model, and heuristics is suggested in solving the decomposed MINLP model to reduce the solution time when handling the crude oil scheduling problem.27 In this work, the integrated MINLP model is decomposed into an MILP model and an NLP model as illustrated in Figure 4, and the heuristic method is incorporated in model reformulation. Pfc,p is introduced to replace continues variable Pc,p and fix Pc,p as a parameter in eq 32 and eqs 35 and 36. Thus, the bilinear relation is eliminated leading the MINLP problem to an MILP problem. xf u,m,t, yf u,t, Afq,u,c,t and Bfq,u,t are introduced to replace discrete 0−1 variables of xu,m,t, yu,t, Aq,u,c,t and Bq,u,t, respectively, and fix them as parameters. Therefore, the 0−1 variables are eliminated, Pc,p remains unchanged as a continuous variable, and the MINLP problem is transformed into an NLP problem. The NLP model determines the blending property Pc,p
considering rigorous bilinear constraints expressed in eq 32, whereas all of the discrete variables are fixed as parameters with value obtained from the MILP model. In the MILP model, the blending property Pc,p remains the same as the value determined by the NLP model, although the other variables including the 0−1 “transfer variables” of xu,m,t, yu,t, Aq,u,c,t and Bq,u,t are optimized. The newly defined MILP model and NLP model are solved by CPLEX solver and CONOPT solver, respectively, in GAMS effectively. The initial estimate determination and value reassignment of Pfc,p in the MILP model and yf u,t, Afq,u,c,t and Bfq,u,t in the NLP model are illustrated in the next subsection. 4.2. Iteration Solution Strategy. Figure 5 presents the flowchart of the solution process based on heuristics, and the solving procedure is given as follows. 9244
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Industrial & Engineering Chemistry Research 1. Generating initial estimate from the sequential optimization method. Based on the product quantity demand, the production model assembling eqs 1−13 as an MILP model is initially solved with the objective maximizing the total profit for material production. Gaining the material flow rates, operation mode on units and product property of the production system, utility demands that are classified as steam and electricity are estimated according to eqs 29 and 30. The operational utility system model assembling eqs 14−28 as an MILP model is then solved by CPLEX solver with the objective minimizing the total operation cost. This hierarchical approach helps generate a good starting point for the NLP model in the next step. 2. Solving the integrated NLP model. The discrete 0−1 variables of xu,m,t, yu,t, Aq,u,c,t, and Bq,u,t determined by the former solving procedure of the sequential approach indicating the detailed operation condition for processing units and energy equipment are sent to the corresponding parameters xf u,m,t, yf u,t, Afq,u,c,t and Bfq,u,t in the NLP model, which is solved by CONOPT solver deriving the continuous variable Pc,p. 3. Iteration condition 1.The iteration is terminated and returns the result to the solution 1 if the relative difference between the objective values of the NLP model and the former MILP model (the sequential model in the beginning for the first iteration and the MILP model for following iterations) is smaller than an appropriate tolerance, or the objective value is lower than the former MILP model and the relative difference exceeds a certain tolerance. The former termination condition indicates that a satisfactory final solution has been obtained, and the latter termination condition stops the iteration result from worsening. Otherwise, the procedure goes on. 4. Continuous variables transfer. The continuous variable Pc,p determined in the NLP model from the former step is assigned to the corresponding parameter Pfc,p in the MILP model in the following step. 5. Solving the integrated MILP model. The MILP model is solved by CPLEX solver by fixing the value of parameter Pfc,p newly assigned by the preceding step while having the discrete variables of xu,m,t, yu,t, Aq,u,c,t, and Bq,u,t to vary. 6. Iteration condition 2. The iteration is terminated, and the results are sent to solution 1 if the discrete variables of xu,m,t, yu,t, Aq,u,c,t, and Bq,u,t determined by the MILP in the former step maintains unchanged from the corresponding parameters xf u,m,t, yf u,t, Afq,u,c,t in the NLP model of step 2 and the objective value obtained by the MILP model is smaller than that of the NLP model in step 2. Because if the solution obtained by the MILP model is greater than the NLP model, it means a better value of 0−1 variables exist that are previously fixed in the NLP and might lead to a better solution for the NLP model. Therefore, the corresponding parameters xf u,m,t, yf u,t, Afq,u,c,t, and Bfq,u,t in the NLP model need to be updated by the better value of the discrete variables xu,m,t, yu,t, Aq,u,c,t , and Bq,u,t newly attained in the MILP model. 7. Discrete variables transfer. If the check of step 6 is not satisfied, discrete variables xu,m,t, yu,t, Aq,u,c,t, and Bq,u,t determined in the MILP model of step 5 are reassigned to the corresponding parameters xf u,m,t, yf u,t, Afq,u,c,t, and Bfq,u,t in the NLP model of step 2 for the next iteration.
8. Iteration condition 3. If the value of FPu,m,c,t in eq 32 from the integrated model finally determined in the solution 1 is equal to the value of FPu,m,c,t in eq 5 attained from the sequential model in step 1, or the objective value obtained in the current loop is lower than or equal to the objective value of the former loop from solution 1, the procedure is terminated and returns to the final solution. 9. Initial estimates assignment for the sequential model. If the check in the previous step is not satisfied, the value of FPu,m,c,t in solution 1 is sent to the corresponding parameter FPf u,m,c,t to replace FPu,m,c,t in the sequential model to make it fixed in step 1. The other variables are assigned as the starting point for the sequential model for the model solving in the next iteration.
5. REMARKS The modeling system GAMS version 24.133 is used to implement the model. The system investigated in this research is a real industrial case of a refinery in China, which consists of two systems including the production and utility system as described in section 2. The planning horizon consists of 8 periods. Four operation modes are included in the CDU, three in the FCC, HT, DCU, and CRU, and two in HDS and DS. Product quality associated with energy consumption, unit yield and blending product involves API, octane number, pour point, sulfur content, and carbon content. The boilers are allowed to consume either fuel gas, fuel oil, or natural gas in each period. The production system is supposed to fulfill given demands of final products in each period. The original MINLP model consists of 5087 continuous variables, 904 discrete variables and 5095 constraints. The decomposed MILP model consists of 5087 continuous variables, 904 discrete variables, and 5105 constraints. The decomposed NLP model consists of 4185 continuous variables and 4919 constraints. Two scenarios are developed to provide further investigation on the effect of the problem solution. Scenario 1 and scenario 2 are presented on the condition that the capacity of energy generation of the utility system is within and exceeds the demand of the production system respectively in certain periods because of varying multiperiods market demand as shown in Figure 6 as the example of gas93.
Figure 6. Different demands situation of two scenarios (example of gas93). 9245
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presents a fast searching ability for a better solution. The procedure takes 2 iterations to obtain the final objective and stops in iteration 3 when the condition 1 is met. Note that the relative difference in iteration condition 1 is calculated between the objective value of the NLP model in current iteration and the MILP model (initial estimate for the first iteration) in former iteration and the relative difference tolerance in iteration condition 1 is set as 1.00 × 10−05. Comparative results of solution effect of scenario 1 are shown in Table 2.
To give a clearer illustration about the optimization results of the proposed model, the traditional sequential optimization method, the commonly used solver DICOPT (also applied in previous work) for MINLP problem as well as the global optimal solver BARON are applied to solve the problem and compared with the proposed decomposition solution strategy on the aspects of solution efficiency and solution results. Scenario 1. The market demand for all products from the production system is assumed within the production capacity of maximum energy generation of the utility system in this example. The detailed computational results are supplemented in the Supporting Information involving unit operation, inventory decision, energy utilization as well as the gas emission of scenario 1. Figure 7 shows the comparative solving results of proposed iterative approach compared with the other three methods.
Table 2. Solution Effect Comparison of Different Methods in Scenario 1 methods
optimality gap
sequential MINLP-BARON
0% 15%
MINLP-DICOPT
5%
proposed decomposition
0%
CPU time
objective
iteration number
5 min about 3 days 18 h 1 h and 34 min 45 min
6154185.3 6220366.2
---
6216366.2
--
6216170.3
4
The decomposed models of MILP model and NLP model in the proposed approach are solved with CPLEX solver and CONOPT solver, respectively. Because the CONOPT is a local solver, it can not necessarily guarantee the global optimum as BARON solver but can return a solution much faster than BARON. It can be concluded that the proposed method can achieve satisfactory solution results compared with DICOPT solver while consuming much less time in searching a considerably feasible solution. Note that although DICOPT can handle nonconvexities, it does not necessarily obtain the global optimum. Although the BARON is a global solver and achieves a little higher objective than the proposed method, it is not recommended for the original MINLP model due to the long solution time of 3 days 18 h consumed. Comparative results for the mixing amount of fuel oil and fuel gas in the blending process are displayed in Figure 8 and Figure 9, respectively.
Figure 7. Iteration objective values of proposed method compared with sequential approach and standard MINLP solvers of scenario 1.
Purple and green dash lines represent the objective value of the traditional sequential method and the former MINLP solver DICOPT, respectively. It can be seen that the proposed method reaches the final results quickly within 3 iterations gaining approximately the same value in profit compared with the standard MINLP solver of DICOPT in GAMS. The solution process of the proposed method stops in iteration 4 because the objective value of the proposed method is equal to the result of the iteration 3 and the termination prevents the deterioration trend. The solution results of proposed method in iteration 2 have an apparent growth due to the obvious increase in the initial estimate. This is because the blending streamflow FPu,m,c,t is updated in iteration 2 by the solution result of proposed method in iteration 1, which helps the sequential model generate a better initial starting point for the solution procedure of the proposed model in iteration 2. To show the solution effect for the inner loop of the proposed method, the solution results of the inner loop from iteration 3 in the outer loop are presented in Table 1 including the objectives of the MILP and NLP models and iteration condition. It can be noticed from Table 1 that the inner loop
Figure 8. Comparative results of mixing amount in fuel oil blender of scenario 1.
Table 1. Example Results of Inner Iteration in Outer Iteration 3 of Scenario 1
iteration 1 iteration 2 iteration 3
NLP objective
MLP objective
relative difference (condition 1)
initial estimate
condition 1
condition 2
6184470 6216165 6216170
6216085 6216170 \
1.32 × 10−03 1.29 × 10−05 0
6176294 \ \
no no yes
no no \
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Figure 10 shows the comparative solving results of the proposed iterative approach compared with the other three
Figure 9. Comparative results of mixing amount in fuel gas tank of scenario 1.
It can be illustrated by Figures 8 and 9 that the mixing decision of the proposed approach is relatively close to that of the previous DICOPT method while attaining more stable ratio in mixing quantity. Both the proposed solution and the standard DICOPT solution derive more production of fuel oil and fuel gas for supplying enough fuel resources to the utility system compared with the traditional sequential method. This is because that the fuel gas is the byproduct in the production system and not treated as products to be sold as gasoline and diesel. Then the production system model is solved upon the products demand of gasoline and diesel as well as fuel oil not taking the consumption of fuel oil and fuel gas from boilers in the utility system for generating steam into account. As a result, to maximize the total profit, the solution results of the sequential method present less fuel oil and fuel gas production while facing the situation when in the procedure of optimizing the utility system, more fuel oil might need to be purchased externally because of unmet fuel demand upon the short fuel supply from the production system. However, the integrated model for the production system and the utility system gives a trade-off between the market demand of products and the fuel production as demanded in the utility system on optimizing the production of both byproduct as fuel gas and products for sale as gasoline, diesel, and fuel oil. It can be also verified from Table 3 that more fuel gas is supplied to the utility system and less fuel oil is consumed in
Figure 10. Iteration objective values of proposed method compared with sequential approach and standard MINLP solvers of scenario 2.
research methods. It can be seen that the proposed method reaches the final solution within 2 iterations, gaining respectably higher value in profit compared with standard MINLP solver of DICOPT in GAMS. The solution process of the proposed method stops in iteration 3 and returns the final result of iteration 2 because the objective value of the proposed model in iteration 3 starts to deteriorate compared with that of the iteration 2. It can be concluded from Table 4 that the procedure in the inner loop of the proposed method takes three iterations for convergence reaching the final solution when the iteration condition 2 is met. Note that the iteration condition 1 in iteration 3 is not satisfied due to the unmet difference between the objective values of the NLP model and that of the MILP model in iteration 2 until the condition 2 in iteration 3 is satisfied when the MILP model obtains the same value. Comparative results of solution effect of scenario 2 in are shown in Table 5. It can be concluded from Table 5 that the proposed method can achieve higher solution results compared with DICOPT solver while consuming much less time in searching a satisfactory feasible solution. Still, it takes a considerably long time for the BARON solver to achieve approximately equal results with a gap of 15% to the global optimum. Comparative results for the mixing amount of fuel oil and fuel gas in the blending process are given in Figure 11 and Figure 12, respectively. In consistence with the results of scenario 1 in Table 3, it can be seen from Table 6 that more fuel gas and less fuel oil are consumed while more LS and EL are generated, and less MS is produced in the proposed method compared with the sequential method in scenario 2. Thus, the fuel gas produced by the production system is largely utilized as a fuel resource, at the same time less fuel oil is consumed in the utility system due to its value for sale in the market.
Table 3. Comparison Results of Fuel and Utility Consumption in Scenario 1 commodities/ton
fuel oil
fuel gas/NM3
MS
LS
EL/MW
sequential proposed
98.7 82.3
35493.2 49033.4
412.6 378.5
1539.3 1552.2
77.3 81.4
the proposed method compared with the sequential method in scenario 1. This is because the fuel gas is produced by the production system and has more value as fuel resource in the utility system than in the production system. Thus, more fuel gas rather than fuel oil is produced to be sent to the utility system. It can be also seen that more LS and EL are generated, and less MS is produced in the utility system in the proposed model than that of the sequential model due to the higher cost of MS. Scenario 2. The market demand for several products from the production system is assumed to exceed the production capacity of maximum energy generation in the utility system in particular periods in this example as explained at the beginning of this section. More computational results of scenario 2 for material and energy distribution are given in the Supporting Information.
6. CONCLUSIONS In this work, a novel model solution strategy is proposed to reduce the computational effort in obtaining better solution results of a large-scale integrated model for the refinery production and utility systems. The original nonconvexity of the integrated MINLP model is mainly caused by the bilinear terms originated from the complex interaction of material and energy flows between the refinery production and utility 9247
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Industrial & Engineering Chemistry Research Table 4. Example Results of Inner Iteration in Outer Iteration 2 of Scenario 2 iteration 1 iteration 2 iteration 3
NLP objective
MLP objective
relative difference (condition 1)
initial estimate
condition 1
condition 2
6839524 6892670 6893764
6890228 6893589 6893764
1.42 × 10−03 3.54 × 10−04 2.54 × 10−05
6.83 × 1006 \ \
no no no
no no yes
be drawn from the results that the proposed strategy can not only achieve satisfactory profitable interest by deciding the distribution of material and energy including the blending ratio for the fuel resources but also reduce the computational effort significantly through the combined utilization of the solution of the sequential model and the decomposed models.
Table 5. Solution Effect Comparison of Different Methods in Scenario 2 methods
optimality gap
sequential MINLP-BARON
0% 15%
MINLP-DICOPT
5%
proposed decomposition
0%
CPU time
objective
iteration number
9 min about 4 days 9h 3 h and 34 min 54 min
6806414.7 6901827.8
---
6871987.5
--
■
ASSOCIATED CONTENT
* Supporting Information S
6893764.2
3
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b00713. The detailed computational results involving unit operation, inventory decision, energy utilization for scenarios 1 and 2 are presented in the Supporting Information (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Tel: 086-87953145. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the National High Technology R&D Program of China (2013AA040701).
Figure 11. Comparative results of mixing amount in fuel oil blender of scenario 2.
■
NOMENCLATURE
Sets
C = set of commodities CC = subset of C of raw materials CE = subset of C of electricity CF = subset of C of fuel oil/fuel gas CIu,m = set of feed material of operation m on unit u COu,m = set of products of operation m on unit u CP = subset of C of production product CS = subset of C of high/medium/low pressure steam CU = subset of C of utility (high/medium/low steam/ electricity) CV = subset of C of inventorial commodities(production product) M = set of operation modes MU = subset of operation mode on unit Q = set of piecewise numbers of efficiency curve for boilers and turbines U = units (processing unit/boilers/turbines) UB = subset of U of boilers UBL = subset of U of blending headers UP = subset of U of processing units UT = subset of U of turbines UU = subset of U of utility equipment
Figure 12. Comparative results of mixing amount in fuel gas tank of scenario 2.
Table 6. Comparison of fuel and utility consumption in scenario 2 commodities/ton
fuel oil
fuel gas/NM3
MS
LS
EL/MW
sequential proposed
111.5 91.9
37530.7 53723.4
467.1 422.1
1686.6 1693.4
86.0 89.9
system. In this approach, the integrated model is decomposed into an NLP model and an MILP model, which are then solved iteratively by variable values fixing, relaxing, and transferring. The MILP model solution of the traditional sequential method is adopted as a heuristic method to gain a feasible solution for the integrated model as a good starting point. The proposed approach is investigated in an industrial example with two scenarios and compared with several common solution approaches for the optimization problem. The conclusion can
Indices
c = commodities lo = lower bound 9248
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Industrial & Engineering Chemistry Research m = operation mode p = property q = piecewise segment of efficiency curve stm = different pressure grade steam t = time period u = units (processing unit/boilers/turbines) up = upper bound wat = water Variables
Aq,u,c,t = 0−1 variable of piecewise segment of boiler Bq,u,t = 0−1 variable of piecewise segment of turbine EDu,m,c,t = amount of utility c consumed of period t by unit u on operation m EGu,m,c,t = amount of utility c generated of period t by unit u on operation m EPc,t = amount of utility purchased of t ETu,t = electricity generated by turbine u FBu,c,t = fuel c consumed by boiler u FCu,m,c,t = amount of commodities c consumed of period t of unit u on operation m FFu,m,t = flow rate of unit u of period t with operation mode m FPu,m,c,t = amount of commodities c produced of period t of unit u on operation m FUu,t = flow rate of unit u of period t LSIc,t = letdown valve flow rate to steam c LSOc,t = letdown valve flow rate out of steam c MIc,t = material inventory of c of period t Pc,p = property p of final product c PCc,t = commodities purchased of period t (crude oil, MTBE, electricity) Pr = overall profit SBu,c,t = steam c generated by boiler u SCc,t = commodities sold in period t (final product) STCu,c,t = steam c consumed by turbine u STGu,c,t = steam c generated by turbine u TC = total cost xu,m,t = 0−1 variable that denote whether processing unit u is on with operation m of t XCu,c,t = CO2 emission of unit u of period t XSu,c,t = SO2 emission of unit u of period t yc,t = 0−1 variable that denote whether production unit or the utility equipment u is on of t βq,u,c,t = continuous piecewise variable of boiler u in segment q from 0 to 1 γq,u,t = continuous piecewise variable of turbine u in segment q from 0 to 1
■
Hc = enthalpy value of fuel c Hstm = enthalpy value of saturated steam Hwat = enthalpy value of water ICc = inventory cost of commodity c pfc,p = parameter corresponding to Pc,p PIc,p = property p of intermediate product c PLc,p = product property p constraints of material c pric,t = price of material c of period t PUc,p = product property p constraints of material c Sec = emission cost coefficient of SO2 SOc = SO2 emission factor of fuel c T = time horizon XETq,u,t = maximum amount of steam that can be extracted from turbine u in q piece of period t yf u,m,t = parameter corresponding to xyu,m,t XFBq,u,c,t = maximum amount of fuel c that can be consumed by boiler u in q piece of period t XSBq,u,c,t = maximum amount of steam c that can be generated by boiler u in q piece of period t XSTq,u,t = maximum amount of steam that can be consumed by turbine u in q piece of period t yf u,t = parameter corresponding to yu,t αu,m,c = yield ratio of the material c of unit u on operation mode m λu,m,c = coefficient of utility consumed by unit u of operation m μu,m,c = coefficient of utility generated by unit u of operation m πu = operation cost of production unit (increasing with the throughput) ρu,c,p = coefficient of feed property p on the demand of utility c of unit u σu,c,p = coefficient of feed property p on the generation of utility c of unit ηu,c = efficiency of boiler u consuming fuel c ωCO2 = coefficient of CO2 emission factor with the carbon content of product ωSO2 = coefficient of SO2 emission factor with the sulfur content of product
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Afq,u,c,t = parameter corresponding to Aq,u,c,t Bfq,u,t = parameter corresponding to Bq,u,t CDu,m,c = the constant for utility material c demanded by operation mode m of unit u CGu,m,c = the constant for utility material c generated by operation mode m of unit u COc = CO2 emission factor of fuel c Cec = emission cost coefficient of CO2 delc = operation cost of letdown valve for steam c DPc,t = market demand of product c of period t Emcu = fixed maintenance cost for either production unit and utility equipment u when the operation is on EPUt = electricity purchased limitation of period t FPfq,m,c,t = parameter corresponding to FPu,m,c,t 9249
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