Algebraic/implicit loops handling by Gekko

Question

I have got a specific question with regards to algebraic / implicit loops handling by Gekko.

I will give examples in the field of Chemical Engineering, as this is how I found the project and its other libraries.

For example, when it comes to multicomponent chemical equilibrium calculations, it is not possible to explicitly work out the equations, because the concentration of one specie may be present in many different equations.

I have been using other paid software in the past and it would automatically propose a resolution procedure based on how the system is solvable (by analyzing dependency and creating automatic algebraic loops).

My question would be:

Does Gekko do that automatically?

It is a little bit tricky because sometimes one needs to add tear variables and iterate from a good starting value.

I know this message may be a little bit abstract, but I am trying to decide which software to use for my work and this is a pragmatic bottle neck that I have happened to find.

Thanks in advance for your valuable insight.

Answer 1

Python Gekko uses a simultaneous solution strategy so that all units are solved together instead of sequentially. Therefore, tear variables are not needed but large flowsheet problems with recycle can be difficult to converge to a feasible solution. Below are three methods that are in Python Gekko to assist in efficient solutions and initialization.

Method 1: Intermediate Variables

Intermediate variables are useful to decrease the complexity of the model. In many models, the temporary variables outnumber the regular variables. This model reduction often aides the solver in finding a solution by reducing the problem size. Intermediate variables are declared with m.Intermediates() in Python Gekko. The intermediate variables may be defined in one section or in multiple declarations throughout the model. Intermediate variables are parsed sequentially, from top to bottom. To avoid inadvertent overwrites, intermediate variable can be defined once. In the case of intermediate variables, the order of declaration is critical. If an intermediate is used before the definition, an error reports that there is an uninitialized value. Here is additional information on Intermediates with an example problem.

Method 2: Lower Block Triangular Decomposition

For large problems that have trouble with initialization, there is a mode that is activated with the option m.options.COLDSTART=2. This mode performs a lower block triangular decomposition to automatically identify independent blocks that are then solved independently and sequentially.

This decomposition method for initialization is discussed in the PhD dissertation (chapter 2) of Mostafa Safdarnejad or also in Safdarnejad, S.M., Hedengren, J.D., Lewis, N.R., Haseltine, E., Initialization Strategies for Optimization of Dynamic Systems, Computers and Chemical Engineering, 2015, Vol. 78, pp. 39-50, DOI: 10.1016/j.compchemeng.2015.04.016.

Method 3: Automatic Model Reduction

Model reduction requires more pre-processing time but can help to significantly reduce the solver time. There is additional documentation on m.options.REDUCE.

Overall Strategy for Initialization

The overall strategy that we use for initializing hard problems, such as flowsheets with recycle, is shown in this flowchart.

Sometimes it does mean breaking recycles to get an initialized solution. Other times, the initialization strategies detailed above work well and no model rearrangement is necessary. The advantage of working with a simultaneous solution strategy is degree of freedom swapping such as downstream variables can be fixed and upstream variables calculated to meet that value.