(409f) Analytical Results for Tiered Intermediate Product Prices for Apc Using a Refinery Lp

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.