Intermediate tiered prices are needed for advanced or multivariable process control (APC) and economic optimization in petroleum refineries.  The objective function of an APC controller requires economic values for intermediate products such as naphtha, gas oils, resids, etc.  These economic values are used directly or indirectly depending on the choice of APC technology.  Controlled variable targets and manipulated variable set-points are directly determined from the once-a-minute (typically) solution of the APC LP or QP. The tiered price of refinery products such as aviation gasoline and motor fuel are established by their market value. Intermediate products like naphtha and heavy oils generally do not have a direct market value and their prices are determined implicitly by variations associated with fuel product prices, yield characteristics, operating costs, capacity limitation of equipment and, crude type.

This paper addresses the use of marginal values generated by the Refinery Planning & Economics plant-wide LP for intermediate product prices in APC controllers.  Solution of a plant-wide LP produces marginal values or shadow prices for all intermediate products by virtue of material balance equality constraints.  As background material, this paper provides a detailed discussion of the unique characteristics, correct interpretation, and theoretical implications of marginal values associated with material balance constraints.  The background discussion (geometric and analytic) that is provided extends beyond the surprisingly limited treatment of marginal values for equality constraints found in most LP texts.  A solid understanding of the implications of material balance equality constraints is required to correctly interpret the results presented in the remainder of the paper.

There is no consensus regarding the potential use of refinery-wide LP marginal values for APC intermediate product prices.  Rigorously speaking, a purist can state two reasons not to use LP marginal values for APC intermediate product prices.  The first is that the model used to generate the marginal values is linear while the actual process is nonlinear.  The second is that the refinery will normally operate at a point different than the optimum LP solution.  In both cases the purist can claim that the LP marginal values will be different from the “true” marginal values.  While true, the practical questions are 1) how large is the difference between the LP and “true” marginal values, and 2) is the relative difference between LP and “true” marginal values for different intermediate products constant?  The APC controller is more concerned with the latter.  While the purist is correct, the APC engineer needs intermediate product prices. 

The results reported in this paper assume the process is linear and investigate the change in LP marginal values for intermediate product prices over varying operating conditions.   The intent is to understand the applicability of intermediate product price over a range of different operating conditions. The concept of tiered price is introduced to address this issue.  We define tiered prices as the marginal values at degenerate points when a maximum capacity constraint is systematically reduced.  In this work tiered prices were generated by varying the crude distillation capacity of a refinery LP with 33-decision variables and 39-constraints obtained from literature.

The feasible region in a refinery LP is characterized by the null space defined by the material balance constraints. While tiered prices were generated by varying a particular constraint, the constraint of interest traverses the null space. The feasibility of the LP solution is maintained as long as the constraint movement is restricted within the null space.  Analytically, the LP basis changes at the optimum solution associated with each degenerate LP solution or tier. Changes in intermediate product prices were not observed at all the tiers. Instead, the marginal values of intermediate products were found to vary only at specific tiers. The rationale behind this occurrence is correlated to the type and behavior of the active constraints at each tier. A physical and theoretical interpretation of tier prices relative to constraint movement and the activity of decision variables are provided and offer a better appreciation for the values and ranges of intermediate product prices. The discussion also addresses the APC engineers concern on when to change the price in APC objective function.  To assist better understating and visualization of these concepts an example 3-D LP problem with algebraic and graphical solution will be provided.