(373aa) A Nonconvex Miqcp Model for Optimal Scheduling of Furnaces Shutdown in a Steam-Cracking Ethylene Plant Under Demand Uncertainty

In competitive and changing environments, efficient operation and planning of the output level is paramount. In the presence of demand fluctuations, the efficient use of the equipment and inventory can be accomplished through simultaneously planning and scheduling, which should deal with tradeoffs such as production and inventory costs, customer responsiveness and maintenance shutdowns. Properly accounting for demand variations is a requirement when aiming to avoid loss of market share, unnecessary inventory costs and customer demand unsatisfaction.

This work addresses the middle-term production planning problem under demand uncertainty of an ethane-fed ethylene plant, taking into account the interactions between the entire plant operation and eight parallel cracking furnaces over a discretized time horizon. The plant exhibits complex behavior due to an important ethane recycle and furnaces decaying performance that leads to higher heat duty requirements and/or lower conversion rates (Jain & Grossmann, 1998). The furnaces have to be frequently shutdown to perform cleaning tasks, with the consequent total loss of furnace production during that period. The proposed model, based on Schulz et al. (2006), represents ethane cracking furnaces and the separation train with surrogate models based on plant data as function of typical operation variables. It represents the decaying performance of the cracking furnaces due to coke deposition on the internal coil walls during operation by means of the coil “roughness”.

The optimal maintenance scheduling for these reaction units is represented with a Generalized Disjunctive Programming (GDP) model and translated into an MINLP using the convex-hull reformulation (Trespalacios and Grossmann, 2014). The decaying performance and the equipment recovery into “good-as-new” condition after maintenance has been performed following the relative inefficiency concept, as defined by Wu et al. 2018 (2). The resulting model is a nonconvex MIQCP problem.

In this work, we also propose a stochastic programing model. Parametric uncertainty is usually dealt as two-stage optimization problems. In the case study, first stage decisions are associated to manufacturing variables (production levels, process units operating conditions and furnaces run lengths), while second stage decisions are related to logistics (inventory levels, product sales and shortage of product). In the present work, equivalent deterministic forms are employed to express the stochastic elements of the model, thus avoiding the use of discretization and sampling techniques. It is widely assumed that demands have a normal distribution not only because of convenience but also this assumption captures the main features of demand uncertainty. Since product demands are typically affected by a large number of stochastic events, on the basis of the central limit theorem supports the normality assumption. In a stochastic model, the amount of product sold is the minimum between the planned sales (PS) and the demand realization (θ).

The stochastic model has as a goal to maximize the expected profit while avoiding overproduction (thus, unnecessary production and inventory costs) and underproduction (missed sales and loss of market share). Therefore, the objective function is the maximization of the expected value of the revenue, minus the expected value of inventory costs and expected value of penalties for underproduction. It is subject to satisfying the product demands with prespecified probability levels, i.e., chance-constraints that are included to guarantee that product demands are met with a certain level of probability. As the demand is assumed to have a normal distribution, these expected value terms are expressed with their corresponding deterministic equivalent forms, that are functions of the expected value of the minimum between PS and θ, E[min(θ,PS] (Petkov & Maranas, 1997). This nonlinear, convex function, which is included in the calculation of expected values for revenues, inventory costs and penalties for underproduction, is closely approximated with a piecewise quadratic surrogate model by solving a separate MIQCP problem to obtain the optimal approximation, where the discontinuity point is also a variable and smoothing constraints are considered.

The deterministic model has 15,447 continuous variables and 128 binary variables, while the stochastic one has 15,574 continuous variables. Both problems have been solved with the nonconvex MIQCP solver in GUROBI 11.0 in GAMS 46.3, and the solutions have been compared to the local solutions obtained with DICOPT. It is shown that, even when preliminary results do not provide the global optimum (close the gap) with GUROBI, a 103% increase in the stochastic objective function is obtained with respect to the local solver.

References

Achkar, V., B.B. Brunaud, Rami Musa, I.E. Grossmann (2024), Efficient computational strategies for a mathematical programming model for multi-echelon inventory optimization based on the guaranteed-service approach, Computers and Chemical Engineering 182, 108567.

GUROBI 11.0 Release notes, https://www.gurobi.com/whats-new-gurobi-11-0/?utm_source=google&utm_med…

Jain, V, I.E. Grossmann, Cyclic scheduling of continuous parallel‑process units with decaying performance (1998), AIChE J, 44, 7, 1623-1636.

Petkov, C.D. Maranas (1997), Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty. Ind.Eng.Chem.Res. 36,4864-4881.

Schulz, E.P., Bandoni, A., M.S. Diaz (2006), Optimal Shutdown Policy for Maintenance of Cracking Furnaces in Ethylene Plants. Ind. Eng. Chem. Res. , 45, 2748-2757.

Trespalacios, F., I.E. Grossmann (2014), Review of mixed-integer nonlinear and generalized disjunctive programming methods, Chemie Ingenieur Technik 86, 991-1012.

Wu, Yaqing; Maravelias, Christos T.; Jiang, Yuqiu; Fan, Jingduo; Wenzel, Michael J.; Mohammad, ElBsat N.; Turney, Robert D. (2018),"Optimization Methods for Predictive Maintenance Scheduling of Building Heating/Cooling Equipment with Performance Decay". International High Performance Buildings Conference. Paper 273. https://docs.lib.purdue.edu/ihpbc/273