Petroleum refinery operational planning using robust optimization

Transcription

1 Engineering Optimization Vol. 42, No. 12, December 2010, Petroleum refinery operational planning using robust optimization A. Leiras a, S. Hamacher a and A. Elkamel b * a Center of Excellence in Optimization Solutions (NExO), Industrial Engineering Department (DEI), Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), CP38097, Rio de Janeiro RJ, Brazil; b Department of Chemical Engineering, University of Waterloo, Ontario N2L 3G1, Canada (Received 11 September 2009; final version received 20 January 2010 ) In this article, the robust optimization methodology is applied to deal with uncertainties in the prices of saleable products, operating costs, product demand, and product yield in the context of refinery operational planning. A numerical study demonstrates the effectiveness of the proposed robust approach. The benefits of incorporating uncertainty in the different model parameters were evaluated in terms of the cost of ignoring uncertainty in the problem. The calculations suggest that this benefit is equivalent to 7.47% of the deterministic solution value, which indicates that the robust model may offer advantages to those involved with refinery operational planning. In addition, the probability bounds of constraint violation are calculated to help the decision-maker adopt a more appropriate parameter to control robustness and udge the tradeoff between conservatism and total profit. Keywords: robust optimization; refinery operational planning; optimization under uncertainty Nomenclature Sets and indices J set of streams,i J feed subset of streams that carry feed flows in the context of a process unit J prod subset of streams that carry finished products T set of time periods t [0, T] Deterministic decision variable x t production flow rate for stream in period t *Corresponding author. aelkamel@uwaterloo.ca ISSN X print/issn online 2010 Taylor & Francis DOI: /

2 1120 A. Leiras et al. Deterministic parameters p t c t cap t η i σ i α i prod t sales prices of finished products in period t purchasing and operating costs in period t production capacity of a process unit that is fed by feed flow in period t [0, 1] production yield of flow from flow i [0, 1] blend coefficient for flow from flow i {0, 1}, 1 if the flow rate x is produced from splitting the flow rate x i, 0 otherwise maximum production requirement for the flow rate x in period t Robust variables μ cost t,μ price t,μ yield i t,μ prod t quantify sensitivity to changes in cost, price, yield, and demand, respectively λ cost,λ price,λ yield,λ prod quantify sensitivity to changes in cost, price, yield, and demand, respectively Robust parameters Ɣ cost,ɣ price,ɣ yield,ɣ prod parameters to adust cost, price, yield, and demand robustness, respectively d cost,d price, ˆη it, prod ˆ t cost, price, yield, and demand deviations, respectively 1. Introduction The development of global competition and the continued aggressive search for cost savings is forcing the refining industry to modify operations in an effort to improve economic performance (Moro 2003). As a result, production planning must be aided by decision-making systems, especially those that employ mathematical programming or optimization for example, RPMS (Refinery and Petrochemical Modelling System) (Bonner and Moore, 1979), OMEGA (Optimization Method for the Estimation of Gasoline Attributes) (Dewitt et al., 1989), and PIMS (Process Industry Modelling System) (Bechtel, 1993). Moro (2003) emphasizes that the use of mathematical programming can increase profits by US$10 per tonne of product refined. Petroleum refineries extract and upgrade the valuable components of crude oil to produce a variety of marketable petroleum products. The refinery topology is defined by sets that specify the connections among units as well as between streams and units. Refineries carry process units and tanks to blend products and produce several streams of intermediate products that can be blended to create distinct commercial offerings. A planning model for oil refineries must allow for the proper selection of oil blending and the manipulation of intermediary streams to obtain the final products in the desired quantities and qualities. At the same time, the production cost must be minimized and profitability must be maximized (Moro 2000). This work focuses on refinery operational planning processes. Production planning in the oil sector clearly involves uncertainties, since it is hard to anticipate some of the parameters that need to be taken into account, such as prices, oil supply and product demand. There are several approaches to incorporate and deal with such uncertainties, including the fuzzy, stochastic, and robust programming techniques (Ravi and Reddy 1998, Li et al. 2004, Neiro and Pinto 2005, Khor et al. 2008, Ribas et al. 2008).

3 Engineering Optimization 1121 The robust optimization approach focuses on models that ensure solution feasibility given the possible outcomes of uncertain parameters. Under this approach, the decision-maker is willing to accept a suboptimal solution for the nominal values in order to ensure that the solution remains feasible and near optimal when the data changes. Many works have developed methodologies based on the robust technique to deal with optimization under uncertainty, as seen in Beyer and Sendhoff (2007). The first step in this direction was taken by Soyster (1973), who proposed a conservative approach that assumes that all random parameters are equal to their worst-case value. The robust Soyster approach has recently been extensively studied and extended. El-Ghaoui and Lebret (1997), El-Ghaoui et al. (1998) and Ben-Tal and Nemirovski (1998, 1999, 2000), presented a robust method that is less conservative, introducing a nonlinear term in the obective function. In this article, the robust optimization methodology proposed by Bertsimas and Sim (2003, 2004) is adopted. These authors proposed an approach that attempts to make the trade-off between solution optimality and solution robustness more attractive. An important aspect of this method is that the new robust formulation does not add complexity to the original problem. No refinery planning models that consider the robust optimization approach of Bertsimas and Sim (2003, 2004) have been found in the literature searches for this work. This article proposes a robust programming model within the framework of Bertsimas and Sim (2003, 2004) that addresses three areas of uncertainty: purchasing and operating costs and prices of saleable products (in the obective function), product demands (in the right-hand-side (RHS) coefficient constraints), and product yields (in the left-hand-side (LHS) coefficient constraints). Without loss of generality, a numerical study based on the refinery planning model of Allen (1971) is used to demonstrate the implementation of the proposed approach. The remainder of this article is organized as follows. In Section 2, the robust optimization framework is explained. This methodology is applied to the refinery operational planning problem in Section 3. Section 4 presents results and discussions in the context of a numerical example. Finally, Section 5 offers the article conclusions. 2. Robust optimization framework The main idea behind Bertsimas and Sim s approach is to control the conservatism of the robust solution by introducing a parameter that can be defined by the decision-maker. Since in practice it is unlikely that all the uncertain coefficients are equal to their worst case value (such as Soyster s method), Bertsimas and Sim propose a less conservative approach such that the decision-maker can choose the number of uncertain factors against which he/she wishes to be protected. Consider the following linear optimization problem with a set of n variables: Minimize Subect to n c x =1 n a i x b i, i =1 l x u, (1) Assume that the coefficients of the technological matrix A are subect to uncertainty and modelled as a symmetric and bounded random variable ã i, Ji a that assume values in the interval

4 1122 A. Leiras et al. [a i â i,a i +â i ]. For every constraint i, a parameter Ɣi a (not necessarily integer) is introduced, which assumes values in the interval [0, Ji a a ], where Ji ={ â i > 0}. Ɣi a can be seen as a parameter to adust the model robustness given the level of solution conservatism. If Ɣi a = 0, the uncertainties in the parameters of constraint i can be ignored (deterministic problem). By contrast, Ɣi a = Ji a represents the most conservative case in which all the uncertainty parameters of the constraint i are considered (Soyster s model). Accordingly, this parameter limits the number of coefficients that are simultaneously worst-case valued. If there exists uncertainty regarding the independent coefficients of the constraints (b i ),anew variable x n+1 can be introduced into the model, and the model can be rewritten as n =1 a ix b i x n+1 0, l x u,1 x n+1 1 (Bertsimas and Sim 2003). Problem (1) has an equivalent formulation as follows: Minimize n c x =1 Subect to n a i x b i x n+1 0, i =1 l x u, x n+1 = 1, i (2) The costs associated with the obective function may also be subect to uncertainty and can be modelled as a symmetric and bounded random variable c, J c that takes values in the range [c,c + d ], where J c ={ d > 0} is the set of indices with a positive deviation d. In this case, Ɣ c [0, J c ] is the parameter used to control robustness in the obective function. To deal with randomness in the obective function coefficients, RHS and LHS coefficients of the constraints, Model (1) has an equivalent robust linear counterpart as follows see Bertsimas and Sim (2003) for proofs: Minimize n c x + λ c Ɣ c + μ c J c =1 Subect to a i x b i x n+1 + λ a i Ɣa i + J a i μ a i 0 i λ c + μ c d w J c λ a i + μ a i â iw l x u, i, J a i J w x w, J 1 x n+1 1 i (3) w 0 J λ a i 0 i μ a i 0 a i, J μ c 0 c i, J λ c 0

5 Engineering Optimization 1123 The variables λ c and μ c, λa i and μ a i, quantify the system s sensitivity to changes in the uncertain parameters c, a i and b i. The sum of the variables λ and μ represents the minimum deviation. At optimality, w = x for all, where x is an optimal solution of problem (3). In fact, only deviations with negative impact in the obective function are controlled, because all others automatically lead to feasible solutions Probability bounds of constraint violation The parameter Ɣi a controls the trade-off between the probability of violation and the effect on the obective function of the nominal problem. If Ɣi a [0, Ji a ], then the robust solution will be deterministically feasible. Even if more than Ɣi a change, the robust solution will remain feasible with very high probability. Assuming a symmetrical distribution of random variables, Bertsimas and Sim (2003, 2004) calculated the probability that the ith constraint will be violated, if more than Ɣi a coefficients vary. This probability can be approximated by the following expression: Pr ( Ɣ ã i x a ) >b i 1 i 1 ni (4) where n = Ji a, and (θ) is the cumulative distributive function of a standard normal. The limiting bound (Equation (4)) is particularly interesting because it leads us towards a more appropriate choice for Ɣi a, as shown below in the computational results. 3. Problem statement The operational refinery production planning problem addressed in this article can be stated as follows. The physical resources of the plant are assumed to be fixed and the associated prices, costs, and demands are externally imposed (Reklaitis 1982). The obective is to maximize profitability and to determine the optimal planning traectory by computing the amount of material that is processed at each time interval within each unit (Khor et al. 2008). The deterministic operational refinery planning model that was originally proposed by Allen (1971) is now considered. This model was revisited by Khor et al. (2008) and Ravi and Reddy (1998) which employed, respectively, robust stochastic programming and fuzzy programming to account for uncertainty. The refinery topology proposed by Allen (1971) is presented in Figure 1. The refinery begins with the primary distillation unit (PDU) that fractionates crude oils into separate hydrocarbon groups. The resulting products are directly related to the characteristics of the crude processed. Most distillation products are further converted into more useable products by changing the size and structure of the hydrocarbon molecules through cracking. As shown in Figure 1, the linear model case study represents a refinery that consists of three units: PDU, cracking and blending. The refinery processes crude oil (x 1 ) to produce gasoline (x 2 ), naphtha (x 3 ), et fuel (x 4 ), heating oil (x 5 ), and fuel oil (x 6 ), where x 7 to x 20 are intermediary streams. This article proposes a multi-period linear programming model in order to generalize the model developed by Allen (1971). Furthermore, in the next section the model is extended to account for uncertainties. The deterministic model is given below. The definitions of the variables and parameters are given in the nomenclature section at the end of this article.

6 1124 A. Leiras et al. Figure 1. Refinery flowchart case study (Allen 1971). Maximize t T J feed J prod (p t x t c t x t ) Subect to Limits in terms of plant capacity: x t cap t J feed J,t T, (5.1) Production yields: x t i η i x it J,t T, (5.2) Fixed proportion blends: x it = σ i x t i I,t T, (5.3) Production balance: x it = α i x t i I,t T, (5.4) Maximum production requirement: x t prod t J prod J,t T, (5.5) Non-negativity: x t 0 J,t T. (5.6) The obective function (OF) aims to maximize total daily profit. Besides the refinery product prices (p t ), the OF considers the purchasing cost of crude oil and operating costs of the PDU and the cracker unit (both represented by c t ). Prices and costs can vary in different periods of time (t). There is no cost associated with blending. The feed rates of PDU and cracking can be anything from zero to the maximum plant capacity, as shown in the constraint (5.1).

7 Engineering Optimization 1125 There are three types of mass balance constraints: production yields, fixed proportion blends, and production balances. The process units (PDU and cracking) split the crude or products into other products, according to production yield coefficients (constraint (5.2)). For example, crude oil (x 1 ) produces a maximum of 13 % of naphtha (x 3 ), 15% of et fuel (x 4 ), 22% of gas oil (x 8 ), 20% of cracker feed (x 9 ), and 30% of residuum (x 10 ). Constraint (5.3) represents a fixed proportion blend, for example, gasoline (x 2 ) is blended from naphtha (x 11 ) and cracked blend (x 16 ) in equal proportions (in Figure 1, x 11 = 0.5 x 2 and x 16 = 0.5 x 2 ), while heating oil (x 5 ) is a blend of 75% gas oil (x 12 ) and 25% cracked oil (x 18 ). Constraint (5.4) establishes mass balances. For example, fuel oil (x 6 ) is produced from residuum (x 10 ), cracked feed (x 15 ), gas oil (x 13 ), and cracked oil (x 19 ) in any proportions (in Figure 1: x 6 = x 10 + x 15 + x 13 + x 19 ). The production requirement constraints are directly impacted by the market demand for refinery products (constraint (5.5)). Therefore, in the robust model, several random decision variables will be introduced into these constraints. Under this model, there are no restrictions on crude oil availability or minimum production. Finally, constraint (5.6) denotes that the decision variables cannot be negative. 4. Robust optimization model for refinery operational planning The robust formulation for Model (5) to account for uncertainty in prices and costs (economic risk) and in product yields and market demand (operational risk) is presented below. Costs/price randomness (in the obective function) The parameters Ɣ price and Ɣ cost control for uncertainty in the obective function and assume values in the intervals [0, J price ] and [0, J cost ], where J is the set of uncertain coefficients with a deviation d. Yield randomness (in LHS constraint coefficients) Yield uncertainties introduce randomness in terms of the mass balance for yield constraint (5.2). The parameter Ɣ yield assumes values in the interval [0, J yield ], where J yield equals the number of different types of feed flows within the process unit. The yield deviation is represented by ˆη it. Demand randomness (in RHS constraint coefficients) Finally, incorporating uncertainty into market demand introduces constraint randomness in terms of maximum production requirements (5.5). The parameter Ɣ prod adusts the demand robustness and prod ˆ t represents the demand deviation, adopting values in the interval [0, 1]. Model (5) has a robust formulation as follows: Maximize t T (p t x t c t x t ) λ price Ɣ price J feed J prod t T t T μ cost t J feed J prod μ price t J feed J prod λ cost Ɣ cost

8 1126 A. Leiras et al. Subect to x t prod t x i + λ prod Ɣ prod + μ prod t 0 J prod,i = J prod + 1,t T, (6.1) i η i t x it x t + λ yield Ɣ yield + i μyield i t J yield,t T, (6.2) λ prod + μ prod prod ˆ t t x i J prod,i = J prod + 1,t T, (6.3) λ yield + μ yield i t ˆη i t x it J yield,t T, (6.4) λ price + μ price t d price x t J price,t T, (6.5) λ cost + μ cost t d cost x t J cost,t T, (6.6) x 0 J, (6.7) x i = 1 i = Jprod + 1, (6.8) μ price 0 t J price,t T, (6.9) μ cost 0 t J cost,t T, (6.10) μ prod 0 t J prod,t T, (6.11) μ yield 0 i t J yield,t T, (6.12) λ price,λ cost,λ prod,λ yield 0 (6.13) Given constraints (5.1), (5.3), (5.4), and (5.6). 5. Numerical example The numerical study is based on the data proposed by Allen (1971) and also adopted in the works of Khor et al. (2008) and Ravi and Reddy (1998). This example is presented and solved to optimality to demonstrate the effectiveness of the robust model. The model of Allen was first solved deterministically and was then reformulated to address uncertainties as mentioned earlier. Unlike the approach of stochastic optimization adopted by Khor et al. (2008), in the robust optimization technique there is no necessity of specifying scenarios and their probabilities or expected recourse costs, which are often cumbersome to estimate. The method adopted in the present article assumes only limited information about the distributions of the underlying uncertainties, such as known mean value and its range. Moreover, to handle randomness in the obective function coefficients, Khor et al. (2008) reformulated the problem as a robust stochastic model by adding a risk measure (variance) in the obective function. Unfortunately, this approach requires the introduction of non-linearities to the model. Since the robust programming approach applied in the present work does not add complexity to the problem, it is the main advantage of the Bertsimas and Sim s methodology. The robust model leads to total profits lower than the stochastic model in order to achieve robustness, i.e. to ensure that the solution remains feasible and near optimal when the data changes. In the case study, the robust formulation to account for uncertainty in the obective function coefficients considers 10% of the positive deviation related to cost coefficients and 10% of the negative deviation related to price coefficients from the expected values. The parameters Ɣ price and Ɣ cost take values in the intervals [0, 5] and [0, 2], respectively.

9 Engineering Optimization 1127 In addition, the robust dimensions related to demand and yield uncertainty are constructed based on 5% and 1% of the negative deviation from the nominal value. For these approaches, J is equal to 1. In this work, only the yields of products from the distillation unit were controlled. To ensure that material balances are satisfied, yield for the residuum is determined by subtracting the sum of yields for the other four products from the total yield of the distillation unit Computational results and discussion The model was implemented using theaimms (Advanced Integrated Multidimensional Modeling Software) software package (Bisschop and Roelofs 2007) and solved with CPLEX The deterministic solution value (maximum profit) is equal to US$23,387.50/day. First, experiments were conducted for each of the uncertain parameters (yield, demand, cost, and price) separately. The robust parameter, Ɣ, was varied in integer values over the interval [0, J ]. Subsequently, the four uncertain parameters were combined. From Table 1, comparing the obective function values for the four types of uncertainties in the Soyster case (Ɣ i = J i ), it can be concluded that price deviation exerts the greatest impact on profit, followed by cost, yield and demand, respectively. The computational results also suggest that yield deviation has a more pronounced impact in the obective function (17.44% lower than the deterministic solution) than in the case of demand deviation (only 3.09%). This can be seen in Figures 2a and 2b which show computational results in terms of price and cost deviations only, and for the same parameters when combined with yield and demand uncertainties. From these figures, it can be concluded that price deviations may impact the obective function more strongly than cost deviations. In fact, considering only price and cost uncertainties, the solution for Ɣ price = 5 is 49.01% worse than the deterministic solution, while for Ɣ cost = 2 is 41% worse. Table 1. Obective function values for four types of uncertainties in the Soyster case (Ɣ i = J i ). Uncertain parameter Yield Demand Cost Price Obective function value 19, , , , Figure 2a. Obective function variation consistent with Ɣ price.

10 1128 A. Leiras et al. Figure 2b. Obective function variation consistent with Ɣ cost. The solutions for Ɣ cost = 1 and Ɣ cost = 2 are almost constant, because the costs related to the crude distillation unit impact the obective function more strongly than those related to the cracker unit. Yield and demand randomness are also considered in order to combine these results with solutions in terms of price and cost deviations. This process will gradually cause deterioration of the robust solution. Moreover, it is interesting to note that the obective function value is marginally affected by the increased level of protection. Specifically, the increments decrease each time Ɣ increases. This is a feature of the robust formulation and is entirely independent of the problem treated. Additionally, when Ɣ yield = 0 the additional production to protect against the largest deviation in the uncertain parameters can be attributed to λ yield. Since λ yield is multiplied by zero in the robust yield constraint (6.2), this variable can take any value to satisfy the minimum deviation constraint (6.4). A similar analysis can be performed for demand, price and cost uncertainties. Uncertainties in both costs and prices were also tested. Figure 3 presents the computational results for Ɣ cost = 2. The solutions with Ɣ cost = 1 were around 1% closer to the nominal solution. Considering the four types of randomness, results suggest a strong impact of the combination of prices and costs, indicating a high correlation between these two uncertain parameters. Figure 4 shows computational results for variations in price deviations in the case study considering the four different types of randomness (case (m) of Figure 3). These results seem to confirm the model s sensitivity to deviations from the nominal value. From the results above, it can be concluded that taking into account uncertain parameters gradually deteriorates the obective function value. However, ignoring parameter uncertainty in planning problems can lead to suboptimal or infeasible outcomes. Thus, in order to properly evaluate the added value of including uncertainty within the model, the robust solution approach was applied to the deterministic obective function as a means of calculating the total profit. The obective was to focus on the difference between the deterministic solution given the worst-case value of the uncertain parameters and the robust solution that uses the deterministic obective function. This difference quantifies the benefit of incorporating uncertainty in the model parameters for operational planning purposes. This result can also be evaluated in terms of the cost of ignoring uncertainty in the problem. The calculations suggest that these benefits amount to US$ for the case that includes four types of uncertainty (case m). This is equivalent to 7.47% of the deterministic solution in the context of the original problem,

11 Engineering Optimization 1129 Figure 3. Obective function variation consistent with Ɣ price (Ɣ cost = 2). Figure 4. Obective function variation with different price deviations (Ɣ cost = 2). which in turn indicates that the robust model may offer advantages to those involved with refinery operational planning. Probability bounds of constraint violation The robust model approach does not account for a decision-maker s risk-taking behaviour. For this reason, a more realistic approach should include a measure of the degree of solution conservatism/reliability. One relevant measure of interest is the probability of constraint violation, which can lead to a more appropriate choice for Ɣ i, as shown in expression (4). Table 2 presents the results for different choices of Ɣ price as a function of the probability bound.a range of probability levels was evaluated to offer the decision-maker a tradeoff between robustness and profit. Given that the maximum profit decreases when the degree of conservatism increases. The problem can be solved for different integer values of the price parameter (Ɣ price [0, 5]).

12 1130 A. Leiras et al. Table 2. Choice of Ɣ price values as a function of the probability of constraint violation. Probability of constraint violation Ɣ price The decision-maker can subsequently udge the tradeoff between conservatism and total profit in order to choose the appropriate Ɣ price value. From Table 2, if the decision-maker only accepts a 2% probability of constraint violation, then he has to use Ɣ price = 5 to protect himself against price parameter deviations. On the other hand, if the decision-maker accepts up to 60% probability, there is no need to secure additional protection from the nominal problem and Ɣ price can be set equal to zero. 6. Conclusions In this article, a systematic methodology was reported for developing robust programming models for the operational refinery planning problem by simultaneously accounting for uncertainties in costs, prices, product demands, and product yields. The conclusion is that the Bertsimas and Sim approach may be a useful tool for modelling planning problems without introducing additional computational complexity. In addition, probability bounds of constraint violation were calculated in order to help the decision-maker make better choices with regard to parameter choices to control robustness. A numerical example was used to demonstrate how to implement the proposed approach. In this article, the robust method was only applied to a small case study however, extensions to real world systems with similar structures should be straightforward. As a future work, this research is going to be extended to large scale problems and networks of refineries. Discrete decisions and nonlinearities in the context of the unit s operational variables and related properties such as product flow rates will also be investigated. Acknowledgements The Brazilian authors would like to thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) (Coordination for the Improvement of Higher Education Personnel in Brazil), and CNPq (research proect number /2006-3). References Allen, D.H., Linear programming models for plant operations planning. British Chemical Engineering, 16, Bechtel Corp., PIMS (Process Industry Modeling System) user s manual. Version 6.0. Houston, TX: Bechtel Corp. Ben-Tal, A. and Nemirovski, A., Robust convex optimization. Mathematics of Operations Research, 23, Ben-Tal, A. and Nemirovski, A., Robust solutions of uncertain linear programs. OR Letters, 25, Ben-Tal, A. and Nemirovski, A., Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88, Bertsimas, D. and Sim, M., Robust discrete optimization and network flows. Mathematical Programming, 98, Bertsimas, D. and Sim, M., The price of robustness. Operations Research, 52, Beyer, H.-G. and Sendhoff, B., Robust optimization a comprehensive survey. Computers Methods in Applied Mechanics Engineering, 196, Bisschop, J. and Roelofs, M., AIMMS: The user s guide. The AIMMS 3.8 Language Reference Kirkland, WA: Paragon Decision Technology.

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