Process Optimization via Constraints Adaptation

In the framework of real-time optimization, measurement-based schemes have been developed to deal with plant-model mismatch and process variations. These schemes differ in how the feedback information from the plant is used to adapt the inputs. A recent idea therein is to use the feedback information to adapt the constraints of the optimization problem instead of updating the model parameters. These methods are based on the observation that, for many problems, most of the optimization potential arises from activating the correct set of constraints. In this paper, we provide a theoretical justification of these methods based on a variational analysis. Then, various aspects of the constraint-adaptation algorithm are discussed, including the detection of active constraints and convergence issues. Finally, the applicability and suitability of the constraint-adaptation algorithm is demonstrated with the case study of an isothermal stirred-tank reactor.

On the Use of Transient Information for Static Real-Time Optimization

Dominique Bonvin, Grégory François, Alejandro Gabriel Marchetti

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 leads to operating conditions that typically are not optimal for the plant and may violate constraints. Furthermore, even with an accurate model, the presence of disturbances generally leads to a drift of the optimal operating conditions, and adaptation based on measurements 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 typically used to help achieve 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, while implicit schemes adjust the inputs on-line in a control inspired manner. Explicit RTO schemes solve a numerical optimization problem repeatedly. For example, the two-step approach uses (i) measurements to update the model parameters, and (ii) the updated model to perform the numerical optimization [2]. 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 it shares the first-order optimality condition with the plant. The main advantage of the technique, labeled modifier adaptation (MA), lies in its proven ability to converge to the plant optimum, even in the presence of structural plant-model mismatch [3]. Furthermore, MA is capable of detecting the correct set of active plant constraints without additional assumptions. As a static optimization method applicable to continuous plants, MA requires waiting for steady state before taking measurements, updating the correction terms and repeating the numerical optimization. Hence, several iterations are generally required to achieve convergence. The main difficulty lies in the estimation of the steady-state plant gradient at each iteration. In contrast, implicit RTO schemes, such as extremum-seeking control [4], self optimizing control [5] and NCO tracking [6], propose to adjust the inputs on-line in a control-inspired manner. In the absence of constraints, or when assumptions can be made regarding the set of plant constraints that are active at the optimum, implicit RTO methods reduce to gradient control, as the degrees of freedom are adjusted in real time to drive the plant cost gradient to zero. Here again, the difficulty lies in the estimation of the steady-state plant gradient, which, in addition, must be performed during transient operation. This is achieved via either low-frequency plant excitation and corresponding cost measurements (as in extremum-seeking control) or the use of transient measurements together with a model of the steady-state gradient (as in self optimizing control and NCO tracking via neighboring extremals, where the required steady-state measurements are simply replaced by the corresponding transient measurements) [7]. Implicit RTO is much more challenging when the set of active constraints is unknown, as not only the cost gradient has to be inferred from the measurements but also the set of active constraints and the constraint gradients. This contribution proposes a framework for using MA during the transient phase toward steady state, thereby attempting to reach optimality in a single iteration to steady state. With this approach, a modified optimization problem is solved repeatedly at each optimization instant during transient, with the input-affine correction terms, which theoretically depend on steady-state plant quantities, being estimated on the basis of transient measurements. Note that such an attempt has already been documented in the literature but, as for the aforementioned implicit methods, the "steady-state" gradients were estimated using transient information in the framework of both multiple units and neighboring extremals [8]. In contrast, this work estimates the steady-state outputs from transient outputs and then uses these estimates "correctly" in the expressions for computing the steady-state gradients. For this, we propose to use the best available dynamic model and perform state estimation using an Extended Kalman Filter (EKF) framework [9]. Since the model is typically not perfect, one key parameter related to the static gain is made adjustable for each input-output pair. This way, the EKF feeds on the transient plant outputs and estimates, at the current time t, the corresponding steady-state outputs, which leads to the computation of the staticgradient. The dynamic model at hand can be seen as a surrogate model that, although not sufficiently accurate globally for process optimization, can process measurements to generate an estimate of the local gradients. The approach will be illustrated on various numerical examples and then applied to the optimization of a continuous stirred-tank reactor. References : [1] G. François 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] 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). [3] A. Marchetti, B. Chachuat and D. Bonvin, Modifier-Adaptation Methodology for Real-Time Optimization, Industrial & Engineering Chemistry Research, 48(13), 6022-6033 (2009). [4] K. Ariyur and M. Krstic, Real-Time Optimization by Extremum-Seeking Control, John Wiley, New York (2003). [5] S. Skogestad, Plantwide Control: The Search for the Self-Optimizing Control Structure, J. Process Control, 10, 487-507 (2000). [6] G. Francois, B. Srinivasan and D. Bonvin, Use of Measurements for Enforcing the Necessary Conditions of Optimality in the Presence of Constraints and Uncertainty, J. Process Control, 15, 701-712 (2005). [7] G. Francois, B. Srinivasan and D. Bonvin, Comparison of Six Implicit Real-Time Optimization Schemes, J. Européen des Systèmes Automatisés, 46, 291-305 (2012). [8] G. Francois and D. Bonvin, Use of Transient Measurements for the Optimization of Steady-State Performance via Modifier Adaptation, Industrial & Engineering Chemistry Research, 53(13), 5148-5159 (2014). [9] A. H. Jazwinski, Stochastic Processes and Filtering, Mathematics in Science and Engineering, Academic Press (1970).

2015

On enforcing the necessary conditions of optimality under plant-model mismatch - What to measure and what to adapt?

Martand Singhal

Industrial processes are run with the aim of maximizing economic profit while simultaneously meeting process-critical constraints. To this end, model-based optimization can be performed to ensure optimal plant operations. Usually, inevitable model inaccuracies are dealt by collecting the plant measurements at the local operating conditions in order to adapt model parameters, followed by numerical re-optimization. This iterative two-step procedure often results in a sub-optimal solution, since the models are typically not designed for optimization. Modifier Adaptation (MA) is a Real-Time Optimization (RTO) technique that directly adds the affine-correction terms to the model. The affine corrections are parametrized in modifiers that are tailored to the optimization needs. This enables modifier adaptation to guarantee, upon convergence, matching the plant and the modified model's optimality conditions. However, computing the modifiers requires estimates of the plant gradients that are obtained via expensive plant experiments. The experimental cost can be reduced by relying more on the model of the considered plant. For example, Directional Modifier Adaptation (DMA) relies on offline-computed local parametric sensitivity analysis performed on the gradient of the Lagrangian function of the model resulting in reduced number of input directions that describe the gradient uncertainty in the model. Thereby, plant gradients are estimated only in a low-dimensional space of privileged input directions considerably reducing the experimental costs. However, local sensitivity analysis is often ineffective when the gradient of the model is considerably nonlinear in parameters. This thesis proposes an online procedure based on global sensitivity analysis for finding the most promising privileged directions that adequately compensates for the model deficiencies in predicting the plant optimality conditions. The discovered privileged directions are such that, upon parametric perturbations, the gradient varies a lot along the privileged directions and varies only a little along the remaining input directions. Consequently, the gradients of the model cost and constraints are corrected only along the privileged directions by adapting modifiers. The resulting methodology is named as Active Directional Modifier Adaptation (ADMA). Several simulation studies conducted show that the proposed approach reaches the near-optimality conditions at a considerably reduced experimental cost. In addition, this thesis attempts to establish a direct relation between the optimality conditions and the parameters of a given model. Model parameters are analyzed to discover mirror parameters that mimic the behavior of modifiers in influencing the optimality conditions. It is proposed to adapt mirror parameters instead of modifiers yielding the benefit of both, modifier adaptation in enforcing optimality conditions and of parameter adaptation in better noise handling and convergence. Moreover, it is investigated how to establish the synergies between privileged input directions with model parameters in order to reduce experimental efforts. The steady-state optimization of a simulated chemical process shows that the privileged directions and the selected parameters work together to reach near-optimal performance. Finally, the study on the power maximization of flying kites leads to the development of trust-region based ADMA method to better control the input step size.

EPFL2018

Real-time optimization of continuous processes via constraints adaptation

Dominique Bonvin, Benoît Chachuat, Alejandro Gabriel Marchetti

In the framework of process optimization, measurements can be used to compensate for the effect of uncertainty. The method studied in this paper combines a process model and measurements to iteratively improve the operation of continuous processes. Unlike many existing real-time optimization schemes, the measurements are not used to update the process model, but to adapt the constraints in the optimization problem. Upon convergence, all the constraints are respected even in the presence of large model mismatch. Moreover, it is shown that constraints adaptation can handle changes in the set of active constraints. The approach is illustrated, via numerical simulation, for the optimization of a continuous stirred-tank reactor.