The presence of uncertainty in process systems is one of the key reasons for deviation from set operation policies, resulting in suboptimal or even infeasible operation. As these uncertainties realize themselves on different time scales such as on a control, scheduling or design level, an integrated, comprehensive approach to consider uncertainty is required. Thus, in this contribution we demonstrate PAROC (PARametric Optimization and Control), a novel unified framework for the design, operational optimization and advanced model-based control of process systems, which decomposes this challenging problem into a series of steps, shown in the Figure below [1].
The first step comprises the formulation of a high-fidelity dynamic model of the original process, as well as its validation using various techniques such as parameter estimation and dynamic optimization. This model does not only serve as the first step in translating a real-world system into a set of equations, but also as a platform for the validation of any receding horizon policy developed.
While the high-fidelity model is in general applicable to design purposes, its complexity may render its use for the development of receding horizon policies computationally infeasible. Thus, in the second step, the validated high-fidelity model is reduced in complexity and size using system identification or advanced model reduction techniques, aiming at compromising the accuracy of the original model as little as possible [2]. This approach results in a discretized state-space model, which is used in the next step for the development of receding horizon policies such as control laws and scheduling policies [3, 4].
At this step, based on the discretized state-space model, the problem of devising a suitable receding horizon policy is formulated as a constrained optimization problem. Within our framework, this problem is solved offline employing multi-parametric programming, where the states of the system are treated as parameters and the constrained optimization problem is solved as a function thereof. Due to the parameter-dependence of the constraints, different solutions might be optimal in different parts of the parameter space. This results in a partition of the parameter space into different regions, called critical regions, and each region is associated with a corresponding optimal solution of the optimization problem as a function of the parameters. As a result we obtain the receding horizon policies explicitly as a function of the states of the system[1], and reduce the computational effort of their evaluation to a point location in the parameter space and a function evaluation.
However, when solving the receding horizon policies it is assumed that the values of the state vector are exactly known. As this might not be the case, e.g. due to noise, it is necessary to infer the state information from the available output measurements using a state estimator. While a long existing model-based technique for unconstrained state estimation is the Kalman filter, the use of constrained estimation techniques such as the moving horizon estimator (MHE) can lead to significant improvements of the estimation result by adding system knowledge [5, 6]. MHE is an estimation method that obtains the estimates by solving a constrained optimization problem given a horizon of past measurements. Thus similarly to the problem of receding horizon policies, the presented framework solves the MHE problem in a multi-parametric fashion, where the past and current measurements and inputs and the initial guess for the estimated states are the parameters of the problem [5, 7].
As a last step, the obtained receding horizon policies are validated 'in-silico' using the original high-fidelity model, thus closing the loop. This validation is of crucial importance as iterative experiments on real plants might be too costly or dangerous to run. In particular in the case of multiple objectives such as minimization of error, safe operation and economically optimal performance, the possibility of performing 'in-silico' tests of a developed control strategy allows for the fine-tuning and optimal design of the control strategy.
In order to apply the afore-described framework, we also present software solutions for the different aspects of the framework. Due to its modeling and dynamic optimization capabilities, we employ PSE's gPROMS® ModelBuilder to formulate and validate the high-fidelity model of the process. Similarly, due to its wide-spread use and numerous in-build functions, the steps of model approximation as well as formulation and solution of multi-parametric programming problems is performed in MATLAB® using state-of the art software [8, 9] based on the POP® toolbox [10]. Lastly, the solution of the multi-parametric programming problem is integrated into gPROMS® using a specifically designed foreign process written in C++. This approach avoids the use of tools such as gO:MATLAB, and thus enables the use of the dynamic simulation and optimization capabilities of PSE's gPROMS®.
The applicability of this novel framework will be demonstrated on a wide range of problems such as industrial processes [11], bio-medical [5, 12, 13], cogeneration heat and power systems [14].
References
10. ParOs, Parametric Optimization Programming (POP). 2004, ParOS.
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