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Ind. Eng. Chem. Res. 2008, 47, 8282–8285
Optimal Solution-Range Analysis in Production Planning: Refinery Feedstock Selection Dimitrios K. Varvarezos* Aspen Technology, Houston, Texas 77042
This paper presents a new way to expand and analyze the optimal solution of a process planning model. Feedstock selection in a refinery is a complex process, accomplished in many steps that typically involve numerous optimization executions that aim at providing a “value” for each potential crude oil feedstock to the refinery for a given operational period. Traditionally, the optimal crude oil slate as determined by the planning system is a point solution that does not provide detailed information regarding potential (finite) changes in the optimal crude oil selection without degradation of the overall plant economics., In this work we define the optimal selection range for each feedstock as the result of a Pareto-type analysis including all feedstocks. Two flexibility indices are defined for each crude oil feed providing a quantitative flexibility metric in the context of optimal economics. The proposed approach provides a useful and practical way to quantitatively evaluate each potential crude feed. This is expressed in terms of an optimal range as defined by the ability of a feedstock to replace and be replaced in the optimal solution without significant economic change. In addition, this analysis provides a map of the optimal surface in terms of near-optimal points around the base solution. Introduction Although crude oil feed acquisition is one of the most important cost factors in refining operations, the tools available for assessing the optimal crude oil selection are mainly limited to case management utilities. Part of the reason for lack of more sophisticated tools has to do with the structure of the models and solver packages in traditional industrial planning applications. Planning software systems such as PIMS (Aspen Technology), GRTMPS (Haverly Systems), and RPMS (Honeywell HiSpec Solutions) are being used today for optimizing the refinery planning process. Only recently, however, commercially available planning systems (such as PIMS) have been outfitted with the ability to internally construct (and export if needed) an explicit nonlinear algebraic version of the problem1 and thus effectively decouple the modeling from the solution process. Such infrastructure technology serves as the backbone that enables sophisticated optimization analysis technologies to be developed and deployed. Despite the extensive use of optimization technologies in the area of production planning (since the 1970s), there has been very little progress in the area of providing rigorous analysis of the optimal solution and in particular the flexibility aspect. The extent of traditional planning analysis is usually limited to either a-priori case studies or use of the traditional linear programming parametric analysis2 applicable to the basic solution. For reasons related to the complexity and size of planning applications, formal methods available for addressing solution robustness, flexibility, and uncertainty in the planning process (both deterministic and stochastic) are not used in planning applications in the refining industry. Although there is a renewed interest in research in the planning and scheduling aspects of the refining and petrochemical processes,3-6 there is no comprehensive framework for the incorporation of all the modeling, optimization, and analysis aspect of refinery planning of feed acquisition and process operations. * To whom correspondence should be addressed. E-mail: dimitri@ aspentech.com.
Traditionally, the planning process consists of several steps. First, there is the phase of gathering data such as available supply and expected demand, pricing information, operating constraints, and updated process unit information (such as process yields). Validated data are then entered into a planning software system where a single execution is performed that results in an optimal base solution. The extent of the postoptimal analysis typically performed includes optimization executions with different data such as prices, supply, and demand. One thing that is not being investigated for the basesas well as all the other solutionssis the characteristics of the optimum. The optimal solution is a point in an n-dimensional space (where n ranges in the tens of thousands of variables). Without further optimization analysis, there is no way of knowing whether the solution is “flat” or “steep” in any of the critical decision directions such as feedstock variables. In other words, there is no understanding of the impact of minor to modest deviations from the optimal solution profile. Given the nature of the decision making process in a refining organization, more often than not the actual solution is never implemented “exactly” (for example, nobody purchases exactly 34156.7 bbl of Arabian light crude oil as the solution might indicate). It is imperative, therefore, to provide an analysis framework for validating the solution and exploring the solution surface around the optimal point. In this paper we formulate the concepts of flexibility indices in the context of feedstock selection. We describe the impact of those quantitative measures in the frame of risk management. Then we present an analysis and application of this approach illustrated by an example of how those indices are evaluated and utilized in the planning process. We then present this technique in the context of risk assessment and management. Finally, we conclude with a summary of the benefits of using this analytic framework in refinery planning. Feedstock Range Analysis Index We formally define two flexibility indices for each feedstock that (combined) define the optimal range for each feed.
10.1021/ie800078e CCC: $40.75 2008 American Chemical Society Published on Web 10/09/2008
Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8283
The first index is the utility index (maximum feedstock addition). For a given feedstock, this metric is defined as the fraction of the remaining total feed that this particular crude oil can displace for a predefined marginal drop in the economic objective function (typically 1%). This index ranges from 0 to 1. The formula for this is utility index )
XMAX - XBASE 1 - XBASE
(1)
where XMAX is the maximum fraction of the total feed this feedstock can reach at near optimal conditions and XBASE is the fraction of total feed this feedstock represents at the base optimal solution. (If the particular crude oil feedstock constitutes the ENTIRE optimum feed, i.e., XBASE ) 1, the index is by definition 1). The second index is the flexibility index (maximum feedstock removal). For a given feedstock, this metric is defined as the fraction (in terms of the total feed) of the optimum amount that can be displaced for a predefined marginal drop in the economic objective function (typically 1%). This index ranges from 0 to 1. flexibility index )
XBASE - XMIN XBASE
(2)
where XMIN is the minimum fraction of the total feed feedstock F can reach at near optimal conditions. (If the particular crude oil feedstock is NOT part of the optimum feed, i.e., XBASE ) 0, the index is by definition 1). The combination of those indices provides the maximum operational range under optimality conditions for each feedstock. Each index is calculated by the solution of two separate optimization problems. We start by defining the base optimization solution as the optimal solution to problem P0 as described by the following standard nonlinear programming7 formulation: max f(x)
h(x) ) 0 xL e x e xU x ∈ Rn (P0)
where f(x) is the objective function that typically is a measure of profit, x is the vector of all the variables in the problem with xL and xU the lower and upper bounds, respectively. If we denote as f 0, x 0 the optimal solution to P1, then the range analysis indices are found as the solution to the following minimization (P1) and maximization (P2) problems. min xF h(x) ) 0 f(x) g f 0(1 - εM) xL e x e xU x ∈ Rn (P1) Note that M is a user-defined constant (typically 0.01) that represents the acceptable marginal relaxation of the economic objective function for the range analysis. It is clear that as the relaxation M f 0 the economic solution value f (x) f f 0. max xF h(x) ) 0 f(x) g f 0(1 - εM) xL e x e xU x ∈ Rn (P2)
In problems P1 and P2 xF is the scalar variable that represents the purchased quantity of the feedstock to be evaluated. Analysis and Application The importance of the range analysissand the utility and flexibility indices in particularsis based on the fact that the entire analysis is done on a surface of a multitude of (near) optimal solutions with (approximately) equal value (a Pareto surface). This analysis provides a quantitative metric for each crude oil feedstock in terms of its relative importance. In an uncertain environment it is often the case that the optimal feed slate cannot be acquired at the exact amounts or there may be opportunities that need to be assessed quickly. As an example, the information that crude oil A has a flexibility index of 1 means that the rest of the crude oil feeds can substitute this particular crude oil up to 100% without compromising the optimal solution economics (in other words crude oil A can be completely replaced by a combination of the other feedstocks). Conversely, if there is additional availability (with advantageous terms) of a given crude oil B that has a utility index of 0.6 it means that this crude oil can substitute the original (remaining) crude oil slate at up to 60% with no deterioration in the overall profitability. The value of this information is evident and applicable at two levels. On the one hand, the feedstock range can be used as a way to address uncertainty in the feedstock availability. On the other hand, it can be utilized as a strategic tool for trading and evaluation decisions. As a result, the risk of having a crude oil with little flexibility (small flexibility index) can be assessed and utilized at the crude oil acquisition level. The user can then minimize the overall risk by prioritizing and emphasizing the acquisition of the crude oil feedstock with the lower flexibility index, thus minimizing the risk of an economic upset due to delay or unavailability. The user can also identify the feedstock with the higher utility index as the “back-up” feed since that feed can replace most others if needed. Utilizing this information (by creating inventory of this particular crude oil for example) will minimize the risk of operating suboptimally in the case of an unforeseen change in the crude oil slate. From an optimal solution analysis viewpoint, this solution provides a surface map of the optimal solution as opposed to a point. The analogy is to define a mountain peak purely by its coordinates (base optimal solution) as opposed to providing a map that includes the peak as well as the surrounding peaks and plateau areas. The practical advantage of the latter is the fact that the -optimal cases around the discrete base case can serve as fallback positions or as alternative operation policies. As a result of the problem formulation in P2 and P3, the additional operating points generated around the base solution are two times the number of feedstocks (2NFEEDS). A key distinction between the proposed approach and a sensitivity analysis framework is the fact that the flexibility analysis here does not depend on dual information (and the known associated issues mainly around accuracy). In addition, the information derived by this methodology is not localized at the optimal point but rather extends well beyond it. As it will be clear in the following session, the optimality range in terms of values for key decision variables can be more than 100%. Note also that there are no implied approximations of any sort (linearity or otherwise). In a complex decision making environment and the presence of large numbers of nonlinearity constraints, as it is the case in refinery planning, the proposed approach provides another level of actionable information that is very beneficial (both from a purely economic as well as from
8284 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 Table 1. Feedstock Properties feedstock name
code
price, $/Bbl
Arabian Heavy Arabian Light Alaskan North Slope Bachequero Kuwaiti Export North Sea Forties Tiajuana Light
ARH ARL ANS BAC KUW NSF TJL
82.50 85.50 86.25 82.13 84.38 27.45 86.55
Table 2. Base Optimal Solution code
base optimal value (KBbl)
base optimal value (% Feed)
ARH ARL ANS BAC KUW NSF TJL Total
7.6 40.0 30.8 1.0 1.4 6.1 3.1 90.0
8.4 44.4 34.2 1.1 1.6 6.8 3.4 100.0
a risk management viewpoint) to the trading organization that supports feedstock acquisitions. One practical aspect of this approach lies in the initialization and pruning strategy for solving the NLP subproblems P1 and P2. As we have stated, the upper bound of the number of optimization subproblems to be solved is (2NFEEDS). However, in practice this is very rarely the case. By inspecting the solutions of prior subproblems, with respect to the key optimization variables (crude oil feedstocks), we can deduce what certain cases need not be executed simply because the corresponding feeds have reached one or both bounds in a prior optimization case. By utilizing this property of subproblems P1 and P2 along with the natural “warm start” afforded by prior solutions, we have realized very significant performance benefits to the point that this methodology is automated and computationally efficient. Those attributes make this approach very practical even for routine planning operations. Examples We consider the following model of a refinery with a crude oil processing capacity of 90 KBPD. The eight potential feedstocks to be evaluated are listed in Table 1. The base optimal solution is summarized in Table 2. The results of the feedstock range analysis are summarized and displayed in Figures 1, 2, and 3. In Figure 1, the operating range for each individual feedstock is represented as a percent of the total feed. The vertical bar represents the band of optimal operation with x-mark indicating the base optimal solution. From this graph it is evident that the ARH feed can be completely
Figure 2. Feedstock indices.
Figure 3. Optimal feedstock allocation scenarios.
replaced by the rest and can optimally vary in the range between 0% and 19% of the total feed. Feedstock ANS has the maximum range (21 percentage points) although it cannot be replaced in an optimal fashion (it has a minimum of 15%). On the other side of the spectrum, feed ARL has the least flexibility and thus, from a risk management perspective, it is the feed that has the highest economic impact if not acquired. Flexibility Analysis and Risk Management On the basis of the above and considering the different crude oil acquisitions as independent events, there is a direct correlation between the operational risk R (to be forced to operate in a significantly suboptimal fashion) and the flexibility index as follows. Let P(ROPT) be the probability to operate near optimality as determined only by the probability of acquiring all individual feedstocks on time, P(Fi). Using the notion of independent events the above probability is P1(ROPT) )
∏
P(Fi)
(3)
i∈FEEDS
This probability is increased if we use the results of the feedstock analysis presented above for prioritizing the acquisition process according to the flexibility index. In this case for the feeds that have flexibility index of 1, the above becomes P2(ROPT) )
∏
P(F)
i ∈ FEEDS
(4)
utility index
