Real-Time Optimization Via Modifier Adaptation – on Updating the Model Outputs

Optimal operation of chemical processes is key for meeting productivity, quality, safety and environmental objectives. Both model-based and data-driven schemes are used to compute optimal operating conditions [1]:

-       The model-based techniques are intuitive and widespread, but they suffer from the effect of plant-model mismatch.  For instance, an inaccurate plant model can lead to operating conditions that are not optimal for the plant and may violate constraints. Furthermore, even with an accurate model, the presence of disturbances typically leads to a drift of the optimal operating conditions, so that measurement-based adaptation is needed to maintain plant optimality.

-       The data-driven optimization techniques rely on measurements to adjust the optimal inputs in real time. Consequently, real-time measurements are used to help reach plant optimality. This field, which is labeled real-time optimization (RTO), has received growing attention in recent years. RTO schemes can be of two types: explicit schemes solve a numerical optimization problem repeatedly at each iteration, while implicit schemes adjust the inputs on-line in a control-inspired manner [2].

We consider here model-based explicit RTO schemes that solve an updated optimization problem at each iteration. For example, the two-step approach uses (i) measurements to update the model parameters, and (ii) the updated model to compute optimal inputs [3]. It has also been proposed to update the model differently. Instead of adjusting the model parameters, input-affine correction terms can be added to the cost and constraint functions of the optimization problem so that the modified model shares the first-order optimality condition with the plant [4]. The main advantage of this approach, labeled modifier adaptation (MA), lies in its proven ability to converge to the plant optimum, even in the presence of structural plant-model mismatch [5]. The ability of the algorithm to converge to the plant optimum is linked to the quality of the model, the filtering used, and the initial operating point. Methods for ensuring global convergence tend to slow down convergence [6], which makes them unsuited for some practical applications. The present contribution proposes a different correction to the model that can both increase the convergence speed and enlarge the set of acceptable initial operating points in some applications.

Standard MA introduces input-affine corrections of the cost and constraint functions, since these functions are linked to the KKT optimality conditions. The proposal here is to update the model outputs y instead, thus leading to the approach labeled MA-y. We will demonstrate that this approach results in input-affine corrections of the cost and constraint functions as well. Consequently, MA-y also leads to plant optimality upon convergence, provided perfect output measurements and gradient estimates are available. Furthermore, it turns out that, in cases where the cost and constraint functions are nonlinear functions of the output variables, the input-affine corrections of the outputs also lead to partial correction of the model Hessians, which results in the aforementioned improvements in convergence properties. For example, it can be shown that, if the outputs are affine in the inputs u, MA-y converges to the plant optimum in a single iteration, while the standard MA might take several iterations to do the same. The approach will be illustrated on the performance optimization of the Williams-Otto continuous stirred-tank reactor.   

References

 

[1]       G. Franois and D. Bonvin, Measurement-Based Real-Time Optimization of Chemical Processes, In S. Pushpavanam, editor, Advances in Chemical Engineering, Vol. 43, 1-50, Academic Press (2013).

 

[2] B. Chachuat, B. Srinivasan and D. Bonvin, Adaptation Strategies for Real-Time Optimization, Comput. Chem. Engng, 33(10), 1557-1567 (2009).

[3] T. E. Marlin and A. N. Hrymak, Real-Time Operations Optimization of Continuous Processes, In AIChE Symposium Series - CPC-V, Vol. 93, 156-164 (1997).

[4] A.G. Marchetti, B. Chachuat and D. Bonvin, Modifier-Adaptation Methodology for Real-Time Optimization, Industrial & Engineering Chemistry Research, 48(13), 6022-6033 (2009).

 [5] A.G. Marchetti, G. Franois, T. Faulwasser and D. Bonvin, Modifier Adaptation for Real-Time Optimization -- Methods and Applications, Processes, 4(4), 55 (2016) doi:10.3390/pr4040055.

 [6] A.G. Marchetti, T. Faulwasser and D. Bonvin, A Feasible-Side Globally Convergent Modifier-Adaptation Scheme, J. of Process Control, in press (2017).