Victor M. Zavala

The advanced-step NMPC controller: Optimality, stability and robustness

Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear programming (NLP) sensitivity have reduced on-line computational costs and can lead to significantly improved performance. In this study, we extend this concept through a simple reformulation of the NMPC problem and propose the advanced-step NMPC controller. The main result of this extension is that the proposed controller enjoys the same nominal stability properties of the conventional NMPC controller without computational delay. In addition, we establish further robustness properties in a straightforward manner through input-to-state stability concepts. A case study example is presented to demonstrate the concepts.

Large-scale nonlinear programming using IPOPT : An integrating framework for enterprise-wide dynamic optimization

Integration of real-time optimization and control with higher level decision-making (scheduling and planning) is an essential goal for profitable operation in a highly competitive environment. While integrated large-scale optimization models have been formulated for this task, their size and complexity remains a challenge to many available optimization solvers. On the other hand, recent development of powerful, large-scale solvers leads to a reconsideration of these formulations, in particular, through development of efficient large-scale barrier methods for nonlinear programming (NLP). As a result, it is now realistic to solve NLPs on the order of a million variables, for instance, with the IPOPT algorithm. Moreover, the recent NLP sensitivity extension to IPOPT quickly computes approximate solutions of perturbed NLPs. This allows on-line computations to be drastically reduced, even when large nonlinear optimization models are considered. These developments are demonstrated on dynamic real-time optimization strategies that can be used to merge and replace the tasks of (steady-state) real-time optimization and (linear) model predictive control. We consider a recent case study of a low density polyethylene (LDPE) process to illustrate these concepts.

A Computational Framework for Uncertainty Quantification and Stochastic Optimization in Unit Commitment With Wind Power Generation

We present a computational framework for integrating a state-of-the-art numerical weather prediction (NWP) model in stochastic unit commitment/economic dispatch formulations that account for wind power uncertainty. We first enhance the NWP model with an ensemble-based uncertainty quantification strategy implemented in a distributed-memory parallel computing architecture. We discuss computational issues arising in the implementation of the framework and validate the model using real wind-speed data obtained from a set of meteorological stations. We build a simulated power system to demonstrate the developments.

Optimization-based strategies for the operation of low-density polyethylene tubular reactors: nonlinear model predictive control

In this work, we present a general nonlinear model predictive control (NMPC) framework for low-density polyethylene (LDPE) tubular reactors. The framework is based on a first-principles dynamic model able to capture complex phenomena arising in these units. We first demonstrate the potential of using NMPC to simultaneously regulate and optimize the process economics in the presence of persistent disturbances such as fouling. We then couple the NMPC controller with a compatible moving horizon estimator (MHE) to provide output feedback. Finally, we discuss computational limitations arising in this framework and make use of recently proposed advanced-step MHE and NMPC strategies to provide nearly instantaneous feedback.

Stability of multiobjective predictive control

We propose a utopia-tracking strategy to handle multiple conflicting objectives in model predictive control. The controller minimizes the distance of its vector of objectives to that of the compromise solution: the point along the steady-state Pareto front closest to the utopia point, where all the objectives are independently minimized. We establish conditions for asymptotic stability and propose numerical implementation variants. One of the key advantages of the approach is that it avoids the computation of Pareto fronts in real-time environments. In addition, the approach can handle general objectives of different nature such as economic and regularization.

Nonlinear Programming Strategies for State Estimation and Model Predictive Control

Sensitivity-based strategies for on-line moving horizon estimation (MHE) and nonlinear model predictive control (NMPC) are presented both from a stability and computational perspective. These strategies make use of full-space interior-point nonlinear programming (NLP) algorithms and NLP sensitivity concepts. In particular, NLP sensitivity allows us to partition the solution of the optimization problems into background and negligible on-line computations, thus avoiding the problem of computational delay even with large dynamic models. We demonstrate these developments through a distributed polymerization reactor model containing around 10,000 differential and algebraic equations (DAEs).

Real-Time Nonlinear Optimization as a Generalized Equation

We establish results for the problem of tracking a time-dependent manifold arising in real-time optimization by casting this as a parametric generalized equation. We demonstrate that if points along a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by solving a single linear complementarity problem (LCP) at each time step. We derive sufficient conditions guaranteeing that the tracking error remains bounded to second order with the size of the time step even if the LCP is solved only approximately. We use these results to derive a fast, augmented Lagrangian tracking algorithm and demonstrate the developments through a numerical case study.

Optimization-based strategies for the operation of low-density polyethylene tubular reactors: Moving horizon estimation

Abstract We present a moving horizon estimation (MHE) application for multi-zone low-density (LDPE) polyethylene tubular reactors. The strategy incorporates a first-principles dynamic model comprised of large sets of nonlinear partial, differential and algebraic equations (PDAEs). It was found that limited temperature measurements distributed along the reactor are sufficient to infer all the model states in space and time and to track uncertain time-varying phenomena such as fouling. A full discretization strategy and a state-of-the-art nonlinear programming (NLP) solver are used to enable the computational feasibility of the approach. It is demonstrated that the MHE estimator exhibits fast performance and is well suited for applications of industrial interest.

Stochastic optimal control model for natural gas networks

Abstract We present a stochastic optimal control model to optimize gas network inventories in the face of system uncertainties. The model captures detailed network dynamics and operational constraints and uses a weighted risk-mean objective. We perform a degrees-of-freedom analysis to assess operational flexibility and to determine conditions for model consistency. We compare the control policies obtained with the stochastic model against those of deterministic and robust counterparts. We demonstrate that the use of risk metrics can help operators to systematically mitigate system volatility. Moreover, we discuss computational scalability issues and effects of discretization resolution on economic performance.

Advances in moving horizon estimation for nonlinear systems

In the last decade, moving horizon estimation (MHE) has emerged as a powerful technique for tackling the problem of estimating the state of a dynamic system in the presence of nonlinearities and disturbances. MHE is based on the idea of minimizing an estimation cost function defined on a sliding window composed of a finite number of time stages. The cost function is usually made up of two contributions: a prediction error computed on a recent batch of inputs and outputs; an arrival cost that serves the purpose of summarizing the past data. However, the diffusion of such techniques has been hampered by: i) the difficulty in choosing the arrival cost so as to ensure stability of the overall estimation scheme; ii) the request of an adequate computational effort on line. In this paper, both problems are addressed and possible solutions are proposed. First, by means of a novel stability analysis, it is constructively shown that under very general observability conditions a quadratic arrival cost is sufficient to ensure the stability of the estimation error provided that the weight matrix is adequately chosen. Second, a novel approximate MHE algorithm is proposed that is based on nonlinear programming sensitivity calculations. The approximate MHE algorithm has the same stability properties of the optimal one which make the overall approach suitable to be applied in real settings. Preliminary simulation results confirm the effectiveness of proposed method.