Model Adequacy for Real-Time Optimization

Optimization is important in science and engineering as a way of finding ”optimal” situations, designs or operating conditions. Optimization is typically performed on the basis of a mathematical model of the process under investigation. In practice, optimization is complicated by the presence of uncertainty in the form of plant-model mismatch and unknown disturbances. Without uncertainty, one could use the model at hand, optimize it numerically off-line and implement the optimal inputs in an open-loop fashion. However, because of uncertainty, the inputs need to be re-computed or adjusted in real-time based on measurements. This is the field of real-time optimization, which is labeled RTO for static optimization problems [1]. A standard way of implementing RTO is via the so-called two-step approach of repeated parameter estimation and optimization. In this scheme, measurements are used to adapt the model parameters, and the updated model is used for optimization. The fixed point of this iterative scheme is an important issue. The quality of the model is often overlooked in real-time optimization. The model is typically inaccurate due to lack of time for detailed modeling or shear complexity. More surprising is the fact that a model can predict the plant outputs well but be inadequate to push the plant to its optimum, which means that the model does not adequately represent the optimality conditions of the plant. The problem of model selection in the two-step RTO approach has been discussed in [2]. If the model is structurally correct and the parameters are identifiable, convergence to the plant optimum can be achieved in a single iteration. However, in the presence of plant-model mismatch, whether the scheme converges, or to which point it does converge, becomes anyone’s guess. This is due to the fact that the objective for parameter adaptation might be unrelated to the cost and constraint values and gradients that drive optimality in the optimization problem. Hence, minimizing the mean-square error of the plant outputs may not help in our quest for feasibility and optimality. Convergence under plant-model mismatch has been addressed in [3] and [4], where it has been shown that optimal operation is reached if model adaptation leads to matched KKT conditions for the model and the plant. This contribution addresses the convergence of two-step RTO schemes in the presence of structural plant-model mismatch. We propose to investigate the parameter estimation and optimization problems in the light of the second-order sufficient conditions of optimality to show that, in general, there are too few degrees of freedom to be able to reach plant optimality. A possible solution to this problem consists in reconciling the objectives of the parameter estimation and optimization problems along the lines of “modeling for optimization” [5,6]. One could for example use as “measured outputs” in the parameter estimation problem estimates of the necessary conditions of optimality for the optimization problem. These issues will be investigated both theoretically and via the simulation of chemical processes at steady state. References [1] Marlin, T.E. and Hrymak, A.N. (1997). “Real-Time Operations Optimization of Continuous Processes”, In AIChE Symposium Series - CPC-V. Vol. 93. 156–164. [2] Forbes, J.F. and Marlin, T.E. (1996). “Design cost: A Systematic Approach to Technology Selection for Model-Based Real-Time Optimization Systems”, Comp. Chem. Eng. 20, 717–734. [3] Biegler, L.T., Grossmann, I.E. and Westerberg, A.W. (1985). “A Note on Approximation Techniques Used for Process Optimization”, Comp. Chem. Eng. 9, 201–206. [4] Forbes, J.F., Marlin, T. E. and MacGregor, J.F. (1994). “Model Adequacy Requirements for Optimizing Plant Operations”, Comp. Chem. Eng. 18(6), 497–510. [5] Srinivasan, B. and Bonvin, D. (2002). “Interplay Between Identification and Optimization in Run-to-Run Optimization Schemes”, ACC, Anchorage 2174–2179. [6] Bonvin, D. and Srinivasan, B. (2012). “On the Role of the Necessary Conditions of Optimality in Structuring Dynamic Real-Time Optimization Schemes”, submitted to Comp. Chem. Eng.

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

Real-Time Optimization of Solid Oxide Fuel Cell Systems

Tafarel de Avila Ferreira

Optimization of industrial processes aims at minimizing operating cost or maximizing economic profit while respecting plant constraints. In process industry, real-time optimization (RTO) is often considered to ensure optimal plant operation and constraint satisfaction. Solid-oxide fuel-cell (SOFC) devices oxidize fuels such as hydrogen and methane to produce electric energy through electrochemical reactions at very high efficiency. To reduce the emission of carbon dioxide and the high cost of non-renewable energies, fuel cells have become popular as an alternative energy source with a wide range of applications. The optimal operation of SOFC systems, while being subjected to changing operating conditions such as variations in the electrical power demand, remains a challenging problem in the field of control and optimization. In addition, these types of devices are characterized by slow dynamics and drifts such as degradation. This motivates the use of real-time optimization for optimizing and controlling the operation of SOFC systems. The most popular RTO technique in industry is the so-called two-step approach. This approach consists in estimating the model parameters based on measurements by solving a parameter estimation problem, and computing the next operating point by solving an optimization problem using the updated model. However, this technique is highly dependent on model accuracy and tends to fail in the presence of structural plant-model mismatch. As an alternative, modifier adaptation has been developed to enforce convergence to the plant optimum even in presence of structural plant-model mismatch. Since plant optimality is often characterized by active constraints, it is possible to use a simpler version of modifier adaptation, called constraint adaptation (CA). This approach corrects the constraints in the model-based optimization problem by means of bias terms computed from the differences between the measured and predicted values of the constraints. The active plant constraintsâi.e. the intersection of the constraints where the plant opti- mum liesâfor each mode of operation is typically known from experience. Therefore, as the optimum of SOFC systems typically lies on active plant constraints, we can apply constraint adaptation. One of the drawbacks of static RTO is the fact that it is necessary to wait until the plant reaches its steady state. Since this may be inappropriate for slow processes, we propose an RTO approach that can speed up the procedure by using transient measurements combined with a dynamic tendency model. In this work, we apply constraint adaptation combined with transient measurements and a dynamic model to experimental SOFC systems. The proposed approach is applied to two SOFC systems: (i) an SOFC system with 6 cells that consists of both hardware and software components; (ii) a commercial SOFC system named BlueGEN. The results obtained are very promising. For both systems, the proposed approach can reach the power demand much faster than the time it takes for the process to settle to steady state. The optimizer is quick at finding and tracking the correct set of active constraints, thus avoiding large constraint violations. In addition, we study the effect of the RTO period on the convergence rate. Finally, we show that CA acts simultaneously as a controller and an optimizer that is able to efficiently detect and track the correct set of active constraints.

EPFL2019

Real-Time Optimization of Solid Oxide Fuel Cell Systems

Tafarel de Avila Ferreira

EPFL2019

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

Dominique Bonvin, Grégory François, Alejandro Gabriel Marchetti

2015