Este trabalho apresenta uma abordagem do método da função barreira logarítmica (MFBL) para a resolução do problema de fluxo de potência ótimo (FPO). A pesquisa fundamenta-se metodologicamente na função barreira logarítmica e nas condições de primeira ordem de Karush-Kuhn-Tucker (KKT). Para a solução do sistema de equações resultantes das condições de estacionaridade, da função Lagrangiana, utiliza-se o método de Newton. Na implementação computacional utiliza-se técnicas de esparsidade. Através dos resultados numéricos dos testes realizados em 5 sistemas (3, 8, 14, 30 e 118 barras) evidencia-se o potencial desta metodologia na solução do problema de FPO. / This work describes an approach on logarithmic barrier function method to solving the optimal power flow (OPF) problem. Search was based on the logarithmic barrier function and first order conditions of Karush-Kuhn-Tucker (KKT). To solve the equation system, obtained from the stationary conditions of the Lagrangian function, is used the Newton method. Implementation is performed using sparsity techniques. The numerical results, carried out in five systems (3, 8, 14, 30 and 118 bus), demonstrate the reliability of this approach in the solution OPF problem.

Este trabalho descreve uma abordagem do método primal-dual barreira logarítmica (MPDBL) associado ao método de Newton modificado para a resolução do problema de fluxo barreira logarítmica e nas condições de primeira ordem de Karush-Kuhn-Tucker (KKT). O sistema de equações resultantes das condições de estacionaridade, da função Lagrangiana, foi resolvido pelo método de Newton modificado. Na implementação computacional foram usadas as técnicas de esparsidade. Os resultados numéricos dos testes realizados em 5 sistemas (3, 14, 30, 57 e 118 barras) evidenciam o potencial desta metodologia na solução do problema de FPO. / This work describes an approach on primal-dual logarithmic barrier for solving the optimal power flow problem (OPF). The investigation was based on the logarithmic barrier function and Karush-Kuhn-Tucker (KKT) first-order necessary conditions. The equation system, obtained from the stationary conditions of the Lagrangian function, was solved using the Newton\'s modified method. The implementation was performed using sparsity techniques. The numerical results, carried out in five systems (3, 14,30, 57 and 118 bus), demonstrate the reliability of this approach in the solution OPF problem.

Die Arbeit abstrahiert bekannte Konzepte der endlichdimensionalen Optimierung im Hinblick auf deren Anwendung in Banachräumen. Hierfür werden zunächst grundlegende Elemente der Funktionalanalysis wie schwache Konvergenz, Dualräume und Reflexivität vorgestellt. Anschließend erfolgt eine kurze Einführung in die Thematik der Fréchet-Differenzierbarkeit und eine Abstraktion des Begriffs der partiellen Ordnungsrelation in normierten Räumen. Nach der Formulierung eines allgemeinen Existenzsatzes für globale Optimallösungen von abstrakten Optimierungsaufgaben werden notwendige Optimalitätsbedingungen vom Karush-Kuhn-Tucker-Typ hergeleitet. Abschließend wird eine hinreichende Optimalitätsbedingung vom Karush-Kuhn-Tucker-Typ unter verallgemeinerten Konvexitätsvoraussetzungen verifiziert.

Este trabalho descreve uma abordagem do método primal-dual barreira logarítmica (MPDBL) associado ao método de Newton modificado para a resolução do problema de fluxo barreira logarítmica e nas condições de primeira ordem de Karush-Kuhn-Tucker (KKT). O sistema de equações resultantes das condições de estacionaridade, da função Lagrangiana, foi resolvido pelo método de Newton modificado. Na implementação computacional foram usadas as técnicas de esparsidade. Os resultados numéricos dos testes realizados em 5 sistemas (3, 14, 30, 57 e 118 barras) evidenciam o potencial desta metodologia na solução do problema de FPO. / This work describes an approach on primal-dual logarithmic barrier for solving the optimal power flow problem (OPF). The investigation was based on the logarithmic barrier function and Karush-Kuhn-Tucker (KKT) first-order necessary conditions. The equation system, obtained from the stationary conditions of the Lagrangian function, was solved using the Newton\'s modified method. The implementation was performed using sparsity techniques. The numerical results, carried out in five systems (3, 14,30, 57 and 118 bus), demonstrate the reliability of this approach in the solution OPF problem.

Este trabalho apresenta uma abordagem do método da função barreira logarítmica (MFBL) para a resolução do problema de fluxo de potência ótimo (FPO). A pesquisa fundamenta-se metodologicamente na função barreira logarítmica e nas condições de primeira ordem de Karush-Kuhn-Tucker (KKT). Para a solução do sistema de equações resultantes das condições de estacionaridade, da função Lagrangiana, utiliza-se o método de Newton. Na implementação computacional utiliza-se técnicas de esparsidade. Através dos resultados numéricos dos testes realizados em 5 sistemas (3, 8, 14, 30 e 118 barras) evidencia-se o potencial desta metodologia na solução do problema de FPO. / This work describes an approach on logarithmic barrier function method to solving the optimal power flow (OPF) problem. Search was based on the logarithmic barrier function and first order conditions of Karush-Kuhn-Tucker (KKT). To solve the equation system, obtained from the stationary conditions of the Lagrangian function, is used the Newton method. Implementation is performed using sparsity techniques. The numerical results, carried out in five systems (3, 8, 14, 30 and 118 bus), demonstrate the reliability of this approach in the solution OPF problem.

Real-time optimization (RTO) is a supervisory strategy in the hierarchical industrial process automation architecture in which economically optimal set-point targets are computed for the lower level advanced control system, which is typically model predictive control (MPC). Due to highly volatile market conditions, recent developments have considered transforming the conventional steady-state RTO to dynamic RTO (DRTO) to permit economic optimization during transient operation. Published DRTO literature optimizes plant input trajectories without taking into account the presence of the plant control system, constituting an open-loop DRTO (OL-DRTO) approach. The goal of this research is to develop a design framework for a DRTO system that optimizes process economics based on a closed-loop response prediction. We focus, in particular, on DRTO applied to a continuous process operation regulated under constrained MPC. We follow a two-layer DRTO/MPC configuration due to its close tie to the industrial process automation architecture. We first analyze the effects of optimizing MPC closed-loop response dynamics at the DRTO level. A rigorous DRTO problem structure proposed in this thesis is in the form of a multilevel dynamic optimization problem, as it embeds a sequence of MPC optimization subproblems to be solved in order to generate the closed-loop prediction in the DRTO formulation, denoted here as a closed-loop DRTO (CL-DRTO) strategy. A simultaneous solution approach is applied in which the convex MPC optimization subproblems are replaced by their necessary and sufficient, Karush-Kuhn-Tucker (KKT) optimality conditions, resulting in the reformulation of the original multilevel problem as a single-level mathematical program with complementarity constraints (MPCC) with the complementarities handled using an exact penalty formulation. Performance analysis is carried out, and process conditions under which the CL-DRTO strategy significantly outperforms the traditional open-loop counterpart are identified. The multilevel DRTO problem with a rigorous inclusion of the future MPC calculations significantly increases the size and solution time of the economic optimization problem. Next, we identify and analyze multiple closed-loop approximation techniques, namely, a hybrid formulation, a bilevel programming formulation, and an input clipping formulation applied to an unconstrained MPC algorithm. Performance analysis based on a linear dynamic system shows that the proposed approximation techniques are able to substantially reduce the size and solution time of the rigorous CL-DRTO problem, while largely retaining its original performance. Application to an industrially-based case study of a polystyrene production described by a nonlinear differential-algebraic equation (DAE) system is also presented. Often large-scale industrial systems comprise multi-unit subsystems regulated under multiple local controllers that require systematic coordination between them. Utilization of closed-loop prediction in the CL-DRTO formulation is extended for application as a higher-level, centralized supervisory control strategy for coordination of a distributed MPC system. The advantage of the CL-DRTO coordination formulation is that it naturally considers interaction between the underlying MPC subsystems due to the embedded controller optimization subproblems while optimizing the overall process dynamics. In this case, we take advantage of the bilevel formulation to perform closed-loop prediction in two DRTO coordination schemes, with variations in the coordinator objective function based on dynamic economics and target tracking. Case study simulations demonstrate excellent performance in which the proposed coordination schemes minimize the impact of disturbance propagation originating from the upstream subsystem dynamics, and also reduce the magnitude of constraint violation through appropriate adjustment of the controller set-point trajectories. / Thesis / Doctor of Philosophy (PhD)

In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

Die Arbeit abstrahiert bekannte Konzepte der endlichdimensionalen Optimierung im Hinblick auf deren Anwendung in Banachräumen. Hierfür werden zunächst grundlegende Elemente der Funktionalanalysis wie schwache Konvergenz, Dualräume und Reflexivität vorgestellt. Anschließend erfolgt eine kurze Einführung in die Thematik der Fréchet-Differenzierbarkeit und eine Abstraktion des Begriffs der partiellen Ordnungsrelation in normierten Räumen. Nach der Formulierung eines allgemeinen Existenzsatzes für globale Optimallösungen von abstrakten Optimierungsaufgaben werden notwendige Optimalitätsbedingungen vom Karush-Kuhn-Tucker-Typ hergeleitet. Abschließend wird eine hinreichende Optimalitätsbedingung vom Karush-Kuhn-Tucker-Typ unter verallgemeinerten Konvexitätsvoraussetzungen verifiziert.

We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems. / text