Les travaux de cette thèse traite de la modélisation de systèmes par fonctions de transfert non entières à partir de données fréquentielles incertaines et bornées. A cet effet, les définitions d'intégration et de dérivation non entières sont d'abord étendues aux intervalles. Puis des approches ensemblistes sont appliquées pour l'estimation de l'ensemble des coefficients et des ordres de dérivation sous la forme d'intervalles. Ces approches s'appliquent pour l'estimation des paramètres de systèmes linéaires invariants dans le temps (LTI) certains, systèmes LTI incertains et systèmes linéaires à paramètres variant dans le temps (LPV). L'estimation paramétrique par approche ensembliste est particulièrement adaptée à la modélisation de la dynamique du conducteur, car les études sur un, voire plusieurs, individus montrent que les réactions recueillies ne sont jamais identiques mais varient d'une expérience à l'autre, voire d'un individu à l'autre. / This thesis deals with system identification and modeling of fractional transfer functions using bounded and uncertain frequency responses. Therefor, both of fractional differentiation and integration definitions are extended into intervals. Set membership approaches are then applied to estimate coefficients and derivative orders as intervals. These methods are applied to estimate certain Linear Time Invariant systems (LTI), uncertain LTI systems and Linear Parameter Varying systems (LPV). They are notably adopted to model driver's dynamics, since most of studies on one or several individuals shave shown that the collected reactions are not identical and are varying from an experiment to another.

Dans ce travail de thèse, nous nous focalisons sur le problème de l'intégration d'incertitude à erreurs bornées pour les systèmes dynamiques, dont les entrées et les états initiaux doivent être optimaux afin de réaliser certaines fonctionnalités.Le document comporte 5 chapitres: le premier est une introduction présentant le panorama du travail. Le deuxième chapitre présente les outils de base de l'analyse par intervalle. Le chapitre 3 est dédié à l'estimation d'états et de paramètres. Nous décrivons d'abord une procédure pour résoudre un système d'équations différentielles ordinaires avec l'aide de cet outil. Ainsi, une estimation des états à partir des conditions initiales peut être faite. Les systèmes différentiels considérés dépendent de paramètres qui doivent être estimés. Ce problème inverse pourra être résolu via l'inversion ensembliste. L'approche par intervalle est une procédure déterministe naturelle sans incertitude, tous les résultats obtenus sont garantis. Néanmoins, cette approche n'est pas toujours efficace, ceci est dû au fait que certaines opérations ensemblistes conduisent à des temps de calcul important. Nous présentons quelques techniques, par cela, nous nous plaçons dans un contexte à erreurs bornées permettant d'accélérer cette procédure. Celles-ci utilisent des contracteurs ciblés qui permettent ainsi une réduction de ce temps. Ces algorithmes ont été testés et ont montré leur efficacité sur plusieurs applications: des modèles pharmacocinétiques et un modèle du vol longitudinal d'avion en atmosphère au repos.Le chapitre 4 présente la recherche d'entrées optimales dans le cadre analyse par intervalle, ce qui est une approche originale. Nous avons construit plusieurs critères nouveaux permettant cette recherche. Certains sont intuitifs, d'autres ont nécessité un développement théorique. Ces critères ont été utilisés pour la recherche d'états initiaux optimaux. Des comparaisons ont été faites sur plusieurs applications et l'efficacité de certains critères a été mise en évidence.Dans le chapitre 5, nous appliquons les approches présentées précédemment au diagnostic via l'estimation de paramètres. Nous avons développé un processus complet pour le diagnostic et aussi formulé un processus pour le diagnostic actif avec une application en aéronautique. Le dernier chapitre résume les travaux réalisés dans cette thèse et essaye de donner des perspectives à la recherche.Les algorithmes proposés dans ce travail ont été développés en C++ et utilisent l'environnement du calcul ensembliste. / In this work, we will study the uncertainty integration problem in a bounded error context for the dynamic systems, whose input and the initial state have to be optimized so that some other operation could be more easily and better obtained. This work is consisted of 6 chapters : the chapter 1 is an introduction to the general subject which we will discuss about. The chapter 2 represents the basic tools of interval analysis.The chapter 3 is dedicated to state estimation and parameter estimation. We explain at the first, how to solve the ordinary differential equation using interval analysis, which will be the basic tool for the state estimation problem given the initial condition of studied systems. On the other ride, we will look into the parameter estimation problem using interval analysis too. Based on a simple hypothesis over the uncertain variable, we calculate the system's parameter in a bounded error form, considering the operation of intervals as the operation of sets. Guaranteed results are the advantage of interval analysis, but the big time consumption is still a problem for its popularization in many non linear estimation field. We present our founding techniques to accelerate this time consuming processes, which are called contractor in constraint propagation field. At the end of this chapter, différent examples will be the test proof for our proposed methods.Chapter 4 presents the searching for optimal input in the context of interval analysis, which is an original approach. We have constructed several new criteria allow such searching. Some of them are intuitive, the other need a theoretical proof. These criteria have been used for the search of optimal initial States and le better parameter estimation results. The comparisons are done by using multiple applications and the efficiency is proved by evidence.In chapter 5, we applied the approaches proposed above in diagnosis by state estimation and parameter estimation. We have developed a complete procedure for the diagnosis. The optimal input design has been reconsidered in an active diagnosis context. Both state and parameter estimation are implemented using an aeronautical application in literature.The last chapter given a brief summary over the realized subject, some further research directions are given in the perspective section.All the algorithms are written in C/C++ on a Linux based operation system.

<p>Efficient global dynamic optimization (GDO) using spatial branch-and-bound (SBB) requires the ability to construct tight bounds for the dynamic model. This thesis works toward efficient GDO by developing effective convex relaxation techniques for models with ordinary differential equations (ODEs). In particular, a novel algorithm, based upon a verified interval ODE method and the McCormick relaxation technique, is developed for constructing convex and concave relaxations of solutions of nonlinear parametric ODEs. In addition to better convergence properties, the relaxations so obtained are guaranteed to be no looser than their underlying interval bounds, and are typically tighter in practice. Moreover, they are rigorous in the sense of accounting for truncation errors. Nonetheless, the tightness of the relaxations is affected by the overestimation from the dependency problem of interval arithmetic that is not addressed systematically in the underlying interval ODE method. To handle this issue, the relaxation algorithm is extended to a Taylor model ODE method, which can provide generally tighter enclosures with better convergence properties than the interval ODE method. This way, an improved version of the algorithm is achieved where the relaxations are generally tighter than those computed with the interval ODE method, and offer better convergence. Moreover, they are guaranteed to be no looser than the interval bounds obtained from Taylor models, and are usually tighter in practice. However, the nonlinearity and (potentially) nonsmoothness of the relaxations impedes their fast and reliable solution. Therefore, the algorithm is finally modified by incorporating polyhedral relaxations in order to generate relatively tight and computationally cheap linear relaxations for the dynamic model. The resulting relaxation algorithm along with a SBB procedure is implemented in the MC++ software package. GDO utilizing the proposed relaxation algorithm is demonstrated to have significantly reduced computational expense, up to orders of magnitude, compared to existing GDO methods.</p> / Doctor of Philosophy (PhD)

Cette thèse est une partie d’un projet français qui s’appelle AEROSTRIP, un partenariat entre l’Institut Pascal, Sigma, SAPPI et Air-France industries, il est financé par le gouvernement français par le programme FUI (20 eme appel). Le projet AEROSTRIP consiste à développer le premier système automatique qui nettoie écologiquement les surfaces des avions et les pièces de rechange en utilisant un abrasif écologique projeté à grande vitesse sur la surface des avions (maïs). Ma thèse consiste à optimiser les trajectoires du système robotique total de telle façon que le décapage de l’avion soit optimal. Le déplacement des robots est nécessaire pour assurer une couverture totale de la surface à décaper parce que ces surfaces sont trop grandes et elles ne peuvent pas être décapées d’une seule position. Le but de mon travail est de trouver le nombre optimal de robots avec leur positions optimales pour décaper totalement l’avion. Une fois ce nombre est déterminé, on cherche les trajectoires des robots entre ces différentes positions. Alors, pour atteindre ce but, j’ai défini un cadre général composant de quatre étapes essentiels: l’étape pre-processing, l’étape optimization algorithm, l’étape generation of the end-effector trajectories et l’étape robot scheduling, assignment and control.Dans ma thèse, j’ai deux contributions dans deux différentes étapes du cadre général: l’étape pre-processing et l’étape optimization algorithm. Le calcul de l’espace de travail du robot est nécessairedans l’étape pre-processing: on a proposé l’Analyse par Intervalles pour trouver cet espace de travail parce qu’il garantie le fait de trouver des solutions dans un temps de calcul raisonnable. Alors, ma première contribution est une nouvelle fonction d’inclusion qui réduit le pessimisme, la surestimation des solutions qui est le principal inconvénient de l’Analyse par Intervalles. La nouvelle fonction d’inclusion est évaluée sur des problèmes de satisfaction de contraintes et des problèmes d’optimisation des contraintes. En plus, on a proposé un algorithme d’optimisation hybride pour trouver le nombre optimal de robots avec leur positions optimales: c’est notre deuxième contribution qui est dans l’étape optimization algorithm. Pour évaluer l’algorithme d’optimisation, on a testé cet algorithme sur des surfaces régulières, comme un cylindre et un hémisphère, et sur un surface complexe: une voiture. / This PHD is a part of a French project named AEROSTRIP, (a partnership between Pascal Institute,Sigma, SAPPI, and Air-France industries), it is funded by the French Government through the FUIProgram (20th call). The AEROSTRIP project aims at developing the first automated system thatecologically cleans the airplanes surfaces using a process of soft projection of ecological media onthe surface (corn). My PHD aims at optimizing the trajectory of the whole robotic systems in orderto optimally strip the airplane. Since a large surface can not be totally covered by a single robot base placement, repositioning of the robots is necessary to ensure a complete stripping of the surface. The goal in this work is to find the optimal number of robots with their optimal positions required to totally strip the air-plane. Once found, we search for the trajectories of the robots of the multi-robot system between those poses. Hence, we define a general framework to solve this problem having four main steps: the pre-processing step, the optimization algorithm step, the generation of the end-effector trajectories step and the robot scheduling, assignment and control step.In my thesis, I present two contributions in two different steps of the general framework: the pre-processing step, the optimization algorithm step. The computation of the robot workspace is required in the pre-processing step: we proposed Interval Analysis to find this workspace since it guarantees finding solutions in a reasonable computation time. Though, our first contribution is a new inclusion function that reduces the pessimism, the overestimation of the solution, which is the main disadvantage of Interval Analysis. The proposed inclusion function is assessed on some Constraints Satisfaction Problems and Constraints Optimization problems. Furthermore, we propose an hybrid optimization algorithm in order to find the optimal number of robots with their optimal poses: it is our second contribution in the optimization algorithm step. To assess our hybrid optimization algorithm, we test the algorithm on regular surfaces, such as a cylinder and a hemisphere, and on a complex surface: a car.

This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.

Σε αυτή την εργασία, ασχολούμαστε με το πρόβλημα της επίλυσης συστημάτων μη γραμμικών αλγεβρικών ή/και υπερβατικών εξισώσεων και συγκεκριμένα αναφερόμαστε σε βελτιωμένες αλγοριθμικές τεχνικές επίλυσης τέτοιων συστημάτων. Μη γραμμικά συστήματα υπάρχουν σε πολλούς τομείς της επιστήμης, όπως στη Μηχανική, την Ιατρική, τη Χημεία, τη Ρομποτική, τα Οικονομικά, κ.τ.λ. Υπάρχουν πολλές μέθοδοι για την επίλυση συστημάτων μη γραμμικών εξισώσεων. Ανάμεσά τους η μέθοδος Newton είναι η πιο γνωστή μέθοδος, λόγω της τετραγωνικής της σύγκλισης όταν υπάρχει μια καλή αρχική εκτίμηση και ο Ιακωβιανός πίνακας είναι nonsingular. Η μέθοδος Newton έχει μερικά μειονεκτήματα, όπως τοπική σύγκλιση, αναγκαιότητα υπολογισμού του Ιακωβιανού πίνακα και ακριβής επίλυση του γραμμικού συστήματος σε κάθε επανάληψη. Σε αυτή τη μεταπτυχιακή διπλωματική εργασία αναλύουμε τη μέθοδο Newton και κατηγοριοποιούμε μεθόδους που συμβάλλουν στην αντιμετώπιση των μειονεκτημάτων της μεθόδου Newton, π.χ. Quasi-Newton και Inexact-Newton μεθόδους. Μερικές πιο πρόσφατες μέθοδοι που περιγράφονται σε αυτή την εργασία είναι η μέθοδος MRV και δύο νέες μέθοδοι Newton χωρίς άμεσες συναρτησιακές τιμές, κατάλληλες για προβλήματα με μη ακριβείς συναρτησιακές τιμές ή με μεγάλο υπολογιστικό κόστος. Στο τέλος αυτής της μεταπτυχιακής εργασίας, παρουσιάζουμε τις βασικές αρχές της Ανάλυσης Διαστημάτων και τη Διαστηματική μέθοδο Newton. / In this contribution, we deal with the problem of solving systems of nonlinear algebraic or/and transcendental equations and in particular we are referred to improved algorithmic techniques of such kind of systems. Nonlinear systems arise in many domains of science, such as Mechanics, Medicine, Chemistry, Robotics, Economics, etc. There are several methods for solving systems of nonlinear equations. Among them Newton's method is the most famous, because of its quadratic convergence when a good initial guess exists and the Jacobian matrix is nonsingular. Newton's method has some disadvantages, such as local convergence, necessity of computation of Jacobian matrix and the exact solution of linear system at each iteration. In this master thesis we analyze Newton's method and we categorize methods that contribute to the treatment of drawbacks of Newton's method, e.g. Quasi-Newton and Inexact-Newton methods. Some more recent methods which are described in this thesis are the MRV method and two new Newton's methods without direct function evaluations, ideal for problems with inaccurate function values or high computational cost. At the end of this master thesis, we present the basic principles of Interval Analysis and Interval Newton's method.

Apresenta-se um estudo em Teoria de Domínios das equações integrais da forma geral f (x) = h(x)+g Z b(x) a(x) g(x, y, f (y))dy com h, a e b definidas para x &#8712; [a0,b0], a0 &#8804;a(x)&#8804;b(x)&#8804;b0 e g definida para x, y &#8712; [a0,b0], cujo lado direito define uma contração sobre o espaço métrico de funções reais contínuas limitadas. O ponto de partida desse trabalho é a reescrita da Análise Intervalar para Teoria de Domínios do problema de valor incial em equações diferenciais ordinárias que possuem solução como ponto fixo do operador de Picard. Com o conjunto dos números reais interpretados pelo Domínio Intervalar, as funções reais são estendidas para operarem no domínio de funçoes intervalares de variável real. Em particular, faz-se a extensão canônica do campo vetorial em relação à segunda variável. Nesse contexto, pela primeira vez tem-se o estudo das equações integrais de Fredholm e Volterra sobre o domínio de funções intervalares de variável real definida pelo operador integral intervalar com a participação da extensão canônica de g em relação à terceira variável. Adicionando ao domínio de funções intervalares sua função medição, efetua-se a análise da convergência do operador intervalar de Fredholm e Volterra em Teoria de Domínios com o cálculo da sua derivada informática em relação à medição no seu ponto fixo. Com a representação das funções intervalares em função passo constante a partir da partição do intervalo [a0,b0], reescrevese o algoritmo da Análise Intervalar em Teoria de Domínios com a introdução do cálculo da aproximação da extensão canônica de g e com o comprimento do intervalo da partição tendendo para zero. Estende-se essa abordagem mais completa do estudo das equações integrais na resolução de problemas de valores iniciais e valor de contorno em equações diferenciais ordinárias e parciais. Uma vez que para uma pequena variação do campo vetorial v ou do valor inicial y0 da equação diferencial f &#8242;(x) = v(x, f (x)) com a condição inicial f (x0) = y0, pode-se ter uma solução tão próxima da solução f da equação quanto possível, formaliza-se pela primeira vez em Teoria de Domínios um algoritmo na resolução do problema inverso em que, conhecendo a função f , determina-se uma equação diferencial ordinária com o cálculo de um campo vetorial v tal que o operador de Picard associado mapeia f tão próxima quanto possível a ela mesma. / We present a study in Domain Theory of integral equations of the form f (x) = h(x)+g Z b(x) a(x) g(x, y, f (y))dy for a0 &#8804; a(x) &#8804; b(x) &#8804; b0 with h, a, b defined for x &#8712; [a0,b0] and g defined for x, y &#8712; [a0,b0], in which the right-hand side defines a contraction on the metric space of continuous realvalued functions on [a0,b0]. The starting point of this work is to revisit Interval Analysis in Domain Theory for the initial-value problem in ordinary differential equations where a solution is expressed as a fixed point of the Picard operator. With the set of real numbers interpreted as the interval domain, real-valued functions are extended to work in the space of interval-valued functions of the real variable domain. In particular, the vector field is extended in the second argument. Under these conditions, for the first time Fredholm and Volterra integral equations have solutions expressed as fixed points of a contraction mapping in terms of the splitting on interval-valued functions of the real variable domain. The measurement for interval-valued functions of the real variable domain is considered where we can asssess the convergence properties of the interval integral operator by means of the informatic derivative. The proposed techniques are applied to more general methods in ordinary differencial equations (ODEs) and partial differential equations (PDEs). For the first time, an algorithm is proposed to provide solutions to the inverse problem for Odinary Differential Equation where, given a function f , it is found a vector field v that defines a Picard operator which maps the solution f as close as possible to itself, such that the ODE f &#8242;(x) = v(x, f (x)) admits f as either an exact or, as closely as desired, an approximate solution.

Apresenta-se um estudo em Teoria de Domínios das equações integrais da forma geral f (x) = h(x)+g Z b(x) a(x) g(x, y, f (y))dy com h, a e b definidas para x &#8712; [a0,b0], a0 &#8804;a(x)&#8804;b(x)&#8804;b0 e g definida para x, y &#8712; [a0,b0], cujo lado direito define uma contração sobre o espaço métrico de funções reais contínuas limitadas. O ponto de partida desse trabalho é a reescrita da Análise Intervalar para Teoria de Domínios do problema de valor incial em equações diferenciais ordinárias que possuem solução como ponto fixo do operador de Picard. Com o conjunto dos números reais interpretados pelo Domínio Intervalar, as funções reais são estendidas para operarem no domínio de funçoes intervalares de variável real. Em particular, faz-se a extensão canônica do campo vetorial em relação à segunda variável. Nesse contexto, pela primeira vez tem-se o estudo das equações integrais de Fredholm e Volterra sobre o domínio de funções intervalares de variável real definida pelo operador integral intervalar com a participação da extensão canônica de g em relação à terceira variável. Adicionando ao domínio de funções intervalares sua função medição, efetua-se a análise da convergência do operador intervalar de Fredholm e Volterra em Teoria de Domínios com o cálculo da sua derivada informática em relação à medição no seu ponto fixo. Com a representação das funções intervalares em função passo constante a partir da partição do intervalo [a0,b0], reescrevese o algoritmo da Análise Intervalar em Teoria de Domínios com a introdução do cálculo da aproximação da extensão canônica de g e com o comprimento do intervalo da partição tendendo para zero. Estende-se essa abordagem mais completa do estudo das equações integrais na resolução de problemas de valores iniciais e valor de contorno em equações diferenciais ordinárias e parciais. Uma vez que para uma pequena variação do campo vetorial v ou do valor inicial y0 da equação diferencial f &#8242;(x) = v(x, f (x)) com a condição inicial f (x0) = y0, pode-se ter uma solução tão próxima da solução f da equação quanto possível, formaliza-se pela primeira vez em Teoria de Domínios um algoritmo na resolução do problema inverso em que, conhecendo a função f , determina-se uma equação diferencial ordinária com o cálculo de um campo vetorial v tal que o operador de Picard associado mapeia f tão próxima quanto possível a ela mesma. / We present a study in Domain Theory of integral equations of the form f (x) = h(x)+g Z b(x) a(x) g(x, y, f (y))dy for a0 &#8804; a(x) &#8804; b(x) &#8804; b0 with h, a, b defined for x &#8712; [a0,b0] and g defined for x, y &#8712; [a0,b0], in which the right-hand side defines a contraction on the metric space of continuous realvalued functions on [a0,b0]. The starting point of this work is to revisit Interval Analysis in Domain Theory for the initial-value problem in ordinary differential equations where a solution is expressed as a fixed point of the Picard operator. With the set of real numbers interpreted as the interval domain, real-valued functions are extended to work in the space of interval-valued functions of the real variable domain. In particular, the vector field is extended in the second argument. Under these conditions, for the first time Fredholm and Volterra integral equations have solutions expressed as fixed points of a contraction mapping in terms of the splitting on interval-valued functions of the real variable domain. The measurement for interval-valued functions of the real variable domain is considered where we can asssess the convergence properties of the interval integral operator by means of the informatic derivative. The proposed techniques are applied to more general methods in ordinary differencial equations (ODEs) and partial differential equations (PDEs). For the first time, an algorithm is proposed to provide solutions to the inverse problem for Odinary Differential Equation where, given a function f , it is found a vector field v that defines a Picard operator which maps the solution f as close as possible to itself, such that the ODE f &#8242;(x) = v(x, f (x)) admits f as either an exact or, as closely as desired, an approximate solution.

The design of hydropower projects requires a comprehensive planning process in order to achieve the objective to maximise exploitation of the existing hydropower potential as well as future revenues of the plant. For this purpose and to satisfy approval requirements for a complex hydropower development, it is imperative at planning stage, that the conceptual development contemplates a wide range of influencing design factors and ensures appropriate consideration of all related aspects. Since the majority of technical and economical parameters that are required for detailed and final design cannot be precisely determined at early planning stages, crucial design parameters such as design discharge and hydraulic head have to be examined through an extensive optimisation process. One disadvantage inherent to commonly used deterministic analysis is the lack of objectivity for the selection of input parameters. Moreover, it cannot be ensured that the entire existing parameter ranges and all possible parameter combinations are covered. Probabilistic methods utilise discrete probability distributions or parameter input ranges to cover the entire range of uncertainties resulting from an information deficit during the planning phase and integrate them into the optimisation by means of an alternative calculation method. The investigated method assists with the mathematical assessment and integration of uncertainties into the rational economic appraisal of complex infrastructure projects. The assessment includes an exemplary verification to what extent the Random Set Theory can be utilised for the determination of input parameters that are relevant for the optimisation of hydropower projects and evaluates possible improvements with respect to accuracy and suitability of the calculated results. / Die Auslegung von Wasserkraftanlagen stellt einen komplexen Planungsablauf dar, mit dem Ziel das vorhandene Wasserkraftpotential möglichst vollständig zu nutzen und künftige, wirtschaftliche Erträge der Kraftanlage zu maximieren. Um dies zu erreichen und gleichzeitig die Genehmigungsfähigkeit eines komplexen Wasserkraftprojektes zu gewährleisten, besteht hierbei die zwingende Notwendigkeit eine Vielzahl für die Konzepterstellung relevanter Einflussfaktoren zu erfassen und in der Projektplanungsphase hinreichend zu berücksichtigen. In frühen Planungsstadien kann ein Großteil der für die Detailplanung entscheidenden, technischen und wirtschaftlichen Parameter meist nicht exakt bestimmt werden, wodurch maßgebende Designparameter der Wasserkraftanlage, wie Durchfluss und Fallhöhe, einen umfangreichen Optimierungsprozess durchlaufen müssen. Ein Nachteil gebräuchlicher, deterministischer Berechnungsansätze besteht in der zumeist unzureichenden Objektivität bei der Bestimmung der Eingangsparameter, sowie der Tatsache, dass die Erfassung der Parameter in ihrer gesamten Streubreite und sämtlichen, maßgeblichen Parameterkombinationen nicht sichergestellt werden kann. Probabilistische Verfahren verwenden Eingangsparameter in ihrer statistischen Verteilung bzw. in Form von Bandbreiten, mit dem Ziel, Unsicherheiten, die sich aus dem in der Planungsphase unausweichlichen Informationsdefizit ergeben, durch Anwendung einer alternativen Berechnungsmethode mathematisch zu erfassen und in die Berechnung einzubeziehen. Die untersuchte Vorgehensweise trägt dazu bei, aus einem Informationsdefizit resultierende Unschärfen bei der wirtschaftlichen Beurteilung komplexer Infrastrukturprojekte objektiv bzw. mathematisch zu erfassen und in den Planungsprozess einzubeziehen. Es erfolgt eine Beurteilung und beispielhafte Überprüfung, inwiefern die Random Set Methode bei Bestimmung der für den Optimierungsprozess von Wasserkraftanlagen relevanten Eingangsgrößen Anwendung finden kann und in wieweit sich hieraus Verbesserungen hinsichtlich Genauigkeit und Aussagekraft der Berechnungsergebnisse ergeben.

The design of hydropower projects requires a comprehensive planning process in order to achieve the objective to maximise exploitation of the existing hydropower potential as well as future revenues of the plant. For this purpose and to satisfy approval requirements for a complex hydropower development, it is imperative at planning stage, that the conceptual development contemplates a wide range of influencing design factors and ensures appropriate consideration of all related aspects. Since the majority of technical and economical parameters that are required for detailed and final design cannot be precisely determined at early planning stages, crucial design parameters such as design discharge and hydraulic head have to be examined through an extensive optimisation process. One disadvantage inherent to commonly used deterministic analysis is the lack of objectivity for the selection of input parameters. Moreover, it cannot be ensured that the entire existing parameter ranges and all possible parameter combinations are covered. Probabilistic methods utilise discrete probability distributions or parameter input ranges to cover the entire range of uncertainties resulting from an information deficit during the planning phase and integrate them into the optimisation by means of an alternative calculation method. The investigated method assists with the mathematical assessment and integration of uncertainties into the rational economic appraisal of complex infrastructure projects. The assessment includes an exemplary verification to what extent the Random Set Theory can be utilised for the determination of input parameters that are relevant for the optimisation of hydropower projects and evaluates possible improvements with respect to accuracy and suitability of the calculated results. / Die Auslegung von Wasserkraftanlagen stellt einen komplexen Planungsablauf dar, mit dem Ziel das vorhandene Wasserkraftpotential möglichst vollständig zu nutzen und künftige, wirtschaftliche Erträge der Kraftanlage zu maximieren. Um dies zu erreichen und gleichzeitig die Genehmigungsfähigkeit eines komplexen Wasserkraftprojektes zu gewährleisten, besteht hierbei die zwingende Notwendigkeit eine Vielzahl für die Konzepterstellung relevanter Einflussfaktoren zu erfassen und in der Projektplanungsphase hinreichend zu berücksichtigen. In frühen Planungsstadien kann ein Großteil der für die Detailplanung entscheidenden, technischen und wirtschaftlichen Parameter meist nicht exakt bestimmt werden, wodurch maßgebende Designparameter der Wasserkraftanlage, wie Durchfluss und Fallhöhe, einen umfangreichen Optimierungsprozess durchlaufen müssen. Ein Nachteil gebräuchlicher, deterministischer Berechnungsansätze besteht in der zumeist unzureichenden Objektivität bei der Bestimmung der Eingangsparameter, sowie der Tatsache, dass die Erfassung der Parameter in ihrer gesamten Streubreite und sämtlichen, maßgeblichen Parameterkombinationen nicht sichergestellt werden kann. Probabilistische Verfahren verwenden Eingangsparameter in ihrer statistischen Verteilung bzw. in Form von Bandbreiten, mit dem Ziel, Unsicherheiten, die sich aus dem in der Planungsphase unausweichlichen Informationsdefizit ergeben, durch Anwendung einer alternativen Berechnungsmethode mathematisch zu erfassen und in die Berechnung einzubeziehen. Die untersuchte Vorgehensweise trägt dazu bei, aus einem Informationsdefizit resultierende Unschärfen bei der wirtschaftlichen Beurteilung komplexer Infrastrukturprojekte objektiv bzw. mathematisch zu erfassen und in den Planungsprozess einzubeziehen. Es erfolgt eine Beurteilung und beispielhafte Überprüfung, inwiefern die Random Set Methode bei Bestimmung der für den Optimierungsprozess von Wasserkraftanlagen relevanten Eingangsgrößen Anwendung finden kann und in wieweit sich hieraus Verbesserungen hinsichtlich Genauigkeit und Aussagekraft der Berechnungsergebnisse ergeben.