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Publication# Real-Time Optimization of Interconnected Systems via Modifier Adaptation, with Application to Gas-Compressor Stations

Abstract

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.

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Optimization arises naturally when process performance needs improvement. This is often the case in industry because of competition – the product has to be proposed at the lowest possible cost. From the point of view of control, optimization consists in designing a control policy that best satisfies the chosen objectives. Most optimization schemes rely on a process model, which, however, is always an approximation of the real plant. Hence, the resulting optimal control policy is suboptimal for the real process. The fact that accurate models can be prohibitively expensive to build has triggered the development of a field of research known as Optimization under Uncertainty. One promising approach in this field proposes to draw a strong parallel between optimization under uncertainty and control. This approach, labeled NCO tracking, considers the Necessary Conditions of Optimality (NCO) of the optimization problem as the controlled outputs. The approach is still under development, and the present work is today's most recent contribution to this development. The problem of NCO tracking can be divided into several subproblems that have been studied separately in earlier works. Two main categories can be distinguished : (i) tracking the NCO associated with active constraints, and (ii) tracking the NCO associated with sensitivities. Research on the former category is mature. The latter problem is more difficult to solve since the sensitivity part of the NCO cannot be directly measured on the real process. The present work proposes a method to tackle these sensitivity problems based on the theory of Neighboring Extremals (NE). More precisely, NE control provides a way of calculating a first-order approximation to the sensitivity part of the NCO. This idea is developed for static and both nonsingular and singular dynamic optimization problems. The approach is illustrated via simulated examples: steady-state optimization of a continuous chemical reactor, optimal control of a semi-batch reactor, and optimal control of a steered car. Model Predictive Control (MPC) is a control scheme that can accommodate both process constraints and nonlinear process models. The repeated solution of a dynamic optimization problem provides an update of the control variables based on the current state, and therefore provides feedback. One of the major drawbacks of MPC lies in the expensive computations required to update the control policy, which often results in a low sampling frequency for the control loop. This limitation of the sampling frequency can be dramatic for fast systems and for systems exhibiting a strong dispersion between the predicted and the real state such as unstable systems. In the MPC framework, two main methods have been proposed to tackle these difficulties: (i) The use of a pre-stabilizing feedback operating in combination with the MPC scheme, and (ii) the use of robust MPC. The drawback of the former approach is that there exists no systematic way of designing such a feedback, nor is there any systematic way of analyzing the interaction between the MPC controller and this additional feedback. This work proposes to use the NE theory to design this additional feedback, and it provides a systematic way of analyzing the resulting control scheme. The approach is illustrated via the control of a simulated unstable continuous stirred-tank reactor and is applied successfully to two laboratory-scale set-ups, an inverted pendulum and a helicopter model called Toycopter. The stabilizing potential of NE control to handle fast and unstable systems is well illustrated. In the case of a strong dispersion between the state trajectories predicted by the model and the real process, robust MPC becomes infeasible. This problem can be addressed using robust MPC based on multiple input profiles, where the inherent feedback provided by MPC is explicitly taken into account, thereby increasing the size of the set of feasible inputs. The drawback of this scheme is its very high computational complexity. This work proposes to use the NE theory in the robust MPC framework as an efficient way of dealing with the feasibility issue, while limiting the computational complexity of the approach. The approach is illustrated via the control of a simulated unstable continuous stirred-tank reactor, and of an inverted pendulum.

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).

2015Dominique Bonvin, Predrag Milosavljevic, René Uwe Schneider

The desire to operate chemical processes in a safe and economically optimal way has motivated the development of so-called real-time optimization (RTO) methods [1]. For continuous processes, these methods aim to compute safe and optimal steady-state setpoints for the lower-level process controllers. A key challenge for this task is plant-model mismatch. For example, in the case of a model that is assumed to be structurally identical with the plant but has unknown parameters, the so-called two-step approach [2-4] has been proposed. It repeats two steps: In the first step, plant measurements are used to identify the parameters of the model. In the second step, the economically optimal setpoints for the updated process model are determined by solving an optimization problem. Unfortunately, a structurally correct process model is rarely available in practice. In that case, the optimal setpoints for the model determined by the two-step approach may not be optimal for the plant. To overcome this problem, the so-called modifier-adaptation (MA) methods have been developed [5]. In MA, no structurally correct model is required. Instead, plant measurements are used to formulate and solve a modified optimization problem at each iteration, such that, upon convergence, the first-order optimality conditions of the plant are guaranteed to be satisfied [5]. These and other available RTO methods usually treat the plant as a single entity, and compute the optimal setpoints in a centralized manner. However, this approach may be suboptimal or even infeasible for an increasing number of applications involving so-called interconnected systems. Interconnected systems are here defined as systems composed of subsystems that exchange material, energy or information, such as compressor networks, teams of autonomous vehicles or large industrial parks, in which different business units of a chemical company share certain resources. In these cases, distributed RTO methods can be employed, which utilize the available interconnection variables and exploit the inherent interconnection structure of the particular system. Only a few distributed RTO methods have been reported in the literature, including the methods proposed by Brdys and Tatjewski [6]. Just as in the two-step approaches, structurally correct models are assumed. In addition to identifying the model parameters, the methods also try to estimate the values of the interconnection variables. Consequently, these methods may not yield the plant optimum in the presence of structural plant-model mismatch. In this contribution, we propose a set of distributed RTO methods based on the modifier-adaptation framework for interconnected systems in the presence of structural plant-model mismatch. Thanks to the modifier-adaptation framework, all proposed distributed RTO methods are able to reach the plant optimum upon convergence despite possible plant-model mismatch. The proposed schemes employ different types of models, use different measurements, and differ in their algorithmic structure and required controller hierarchy, as well as in their communication topology, as detailed below. The first method utilizes a model of the local objective function, a model for the dependence of local outputs on the local setpoints and outputs of other subsystems, and a model for the interconnection structure of the system. The algorithm resembles a double-loop structure: In the simulation-based inner loop, the local MA-problems are solved in parallel until the interconnection constraints are satisfied. As soon as the inner loop has converged, the computed setpoints are applied to the plant in the outer loop. When the plant has reached a new steady state, measurements are taken to improve the performance at the next iteration. The second and third methods do not require a model of the interconnection structure of the system. Consequently, a single-loop algorithmic structure is sufficient. At every iteration, each subsystem computes local setpoints, which are immediately applied to the plant. At the corresponding steady state, plant measurements are taken to improve the performance at the next iteration. At this point, the second and third method proceed differently: The second method uses local measurements of the interconnection variables to each subsystem, whereas the third method additionally measures the local outputs. Consequently, the second method still uses a model describing the dependency of the local outputs on the local setpoints and local interconnection variables, whereas no such relationship is needed for the third method. Because of their different characteristics, each of these algorithms has its specific advantages regarding applications. For example, the first method requires each subsystem to have a complete model of the full system and its interconnection topology. If these models are good, then fast convergence of this scheme with few setpoint changes can be expected. In applications, where providing a full model of the system to every subsystem is feasible and does not raise any privacy concerns, the first scheme may be the method of choice. The second and third schemes, in contrast, do not require an interconnection model. Therefore, they may be preferred for applications where different subsystems do not want to disclose their models to other subsystems. This could be the case when the different subsystems are owned by competing companies. Moreover, the lack of an interconnection model may be advantageous in certain applications with changing interconnection topologies, such as power system networks. Another advantage of the second and third method is their reduced local modeling effort if accurate measurements are available. On the other hand, these methods may need significantly more iterations to converge than the first method if accurate models are available. In our contribution, we finally apply the proposed distributed modifier-adaptation schemes to numerical examples. The main features of each method are illustrated, revealing their potential for real-time optimization of interconnected systems with structural plant-model mismatch. References [1] C. R. Cutler and R. T. Perry. Real time optimization with multivariable control is required to maximize profits. Comput. Chem. Eng. 7(5):663-667, 1983. [2] T. E. Marlin and A. N. Hrymak. Real-time operations optimization of continuous processes. In Proceedings of the 5th International Conference on Chemical Process Controls-V: Assessment and New Directions for Research; J. C. Kantor, C. E. Garcia, and B. Carnahan, Eds.; AIChE Symposium Series, No. 316; American Institute of Chemical Engineers (AIChE): New York, 1997, 156-164. [3] S.-S. Jang, B. Joseph, and H. Mukai. On-line optimization of constrained multivariable chemical processes. AIChE J., 33(1): 26–35, 1987. [4] C. Y. Chen and B. Joseph. On-line optimization using a two-phase approach: An application study. Ind. Eng. Chem. Res., 26:1924-1930, 1987. [5] A. Marchetti, B. Chachuat, and D. Bonvin. Modifier-Adaptation Methodology for Real-Time Optimization. Ind. Eng. Chem. Res., 48: 6022-6033, 2009. [6] M. A. Brdys and P. Tatjewski. Iterative Algorithms for Multilayer Optimizing Control. Imperial College Press, Imperial College Press: London, U.K., 2005.

2016