Certificates of infeasibility via nonsmooth optimization

An important aspect in the solution process of constraint satisfaction problems is to identify exclusion boxes which are boxes that do not contain feasible points. This paper presents a certificate of infeasibility for finding such boxes by solving a linearly constrained nonsmooth optimization problem. Furthermore, the constructed certificate can be used to enlarge an exclusion box by solving a nonlinearly constrained nonsmooth optimization problem.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0802

A feasible second order bundle algorithm for nonsmooth, nonconvex optimization problems with inequality constraints and its application to certificates of infeasibility

Diese Arbeit erweitert den SQP-Zugang des Bundle-Newton-Verfahrens für nichtglatte, unrestringierte Optimierungsprobleme zu einem zulässigen Bundle-Algorithmus zweiter Ordnung für nichtglatte, nichtkonvexe Optimierungsprobleme mit Ungleichungsnebenbedingungen. An Stelle der Verwendung einer Straffunktion oder eines Filters oder einer Improvement-Funktion zur Behandlung der Nebenbedingungen, wird die Suchrichtung durch Lösen eines konvexen quadratischen Optimierungsproblems mit quadratischen Nebenbedingungen bestimmt, um gute Iterationspunkte zu erhalten. Außerdem untersuchen wir einige Varianten des Suchrichtungsproblems, wir geben eine numerische Rechtfertigung für die Anwendbarkeit des vorgestellten Zugangs, indem wir die Effektivität von verschiedener Lösungssoftware für die Berechnung der Suchrichtung vergleichen, und wir weisen die globale Konvergenz der Methode unter bestimmten Voraussetzungen nach. Weiters stellen wir eine wichtige Anwendung der nichtglatten Optimierung für Zulässigkeitsprobleme vor: Dazu führen wir ein Unzulässigkeitszertifikat ein, welches das Auffinden von Ausschlussboxen durch Lösen eines nichtglatten Optimierungsproblems mit linearen Nebenbedingungen ermöglicht. Zusätzlich kann dieses Zertifikat verwendet werden, um eine Ausschlussbox durch Lösen eines nichtglatten Optimierungsproblems mit nichtlinearen Nebenbedingungen zu vergrößern. Schließlich besprechen wir noch die im Vergleich zu anderer Lösungssoftware guten Testergebnisse von unserem Bundle-Algorithmus zweiter Ordnung für einige Hock-Schittkowski-Beispiele, für Beispiele die im Zusammenhang mit der Auffindung von Ausschlussboxen in Zulässigkeitsproblemen auftreten und für höher dimensionale stückweise quadratische Beispiele.This thesis extends the SQP-approach of the well-known bundle-Newton method for nonsmooth unconstrained minimization to a feasible second order bundle algorithm for nonsmooth, nonconvex optimization problems with inequality constraints: Instead of using a penalty function or a filter or an improvement function to deal with the presence of constraints, the search direction is determined by solving a convex quadratically constrained quadratic program to obtain good iteration points. Moreover, we investigate certain versions of the search direction problem, we justify the applicability of this approach numerically by using different solvers for the computation of the search direction and we show global convergence of the method under certain assumptions. Furthermore, we present an important application of nonsmooth optimization to constraint satisfaction problems: We introduce a certificate of infeasibility for finding exclusion boxes by solving a linearly constrained nonsmooth optimization problem. Additionally, the constructed certificate can be used to enlarge an exclusion box by solving a nonlinearly constrained nonsmooth optimization problem. Finally, the good performance of the second order bundle algorithm is demonstrated by comparison with test results of other solvers on examples of the Hock-Schittkowski collection, on custom examples that arise in the context of finding exclusion boxes for constraint satisfaction problems, and on higher dimensional piecewise quadratic examples

Analysis and management of security constraints in overstressed power systems

Management of operational security constraints is one of the important tasks performed by system operators, which must be addressed properly for secure and economic operation. Constraint management is becoming an increasingly complex and challenging to execute in modern electricity networks for three main reasons. First, insufficient transmission capacity during peak and emergency conditions, which typically result in numerous constraint violations. Second, reduced fault levels, inertia and damping due to power electronic interfaced demand and stochastic renewable generation, which are making network more vulnerable to even small disturbances. Third, re-regulated electricity markets require the networks to operate much closer to their operational security limits, which typically result in stressed and overstressed operating conditions. Operational security constraints can be divided into static security limits (bus voltage and branch thermal limits) and dynamic security limits (voltage and angle stability limits). Security constraint management, in general, is formulated as a constrained, nonlinear, and nonconvex optimization problem. The problem is usually solved by conventional gradient-based nonlinear programming methods to devise optimal non-emergency or emergency corrective actions utilizing minimal system reserves. When the network is in emergency state with reduced/insufficient control capability, the solution space of the corresponding nonlinear optimization problem may be too small, or even infeasible. In such cases, conventional non-linear programming methods may fail to compute a feasible (corrective) control solution that mitigate all constraint violations or might fail to rationalize a large number of immediate post-contingency constraint violations into a smaller number of critical constraints. Although there exists some work on devising corrective actions for voltage and thermal congestion management, this has mostly focused on the alert state of the operation, not on the overstressed and emergency conditions, where, if appropriate control actions are not taken, network may lose its integrity. As it will be difficult for a system operator to manage a large number of constraint violations (e.g. more than ten) at one time, it is very important to rationalize the violated constraints to a minimum subset of critical constraints and then use information on their type and location to implement the right corrective actions at the right locations, requiring minimal system reserves and switching operations. Hence, network operators and network planners should be equipped with intelligent computational tools to “filter out” the most critical constraints when the feasible solution space is empty and to provide a feasible control solution when the solution space is too narrow. With an aim to address these operational difficulties and challenges, this PhD thesis presents three novel interdependent frameworks: Infeasibility Diagnosis and Resolution Framework (IDRF), Constraint Rationalization Framework (CRF) and Remedial Action Selection and Implementation Framework (RASIF). IDRF presents a metaheuristic methodology to localise and resolve infeasibility in constraint management problem formulations (in specific) and nonlinear optimization problem formulations (in general). CRF extends PIDRF and reduces many immediate post-contingency constraint violations into a small number of critical constraints, according to various operational priorities during overstressed operating conditions. Each operational priority is modelled as a separate objective function and the formulation can be easily extended to include other operational aspects. Based on the developed CRF, RASIF presents a methodology for optimal selection and implementation of the most effective remedial actions utilizing various ancillary services, such as distributed generation control, reactive power compensation, demand side management, load shedding strategies. The target buses for the implementation of the selected remedial actions are identified using bus active and reactive power injection sensitivity factors, corresponding to the overloaded lines and buses with excessive voltage violations (i.e. critical constraints). The RASIF is validated through both static and dynamic simulations to check the satisfiability of dynamic security constraints during the transition and static security constraints after the transition. The obtained results demonstrate that the framework for implementation of remedial actions allows the most secure transition between the pre-contingency and post-contingency stable equilibrium points