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Uncertainty Analysis for Refinery Production Planning Abdallah Al-Shammari* and Mohammed S. Ba-Shammakh Chemical Engineering Department and Center of Research Excellence in Petroleum Refining and Petrochemicals, King Fahd University of Petroleum & Minerals, P.O. BOX 5050, Dhahran 31261, Saudi Arabia ABSTRACT: Nowadays, oil refineries are facing a challenging task to optimize their production levels and increase their incomes. The objective of refinery optimization is to determine the best production levels that can maximize profit and minimize operating costs with respect to operational and environmental constraints. The existence of uncertainty or variations in some parameters such as the prices of raw materials and products can seriously affect the optimization results and lead to inefficient operation. Many techniques of stochastic programming were proposed to handle the effect of variations in process parameters in order to determine a robust operation conditions; however, applications of postoptimality analysis have received less attention, especially in the refinery industry. In this work, postoptimality analysis is used to study the effect of such variations on the optimal solution of refinery process that is simplified and formulated as a LP model. We used a modified method of postoptimality analysis that jointly use sensitivity relations and stability region calculations to provide the decision maker in the refinery with valuable and easy-to-use information that helps in handling the effect of variation in process parameters. The results of this study can help the decision maker to identify sensitive parameters that need accurate estimate or intensive monitoring; and compute their stability limits, allowable variation ranges, within which the operation remains optimum. Independent and simultaneous variations in the products prices or in the capacities of process units are considered in this study; and it is shown how the refinery profit can be increased up to 4.2% by utilizing sensitivity and stability results.
1. INTRODUCTION Petroleum refining is one of the most important industries, comprising many different and complicated processes with various possible configurations. Optimization of refinery production levels is an essential task to maximize company profit margins and to remain in the competitive market. Since 1955, many refinery models and optimization techniques have been proposed to handle this task. Symonds1 developed a linear programming (LP) model for solving a simplified gasoline refining and blending problem. The advantage of LP is its quick convergence and ease of implementation. Allen2 presented a more detailed model for a simple refinery that consists mainly of three units; distillation, cracking and blending. A nonlinear planning model for refinery production was developed by Moro et al.3 The model represented a general refinery topology and it gave better results compared to the current situation. Pinto and Moro4 also developed a nonlinear planning model for refinery production. The model described a general petroleum refinery and its framework allows for the implementation of nonlinear process models for few units as well as blending relations. Alhajri et al.5 presented a nonlinear model for refinery planning and optimization that offers a good level of accuracy for refinery processes and able to predict the optimum operation variables. A different refinery optimization strategy was proposed by Zhang et al.6 They presented a decomposition strategy for overall refinery optimization to tackle large scale optimization problems by decomposing the model into two levels. These levels were a site level (master model) and a process model (submodels). Li et al.7 presented a refinery planning model that utilizes simplified empirical nonlinear process models with considerations for crude characteristics, products’ yields, and qualities. The study integrated crude distillation, FCC, and product blending modules into refinery planning models. r 2011 American Chemical Society
Moreover, different optimization techniques were used to solve various types of refinery models. A novel decomposition strategy for solving mixed-integer linear problems was proposed by Shah et al.8 They decomposed the general mixed integer linear programming (MILP) problem into small subproblem to ease finding the global optimal solution. A nonconvex mixed-integer nonlinear refinery model (MINLP) was studied by Karuppiah et al.9 They presented an out-approximation algorithm to obtained the global optimal operation conditions of such complicated problems. Simo et al.10 used genetic algorithms in refinery scheduling optimization. On the other hand, other refinery optimization problems were concerned with different scopes such as environmental issues. Elkamel et al.11 developed a refinery planning model taking into consideration meeting the demand with CO2 mitigation and Ba-shammakh12 studied optimizing the refinery production with reduction of SO2 emission. However, it was recognized that most of these optimization results can suffer from the existence of uncertainty in model parameters because of inaccurate estimates of some parameters. In addition, variations in input data, such as supply and cost of raw materials, can easily affect the process profitability. Kraslawski13 classified uncertainty or variation in process design and operation into two types: ambiguity and imprecision. Also, uncertainty can be classified on the basis of the nature of the source of uncertainty in the process. Pistikopoulos et al. classified uncertainty as follows:14 • Model-inherent uncertainty due to inaccurate estimation of model parameters such as kinetic constants. Received: February 14, 2011 Accepted: April 19, 2011 Revised: April 10, 2011 Published: April 19, 2011 7065
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Industrial & Engineering Chemistry Research • Process-inherent uncertainty due to variations in process parameters such as temperature and flow rate. • External uncertainty such as changes in feedstream availability, as well as in demand and price of product. • Discrete uncertainty such as equipment availability. Many studies were conducted to understand the effect of such uncertainty or variation on the optimized model. There are two general approaches for dealing with the existence of uncertainty or variations in optimization problem: incorporating the uncertainty directly into the optimization problem formulation, such as stochastic programming, and analysis of the effect of uncertainty on the optimal solution, postoptimality analysis. Many techniques of stochastic and fuzzy programming were proposed to handle the effect of variation in process parameters in order to determine a robust operation conditions.15 Some of these techniques were applied in refinery optimization such as Leiras et al.16 In contrast, applications of postoptimality analysis have received less attention especially in refinery industry. Postoptimality analysis can help the decision maker to determine how much actual values of parameters may differ from the estimates used in the model before the optimal results become irrelevant. Generally, postoptimality analysis can provide the decisionmaker with valuable information about the process and answer many crucial questions such as: • How the optimal value and solution are affected by individual or simultaneous change(s) in process parameters? • What are the variation limits that do not change the optimality of the solution? • What are the sensitive constraints that need further monitoring? • What are the sensitive parameters that need accurate estimation? • How can we modify the process to make it more profitable or stable? In addition, many “what if” type questions can be answered by performing postoptimality analysis. Despite its value, applications of the postoptimality analysis in production planning have received less attention compared with stochastic programming. This may be due to the challenge of studying the effect of simultaneous variations in the model parameters. In addition, current state-of-art postoptimality analysis methods for different linear optimization problems (e.g., LP and MILP) are rarely used because of the high computational complexity. In this work, we proposed to use postoptimality analysis, mainly stability analysis, in order to determine how variations in model parameters can affect the optimal production levels of the refinery. A LP model of refinery is used as a case study. A stability analysis technique, modified tolerance approach, is applied to determine sensitive parameters that need accurate estimate or intensive monitoring; and compute their stability limits, allowable variation ranges, in order to maintain efficient plant operation.
2. POSTOPTIMALITY ANALYSIS After the optimal solution has been computed for a given model, it is important to know how the solution behaves under different variations in problem parameters. Analysis of the effect of uncertainty or variation on the obtained solution has been considered in many fields of engineering and operations research; therefore, different definitions are used to indicate such
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analysis. For example, the terms “post-optimality analysis”, “uncertainty analysis”, “sensitivity analysis”, and ”stability analysis”, are generally used to study the behavior of an optimal solution with respect to changes in problem parameters. Postoptimality analysis is a general term for understanding the effect of perturbations in problem parameters and it is taken to include both sensitivity and stability analysis. In contrast, sensitivity and stability analysis are more specific depending on the purpose of analysis and the character of variables. There are many definitions of sensitivity analysis and stability analysis in the literature and both are used to evaluate the effects of variations on the optimal solution or basis of the LP problem.17,18 2.1. Sensitivity Analysis. Consider a LP problem in form maxfP ¼ cT x : Ax e b, x g 0, x ∈ R n g
ð1Þ
sensitivity analysis is usually associated with the determination of the values of the Lagrange multipliers, λ, that describe the change in the optimal solution with respect to the variations in righthand side (RHS) coefficients. The sensitivity relations include the following: • for variations in vector c (objective coefficients) rc P ¼ x rc x ¼ 0 rc λ ¼ rc ðcT A A 1 Þ¼ A A 1 rc s ¼ 0
ð2Þ
where s is the vector of slack variables in the primal problem, s = b Ax*. • for variations in vector b (RHS coefficients) rb P ¼ λ ¼ cT A A 1 rb x ¼ A A 1 r c λ ¼ 0
ð3Þ
These sensitivity relations are important and useful for the decision maker, but the major challenge is to determine when they are valid. For example, the Lagrange multiplier, λi, presents the increase in the optimal value for a maximization problem when the associated RHS coefficient, i.e., bi, is increased by one unit; however, we do not know by how much the coefficient can be increased under simultaneous variations in vector b before the optimal basis changes and the value of the Lagrange multiplier becomes invalid. This shows the importance of computing stability limits for each coefficient under simultaneous variations and within which the optimal basis remains unchanged. 2.2. Stability Analysis. Stability analysis of an optimization problem concerns with computing the maximum variation limits in the problem parameters before the optimal solution or basis are changed. Computing these limits can help in understanding the effect of uncertainty or variation on the problem and utilizing available sensitivity information. During the last few decades, many stability approaches have been proposed for perturbations in vectors b, c, or/and matrix A. To date there is no single approach that dominates. Each approach has its strengths and weaknesses. The main challenge in stability analysis is considering simultaneous variations in different directions and computing a maximum allowable perturbation range for each parameter. Saaty and Gass19 introduced a one-dimensional cut approach, 7066
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known as ordinary stability analysis, that computes the maximum perturbation limit when only one coefficient is perturbed at a time. They also proposed higher-dimensional cuts that consider simultaneous perturbations, but it was be difficult to represent and construct the stability region of more than three-dimensional cuts. Bradley et al.20 indirectly used the concept of one-dimensional cut to propose the 100% rule that depends on the direction of change. The 100% rule is impractical for large problems because it considers only one direction of change at each computation. Gal21 introduced the concept of a critical region of perturbations, within which the optimal solution remains unchanged and used the Simplex method to compute the stability radius. This method also considers a single direction of changes. Many other approaches have been proposed for different types of perturbations or variations. Wendell22,23 proposed a Tolerance Approach (TA) that estimates the stability region and showed how this region can be expanded for variations in vector b or c of a LP problem. Wondolowski24 and Filippi25 modified the tolerance approach such that it computes a larger stability region for variation in vector b or c. Arsham26 proposed a different approach to represent the stability region as a set of parametric equations in order to compute the largest stability region for variation in RHS and objective coefficients. In contrast to other approaches, the tolerance approach leads to easy-to-use results within which the coefficients under the study can vary independently and simultaneously in any direction without changing the optimal basis. Al-Shammari and Forbes27,28 introduced another stability analysis approach that compute similar results to Filippi’s approach but in a more efficient and easy-to-use manner. In this project, we used the postoptimality approach proposed by Al-Shammari and Forbes27,28 to investigate the effect of variations in process parameters of a LP refinery model. In fact, most of the researches done on uncertainty or variation in refinery process were conducted using stochastic programming or other similar strategies; however, none of them investigate the effect of variation on the optimal solution using postoptimality analysis. To the best of our knowledge, no research has been found in the literature that dealt with postoptimality as applied to a refinery planning model. The proposed method provides a new perspective on the problem and has two steps for computing the stability region or limits. First, it defines the entire stability region as a cone and studies the relation between the sensitivity information, Lagrange multipliers, and model parameters. Second, it determines maximum stability limits presented by the maximum rectangular parallelepiped or hyperbox that can be built inside the cone. This hyperbox offers flexible and easy-to-use allowable variation limits as shown later on for variation in objective and RHS coefficients such as prices of raw materials and products. 2.2.1. Variations in Objective Coefficients. To demonstrate the approach, consider eq 1 that has a unique optimal solution. To define the entire stability region or stability cone for variations in the coefficients of the objective function, duality information or Lagrange multipliers are used rb P ¼ λ¼ cT A 1 A
ð4Þ
where AA is the matrix of active constraints. By introducing the perturbations vector, ΔcT λ0 ¼ ðcT þ ΔcT ÞA 1 A
ð5Þ
Figure 1. Obtained stability limits inside the stability cone.
by using the non-negativity condition on the optimal solution, there is no change in optimal solution if λ0 g 0. By substituting and rearranging ΔcT A 1 A e λ
ð6Þ
This inequality relation represents the stability region. This stability region can be defined as a stability cone because it satisfies the definition of a cone. In other words, the optimal solution and basis (not objective value) remain optimal under any scalar positive multiplication in the objective function. The solution remains optimal for any variations that satisfy eq 6. The stability cone is shown in Figure 1 for a maximization problem with two variables. Next step is defining the largest possible stability ranges starting with computing ordinary (individual) stability limits for Δci as follows: ! ! λg λg ord max Δch ¼ 1 :Δch < 0 < Δch < min Δch ¼ 1 :Δch >0 h h A Agh A Agh ð7Þ where h and g are indices of objective coefficients and constraints, respectively. In Figure 1, the ordinary stability limits of the coefficient ci are the intersections between the cone’s constraints and Δci axis. The main challenge in LP stability analysis is the presentation of this cone to the decision maker in a simple and useful way, especially for simultaneous variations. The most useful approach is to construct the largest possible hyperbox inside the cone that it presents the independent numerical limits for each coefficient. The analysis starts with constructing a number of hypercubes each one is associated with a Lagrange multiplier of an active constraint, λg, using τg ¼
λg n
∑ khjA1 A j h¼1
ð8Þ
gh
where τg is the stability limit and kh is a weighting factor that represents the relative variation ratio of the associated coefficient to other coefficients. Then, the work is done to enlarge the values of stability limits by examining the relations objective coefficients and Lagrange multipliers. The detailed description of this analysis and its computation algorithm are shown in Al-Shammari.28 2.2.2. Variations in RHS Coefficients. For variations in the RHS coefficients, a similar stability analysis is employed for the 7067
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Figure 2. Simplified sketch of a typical oil refinery.
∑
dual problem minfλ b : λ A g c λ g 0g T
T
k∈K
ð9Þ
T
to determine the variation limits before the optimal basis changes. In this analysis, the optimal solution and slack variables are used in the same manner as the Lagrange multipliers were used in the variations analysis of vector c.
3. REFINERY OPTIMIZATION MODEL In this project, the production levels of main products are optimized in order to determine the most efficient distributions for the intermediate products in order to maximize the net profit. However, the model does not consider the optimization of individual units or equipments but the interaction between the main units of a refinery. Six main refinery units were considered in modeling the problem. They are fractionation, fluidized catalytic cracking, reforming, isomerization, desulphurization, and blending units, as in Figure 2. Considering the above process units, the objective of optimization problem is to find the optimum production levels (quantities of raw materials, intermediate product, and final products) for each unit such that they lead to the maximum profit with satisfying process and quality constrains. The refinery optimization model is formulated as a LP problem as follows max
∑
j∈M
p j yj
∑
i∈N
c i yi
∑ ∑
l∈L k∈K
ql xl, k
ð10Þ
s.t.
∑
k∈K
ak xl, k ¼ 0
ðfor each componentÞ
ð11Þ
ak xl, k ¼ Dj
for j ∈ M
ð12Þ
∑
yi e Si
ð13Þ
∑
y j g Dj
ð14Þ
i∈N
j∈M
k ak xl, k e vj, max yj ∑ F k∈K k
v
for j ∈ M
k ak xl, k g vj, min yj ∑ F k∈K k
v
∑
k∈K
xl, k e Ol
ð15Þ
ð16Þ
ð17Þ
The economic objective function (eq 10) attempts to maximize the revenues from the products and minimize the costs of raw materials and process operations. The constraints of this optimization problem can be classified into four different types. Balances of components (eqs 11and 12) for each product component around each major unit; demand and supply limits (eqs 13 and 14); products quality constraints where refinery products have certain specifications need to be satisfied such as gasoline octane number, butane content, and also maximum and minimum vapor pressure as in (eqs 15 and 16); and equipments or unit capacity constraints like the capacity of the CDU (eq 17). The model parameters including demands, supplies, prices, and 2931 operations costs of this problem are shown in the appendix. 7068
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Table 1. Stability Limits of Individual and Simultaneous Price Variations in Some Variables individual stability limits ($/ton) selected variables
x*i (1000 ton per month)
ci ($/ton)
feedstock 1
442.2
530
feedstock 2
350.0
500
lower
simultaneous stability limits ($/ton)
upper
530
lower 530
14.5
27.5
upper 8.3
11.7
Inf
Inf
reformer feedstock imported
6.138
604
113.6
119.4
8.3
33.4
cracker feedstock imported
0.0
581
108.3
Inf
52.2
Inf
imported gasoline 95
0.0
715
35.0
Inf
17.5
Inf
exported gasoline 95
128.26
680
7.3
35.0
1.8
17.5
imported jet fuel exported jet fuel
0.0 115.10
657 625
32.0 42.8
Inf 32.0
16.0 8.3
Inf 16.0
imported gas oil
0.0
620
30.0
Inf
28.2
Inf
exported gas oil
227.82
590
39.4
3.7
8.3
1.8
imported HFO
0.0
452
37.0
Inf
27.9
Inf
exported HFO
210.96
415
6.4
37.0
1.8
11.7
Table 2. Stability Limits of Individual and Simultaneous Variations in Some RHS Coefficients individual stability limits (1000 ton/month) selected RHS
bi (1000 ton
variables
per month)
λi ($/ton)
lower
upper
lower
upper
30.42
Inf
availability of crude 1
490
0
47.78
Inf
minimum treat. of crude 2
350
27.5
36.83
61.67
distillation column capacity
920
0
85
201.2
reformer capacity
simultaneous stability limits (1000 ton/month)
127.8
Inf
1.37
7.11
3.22 113.1 0.61
14.23 Inf 3.56
FCC capacity
190
68.2
8.91
5.55
0.61
3.23
desulfurization unit capacity
210
0
8.08
Inf
3.23
Inf
gasoline 95 demand
126
0
126
2.26
126
0.61
jet fuel demand AGO demand
100 210
0 0
100 210
15.10 17.83
100 210
14.23 15.92
HOF demand
200
0
200
10.97
200
3.56
4. RESULTS AND DISCUSSION A refinery with capacity of 210 000 barrels per day and two different feedstocks is considered in this study. The simplified LP problem has 52 continuous decision variables and 53 constraints in addition to nonnegative constraints.30,31 The problem was solved using Matlab and the maximum profit was found to be 3.4082 107 dollars per month without considering the costs of maintenance, storage, transporting, and fixed costs. Optimal solutions of key process variables, production levels, are shown in Table 1; in addition to the results obtained form stability analysis of the objective coefficients. Table 1 shows the allowable range of price changes for each feedstock or product within which the obtained optimal production levels remain optimum. In other words, the optimal solution would change if the price of feedstock 1 increased by more than 14.5 $/ton and the process would consume more from feedstock 2 because it became relatively cheaper. Such information can help the decision maker to understand the interaction between production levels and other factors (e.g., prices); and determine when the process need to be reoptimized. Obviously, some parameters have infinite limits in one direction of changes because they are not purchased and any increase
in their prices would not affect the solution. Nevertheless, this is not the real case for feedstock 2. There is a constraint on the minimum consumption of feedstock 2 and it is active for the current solution. In contrast, individual stability limits for individual variations is grater than those for simultaneous variations since the obtained stability range for each parameter may decrease with increasing the number of uncertain parameters. Presented limits of simultaneous variation were obtained using same weighting factor (ki = 1) for all parameters in Table 1; however, different weighting factors can be used based on the relative important or frequent change of each parameters. Analysis and results obtained for variations in RHS coefficients (e.g., supply and demand of materials and capacity constraints) are shown in Tables 2 and 3. In Table 2, the sensitivity information, λi, shows by how much the objective value or profit will increase by each increase in RHS coefficients. Active constraints have nonzero Lagrange multipliers and upper and lower stability limits such as the constraints of FCC and reformer capacities. These two constraints are the most sensitive constraints that affect the profit. Individual and simultaneous variations ranges for these two constraints presented in Table 2 that shows the limits within which they can be increased in order to 7069
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Table 3. Stability Limits for Simultaneous Variations in Units Capacity at Different Weighting Factors simultaneous stability limits (1000 ton/month)
simultaneous stability limits (1000 ton/month)
variables
ki
lower
upper
ki
lower
distillation column capacity
1
112.2
Inf
1
112.2
Inf
reformer capacity
1
1.19
6.61
1
1.05
11.32
FCC capacity
1
1.19
3.29
2
1.05
1.64
desulfurization unit capacity
1
3.29
Inf
2
1.64
Inf
distillation column capacity
0
Inf
1
reformer capacity
1
1.37
7.11
2
0.68
3.55
FCC capacity desulfurization unit capacity
0 1
8.08
Inf
0 2
4.04
Inf
improve the profit; and at the same time they still active and sensitivity information is still valid. For example, if the reformer capacity was individually increased by 7110 tons per month, the profit would increased by 4.2%, whereas the profit can be increased by 1.1% if the FCC capacity increased by 5550 tons per month. However, under simultaneous changes in RHS coefficients, the stability ranges are decreased for these two constraints and the profit would increase by 2.4% in total. Changes or variation beyond these stability limits do not ensure any improve in the profit because new constraints can get activated like the capacity of the desulfurization unit. Some other stability limits presented in Table 2 show when nonactive constraints, like the availability of Crude 1, become active. Stability limits of such constraints under individual variation can be obtained easily from the slack variables. However, most changes in process constraints can happen simultaneously and need detailed stability analysis. Regarding products demand, all the constraints are inactive in this case study because the demand is modeled as a minimum production level of each component and the optimal results exceed these limits in order to increase the profit. In general, the stability ranges indicate the allowable variations limits in objective and RHS coefficients at which their sensitivity relations are valid. These relations help in improving profit and understanding the effect of different changes in process parameters on the optimal operation conditions. In fact, each process parameter can vary with different ranges depending on the associated weighting factor as presented in Table 3. The decision make can use a different weighting factor for each parameter. Uncertain parameters can be associated with a very small or zero value of weighting factor, which offer wider stability limit for other parameters.
5. CONCLUSION The general objective of this study is to try to bridge the gap between the theory and practice of the postoptimality analysis in process engineering. Postoptimality analysis was applied to a simplified refinery model formulated as LP problem. Then, the effect of uncertainty or variations in model parameters, mainly RHS and objective coefficients on optimal production levels was investigated. Modified tolerance approach was used to compute the allowable variation limits, for individual and simultaneous variations, within which the operation levels remain optimum. The results can supply the decision maker in the refinery with useful and easy-to-use information that can help to understand the interaction between production levels and other factors, such as products prices, and to enable effective use of sensitivity
upper
127.8
Inf
information such as Lagrange multipliers. Moreover, the results show that the refinery profit can be increased by 4.2% based on the obtained sensitivity and stability information.
’ APPENDIX Tables 412 contain further data for the analysis in the paper. Table 4. Components Weight Percentage in Crude Oil crude 1
crude 2
refinery gas
0.1
0.2
liquefied gas light naphtha
1.2 4.0
1.5 4.0
heavy naphtha
14.5
7.5
kerosene
15.0
9.0
gas oil 1
31.0
0.0
gas oil 2
0.0
20.3
vacuum gas oil
21.2
27.5
vacuum residue 1
13.0
0.0
vacuum residue 2 total
0.0 100
30.0 100
Table 5. FCC Feedstock Fractions (% wt) max. motor gas
AGO
refinery gas liquefied gas
1.58 5.3
1.2 4.6
cat. cracked spirit
43.6
38.1
light cycle gas oil
44.6
51.1
total
95
95
5% coke
5% coke
Table 6. Conversion of Feedstock (LN) in Isomerization Unit wt % refinery gas isomerate total 7070
3 97 100
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Table 7. Feedstock Fractions of Desulphurization Unit (wt %)
Table 12. Prices of Raw Materials and Products
feedstock from
$/tonne
GO1
GO2
VGO
refinery gas
98
97
96
crude 1
530.0
desulph. raffinate
2
3
4
crude 2
500.0
crude 2
100
100
100
reformer feed stock
604.0
reformer feed stock
total
prices of crude and products
Table 8. Formulation of Jet Fuel (wt %) (wt %) jet fuel formulation
formulation 1
formulation 2
light naphtha heavy naphtha
5 10
3.5 7.5
kerosene cut
85
89.9
Table 9. Market Demands of Main Refinery Products product
market demand (1000 tonne/month)
liquefied gas
15
light naphtha unleaded 98 motor gas
9 28
unleaded 95 motor gas
90
jet fuel
70
AGO
150
HFO
140
Table 10. Unit Capacity in 1000 Tonnes/Month min distillation capacity
max 920
reforming capacity 95 severity
2.8
total
85
desulphurizatin
210
cracking capacity
190
crude availability crude 1 crude 2
490 350
Table 11. Operating Costs for Each Main Unit unit operation costs ($/tonne) distillation
1.0
reforming 95 severity
5.5
100 severity
6.5
cracking
6.0
isomerization
1.2
desulphurization GO1
2.0
GO2
2.0
CGO
2.8
import
export crude 1
cracker feedstock
581.0
cracker feedstock
95 RON
715.0
95 RON
jet fuel
657.0
jet fuel
AGO/HGO HFO
620.0 452.0
AGO/HGO HFO
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The authors acknowledge Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals for supporting this project (FT090005). Also, they thank the Center of Research Excellence in Petroleum Refining and Petrochemicals at KFUPM. ’ NOMENCLATURE a = molar fraction for each component c = cost of raw materials or imported streams in $/ton i = stream index j = stream index k = index of feed (or product) of each unit l = unit index q = operation cost of each stream entering each unit in $/ton p = price of each product in $/ton ν,νmax = vapor pressure x, y = streams flow rate in 1000 ton per month λ = Lagrange multiplier or shadow price D = demand of products in 1000 ton K = number of streams entering all units L = number of units M = number of final products N = number of raw materials and imported streams S = supply of raw materials ’ REFERENCES (1) Symonds, G. Linear programming solves gasoline refining and blending problems. Ind. Eng. Chem. 1956, 48 (3), 394–401. (2) Allen, D. H. Linear programming models for plant operations planning. Brit. Chem. Eng. 1971, 16, 685–691. (3) Moro, L. L.; Zanin, A. C.; Pinto, J. M. A planning model for refinery diesel production, Computers & Chemical Engineering. Comput. Chem. Eng. 1998, 22, 1039–1042. (4) Pinto, J. M.; Moro, L. L. A planning model for petroleum refineries. Braz. J. Chem. Eng. 2000, 17, 575–586. (5) Alhajri, I.; Elkamel, A; Albahri, T.; Douglas, P. L. A nonlinear programming model for refinery planning and optimization with rigorous process models and product quality specifications. Int. J. Oil Gas Coal Technol. 2008, 1 (3), 283–307. 7071
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’ NOTE ADDED AFTER ASAP PUBLICATION This article was published ASAP on April 29, 2011. Changes were made to the author address and the acknowledgment. The corrected version was posted on May 4, 2011.
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