O objetivo deste trabalho é estudar aproximação na esfera por uma soma com pesos de harmônicos esféricos. Apresentamos condições necessárias e suficientes sobre os pesos para garantir a convergência, tanto no caso contínuo quanto no caso Lp. Analisamos a ordem de convergência dos processos aproximatórios usando um módulo de suavidade esférico relacionado à derivada forte de Laplace-Beltrami. Incluímos provas para vários resultados sobre a derivada forte de Laplace-Beltrami, já que não conseguimos encontrá-las na literatura / The subject of this work is to study approximation on the sphere by weighted sums of spherical harmonics. We present necessary and sufficient conditions on the weights for convergence in both, the continuous and the Lp cases. We analyse the convergence rates of the approximation processes using a modulus of smoothness related to the strong Laplace- Beltrami derivative. We include proofs for several results related to such a derivative, since we were unable to find them in the literature

Made available in DSpace on 2016-12-23T14:07:27Z (GMT). No. of bitstreams: 1 Dissertacao de Dirceu Soares Junior Parte 1.pdf: 2325132 bytes, checksum: 944653981d05fdcc1c69ff97b342de4f (MD5) Previous issue date: 2011-09-22 / Uma planta siderúrgica caracteriza-se pela existência em seu processo produtivo de cargas não lineares de elevada potência. Dentre estas cargas, as siderúrgicas do tipo mini-mill possuem normalmente Forno Elétrico a Arco (FEA) na sua fase inicial de produção do aço, que ocorre a partir da fusão de sucatas metálicas como principal matéria-prima utilizada. Os fornos elétricos absorvem uma corrente distorcida da rede elétrica, causando assim uma distorção de tensão no ponto de acoplamento comum (PAC) e inúmeros problemas de qualidade de energia elétrica. Uma solução muito utilizada para a mitigação harmônica neste tipo de carga elétrica é obtida através do uso de filtros passivos, com vários estágios de filtragem. Porém esta é uma solução que pode levar ao efeito da ressonância harmônica, elevando com isto a distorção harmônica no sistema elétrico e causando sobrecarga no sistema de filtragem. Neste trabalho é apresentado um estudo de caso em uma siderúrgica não integrada (siderúrgica que possui sucata metálica e ferro gusa como matérias-primas principais na fabricação do aço), localizada na região da Grande Vitória. Por meio de medições na subestação principal da usina, são apresentadas as principais formas de onda das correntes e tensões, em conjunto com seus espectros harmônicos, que comprovam a existência de distorções harmônicas, sobretudo provenientes da corrente do forno elétrico a arco. As medições consideradas no trabalho, para simulação dos modelos contendo os filtros híbridos, foram aquelas originadas da condição operacional mais crítica do processo, sob o ponto de vista de momento onde o comportamento da carga acarreta o maior desvio quanto à qualidade de energia nas variáveis elétricas medidas. O modelo utilizado para o sistema em questão apresenta forte grau de correlação com o sistema real, retratado através das medições elétricas efetuadas. De posse do modelo é realizada uma análise comparativa por meio de simulação entre duas topologias de filtragem híbrida, o filtro híbrido série e o filtro híbrido paralelo, para a compensação harmônica e amortecimento da ressonância. Os resultados das simulações e das análises das ressonâncias mostraram que o filtro híbrido paralelo é a topologia que permite a maior redução das distorções harmônicas de corrente e de tensão no PAC, utilizando-se um filtro ativo com a menor potência nominal. Contudo, a filtragem híbrida paralela não provê a eliminação por completo da ressonância paralela. E, quanto à ressonância série, o filtro utilizado não possui a capacidade de interferir na condição de amplificação harmônica existente no sistema elétrico. Finalmente, o estudo propõe que a definição da melhor alternativa de filtragem leve em consideração a escolha do ganho do filtro ativo e, consequentemente, a sua potência, de forma bem alinhada aos objetivos de melhoria na qualidade de energia elétrica que são esperados para o sistema em questão / A steelmaking plant is characterized for having in its productive process non linear loads of high power. For these loads, the steelmaking industries called mini-mill normally have Electric Arc Furnace (EAF) in its initial steel production phase that happen with metallic scrap melting being the principal raw material used. The EAF absorb a distorted current from the electric main, causing voltage distortion in its point of common coupling (PCC) and many problems with respects the electric power quality. A standard solution used for the harmonic mitigation in this application is obtained through the use of composite passive filter. However this is a solution that can cause harmonic resonance, increasing the harmonic distortion in the electrical system and causing overload in the filtering system. In this work, a case in a non integrated steel industry, located at Grande Vitoria region, is presented. Through measurements in the main substation of the plant, the main current and voltage waveforms are presented, along with its harmonic spectrum, that confirm the existence of harmonic resonance phenomenon, mainly from Electric Arc Furnace current. The measurement considered in this work to simulate the hybrid filters models, was that originated from operational condition with higher process criticism, under viewpoint of the moment where the behavior of the load brings to a major deviation regarding the power quality at the measured variables. The model used for the system under study shows strong correlation with the real system portrayed through the electrical measurements done. Using the model, a comparative simulation analysis is done, between two hybrid topologies, the series hybrid filter and the parallel hybrid filter, with the objective of harmonic compensation and resonance damping. The results of simulations and resonance analysis showed that the parallel hybrid filter is the topology that allows the greatest reduction in harmonic distortion of current and voltage in the PCC (Point of Common Coupling), using an active filter with the lowest nominal power rating. However, the parallel hybrid filtering does not provide the complete elimination of parallel resonance, and for the series resonance, the filter used does not have the ability to interfere in the harmonic amplification condition existing in the electrical system. Finally, the study proposes that the best filtering alternative takes into account the choice of the active filter gain and consequently its power, so well aligned with the objectives of improving the power quality that are expected for the system under analysis

As faltas de alta impedância (FAI) ocorrem quando um condutor entra em contato com objetos com uma alta resistência, o que não resulta em incrementos significativos de corrente. Este tipo de falta não pode ser detectada por dispositivos de proteção tradicionais que atuam por sobrecorrente. Neste trabalho uma pesquisa do estado da arte das FAI é apresentada e são implementados alguns algoritmos de detecção presentes na literatura. Nesta pesquisa também são propostas duas metodologias de identificação de FAI as quais são baseadas na identificação de buildup e incrementos de energias na faixa de frequências 1920Hz a 3840Hz. Os algoritmos implementados e as metodologias propostas foram testados com simulações de eventos de FAI e de eventos comuns numa rede simulada de uma concessionária do Estado de São Paulo utilizando o software ATP. Estudos do funcionamento de um medidor de qualidade de energia foram feitos com o propósito de determinar as limitações de uma eventual implementação real das metodologias de identificação propostas neste trabalho. Resultados do desempenho das metodologias propostas utilizando os sinais simulados em ATP apresentaram percentagens de identificação superiores aos 80%. Análise das ocorrências de cabo rompido do circuito em estudo foram feitas com o propósito de identificar possíveis eventos de FAI. O desempenho apresentado pelas metodologias propostas na identificação de FAI em sinais reais pode ser melhorado com a disponibilidade de uma maior quantidade de eventos reais de FAI. / High impedance faults (HIF) occur when a conductor makes contact with high resistance objects, generally not resulting in significant current increases. This type of fault cannot be detected by traditional overcurrent protection devices. In this work, a research of the HIF state-of-art is presented and some detection algorithms in the literature are implemented. This research also proposes two HIF identification methods based on respectively the identification of buildup current and energy increments in the frequency range from 1920Hz to 3840Hz. The implemented algorithms and the proposed methodology have been tested with ATP simulations of FAI events and other common events in a simulated network from an electric utility in the State of Sao Paulo. Studies are carried out in order to assess the operation of a power quality meter to determine the limitations of real implementations of the proposed methodologies. Performance results of the proposed methodologies using simulated ATP signals show identifications rates greater than 80%. Broken conductor occurrences in the studied feeder are analyzed to identify possible HIF events. The performance shown by the proposed methodologies in identifying HIF in real situations can be improved with the availability of a larger number of HIF events.

Series Active Filters (SAF) are designed for harmonic isolation and load voltage regulation of single-phase and three-phase voltage harmonic source type nonlinear loads. The novel Absolute Value Method (AVM) for load voltage harmonic extraction is proposed and applied in the control algorithm of SAF. The SAF compensated systems are represented by simplified linear models such that SAF controller gains can be easily determined. Harmonic isolation and load voltage regulation performances of 2.5 kW single-phase and 10 kW three-phase SAF compensated systems are evaluated by detailed simulations. Laboratory prototype single-phase and three-phase SAFs and loads are designed and manufactured. Digital signal processor based control platform is employed. Exclusive laboratory tests are conducted. Via laboratory experiments and simulations it is shown that AVM yields superior harmonic isolation and load voltage regulation performance compared to the conventional low/high pass filtering method. Theory, simulations, and experiments are well correlated and illustrate the feasibility of the proposed method.

Reactive power is critical to reliable operation of the modern AC power system. There is a plethora of motor-loads, transformers, and power-electronic loads connected to the power grid, which consume reactive power for normal operation. Transmission lines also consume reactive power when they are loaded above their surge impedance loading (SIL). Reactive power can exact opportunity cost due to reduced capacity of the lines to carry real power, which in turn lowers revenue. Most transmission owners (TOs) levy large penalties against load serving entities (LSEs), industrial facilities, and other end-use customers, who consume more than their allotted amount, as measured by their power factor. These penalties are to incentivize their customers to meet their reactive power needs locally as well as to recuperate the TOs' financial losses. Harmonic pollution is another factor that prevents the optimal operation of the grid and the connected loads. Harmonics are attributable to proliferation of the diode-rectifier- or thyristor-rectifier-interfaced loads such as variable speed ac drives and power supplies in server farms, electric arc furnaces, and other non-linear loads, which are widely employed by the industrial sector. With wider adoption of harmonic-rich loads by the consumer sector as well, such as HDTVs and compact fluorescent lamps (CFLs), greater level of triplen harmonics associated with single-phase loads are also increasingly seen on the distribution grid. The increasing penetration of renewable resources and electrification of light-duty vehicles are expected to further aggravate the stresses and congestion on the utility grid. Reactive power compensation is necessary for supporting the AC grid and maintaining a healthy voltage stability margin. Compensation can also enhance the utilization of system capacity, lower system losses, provide fault ride-through, and enable a quick fault recovery. Existing VAR and harmonic compensation technologies are either too expensive or inadequate to meet the dynamic needs of the modern and the future power system. This dissertation presents a novel class of Dynamic VAR and Harmonic Compensators (DVHCs) for supplying or absorbing reactive power and providing harmonic filtering, where the compensation is in shunt with the line and the load. The underlying concept is based on augmenting a static or passive component like a capacitor or an inductor with a direct AC converter and imbuing the passive component with dynamic properties. The direct AC converter can be configured as a buck, a boost, or a buck-boost. A `fail-normal' switch is an integral part of the DVHCs that bypasses the converter when it fails, preserving the original functionality and the reliability of the passive component. The DVHCs are modular and scalable such that they can be employed in applications ranging from residential and industrial with voltages less than 480 V, to power distribution level with voltages as high as 35 kV. The Dynamic Inductor (D-IND) and the Dynamic Capacitor (D-CAP) are subclasses of the DVHCs. As the applications for supplying leading VARs are more prevalent, the primary focus of this work is on the buck, the boost, and the buck-boost configurations of the D-CAP. To understand the characteristics and operation of the DVHCs, this work has developed time-domain models for analyzing the transient and dynamic behavior; frequency-domain models for understanding the harmonic interactions and the steady-state relationships between switch duty and current harmonics; and small-signal models for studying the dynamics of the converter due to various perturbations. The small-signal models also enable extraction of transfer functions in designing controllers and assessing stability margins. Control architectures and techniques are presented for effectively controlling the D-CAP when commutating the semiconductor devices with both high and low switching frequencies. In modularly scaling the DVHCs to higher voltages, three medium-voltage topologies are discussed. They are based on series-connecting fractionally-rated devices, AC flying capacitors, and series cascading multiple two-level cells. These implementations allow direct connect to the medium-voltage grid, thereby obviating the use of transformers, and subsequently reducing the losses, cost, complexity, and footprint. A novel AC snubber concept is proposed to provide safe commutation of the AC switches, fault tolerance by managing the energy trapped in parasitics and filters, and to enable dynamic and static voltage sharing when integrated around the series-connected devices. Design equations for selecting and rating the devices and components in the buck, the boost, and the buck-boost configurations of the D-CAP are presented. Three sets of example designs, with one at low-voltage and two at medium-voltage, are discussed to demonstrate the typical size and ratings of the various components under realistic operating conditions. Measurements and the related discussions of a 40 kVA buck D-CAP prototype built to validate the effectiveness of the proposed concepts are presented.

Electric vehicle (EV) development has gone into an accelerated pace in recent years to address pressing concerns on energy security, the environment, and the sustainability of transportation. The future market success of EVs is still uncertain, but the current shift in the automotive industry is indicating a possible bright future for EVs. Because of its unique load characteristics, an extensive deployment of EVs will not only bring challenges to power systems, but will enable new opportunities as well. The objective of this work is to address the increased third harmonic currents expected with the introduction of EVs and to explore the potential of leveraging flexible EV charging to increase wind power production. Since EV chargers rely on a nonlinear power conversion process to obtain a controllable DC source from the utility AC supply, it is expected that these devices will aggravate third harmonic current issues. In fact, utility harmonic field data show that, even without EVs, distribution feeders are already experimenting elevated levels of third harmonic currents. To address present and future utility harmonic filtering needs, a practical third harmonic hybrid active filter for medium voltage (MV) applications is proposed. Its design is based on strict utility requirements of cost, reliability, and ease of system implementation. The operation and performance of the proposed filter is verified through simulations and two experimental setups, one tested at 7.2 kV. Furthermore, a system impact study of the proposed filter is performed using actual data for a typical residential/small commercial distribution feeder. Because vehicles remain stationary most of the time, EVs have the potential of being flexibly charged, providing a spectrum of opportunities for system operators. The recent increase in wind power penetration in the U.S. is raising concerns on how to accommodate this stochastic renewable energy resource in day-ahead scheduling operations. In this work, a detailed integrated day-ahead scheduling framework is developed to explore the impact of leveraging flexible EV charging to balance out the variability and uncertainty of wind power generation. It is determined that the full benefits of balancing wind power generation with flexible EV charging may not be achieved in congested power systems. A potential solution based on deploying power routers (PRs) to augment the flexibility of the transmission system is proposed. Simulation results are presented for a test system based on the IEEE 39-bus system.

This thesis advances the state of the art in modeling electric machines in electro-magnetic transient simulation programs, particularly in real-time digital simulators. A new tool, developed in this thesis, expands the application of real-time digital simulators to closed-loop testing of protection relays designed to protect synchronous machines during internal faults. To evaluate the inductances of synchronous machines, a winding function approach was developed in this thesis which is capable of taking into account both the actual distribution of windings and the shape of the pole-arc. Factors such as MMF drop in the iron and effects of slots are compensated by evaluating the effective permeance function of the machine using experimentally measured values of d-, q- and 0- axis inductances. In this winding function approach, the effects of magnetic saturation are also included by considering the actual distribution of magneto-motive force in each loading condition of the machine. The inductances of an experimental machine are evaluated using this approach and validated using the finite-element method and laboratory measurements. This thesis also proposes an embedded phase-domain approach for time-domain simulation of the machine model in electromagnetic transients programs. The approach significantly improves the numerical stability of the simulations. Special numerical techniques are introduced, which speed up the execution of the algorithm as needed for real-time simulation. The machine model is validated in healthy and faulted conditions using simulations and laboratory experiments. Effects of damper grid representation on simulating turn-to-turn faults are investigated. The capability of this new real-time synchronous machine model in closed-loop testing of synchronous machines ground- faults protection relays is clearly demonstrated.

Harmonics are created by non-linear devices connected to the power system. Power system harmonics are multiples of the fundamental power system frequency and these harmonic frequencies can create distorted voltages and currents. Distortion of voltages and currents can affect the power system adversely causing power quality problems. Therefore, estimation of harmonics is of high importance for efficiency of the power system network. The problem of harmonic loss evaluation is of growing importance for renewable power system industry by impacting the operating costs and the useful life of the system components. Non-linear devices such as power electronics converters can inject harmonics alternating currents (AC) in the electrical power system. The number of sensitive loads that require ideal sinusoidal supply voltage for their proper operation has been increasing. To maintain the quality limits proposed by standards to protect the sensitive loads, it is necessary to include some form of filtering device to the power system. Harmonics also increases overall reactive power demanded by equivalent load. Filters have been devised to achieve an optimal control strategy for harmonic alleviation problems. To achieve an acceptable distortion, increase the power quality and to reduce the harmonics hence several three phase filter banks are used and connected in parallel. In this thesis, high order harmonics cases have been suppressed by employing variants of Butterworth, Chebyshev and Cauer filters. MATLAB/SIMULINK wind farm model was used to generate and analyze the different harmonics magnitude and frequency. High voltage direct current (HVDC) lines for an electrical grid that is more than50km far away wind farm generation plant was investigated for harmonics. These HVDC lines are also used in offshore wind farm plant. Investigated three-phase harmonics filters are shunt elements that are used in power systems for decreasing voltage distortion and for correcting the power factor. Renewable energy sources are not the stable source of energy generation like wind, solar and tidal e.t.c. Though they are secondary sources of generation and hard to connect with electrical grid. In near future the technique is to use the wave digital filter (WDF) or circulator-tree wave digital filter (CTWDF) for the renewable energy application can be employed to mitigate the harmonics. These WDF and CTWDF can b eused in HVDC lines and smart grid applications. A preliminary analysis is conducted for such a study.

This thesis advances the state of the art in modeling electric machines in electro-magnetic transient simulation programs, particularly in real-time digital simulators. A new tool, developed in this thesis, expands the application of real-time digital simulators to closed-loop testing of protection relays designed to protect synchronous machines during internal faults. To evaluate the inductances of synchronous machines, a winding function approach was developed in this thesis which is capable of taking into account both the actual distribution of windings and the shape of the pole-arc. Factors such as MMF drop in the iron and effects of slots are compensated by evaluating the effective permeance function of the machine using experimentally measured values of d-, q- and 0- axis inductances. In this winding function approach, the effects of magnetic saturation are also included by considering the actual distribution of magneto-motive force in each loading condition of the machine. The inductances of an experimental machine are evaluated using this approach and validated using the finite-element method and laboratory measurements. This thesis also proposes an embedded phase-domain approach for time-domain simulation of the machine model in electromagnetic transients programs. The approach significantly improves the numerical stability of the simulations. Special numerical techniques are introduced, which speed up the execution of the algorithm as needed for real-time simulation. The machine model is validated in healthy and faulted conditions using simulations and laboratory experiments. Effects of damper grid representation on simulating turn-to-turn faults are investigated. The capability of this new real-time synchronous machine model in closed-loop testing of synchronous machines ground- faults protection relays is clearly demonstrated.

An AC system’s voltage unbalance by a fundamental frequency negative sequence system is usually the main cause for the emission of non-characteristic harmonics by current source converters as used in conventional HVDC systems. This emission takes place on both sides of each 12-pulse converter. On the DC side mainly a 2nd harmonic voltage appears driving a 2nd harmonic current. The magnitude of this harmonic current can exceed the magnitudes of the characteristic harmonics even if no low order resonance exists. Further non-characteristic harmonics generated by the converter under such unbalanced supply voltage conditions have frequencies with a frequency distance to the characteristic harmonics of 2 times the fundamental frequency. The main technical drawbacks are the unintended coupling between both AC systems and the risk of thyristor over-stresses by DC current discontinuities at low power transfer levels. On both AC sides the largest 2 non-characteristic current harmonics generated by a 12-pulse HVDC converter under unbalanced supply voltage conditions are a negative sequence system of the fundamental harmonic and a positive sequence system of the 3rd harmonic. Also on the AC sides further harmonics are emitted by the converter with a order number distance of 2 to the orders of the characteristic harmonics. However, in practical AC system operation special attention has to be paid to the 3rd harmonic distortion level, in particular when low order resonance appears between the system impedance and the impedance of the converter station AC filters. In order to avoid the above mentioned problems, large smoothing reactors and sometimes large blocking filters are installed on the DC side and the voltage distortion on the AC sides is reduced by AC filters. However, these filters require an expensive high component rating if they are tuned to the 2nd or 3rd harmonic respectively. The work shows that a modification of the valve firing can reduce the levels of the 2nd and 3rd harmonic without investment into additional primary equipment. Furthermore, this offers the chance to reduce the minimum power transfer level since also the risk of an intermittent DC current can be reduced. A corresponding algorithm and a control strategy are proposed. However, the calculation of an appropriate firing pattern requires a detailed modelling of the processes within the converters, especially the formation of the harmonics and the harmonic transfer between AC and DC sides. The work proposes a component vector model for the calculation of the harmonics. This model assumes that each harmonic consists of a first component representing the ideal conversion process, a 2nd component representing the impact of different commutation angles and in the case of the modified firing a 3rd component considering the impact of the intended non-equidistant firing. The work shows, that the harmonic component vectors resulting from voltage unbalance and from firing modulation can be treated separately and superimposed linearly. The calculation of the harmonic component vectors is performed applying the method of switching functions. For the consideration of the commutation and firing angle differences the modelling of switching functions based on differential impulses is proposed. However, especially an accurate representation of the above mentioned 2nd component vector requires a correct calculation of the commutation angles and their valve-specific differences. The investigations of this work have revealed that the conventional method of calculating the commutation angles – assuming an ideal smoothed DC current - may not produce results of sufficient accuracy. This is especially true in the case of a high ripple of the DC current, e.g. smoothed with a small smoothing reactor. A small smoothing reactor is typical for HVDC back-to-back applications. Therefore a new analytical method for the calculation of the commutation angles has been developed which in particular considers the typical pulse form of the DC current and additionally the impacts of the voltage unbalance and of the proposed modification of the firing on the ripple shape of the DC current. Moreover, as this analytical method requires the instantaneous values of the DC current at the instants of valve firing, a further analytical method for the calculation of these discrete current values has been developed. The equations are valid under the same conditions as the new ones for calculation of the commutation angles, i.e. resistive-inductive AC system fundamental frequency impedances, any degree of DC current smoothing between ideal smoothing and a ripple at the limit for current discontinuities. Symmetrical conditions, supply voltage unbalances and non-equidistant firing as proposed are applied. It is shown that, using this method, also the discrete values of the DC current at the end of the commutation intervals can be determined. In practice one of these discrete current values indicates the minimum value during one period of the fundamental frequency. This offers the chance for a more exact analytical determination of the limit for the appearance of DC current discontinuities. For typical parameters of a back-to-back installation the new methods and the new analytical equations have been compared with simulation results showing excellent correlation for typical voltage unbalances of not more than 1...2% and firing angle differences of not more than 2.5°. This verification is performed for the harmonics, the commutation angles and the discrete values of the DC current at the firing instants as well.