Robust planning of multisite refinery networks: Optimization under uncertainty

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Abstract

This paper considers the problem of multisite integration and coordination strategies within a network of petroleum refineries under uncertainty and using robust optimization techniques. The framework of simultaneous analysis of process network integration, originally proposed by Al-Qahtani & Elkamel [Al-Qahtani, K., & Elkamel, A. (2008). Multisite facility network integration design and coordination: An application to the refining industry. Computers & Chemical Engineering, 32, 2198], is extended to account for uncertainty in model parameters. Robustness is analyzed based on both model robustness and solution robustness, where each measure is assigned a scaling factor to analyze the sensitivity of the refinery plan and integration network due to variations. Parameters uncertainty considered include coefficients of the objective function and right-hand-side parameters in the inequality constraints. The proposed method makes use of the sample average approximation (SAA) method with statistical bounding techniques. The proposed model was tested on two industrial-scale studies of a single refinery and a network of complex refineries. Modeling uncertainty in the process parameters provided a practical perspective of this type of problems in the chemical industry where benefits not only appear in terms of economic considerations, but also in terms of process flexibility.

Introduction

Today the petroleum refining industry is facing a challenging task to remain competitive in a globalized market. High crude oil prices and growing stringent international protocols and regulations force petroleum companies to embrace every opportunity that increases their profit margin. A common solution is to seek integration alternatives not only within a single facility but also on an enterprise-wide scale. This will provide enhanced utilization of resources and improved coordination and therefore achieve a global optimal production strategy within the network (Al-Qahtani & Elkamel, 2008). However, considering such highly strategic planning decisions, particularly in the current volatile market and the ever changing customer requirements, uncertainties play a paramount role in the final decision making.

Different approaches have been devised to tackle optimization under uncertainty including stochastic optimization (two-stage, multistage) with recourse based on the seminal work of Dantzig (1955), chance-constrained optimization (Charnes & Cooper, 1959), fuzzy programming (Bellman & Zadeh, 1970), and design flexibility (Grossmann & Sargent, 1978). These early works on optimization under uncertainty have undergone substantial developments in both theory and algorithms (Sahinidis, 2004). In this paper, we employ stochastic programming with recourse which deals with problems with uncertain parameters of a given discrete or continuous probability distributions. The most common formulation of stochastic programming models for planning problems is the two-stage stochastic program. In a two-stage stochastic programming model, decision variables are cast into two groups: first-stage and second-stage variables. The first-stage variables are decided upon prior to the actual realization of the random parameters. Once the uncertain events have unfolded, further design or operational adjustments can be made through values of the second-stage (alternatively called recourse variables at a particular cost). Stochastic programming with recourse commonly gives rise to large-scale models that require the use of decomposition methods and proper approximation techniques due to the high number of samples encountered (Liu & Sahinidis, 1996). However, recent developments in sampling techniques may help maintain the stochastic program to a manageable size. Additional details and surveys can be found in Dempster (1980), Sahinidis (2004) and the recent textbooks of Kall and Wallace (1994) and Ruszczyński and Shapiro (2003).

More recent applications and developments in the chemical engineering arena include the work by Ierapetritou and Pistikopoulos (1994) who proposed an algorithm for a two-stage stochastic linear planning model. The algorithm is based on design flexibility by finding a feasible subspace of the probability region instead of enumerating all possible uncertainty realizations. They also developed a Benders decomposition scheme for solving the problem without a priori discretization of the random space parameters. This was achieved by means of Gaussian quadrature numerical integration of the continuous density function. In a similar production planning problem, Clay and Grossmann (1997) developed a successive disaggregation algorithm for the solution of two-stage stochastic linear models with discrete uncertainty. Liu and Sahinidis, 1995, Liu and Sahinidis, 1996, Liu and Sahinidis, 1997 studied the design uncertainty in process expansion using sensitivity analysis, stochastic programming and fuzzy programming, respectively. In their stochastic model, they used Monte Carlo sampling to calculate the expected objective function values. Their comparison over the different methodologies was in favor of stochastic models when random the parameter distributions are not available. Ahmed, Sahinidis, and Pistikopoulos (2000) proposed a modification to the decomposition algorithm of Ierapetritou and Pistikopoulos (1994). They were able to avoid solving the feasibility subproblems and instead of imposing constraints on the random space, they developed feasibility cuts on the master problem of their decomposition algorithm. The modification mitigates suboptimal solutions and develops a more accurate comparison to cost and flexibility. Neiro and Pinto (2005) developed a multiperiod MINLP model for production planning of refinery operations under uncertain petroleum and product prices and demand. They were able to solve the model for 19 periods and five scenarios.

Another stream of research considered risk and robust optimization. The representation of risk management using variance as a risk measure was proposed by Mulvey, Vanderbei, and Zenios (1995) in which they referred to this approach as robust stochastic programming. They defined two types of robustness: (a) solution robustness referring to the optimal model solution when it remains close to optimal for any scenario realization, and (b) model robustness representing an optimal solution when it is almost feasible for any scenario realization. Ahmed and Sahinidis (1998) proposed the use of upper partial mean (UPM) as an alternative measure of variability with the aim of eliminating nonlinearities introduced by using variance. In addition to avoiding nonlinearity of the problem, UPM presents an asymmetric measure of risk, as apposed to variance, by penalizing unfavorable risk cases. Bok, Lee, and Park (1998) proposed a multiperiod robust optimization model for chemical process networks with demand uncertainty and applied it to the petrochemical industry in South Korea. They adopted the robust optimization framework by Mulvey et al. (1995) where they defined solution robustness as the model solution when it remains close to optimal for any demand realization, and model robustness when it has almost no excess capacity and unmet demand. More recently, Barbaro and Bagajewicz (2004) proposed a new risk metric to manage financial risk. They defined risk as the probability of not meeting a certain target profit, in the case of maximization, or cost, in the case of minimization. Additional binary variables are then defined for each scenario where each variable assumes a value of 1 in the case of not meeting the required target level; either profit or cost, and zero otherwise. Accordingly, appropriate penalty levels are assigned in the objective function. This approach mitigates the shortcomings of the symmetric penalization in the case of using variance, but on the other hand, adds computational burden through additional binary variables. Lin, Janak, and Floudas (2004) proposed a robust optimization approach based on min–max framework where they consider bounded uncertainty without known probability distribution. The uncertainty considered was both in the objective function coefficients and right-hand-side of the inequality constraints and was then applied to a set of MILP problems. This approach allowed the violation of stochastic inequality constraints with a certain probability and uncertainty parameters were estimated from their nominal values through random perturbations. This approach, however, could result in large infeasibilities in some of the constraints when the nominal data values are slightly changed. This work was then extended by Janak, Lin, and Floudas (2007) to cover known probability distributions and mitigate the big violations of constraints in Lin et al. (2004) via bounding the infeasibility of constraints and finding “better” nominal values of the uncertain parameters. It is worth mentioning that both Lin et al. (2004) and Janak et al. (2007) work is based on infeasibility/optimality analysis and does not consider recourse actions. For recent reviews on scheduling problems under uncertainty, we refer the interested reader to Janak et al. (2007) and for reviews on single and multisite planning and coordination to Al-Qahtani and Elkamel (2008).

In this paper, we extend the deterministic modeling for the design and analysis of multisite integration and coordination within a network of petroleum refineries proposed in Al-Qahtani and Elkamel (2008) to consider uncertainty in raw materials and final product prices as well as products demand. This work also accounts for both model robustness and solution robustness, following the Mulvey et al. (1995) approach. The stochastic model is formulated as a two-stage stochastic MILP problem whereas the robust optimization is formulated as an MINLP problem with nonlinearity arising from modeling the risk components. The proposed approach tackles parameters uncertainty in the coefficients of the objective function and the right-hand-side of inequality constraints. Furthermore, we apply the sample average approximation (SAA) method within an iterative scheme to generate the required samples. The solution quality is then statistically assessed by measuring the optimality gap of the final solution. The proposed approach was applied to two industrial-scale case studies of a single petroleum refinery and a network of refineries.

The remainder of the paper is organized as follows. In the following section we will explain the proposed model formulation for the petroleum refining multisite network planning under deterministic conditions, under uncertainty and using robust optimization. Then we will briefly explain the sample average approximation (SAA) method in Section 3. In Section 4, we will present computational results and the performance of the proposed approach on industrial case studies consisting of a single and a network of petroleum refineries. The paper ends with concluding remarks in Section 5.

Section snippets

Deterministic model

The formulation addresses the problem of determining an optimal integration strategy across multiple refineries and establishing an overall production and operating plan for each individual site (Al-Qahtani & Elkamel, 2008). Consider that a set of products cfrCFR to be produced at multiple refinery sites iI is given. Each refinery consists of different production units mM that can operate at different operating modes pP. An optimal feedstock from different available crudes crCR is desired.

SAA method

The solution of stochastic problems is generally very challenging as it involves numerical integration over the random continuous probability space of the second-stage variables (Goyal & Ierapetritou, 2007). An alternative approach is the discretization of the random space using a finite number of scenarios. This approach has received increasing attention in the literature since it gives rise to a deterministic equivalent formulation which can then be solved using available optimization

Illustrative case studies

This section presents computational results of the models and the sampling scheme proposed in this chapter. The refinery examples considered represent industrial-scale size refineries and an actual configuration that can be found in may industrial sites around the world. In the presentation of the results, we focus on demonstrating the sample average approximation computational results as we vary the sample sizes and compare their solution accuracy and the CPU time required for solving the

Conclusion

In this work, we proposed a two-stage stochastic MILP model for designing an integration strategy under uncertainty and plan capacity expansions, as required, in a multisite refinery network. The proposed method employs the sample average approximation (SAA) method with a statistical bounding and validation technique. In this sampling scheme, a relatively small sample size N is used to make decisions, with multiple replications, and another independent larger sample is used to reassess the

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