Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis

Low-Rank Alternating Direction Implicit Iteration in pyMOR

The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework

Quantifying the Impact of Load-Following on Gas-fired Power Plants

Due to rapid penetration of renewables into the grid, natural gas combined cycle (NGCC) power plants are being forced to cycle their loads more frequently and rapidly than for which they were designed. However, the impact of load-following operation on plant efficiency and equipment health are currently poorly understood. The objective of this work is to quantify the impact of load-following on the gas-fired plants by developing high-fidelity multi-scale dynamic models. There are four main tasks in this project. First, dynamic model of an NGCC power plant has been developed. The main components of the NGCC plants are the gas turbine (GT), heat recovery steam generator (HRSG), and steam turbine (ST). The second task focuses on one of the undesired phenomena known as ‘spraying to saturation’ being faced by the NGCC plants during load-following, where the attemperator spray leads to saturation at the inlet of superheater and/or reheater causing damage and eventual failure of the superheater and/or reheater tubes due to two-phase flow. Different configurations of NGCC plants and operation strategies that can not only eliminate ‘spraying to saturation’ but can maximize the plant efficiency have been developed and evaluated. The third task focuses on modeling the unprecedented damages to the boiler components due to rapid load-following, which is leading to higher operation and maintenance (O&M) costs. Stress and wear models have been developed by accounting for creep and fatigue damages in key HRSG components. Multiple locations at the component junctions have been monitored and the most stressed part has been identified as the constraint in the dynamic optimization of the load-following operation. A multi-objective dynamic optimization algorithm has been developed for maximizing plant efficiency and minimizing deviation from desired ramp rates while satisfying operational constraints such as those due to stress and wear. The fourth task focuses on developing reduced order models. Since the modeling domain of interest includes multiple time scales and multiple spatial scales, it can be computationally intractable to use the iii detailed models for optimization/scheduling/control. Therefore, reduced order dynamic models have been developed for the NGCC system including the health models so that they can be computationally tractable for being used in dynamic optimization while providing desired accuracy

A parallel Schur method for solving continuous-time algebraic Riccati equations

Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a real Schur form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library

Design and development of symmetric reflective compound parabolic concentrator (SRCPC) for power generation

This thesis presents a detailed design, simulation, optical performance, construction and experimental validation carried out on a novel non-imaging static symmetric reflective compound parabolic concentrator (SRCPC). By considering the seasonal variation of the sun’s position, a concentrating Photovoltaic (CPV) system with precise acceptance angle and low concentrating ratio will be an ideal alternative to conventional flat plate photovoltaic (PV) modules in harvesting the power from the sun. The SRCPC is a suitable choice well designed to achieve optimum precise acceptance angles and concentration ratio for this purpose. The optical performance theory study shows that a truncated symmetric reflective CPC with acceptance half-angles of 0° and 10° (termed as SRCPC-10) is the optimum design when compared with the symmetric reflective CPC designs with acceptance half-angles of 0° and 15° and 0° and 20° in Penryn and higher latitudes. An increase in the range of acceptance angles decreases the concentration ratio but an increase in the range of acceptance angles is achieved by truncating the concentrator profile which will reduce its cost as well. Ray tracing simulations indicates that the SRCPC-10 exhibited the maximum optical efficiency and steady slope compared with others. The simulated maximum optical efficiency of the SRCPC was found to be 94%. In addition, the SRCPC-10 was found to have a more uniform intensity distribution at the receiver and a total daily-monthly energy collection compared to the other designs. Thermal modelling of the CPV system with the SRCPC-10 concentrator shows that the solar cell operating temperature can reach up to 70°C for irradiance of 1000W/m2 at an ambient temperature of 25° at a wind velocity of 2.5m/s. The integration of the thermal management system is able to control and maintain the temperature to 29°C. The modelled thermal and electrical efficiencies were 47% and 15% respectively with a heat transfer coefficient of 54.29W/m2K thereby bringing the system efficiency to 62%. The maximum power of the SRCPC-10 when characterised in an indoor controlled environment using solar simulator was 5.96W at 1000W/m2 at a cooling flow rate of 0.0079L/s with average conversion efficiency of 8.97%. The maximum power at 1200W/m2 and 0.031L/s was 7.14W with conversion efficiency of 10.57%. The maximum increase in efficiency from non-cooling to cooling is 2.54%. The efficiency increased because of cooling is relatively 40%. The outdoor characterisation (validation) of the SRCPC-10 shows that the maximum power was 7.4W at 1206W/m2 on a sunny day. The maximum electrical conversion efficiency of the SRCPC-10 in outdoor conditions was found to be 10.96%. These results revealed that this designed SRCPC-10 is capable of collecting both direct and diffuse radiation to generate power. Therefore, the SRCPC-10 could be used to provide a solution to the increasing demand on electricity to the energy mix, leaving a clean environment for future developments

On a family of low-rank algorithms for large-scale algebraic Riccati equations

In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions $X_j$ to the solution $X$ of large-scale continuous-time algebraic Riccati equations (CAREs) $0 = \mathcal{R}(X) := A^HX+XA+C^HC-XBB^HX$ generate the same sequence $X_j$ when used with the same parameters. The Hermitian low-rank approximations $X_j$ are of the form $X_j = Z_jY_jZ_j^H,$ where $Z_j$ is a matrix with only few columns and $Y_j$ is a small square Hermitian matrix. Each $X_j$ generates a low-rank Riccati residual $\mathcal{R}(X_j)$ such that the norm of the residual can be evaluated easily allowing for an efficient termination criterion. Here a new family of methods to generate such low-rank approximate solutions $X_j$ of CAREs is proposed. Each member of this family of algorithms proposed here generates the same sequence of $X_j$ as the four previously known algorithms. The approach is based on a block rational Arnoldi decomposition and an associated block rational Krylov subspace spanned by $A^H$ and $C^H.$ Two specific versions of the general algorithm will be considered; one will turn out to be a rediscovery of the RADI algorithm, the other one allows for a slightly more efficient implementation compared to the RADI algorithm. Moreover, our approach allows for adding more than one shift at a time

Effizientes Lösen von großskaligen Riccati-Gleichungen und ein ODE-Framework für lineare Matrixgleichungen

This work considers the iterative solution of large-scale matrix equations. Due to the size of the system matrices in large-scale Riccati equations the solution can not be calculated directly but is approximated by a low rank matrix ZYZ^*. Herein Z is a basis of a low-dimensional rational Krylov subspace. The inner matrix Y is a small square matrix. Two ways to choose this inner matrix are examined: By imposing a rank condition on the Riccati residual and by projecting the Riccati residual onto the Krylov subspace generated by Z. The rank condition is motivated by the well-known ADI iteration. The ADI solutions span a rational Krylov subspace and yield a rank-p residual. It is proven that the rank-p condition guarantees existence and uniqueness of such an approximate solution. Known projection methods are generalized to oblique projections and a new formulation of the Riccati residual is derived, which allows for an efficient evaluation of the residual norm. Further a truncated approximate solution is characterized as the solution of a Riccati equation, which is projected to a subspace of the Krylov subspace generated by Z. For the approximate solution of Lyapunov equations a system of ordinary differential equations (ODEs) is solved via Runge-Kutta methods. It is shown that the space spanned by the approximate solution is a rational Krylov subspace with poles determined by the time step sizes and the eigenvalues of the matrices of the Butcher tableau of the used Runge-Kutta method. The method is applied to a model order reduction problem. The analytical solution of the system of ODEs satisfies an algebraic invariant. Those Runge-Kutta methods which preserve this algebraic invariant are characterized by a simple condition on the corresponding Butcher tableau. It is proven that these methods are equivalent to the ADI iteration. The invariance approach is transferred to Sylvester equations.Diese Arbeit befasst sich mit der numerischen Lösung hochdimensionaler Matrixgleichungen mittels iterativer Verfahren. Aufgrund der Größe der Systemmatrizen in großskaligen algebraischen Riccati-Gleichung kann die Lösung nicht direkt bestimmt werden, sondern wird durch eine approximative Lösung ZYZ^* von geringem Rang angenähert. Hierbei wird Z als Basis eines rationalen Krylovraums gewählt und enthält nur wenige Spalten. Die innere Matrix Y ist klein und quadratisch. Es werden zwei Wege untersucht, die Matrix Y zu wählen: Durch eine Rang-Bedingung an das Riccati-Residuum und durch Projektion des Riccati-Residuums auf den von Z erzeugten Krylovraum. Die Rang-Bedingung wird durch die wohlbekannten ADI-Verfahren motiviert. Die approximativen ADI-Lösungen spannen einen Krylovraum auf und führen zu einem Riccati-Residuum vom Rang p. Es wird bewiesen, dass die Rang-p-Bedingung Existenz und Eindeutigkeit einer solchen approximativen Lösung impliziert. Aus diesem Ergebnis werden effiziente iterative Verfahren abgeleitet, die eine solche approximative Lösung erzeugen. Bisher bekannte Projektionsverfahren werden auf schiefe Projektionen erweitert und es wird eine neue Formulierung des Riccati-Residuums hergeleitet, die eine effiziente Berechnung der Norm erlaubt. Weiter wird eine abgeschnittene approximative Lösung als Lösung einer Riccati-Gleichung charakterisiert, die auf einen Unterraum des von Z erzeugten Krylovraums projiziert wird. Um die Lösung der Lyapunov-Gleichung zu approximieren wird ein System gewöhnlicher Differentialgleichungen mittels Runge-Kutta-Verfahren numerisch gelöst. Es wird gezeigt, dass der von der approximativen Lösung aufgespannte Raum ein rationaler Krylovraum ist, dessen Pole von den Zeitschrittweiten der Integration und den Eigenwerten der Koeffizientenmatrix aus dem Butcher-Tableau des verwendeten Runge-Kutta-Verfahrens abhängen. Das Verfahren wird auf ein Problem der Modellreduktion angewendet. Die analytische Lösung des Differentialgleichungssystems erfüllt eine algebraische Invariante. Diejenigen Runge-Kutta-Verfahren, die diese Invariante erhalten, werden durch eine Bedingung an die zugehörigen Butcher-Tableaus charakterisiert. Es wird gezeigt, dass diese speziellen Verfahren äquivalent zur ADI-Iteration sind. Der Invarianten-Ansatz wird auf Sylvester-Gleichungen übertragen