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Archived Theses
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Archived Theses
Separability properties of agranovich-vishik type elliptic operators in banach space valued function classes
The main objective of the study is to discuss the BVP for an elliptic equation depending on a complex parameter, and defined on three different vector spaces: Euclidian space R_+^n , half-space R^n_+; and a bounded region Ω є R^n with a smooth (n-1) dimensional boundary Γ є R^(n-1), in E-valued function spaces, respectively. In other words, the purpose of this thesis is to discuss the separability properties (maximal regularity properties) of the general elliptic problem depending on a complex parameter in a bounded region Ω є R^n with a smooth (n-1) dimensional boundary. We consider a linear differential equation of arbitrary order with complex valued coefficients depending smoothly on the variable x and on the polynomially dependent complex parameter q. Let A(x,D,q)u = f (x) be a given elliptic equation depending on a complex parameter in the domain Ω. On the boundary G we are given general boundary conditions B_j(x_0,D,q)u_j|_Γ = g_j(x_0), containing a parameter q, that we will give in detail in the next section. Our main aim is to show the following estimate: ‖u‖_l+|q|^l ‖u‖_0 ≤c{‖f‖_(l-2m)+|q|^(l-2m) ‖f‖_0+∑_(j=1)^m ‖g_j‖_(l-m_j-1/p)^'+|q|^(l-m_j-1/p) ‖g_j‖_0^'}≤ C{‖u‖_l+|q|^l ‖u‖_0 }. The maximal regularity properties of elliptic problem in the Hilbert valued Sobolev space W^(l,2)(Ω;H) has been investigated before. Now, we generalize this to the Banach space valued Sobolev class W^(l,p)(Ω;E), where E is an UMD space. Firstly, we determine the maximal regularity properties of elliptic problems with complex parameter in the Banach space-valued Sobolev class W^(l,p) in the whole space and half space. Further, we analyze the separability properties of the parameter dependent general elliptic problem in a bounded domain with smooth boundary. This property is analyzed with an inequality which is called the coercive estimate. We encounter some limit when proving these theorems and in light of these we give some conditions that restrict us which one calls as Condition I and Condition II. So that there is no confusion, we give them the same name in every chapter.
The effects of communication delay on the operation ofdistributed energy resources in power distribution systems
Main objective of Power Electrical system is providing electrical energy for demanding consumers, as much as possible. The hierarchy of electrification starts from the high voltage generating units. The generated electrical energy reaches to the consumers passing through lines that are responsible for transmitting high level and medium level voltages and distribution systems, respectively. In the recent years, power system is growing in their size and complexity. It is determined by many factors and leads to several consequences. Environmental and economic aspects, as well as deregulation in power industry, are the important reasons that drive power system evolution. Mainly the environmental factors, such as limiting CO2 emission, lead to growing usage of the small renewable energy sources (RES) systems, what pushes currently centralized power generation paradigm towards more distributed approach in the future. On the other hand, the integration of more RES, which are usually intermittent sources of energy, adds some complexity to the monitoring and control of the power system. Micro-grids and Distributed Generation (DG) concepts have been introduced to the power system in order to meet the aims with both higher quality and quantity. However, there are some draw backs or at the more precise meaning of the word, Challenges, that are brought to the power system analysis using distributed generations. The aforementioned challenges in evolution of the power system determine inseparable challenges for diverse communication infrastructure. Those challenges include, for example, the synchronization between different entities in case of deregulated power industry, but most importantly challenges more directly related to integration of the new sources. Due to the intermittent behavior, small-scale and high number generation of distributed RES, their management, control, and protection become a difficult task. In order to provide reliable, economically feasible and safe operation, proper coordination through communication of measurement data and control data is essential. The communication infrastructure can exchange crucial data; however, it also brings new threats in the control and management of the power system. It is well known that the delays such as communication delays in the power system reduce the system damping performance and may even lead to instability if the delay margins are exceeded. Standards such as DNP3 and IEC 61850 communication protocols provide definition of intelligent electronic devices at electrical substations of medium voltages that let the power system operate more reliably and wisely. From which the second one has proved its better performance and application feasibility, interoperability and better scalability. IEC 61850 is widely used all around the world due to its in detail specifications, as well. One vital difference between DNP3 and IEC 61850 is that the second one not only includes the details of sending and receiving data and information, but also includes descriptions about the storages and executions. It will be explained that even IEC 61850 does not guarantee the system operation with absolutely no failure or fault. Nevertheless, it provides the opportunity and necessary information that paves the way of having a robust system. On the other hand, as it is mentioned above, failure analysis is not addressed in these protocol. Hence, it is important to investigate the effects of these failures in the system to predict the system behavior under different conditions. These potential hidden failures are important to analyze since they endanger the robustness, reliability, security and stability in the system with communication infrastructure which may result in black outs our outages in system. Pocket loss, link failure, data corruption and delays, etc. are failures which attract more attention of engineers due to their probability of occurrence. From which, the last one is the most frequent one and it couldn't be ignored in the system. On the other hand of the story, the control system over the distributed energy resources (DER) that aims at the frequency restoration, voltage regulation and active-reactive power sharing operates under the rules of hierarchy control, mostly. Droop control laws are used for the primary level of the control structure. This droop control enhances the performance of the system, however, it may not guarantee the stability and robustness, as well. Hence, the secondary layer should be implemented. That is where communication infrastructure is needed for data transmission. The control methodologies could be classified due to this communication between the DERs. These control methodologies have been studied in this thesis. Pros and cons are investigated and a case study with the most appropriate one and the most frequent failure, i.e. constant delay is going to be discussed. Stability analysis considering primary and secondary control and eventually, communication failure (or more specifically latency) is a challenge in power system. Most studies either include simple structures of the power system or excludes some dynamics within the power system. However, this study tries to be as comprehensive as possible. This comprehensiveness aids to get some instable regions of the frequency domain analysis revealed and have the system better known. The algorithm provided and used for the thesis does not use any simplification, as well. All in All, power system and the communication used in, should be able to preserve stability and reliability of the system, regardless of the frequent transients and communication failures imposed on system.