Real-time Optimization with Estimation of Experimental Gradient

For good performance in practice, real-time optimization schemes need to be able to deal with the inevitable model-mismatch problem. Unlike the two-step schemes combining parameter estimation and optimization, the modifier-adaptation approach uses experimental gradient information and does not require the model parameters to be estimated on-line. The dual modifier-adaptation approach presented in this paper drives the process towards optimality, while paying attention to the accuracy of the estimated gradients. The gradients are estimated from the successive operating points generated by the optimization algorithm. The novelty lies in the development of an upper bound on the gradient estimation error, which is used as a constraint for locating the next operating point. The proposed approach is demonstrated in simulation by the real-time optimization of a continuous reactor.

A Dual Modifier-Adaptation Approach for Real-Time Optimization

Dominique Bonvin, Benoît Chachuat, Alejandro Gabriel Marchetti

For good performance in practice, real-time optimization schemes need to be able to deal with the inevitable plant-model mismatch problem. Unlike the two-step schemes combining parameter estimation and optimization, the modifier-adaptation approach does not require the model parameters to be estimated on-line. Instead, it uses information regarding the constraints and selected gradients to improve the plant operation. The dual modifier-adaptation approach presented in this paper drives the process towards optimality, while paying attention to the accuracy of the estimated gradients. The gradients are estimated from successive operating points generated by the optimization algorithm. The novelty lies in the development of an upper bound on the norm of the gradient errors, which is used as a constraint when determining the next operating point. The proposed approach is demonstrated via numerical simulation for both an unconstrained and a constrained problem.

Elsevier2010

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 Interconnected Systems via Modifier Adaptation, with Application to Gas-Compressor Stations

Predrag Milosavljevic

The process industries are characterized by a large number of continuously operating plants, for which optimal operation is of economic and ecological importance. Many industrial systems can be regarded as an arrangement of several subsystems, where outputs of certain subsystems are inputs to others. This gives rise to the notion of interconnected systems. Plant optimality is difficult to achieve when the model used in optimization is inaccurate or in the presence of process disturbances. However, in the presence of plant-model mismatch, optimal operation can be enforced via specific real-time optimization methods. Specifically, this thesis considers so-called Modifier-Adaptation schemes which achieve plant optimality by direct incorporation of process measurements in the form of first-order corrections. As a first contribution, this thesis proposes a novel problem formulation for modifier adaptation. Specifically, it is focused on plants consisting of multiple interconnected subsystems that allows problem decomposition and application of distributed optimization strategies. The underlying key idea is the use of measurements and global plant gradients in place of an interconnection model. As a second contribution, this thesis investigates modifier adaptation for interconnected systems relying on local gradients by using an interconnection model. We show that the use of local information in terms of model, gradients and measurements is sufficient to optimize the steady-state performance of the plant. Finally, we propose a distributed modifier-adaptation algorithm that, besides the interconnection model and local gradients, employs a coordinator. For this scheme, we prove feasible-side convergence to the plant optimum, where a coordinator ensures that the local optimal inputs computed for each subsystem are consistent with the interconnection model. The experimental effort necessary to estimate the plant gradients increases with the number of plant inputs and may become intractable and sometimes not feasible or reliable for large-scale interconnected systems. The proposed approaches that use the interconnection model and local gradients overcome this problem. As an application case study of industrial relevance, this thesis investigates the problem of optimal load-sharing for serial and parallel gas compressors. The aim of load-sharing optimization is operating compressor units in an energy-efficient way, while at the same time satisfying varying load demands. We show how the structure of both the parallel and serial compressor configurations can be exploited in the design of tailored modifier adaptation algorithms based on efficient estimation of local gradients. Our findings show that the complexity of this estimation is independent of the number of compressors. In addition, we discuss gradient estimation for the case where the compressors are operating close to the surge conditions, which induces discontinuities in the problem.