A Pseudospectral Approach to High Index DAE Optimal Control Problems

Historically, solving optimal control problems with high index differential algebraic equations (DAEs) has been considered extremely hard. Computational experience with Runge-Kutta (RK) methods confirms the difficulties. High index DAE problems occur quite naturally in many practical engineering applications. Over the last two decades, a vast number of real-world problems have been solved routinely using pseudospectral (PS) optimal control techniques. In view of this, we solve a "provably hard," index-three problem using the PS method implemented in DIDO, a state-of-the-art MATLAB optimal control toolbox. In contrast to RK-type solution techniques, no laborious index-reduction process was used to generate the PS solution. The PS solution is independently verified and validated using standard industry practices. It turns out that proper PS methods can indeed be used to "directly" solve high index DAE optimal control problems. In view of this, it is proposed that a new theory of difficulty for DAEs be put forth.Comment: 14 pages, 9 figure

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Development of efficient algorithms for model predictive control of fast systems

Die nichtlineare modellprädiktive Regelung (NMPC) ist ein vielversprechender Regelungsalgorithmus, der auf der Echtzeitlüsung eines nichtlinearen dynamischen Optimie- rungsproblems basiert. Nichtlineare Modellgleichungen wie auch die Steuerungs- und Zustandsbeschränkungen werden als Gleichungs- bzw. Ungleichungsbeschränkungen des Optimalsteuerungsproblems behandelt. Jedoch wurde die NMPC wegen des recht hohen Rechenaufwandes bisher meist auf relativ langsame Prozesse angewendet. Daher bildet die Rechenzeit bei Anwendung der NMPC auf schnelle Prozesse einen gewissen Engpass wie z. B. bei mechanischen und/oder elektrischen Prozessen. In dieser Arbeit wird eine neue Lüsungsstrategie für dynamische Optimierungsprobleme vorgeschlagen, wie sie in NMPC auftreten, die auch auf sog. schnelle Systeme anwendbar ist. Diese Strategie kombiniert Mehrschieß -Verfahrens mit der Methode der Kollokation auf finiten Elementen. Mittels Mehrschieß -Verfahren wird das nichtlineare dynamische Optimierungsproblem in ein hochdimensionales statisches Optimierungsproblem (nonlinear program problem, NLP) überführt, wobei Diskretisierungs- und Parametrisierungstechniken zum Einsatz kommen. Um das NLP-Problem zu lüsen, müssen die Zustandswerte und ihre Gradienten am Ende jedes Diskretisierung-Intervalles berechnet werden. In dieser Arbeit wird die Methode der Kollokation auf finiten Elementen benutzt, um diese Aufgabe zu lüsen. Dadurch lassen sich die Zustandsgrüß en und ihre Gradienten am Ende jedes Diskretisierungs-Intervalls auch mit groß er Genauigkeit berechnen. Im Ergebnis künnen die Vorteile beider Methoden (Mehrschieß -Verfahren und Kollokations-Methoden) ausgenutzt werden und die Rechenzeit lässt sich deutlich reduzieren. Wegen des komplexen Optimierungsproblems ist es im Allgemeinen schwierig, eine Stabilitätsanalyse für das zugehürige NMPC durchzuführen. In dieser Arbeit wird eine neue Formulierung des Optimalsteuerungsproblems vorgeschlagen, durch die die Stabilität des NMPC gesichert werden kann. Diese Strategie besteht aus den folgenden drei Eigenschaften. Zunächst wird ein Hilfszustand über eine lineare Zustandsgleichung in das Optimierungsproblem eingeführt. Die Zustandsgleichungen werden durch Hilfszustände ergänzt, die man in Form von Ungleichungsnebenbedingungen einführt. Wenn die Hilfszustände stabil sind, lässt sich damit die Stabilität des Gesamtsystems sichern. Die Eigenwerte der Hilfszustände werden so gewählt, dass das Optimalsteuerungsproblem lüsbar ist. Dazu benutzt man die Eigenwerte als Optimierungsvariable. Damit lassen sich die Stabilitätseigenschaften in einem stationären Punkt des Systemmodells untersuchen. Leistungsfähigkeit und Effektivität des vorgeschlagenen Algorithmus werden an Hand von Fallstudien belegt. Die Bibliothek Numerische Algorithmus Group (NAG), Mark 8, wird eingesetzt, um die linearen und nichtlinearen Gleichungen, die aus der Kollokation resultieren, zu lüsen. Weiterhin wird zur Lüsung des NLP-Problems der Lüser IPOPT für C/C++- Umgebung eingesetzt. Insbesondere wird der vorgeschlagene Algorithmus zur Steuerung einer Verladebrücke im Labor des Institutes für Automatisierungs- und Systemtechnik angewendet.Nonlinear model predictive control (NMPC) has been considered as a promising control algorithm which is based on a real-time solution of a nonlinear dynamic optimization problem. Nonlinear model equations and controls as well as state restrictions are treated as equality and inequality constraints of the optimal control problem. However, NMPC has been applied mostly in relatively slow processes until now, due to its high computational expense. Therefore, computation time needed for the solution of NMPC leads to a bottleneck in its application to fast systems such as mechanical and/or electrical processes. In this dissertation, a new solution strategy to efficiently solve NMPC problems is proposed so that it can be applied to fast systems. This strategy combines the multiple shooting method with the collocation on finite elements method. The multiple shooting method is used for transforming the nonlinear optimal control problem into nonlinear program (NLP) problem using discretization and parametrization techniques. To solve this NLP problem the values of state variables and their gradients at the end of each shooting need to be computed. We use collocation on finite elements to carry out this task, thus, a high precision approximation of the state variables and their sensitivities in each shoot are achieved. As a result, the advantages of both the multiple shooting and the collocation method can be employed and therefore the computation efficiency can be considerably enhanced. Due to the nonlinear and complex optimal control problem formulation, in general, it is difficult to analyze the stability properties of NMPC systems. In this dissertation we propose a new formulation of the optimal control problem to ensure the stability of the NMPC problems. It consists the following three features. First, we introduce auxiliary states and linear state equations into the finite horizon dynamic optimization problem. Second, we enforce system states to be contracted with respect to the auxiliary state variables by adding inequality constraints. Thus, the stability features of the system states will conform to the stability properties of the auxiliary states, i.e. the system states will be stable, if the auxiliary states are stable. Third, the eigenvalues of the linear state equations introduced will be determined to stabilize the auxiliary states and at the same time make the optimal control problem feasible. This is achieved by considering the eigenvalues as optimization variables in the optimal control problem. Moreover, features of this formulation are analyzed at the stationary point of the system model. To show the effectiveness and performance of the proposed algorithm and the new optimal control problem formulation we present a set of NMPC case studies. We use the numerical algorithm group (NAG) library Mark 8 to solve numerically linear and nonlinear systems that resulted from the collocation on finite elements to compute the states and sensitivities, in addition, the interior point optimizer (IPOPT) and in C/C++ environment. Furthermore, to show more applicability, the proposed algorithm is applied to control a laboratory loading bridge

Nonlinear Programming Approaches for Efficient Large-Scale Parameter Estimation with Applications in Epidemiology

The development of infectious disease models remains important to provide scientists with tools to better understand disease dynamics and develop more effective control strategies. In this work we focus on the estimation of seasonally varying transmission parameters in infectious disease models from real measles case data. We formulate both discrete-time and continuous-time models and discussed the benefits and shortcomings of both types of models. Additionally, this work demonstrates the flexibility inherent in large-scale nonlinear programming techniques and the ability of these techniques to efficiently estimate transmission parameters even in very large-scale problems. This computational efficiency and flexibility opens the door for investigating many alternative model formulations and encourages use of these techniques for estimation of larger, more complex models like those with age-dependent dynamics, more complex compartment models, and spatially distributed data. However, the size of these problems can become excessively large even for these powerful estimation techniques, and parallel estimation strategies must be explored. Two parallel decomposition approaches are presented that exploited scenario based decomposition and decomposition in time. These approaches show promise for certain types of estimation problems

Translating parameter estimation problems from EASY-FIT to SOCS

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems. In this thesis, we discuss the design and implementation of a source-to-source translator called EFtoSOCS used to translate EASY FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EFtoSOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EFtoSOCS and SOCS

Method for Solving State-Path Constrained Optimal Control Problems Using Adaptive Radau Collocation

A new method is developed for accurately approximating the solution to state-variable inequality path constrained optimal control problems using a multiple-domain adaptive Legendre-Gauss-Radau collocation method. The method consists of the following parts. First, a structure detection method is developed to estimate switch times in the activation and deactivation of state-variable inequality path constraints. Second, using the detected structure, the domain is partitioned into multiple-domains where each domain corresponds to either a constrained or an unconstrained segment. Furthermore, additional decision variables are introduced in the multiple-domain formulation, where these additional decision variables represent the switch times of the detected active state-variable inequality path constraints. Within a constrained domain, the path constraint is differentiated with respect to the independent variable until the control appears explicitly, and this derivative is set to zero along the constrained arc while all preceding derivatives are set to zero at the start of the constrained arc. The time derivatives of the active state-variable inequality path constraints are computed using automatic differentiation and the properties of the chain rule. The method is demonstrated on two problems, the first being a benchmark optimal control problem which has a known analytical solution and the second being a challenging problem from the field of aerospace engineering in which there is no known analytical solution. When compared against previously developed adaptive Legendre-Gauss-Radau methods, the results show that the method developed in this paper is capable of computing accurate solutions to problems whose solution contain active state-variable inequality path constraints.Comment: 31 pages, 7 figures, 5 table

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices

Optimal allocation of static and dynamic reactive power support for enhancing power system security

Power systems over the recent past few years, has undergone dramatic revolution in terms of government and private investment in various areas such as renewable generation, incorporation of smart grid to better control and operate the power grid, large scale energy storage, and fast responding reactive power sources. The ongoing growth of the electric power industry is mainly because of the deregulation of the industry and regulatory compliance which each participant of the electric power system has to comply with during planning and operational phase. Post worldwide blackouts, especially the year 2003 blackout in north-east USA, which impacted roughly 50 million people, more attention has been given to reactive power planning. At present, there is steady load growth but not enough transmission capacity to carry power to load centers. There is less transmission expansion due to high investment cost, difficulty in getting environmental clearance, and less lucrative cost recovery structure. Moreover, conventional generators close to load centers are aging or closing operation as they cannot comply with the new environmental protection agency (EPA) policies such as Cross-State Air Pollution Rule (CSAPR) and MACT. The conventional generators are getting replaced with far away renewable sources of energy. Thus, the traditional source of dynamic reactive power support close to load centers is getting retired. This has resulted in more frequently overloading of transmission network than before. These issues lead to poor power quality and power system instability. The problem gets even worse during contingencies and especially at high load levels. There is a clear need of power system static and dynamic monitoring. This can help planners and operators to clearly identify severe contingencies causing voltage acceptability problem and system instability. Also, it becomes imperative to find which buses and how much are they impacted by a severe contingency. Thus, sufficient static and dynamic reactive power resource is needed to ensure reliable operation of power system, during stressed conditions and contingencies. In this dissertation, a generic framework has been developed for filtering and ranking of severe contingency. Additionally, vulnerable buses are identified and ranked. The next task after filtering out severe contingencies is to ensure static and dynamic security of the system against them. To ensure system robustness against severe contingencies optimal location and amount of VAR support required needs to be found. Thus, optimal VAR allocation needs to be found which can ensure acceptable voltage performance against all severe contingency. The consideration of contingency in the optimization process leads to security constrained VAR allocation problem. The problem of static VAR allocation requirement is formulated as minlp. To determine optimal dynamic VAR installation requirement the problem is solved in dynamic framework and is formulated as a Mixed Integer Dynamic Optimization (MIDO). Solving the VAR allocation problem for a set of severe contingencies is a very complex problem. Thus an approach is developed in this work which reduces the overall complexity of the problem while ensuring an acceptable optimal solution. The VAR allocation optimization problem has two subparts i.e. interger part and nonlinear part. The integer part of the problem is solved by branch and bound (B&B) method. To enhance the efficiency of B&B, system based knowledge is used to customize the B&B search process. Further to reduce the complexity of B&B method, only selected candidate locations are used instead of all plausible locations in the network. The candidate locations are selected based upon the effectiveness of the location in improving the system voltage. The selected candidate locations are used during the optimization process. The optimization process is divided into two parts: static optimization and dynamic optimization. Separating the overall optimization process into two sub-parts is much more realistic and corresponds to industry practice. Immediately after the occurrence of the contingency, the system goes into transient (or dynamic) phase, which can extend from few milliseconds to a minute. During the transient phase fast acting controllers are used to restore the system. Once the transients die out, the system attains steady state which can extend for hours with the help of slow static controllers. Static optimization is used to ensure acceptable system voltage and system security during steady state. The optimal reactive power allocation as determined via static optimization is a valuable information. It\u27s valuable as during the steady state phase of the system which is a much longer phase (extending in hours), the amount of constant reactive power support needed to maintain steady system voltage is determined. The optimal locations determined during the static optimization are given preference in the dynamic optimization phase. In dynamic optimization optimal location and amount of dynamic reactive power support is determined which can ensure acceptable transient performance and security of the system. To capture the true dynamic behavior of the system, dynamic model of system components such as generator, exciter, load and reactive power source is used. The approach developed in this work can optimally allocate dynamic VAR sources. The results of this work show the effectiveness of the developed reactive power planning tool. The proposed methodology optimally allocates static and dynamic VAR sources that ensure post-contingency acceptable power quality and security of the system. The problem becomes manageable as the developed approach reduces the overall complexity of the optimization problem. We envision that the developed method will provide system planners a useful tool for optimal planning of static and dynamic reactive power support that can ensure system acceptable voltage performance and security

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p