The framework presented here for the simultaneous design and control optimization via MIDO and multi-parametric programming relies on a series of software tools and mathemat- ical formulations, procedures and solvers. In this subsection the most common limitations are presented per step of the framework.
• Step 1: ‘High fidelity’ model – A commercial package is typically used for this purpose. Although PSE’s gPROMS•is very trustworthy software package, it can oftenR
pose problems in terms of the speed of the simulation, especially in more complex model formulations that include PDAE’s in more than one dimensions. This step is among the most robust steps of the framework as it is rarely the bottle-necking factor. On the contrary to other available simulation software packages it produces robust simulation results even for a large range of scales among the model equations (typically a scaling factor of 1e6 is preferable.)
• Step 2: Model approximation – The ability of representing a ‘high fidelity’ model accurately via a single linear state-space model suitable for multi-parametric program- ming poses a multiple challenges. Primarily (i) the user needs to derive a model where only linear relationships between the inputs, (approximate) states and disturbances
are present and (ii) the size of the model must be small enough to allow for an effi- cient ‘Step 3’ of the framework. The algorithm used throughout the thesis for model approximation is presented in Appendix B and discusses the challenges related to (i). Regarding (ii), the size of the approximate model (i.e. the number of states) affects the framework in the following ways:
– The size of the parametric vector is a function of the size of the state vector (see
next step).
– The approximate model states do not typically have physical meaning, and they
do not necessarily correspond to the ‘high fidelity’ system states, therefore a procedure of approximating the states is necessary on the online application of the framework. This could result into the formulation of a Moving Horizon Estimator. Although an offline solution for such types of problems can be obtained, it creates an extra burden to the framework, especially for a large state vector.
– Every state of the approximate model introduces at least two linear inequality
constraints to the multi-parametric programming problem corresponding to the states allowable lower and upper bounds (see next step).
– The approximation of multiple piecewise linear state space models, although it
alleviates the mismatch burden and possibly the size burden, it introduces binary variables to a continuous system, a mathematical artifact of a bad fit. The result is the formulation of a mp-MIP problem of quadratic of linear nature (see next step).
Based on the above, this step is identified as the greatest bottleneck in the framework. This is an active area of research where different methods are tested and added to the framework.
• Step 3: Multi-parametric programming – There are two major factors that affect this step of the framework. The first is the size of the problem at hand and the sec- ond the type of the problem with respect to available solvers. Despite common belief, the size of the parametric vector plays a small role in the efficiency of the solution of available parametric solvers. On the other hand, the size of the optimization variable vector and the constraints are key. As an example, consider an mp-MILP problem with 24 continuous variables, 16 binary variables, 6 parameters and 120 contraints. Assuming that all possible binary combinations are considered (216 = 65536 combina-
full, non-degenerate solution1 (1120 24
2
= 1.09 ◊ 1025 combinations) the upper bound of
number of critical regions for this problem (without taking into account the parametric bounds) is approximately 7.1◊1029. The POP algorithm is able to fathom a very large
percentage of these regions without explicitly considering the solution (via considering feasibility of the active set and the parametric bounds). This happens through the use of any of the three available solution algorithms [258]. Nevertheless, the offline computational time to acquire a solution of the multi-parametric programming prob- lem becomes significant even for problems the deterministic version of which would be a matter of seconds (via e.g. CPLEX). Given that within the framework the multi- parametric programming problems are formulated based on a state space model the size of the model affects the size of the problem in several ways: (i) all inputs of the state space become optimization variables for the mp-P, (ii) all disturbances are added to the parametric vector, (iii) all states are added to the parametric vector, (iv) every state and output variable introduces at least two constraints in the mp-P problem. Given the effect that the number and optimization variables and constraints have to the size of the solution, described above, it is clear how this step’s efficiency correlates with Step 3. Note that multi-parametric problems with up to 100 parameters can be solved via available solvers but the number of states for which a solution of the result- ing mpP problem can be efficiently obtained is a ten fold lower. Note that the easiest problem to hande here is the mp-QP problem, followed by the mp-LP problem. The existence of binary variables makes the mp-MILP easier to solver than the mp-MIQP as there is no need for a global optimizer and in cases of envelopes of solutions the comparison procedure in mp-MILPs is proven to be linear.
• Step 4: Closed-loop validation – There is a dual approach in this step. The first is via gOMATLAB, a tool commercially available, the application and usage of which falls into the aspects described in Step 1. The second one is the development of a dynamic link library (dll) that contains the muti-parametric rolling horizon policy and is introduced within gPROMS•. Although the implementation of the tool withinR
gPROMS•is quite straightforward, the development of the library can pose two chal-R
lenges: (i) the implementation of an efficient point location algorithm in C++ and (ii) the compilation of a large code in C++. Note that the choice of programming language is predefined by the gPROMS•developers and for the user to comply. Challenge (i) isR
not a problem unless the solution is significantly over a few thousands critical regions in which case the point location algorithm could take a few tens of a second to com-
plete. The completion time for the online point location and affine function evaluation is typically lower than a few hundredths of a second. Challenge (ii) is less manageable in cases where the size of the solution is as large as described before. The compilation time can therefore vary greatly. Typically reducing the floating point accuracy helps with the compilation time without affecting the accuracy of the solution. Furthermore, there is ongoing work to develop the C++ algorithm is a way that the solution will be read through an external text file instead of embedding the solution within the C++ code. This will be advantageous as the dll will be pre-compiled for all possible solutions and will mainly contain the point location algorithm.
• Step 5: Dynamic optimization– The dynamic optimization is the second most frag- ile part of the framework. The efficiency of the algorithm is highly affected by the size and complexity of the model and the optimization horizon and steps to be considered. More specifically, the MIDO algorithm used here guarantees local optimality via an Outer Approximation based methodology and a Sequential Quadratic Programming approach. Therefore, the solution is heavily affected by the initial iteration. Ideally, a number of different initial alternatives should be considered for a better solution to be guaranteed. The size of the problem is the key limiting factor. In cases where the model is large (e.g. the CHP example) the solution of the MIDO can take a significant amount of time or even end with an error (typically a constraint violation). Restarting the optimization from the last known feasible solution is the only way forward in such cases. Also, it is advisable to start the MIDO algorithm considering a few optimization steps as a starting point and increase them gradually.
PAROC is currently the only framework and prototype software platform that offers an integrated approach to the unification of control, operational and design optimization of process systems.