(755f) Inventory Pinch Algorithms for Gasoline Blend Planning and Scheduling and Refinery Production Planning

We introduce inventory pinch points and construct temporal multiscale models that are a basis for a new type of algorithms for planning and scheduling of gasoline blend operations and oil refinery operations.

Current gasoline blend scheduling models use fixed recipes or these are computed using blend indices that blend linearly instead of the actual properties (i.e. octane, Reid vapour pressure, etc.) which behaviour is nonlinear.  Continuous-time formulation is usually preferred since the number of discrete variables is less than the corresponding discrete-time model, thus reducing the search space for the optimum.  Most of the time, blend recipes for different runs of the same product are different as well; such solutions are awkward to apply in practice, since practitioners know intuitively that the same grade of gasoline can be blended by the same recipe for extended lengths of time.  Recently introduced complex EPA formulations for gasoline quality constraints complicate the picture further, since they are highly nonlinear and require highly specialized algorithms to be solved to global optimality.  Solving multi-period MINLP models with these constraints remains in the domain of very specialized experts.

This work presents a three-level iterative algorithm to solve the gasoline blend planning and scheduling optimization problem while reducing the number of different blend recipes.  The first level deals with optimization of the blend recipes, the second level with the blender assignment, and at the third level the product distribution sequence is computed.  The first level is formulated as a discrete-time multi-period NLP, and the second and third level as a discrete and continuous-time multi-period MILPs, respectively. Time grids of the three levels are partially synchronized.  After computing at the top level the lowest cost blend recipes, a production schedule that uses these optimal recipes is calculated.  The algorithm (i) computes the blend schedule that keeps the blend recipes as constant as possible along the blending horizon, (ii) avoids the need to use multi-period MINLP models, and (iii) computes the solutions that have the same cost as the multi-period MINLP optimum.

An inventory pinch point on the cumulative total demand (CTD) curve is a point where cumulative average total production (CATP) curve intersects the CTD curve, so the CATP curve is above the CTD curve and if we extrapolate the CATP curve from this point onwards it will not cross the CTD curve.  The inventory infeasibilities are defined as the missing volume to fulfill the demand of one or more products by using only the fixed recipes and the available amount of blending components.  The inventory pinch points delineate the periods with constant blend recipes in the recipe optimization problem.  Time horizons at the second and third level are divided at an increasing time resolution within the time periods of the recipe optimization level.  The blend recipes and target inventories from the first level are fixed at their corresponding time intervals in the lower levels.

Sometimes, inventory infeasibilities may still appear at the lower levels when using the blend recipes from the top level.  These infeasibilities are encountered because: (i) some logistic constraints (e.g., specific production sequences) may reduce the degrees of freedom to compute a production schedule with a fixed recipe, or (ii) the presence of large variations on the blending components supply profiles.  An iterative procedure based on the inventory infeasibilities and the time when they occur is used to generate a feasible schedule.  Slack variables are introduced in the model to detect the inventory infeasibilities, as well as in the objective function to minimize, multiplied by penalty coefficients which are larger than the blending components costs.  In order to extend the use of a blend recipe, the penalty coefficients decrease with time, moving the infeasibilities as late as possible in the planning horizon.  When inventory infeasibilities are found in the second level, the corresponding first-level time periods are divided at the same point in time where the first infeasibility appears.  Then, new blend recipes are computed and used to calculate the production schedule.  If the second level is feasible, then the third level is solved using the blend recipes and target inventories from the first level and the production sequence from the second level.  If inventory infeasibilities appear, the second level is resolved using that information.  The procedure repeats until no infeasibilities are found, or if infeasibilities appear in the first level which means that there are not enough blending components available to meet the product demand or product specifications.

The results obtained with the inventory pinch algorithms are the same as those provided by the corresponding multi-period MINLP model. In addition, the number of different blend recipes computed by our algorithm is less than the number given by the MINLP solution.  The number of iterations among the three levels of the algorithm is very small.  Execution times are significantly shorter than the current continuous time approaches.

By decomposing the problem into three different hierarchical levels and using the inventory pinch point concept, complex rigorous nonlinear models can be used to compute optimal blend recipes without being solved at each time period of the production planning or scheduling horizon, avoiding the use of MINLP models and reducing the computational requirements to solve the overall problem without incurring in a great deviation from the optimal solution.

Refinery production planning models are formulated as multi-period mixed-integer nonlinear or linear programs and most of the commercial software is based on successive linear programming models.  Using the inventory pinch algorithms, the planning problem is decomposed into two levels: the top level (NLP) uses a big-bucket time grid and it calculates the target production volumes of each equipment for each mode of operation; the lower level (MILP) uses a small-bucket time grid and it computes the production plan with the constraints imposed by the top level.  Inventory pinch points on the gasoline and diesel pools define the time period boundaries of the top level.  If inventory infeasibilities appear at the lower level, then corresponding time period of the top level is subdivided.  As with the scheduling problem, the algorithm stops when there are no inventory infeasibilities in the lower level solution.

Case studies showed that inventory pinch algorithms produce (with nonlinear refinery models) solutions very close to those provided by the corresponding multi-period MINLP planning model.  In addition, the number of different blend recipes and number of iterations are smaller and execution times are shorter than those given by MINLP solution.