(330e) A Modeling Approach to for Chemical Process Synthesis

For more than thirty years, advances in discrete-continuous optimization formulations and solution techniques have contributed to the development of systematic superstructure-based  methodologies for the design of chemical process technology. In theory, such approach has the potential to yield a true optimal solution to the process synthesis problem since it pursues a simultaneous determination of the best process structure and operational conditions, thus accounting for all the complex interactions between design decisions. However, the practical use of superstructure optimization presents its own challenges in terms of the time required to set up the realistic optimization models in a way that is also consistent with the capabilities and limitations of current modeling tools and numerical solvers.

In this work, a refined methodology for the formulation of superstructure optimization models is presented, including a new way to generate superstructures, as well as a modular implementation of models supported by the use of set handling capabilities present in common mathematical programming languages (e.g. GAMS), and the incorporation of surrogate models to reduce the inherent mathematical complexity, allowing the application of current optimization solvers (e.g. DICOPT, BARON). We show how such approach leads to rich and yet solvable models whose final implementation can be created, modified and set up for solution in a very short time.

Superstructure generation

The generation of a proper superstructure is the first key step in the successful application of the mentioned optimization-based process synthesis methods. In fact, since the proposed superstructure supports the formulation of the optimization model,  it ultimately determines its search space as well as its mathematical complexity. The richer the superstructure (in terms of the number of unit operations and interconnections), the wider the search space and the higher the probability of finding the best process configuration; however, richer superstructures also lead to more complex optimization models.

Within our methodology, superstructures are build from three basic element types: Processing units, unit ports and streams. Processing units, include reaction and separation operations only. Here, ports are used as interfaces to connect a unit to the rest of the superstructure. Inlet and outlet unit  ports are considered stream mixers and splitters, respectively, allowing multiple units connected to every port.  Streams are regarded as connections between an outlet unit port and an inlet unit port, being capable of performing a general conditioning operation (i.e. temperature and pressure conditioning). Given an initial set of operations, a high connectivity between ports results in a very rich superstructure. In this work, we discuss guidelines for the selection of a proper set of operations, as well as rules to avoid counterproductive connections. Ultimately, the goal is to create a rich but reasonable superstructure based on engineering judgment.

Effective superstructure modeling

Independent of the graphic representation and the mathematical programming strategy used (e.g. Mixed Integer Non-linear Programming.), a large superstructure model includes the models of its unit operations and streams, along with complementary constraints involving discrete variables that allow the simultaneous evaluation of multiple operating conditions and process configurations. In this way, the practical implementation of a large superstructure optimization model requires for the use of a modular approach, where building block models are developed independently for every superstructure element type (e.g. units, unit ports, and streams) with constrains and variables indexed over proper control sets.

For our work, we have implemented these standard element models using GAMS. A distinct model was build per unit operation type in terms of connection port variables and the unit operational variables. Raw material and unit outlet ports are modeled as multi-stream splitters. Product and unit inlet ports are modeled as multi-stream mixers enforcing pressure balance on their inlet streams. Streams are modeled as general conditioning units combining compression/expansion and heating/cooling capabilities. The stream model can be as simple or complicated as required (e.g. including multiple cooling/heating and isentropic/adiabatic compression/expansion stages) providing flexibility and the possibility of additional modeling developments to account for process heat and power integration. All the constraints (i.e. equations and inequalities) in these models are indexed by one or more of the following fundamental sets: components (user supplied), units (calculated from a user supplied parameter indicating the number of units per type included in the superstructure), ports (calculated automatically from the set of units and a predefined list of in-out ports per unit type) and streams (declared as outlet-inlet port pairs and calculated automatically from a user supplied parameter).

The mentioned modular approach, combined with set indexing, allows the set up and modification of complex superstructure optimization models in a very short time by specifying  a short list of key parameters: the component list, number of units per type, and list of connections.

Mathematical complexity and the use of surrogate models

Most of the mathematical complexity in a superstructure optimization model comes from its discrete nature and the highly non linear equations used to realistically model component mixtures and chemical processing units. In principle, the inherently large number of equations, variables and types of nonlinearities (including non-convexities) lead to models that are difficult to solve, even when using state of the art optimization solvers. To reduce such complexity, we propose the replacement of complicated and highly non-linear equations with surrogate models. This replacement can be done at different levels, going from individual equations (e.g. thermodynamic properties calculations and reaction kinetics) and parts of unit operation models, to entire unit models, and even entire plant subsystem models.

The mentioned approach leads to the creation of unit surrogate models composed by many of the linear equations in the original unit models (e.g. material and energy balances, etc.) and a set of dimensionally reduced mappings which help establishing the correct non-linear relationship between the surrogate model variables. The final surrogate implementation  involves the construction of  non-linear mappings as multivariable surfaces  built by fitting data generated using commercial process simulators. We have also established that the use of  surrogate thermodynamics (e.g. a surrogate model that allow the calculation of key thermodynamic properties from mixture state variables) is key in our approach. For this project, a MATLAB code including a MATLAB-(APEN PLUS) interface has been developed to automatically generate samples of the independent variables, execute APEN PLUS simulation cases, retrieve simulation results and create the mentioned mappings via data fitting. Several examples are included to present the key modeling ideas as well as the final implementation and computational efficiency of the methodology.