Differential-algebraic Constraints Research Articles

A Direct Optimization Algorithm for Problems with Differential-Algebraic Constraints: Application to Heat and Mass Transfer

In this article, an optimization task with nonlinear differential-algebraic equations (DAEs) is considered. As a main result, a new solution procedure is designed. The computational procedure represents the sequential optimization approach. The proposed algorithm is based on a multiple shooting parametrization method. Two main aspects of a generalized parametrization approach are analyzed in detail: a control function and DAE model parametrization. A comparison between the original and modified DAEs is made. The new algorithm is applied to solve an optimization task in heat and mass transfer engineering.

Analytic Deep Learning-Based Surrogate Model for Operational Planning With Dynamic TTC Constraints

The increased penetration of wind power introduces more operational changes of critical corridors and the traditional time-consuming transient stability constrained total transfer capability (TTC) operational planning is unable to meet the real-time monitoring need. This paper develops a more computationally efficient approach to address that challenge via the analytical deep learning-based surrogate model. The key idea is to resort to deep learning for developing a computationally cheap surrogate model to replace the original time-consuming differential-algebraic constraints related to TTC. However, the deep learning-based surrogate model introduces implicit rules that are difficult to handle in the optimization process. To this end, we derive the Jacobian and Hessian matrices of the implicit surrogate models and finally transfer them into an analytical formulation that can be easily solved by the interior point method. Surrogate modeling and problem reformulation allow us to achieve significantly improved computational efficiency and the yielded solutions can be used for operational planning. Numerical results carried out on the modified IEEE 39-bus and 68-bus systems demonstrate the effectiveness of the proposed method in dealing with complicated TTC constraints while balancing the computational efficiency and accuracy.

A Differential-Algebraic Riccati Equation for Applications in Flow Control

We investigate existence and structure of solutions to quadratic control problems with semiexplicit differential-algebraic constraints as they appear in the modeling of flows. We introduce a decoupled representation of the state and the formal adjoint equations to identify the conditions for the existence of solutions. We derive a differential-algebraic Riccati equation formulated in the original coefficients for the efficient numerical computation of the optimal solution.

Modeling phytoplankton community in reservoirs. A comparison between taxonomic and functional groups-based models

In this paper we address the formulation of two mechanistic water quality models that differ in the way the phytoplankton community is described. We carry out parameter estimation subject to differential-algebraic constraints and validation for each model and comparison between models performance. The first approach aggregates phytoplankton species based on their phylogenetic characteristics (Taxonomic group model) and the second one, on their morpho-functional properties following Reynolds' classification (Functional group model). The latter approach takes into account tolerance and sensitivity to environmental conditions. The constrained parameter estimation problems are formulated within an equation oriented framework, with a maximum likelihood objective function. The study site is Paso de las Piedras Reservoir (Argentina), which supplies water for consumption for 450,000 population. Numerical results show that phytoplankton morpho-functional groups more closely represent each species growth requirements within the group. Each model performance is quantitatively assessed by three diagnostic measures. Parameter estimation results for seasonal dynamics of the phytoplankton community and main biogeochemical variables for a one-year time horizon are presented and compared for both models, showing the functional group model enhanced performance. Finally, we explore increasing nutrient loading scenarios and predict their effect on phytoplankton dynamics throughout a one-year time horizon.

Kalman Duality Principle for a Class of Ill-Posed Minimax Control Problems with Linear Differential-Algebraic Constraints

In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence $\hat{u}_{\varepsilon}$ for the dual problem applying Tikhonov method. Finally we represent $\hat{u}_{\varepsilon}$ in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.

Optimal Control Conditions with Differential-Algebraic Equation Constraints

Optimal Control Conditions with Differential-Algebraic Equation Constraints

An Extended Ant Colony Optimization Algorithm for Integrated Process and Control System Design

The problem of integrated process and control system design is discussed in this paper. We formulate it as a mixed integer nonlinear programming problem subject to differential-algebraic constraints. This class of problems is frequently nonconvex and, therefore, local optimization techniques usually fail to locate the global solution. Here we propose a global optimization algorithm, based on an extension of the ant colony optimization metaheuristic, in order to solve this challenging class of problems in an efficient and robust way. The ideas of the methodology are explained and, on the basis of different full-plant case studies, the performance of the approach is evaluated. The first set of benchmark problems deal with the integrated design and control of two different wastewater treatment plants, consisting on both NLP and MINLP formulations. The last case study is the well-known Tennessee Eastman process. Numerical experiments with our new method indicate that we can achieve an improved performance in all cases. Additionally, our method outperforms several other recent competitive solvers for the challenging case studies considered.

Global Optimization for Integrated Design and Control of Computationally Expensive Process Models

The problem of integrated design and control optimization of process plants is discussed in this paper. We consider it as a nonlinear programming problem subject to differential-algebraic constraints. This class of problems is frequently multimodal and “costly” (i.e., computationally expensive to evaluate). Thus, on the one hand, local optimization techniques usually fail to locate the global solution, and, on the second hand, most global optimization methods require many simulations of the model, resulting in unaffordable computation times. As an alternative, one may consider global optimization methods which employ surrogate-based approaches to reduce computation times and which require no knowledge of the underlying problem structure. A challenging wastewater treatment plant (WWTP) benchmark model1 is used here to evaluate the performance of these techniques. Numerical experiments with different optimization solvers indicate that the proposed benchmark optimization problem is indeed multimodal, and that via global optimization we can achieve an improvement of the controllers' performance compared to the best tuned controllers' settings available in the literature. Moreover, these results show that surrogate-based methods may reduce computation times while ensuring convergence to the best known solutions.

A Tabu search-based algorithm for mixed-integer nonlinear problems and its application to integrated process and control system design

In this contribution, we consider mixed-integer nonlinear programming problems subject to differential-algebraic constraints. This class of problems arises frequently in process design, and the particular case of integrated process and control system design is considered. Since these problems are frequently non-convex, local optimization techniques usually fail to locate the global solution. Here, we propose a global optimization algorithm, based on extensions of the metaheuristic Tabu Search, in order to solve this challenging class of problems in an efficient and robust way. The ideas of the methodology are explained and, on the basis of two case studies, the performance of the approach is evaluated. The first benchmark problem is a Wastewater Treatment Plant model [Alex, J., Bteau, J. F., Copp, J. B., Hellinga, C., Jeppsson, U., Marsili-Libelli, S., et al. (1999). Benchmark for evaluating control strategies in wastewater treatment plants. In Proceedings of the ECC’99 conference] for nitrogen removal and the second case study is the well-known Tennessee Eastman Process [Downs, J. J., & Vogel, E. F. (1993). A plant-wide industrial process control problem. Computers & Chemical Engineering, 17, 245-255]. Numerical experiments with our new method indicate that we can achieve an improved performance in both cases. Additionally, our method outperforms several other recent competitive solvers for the two challenging case studies considered.

Nonlinear Multi-Criteria Optimization for the Integrated Design and Control of Bioprocesses

The simultaneous design and control of continuous bioprocesses is considered here as a multi-objective optimization problem subject to non-linear differential-algebraic constraints. This type of problem is very challenging to solve due to its non-convexity (multi-modality). We present two novel solution approaches based on extensions of an stochastic global optimization method. The robustness and efficiency of these approaches are illustrated with a wastewater treatment plant case study.

Constrained Optimization of Nonlinear, Dynamic Chemical Processes – A Normalized Form Approach

In chemical manufacturing processes, optimization problems with differential-algebraic constraints are frequently encountered. In general, these problems are difficult to solve and solution approaches are usually based on discretization schemes. This paper proposes an alternative method to determine optimal setpoint trajectories for a class of dynamic systems with differential-algebraic constraints. The method exploits di ff erential flatness to explicitly eliminate the differential state equations. The resulting optimization problem is an algebraic, nonlinearly constrained optimization problem, and can be solved by nonlinear programming. The proposed approach has the potential to reduce the amount of on-line computation required to solve a specific class of these problems in real-time.

Anholonomy in the Control of Kinematically Redundant Mechanisms with Flexible Components

We study robot mechanisms which feature kinematic redundancy together with elastic components. For instance, we discuss manipulators in which there are more controlled (joint space) degrees of freedom than task space degrees of freedom and in which at least one of these controlled joints is connected to an adjacent link through an elastic coupling. It will be shown that there is a wide class of examples for which it is impossible to find joint space motions which track a given end effector trajectory and do not at the same time excite elastic dynamics. No assumptions on the smoothness of the prescribed end effector trajectory or the degree of redundancy alter this conclusion. For certain congenial mechanism geometries, however, it is shown that it is possible to plan motions which store no elastic energy. These motions involve differential-algebraic constraints relating accelerations of the end effector along the work space path to joint space velocity and configuration variables. Planning motions which interpolate between given points in the work space for kinematically redundant flexible robots is similar to recently studied problems in the control theory of nonholonomic systems. Interesting open questions involve the existence and multiplicity of interpolating work space paths, the geometry of the reachable work space, and how closely arbitrarily prespecified paths in the work space may be tracked.