<p>Moving boundary problems such as solidification, crack propagation, multi-body contact or shape optimal design represent an important class of engineering problems. Common to these problems are one or more moving interfaces or boundaries. One of the main challenges associated with boundary evolution is the difficulty that arises when the topology of the geometry changes. Other geometric issues such as distance to the boundary, projected point on the boundary and intersection between surfaces are also important and need to be efficiently solved. In general, the present thesis is concerned with the geometric arrangement and behavioral analysis of evolving parametric boundaries immersed in a domain. </p> <p>The first problem addressed in this thesis is the packing of ellipsoidal fillers in a regular domain and to estimate their effective physical behavior. Particle packing problem arises when one generates simulated microstructures of particulate composites. Such particulate composites used as thermal interface materials (TIMs) motivates this work. The collision detection and distance calculation between ellipsoids is much more difficult than other regular shapes such as spheres or polyhedra.  While many existing methods address the spherical packing problems, few appear to achieve volume loading exceeding 60%. The packing of ellipsoidal particles is even more difficult than that of spherical particles due to the need to detect contact between the particles. In this thesis, an efficient and robust ultra-packing algorithm termed Modified Drop-Fall-Shake is developed. The algorithm is used to simulate the real mixing process when manufacturing TIMs with hundreds of thousands ellipsoidal particles. The effective thermal conductivity of the particulate system is evaluated using an algorithm based on Random Network Model. </p> <p><br></p> <p>In problems where general free-form parametric surfaces (as opposed to the ellipsoidal fillers) need to be evolved inside a regular domain, the geometric distance from a point in the domain to the boundary is necessary to determine the influence of the moving boundary on the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge,  an algebraic procedure is presented here to find the level sets of rational parametric surfaces commonly utilized by commercial CAD systems. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bezier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain can also be used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G₀ continuity---problems where the Newton--Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively. </p> <p><br></p> <p>Next, ALS is also extended for boundary representation (B-rep) models that are popularly used in CAD systems for modeling solids. B-rep model generally contains multiple NURBS patches due to the trimming feature used to construct such models, and as a result are not ``watertight" or mathematically compatible at patch edges. A time consuming geometry clean-up procedure is needed to preprocess geometry prior to finite element mesh generation using a B-rep model, which can take up to 70% of total analysis time according to literature. To avoid the need to clean up geometry and directly provide link between CAD and CAE integration,  signed algebraic level sets using novel inner/outer bounding box strategy is proposed for point classification of B-rep model. Several geometric examples are demonstrated, showing that this technique naturally models single patch NURBS geometry as well, and can deal with multiple patches involving planar trimming feature and Boolean operation. During the investigation of algebraic level sets, a complex self-intersection problem is also reported, especially for three-dimensional surface. The self-intersection may occur within an interval of interest during implicitization of a curve or surface since the implicitized curve or surface is not trimmed and extends to infinity. Although there is no robust and universal solution the problem, two potential solutions are provided and discussed in this thesis.</p> <p><br></p> <p>In order to improve the computational efficiency of analysis in immersed boundary problems, an efficient local refinement technique for both mesh and quadrature  using the kd-tree data structure is further proposed. The kd-tree sub-division is theoretically proved to be more efficient against traditional quad-/oct-tree subdivision methods. In addition, an efficient local refinement strategy based on signed algebraic level sets is proposed to divide the cells. The efficiency of kd-tree based mesh refinement and adaptive quadrature is later shown through numerical examples comparing with oct-tree subdivision, revealing significant reduction of degrees of freedom and quadrature points.</p> <p><br></p> <p>Towards analysis of moving boundaries problems, an explicit interface tracking method termed enriched isogeometric analysis (EIGA) is adopted in this thesis. EIGA utilizes NURBS shape function for both geometry representation and field approximation. The behavior field is modeled by a weighted blending of the underlying domain approximation and enriching field, allowing high order continuity naturally. Since interface is explicitly represented, EIGA provides direct geometric information such as normals and curvatures. In addition, the blending procedure ensures strong enforced boundary conditions. An important moving boundary problem, namely, reaction-diffusion problem, is investigated using EIGA. In reaction-diffusion problems, the phase interfaces evolve due to chemical reaction and diffusion under multi-physics driven forces, such as mechanical, electrical, thermal, etc. Typical failure phenomenon due to reaction-diffusion problems include void formation and intermetallic compound (IMC) growth. EIGA is applied to study factors and behavior patterns in these failure phenomenon, including void size, current direction, current density, etc. A full joint simulation is also conducted to study the degradation of solder joint under thermal aging and electromigration. </p>

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Background: Several previous studies have reported that dyslipidemia is associated with the risk of hypertension, but these studies are mainly conducted in European and US populations, with a very few studies in the Asian population. Moreover, the effects of atherosclerotic indices, including atherogenic coefficient (AC) and atherogenic risk of plasma (AIP), on hypertension in Asians have not been well described so far. Methods: From 2010 to 2016, altogether 211,833 Chinese adults were ultimately recruited at the health centers in 11 Chinese cities (including Shanghai, Beijing, Nanjing, Suzhou, Shenzhen, Changzhou, Chengdu, Guangzhou, Hefei, Wuhan, and Nantong). Differences in continuous variables between the two groups were analyzed by the Mann–Whitney test, while those in categorical variables were examined by the Chi-squared test. Logistic regression was applied to evaluate the association between lipid profiles and the risk of hypertension. The predictive values of AC and AIP for the incidence of hypertension were analyzed using the area under the receiver operating characteristic (ROC) curve. Meanwhile, Bayesian network (BN) models were performed to further analyze the associations between the different covariates and the incidence of hypertension. Results: A total of 117,056 participants were included in the final analysis. There were significant differences in baseline characteristics between normotension and hypertension groups (p < 0.001). In multivariate logistic regression, the risk of hypertension increased by 0.2% (1.002 [1.001–1.003]), 0.2% (1.002 [1.001–1.003]), and 0.2% (1.002 [1.001–1.003]) per 1 mg/dl increase in total cholesterol (TC), low-density lipoprotein (LDL), and non-high-density lipoprotein cholesterol (non-HDL-c), respectively. However, after adjusting for body mass index (BMI), an increase in HDL level was associated with a higher risk of hypertension (p for a trend < 0.001), and the risk of hypertension increased by 0.6% per 1 mg/dl increase in HDL-c (1.006 [1.003–1.008]). In women, AC had the highest predictive value for the incidence of hypertension with an area under the curve (AUC) of 0.667 [95% confidence interval (CI): 0.659–0.674]. BN models suggested that TC and LDL were more closely related to the incidence of hypertension. Conclusions: Overall, lipid profiles were significantly abnormal in the hypertensive population than in the normotensive population. TC and LDL were strongly associated with the incidence of hypertension. TC, LDL, and non-HDL-c levels show a positive association, HDL-c shows a negative association, while TG is not significantly associated with the risk of hypertension. After adjusting for BMI, HDL-c turns out to be positively associated with the risk of hypertension. In addition, AC has a good predictive value for the incidence of hypertension in women.

[pt] A formação e o acúmulo de condensado em reservatórios de gás retrógrado, especialmente na vizinhança de poços de produção, obstruem parcialmente o fluxo de gás e afetam negativamente a composição dos fluidos produzidos. Entretanto, a previsão de bloqueio por condensado é comumente imprecisa, visto que experimentos raramente reproduzem as condições extremas e composições complexas dos fluidos dos reservatórios, enquanto a maioria dos modelos em escala de poros simplificam demasiadamente os fenômenos físicos associados à transição de fases entre gás e condensado. Para corrigir essa lacuna, um modelo de rede de poros isotérmico composicional e totalmente implícito é apresentado. As redes de poros propostas consistem em estruturas tridimensionais de capilares constritos circulares. Modos de condensação e padrões de escoamento são atrubuídos aos capilares de acordo com a molhabilidade do meio, as saturações locais e a influência de forças viscosas e capilares. Nos nós da rede, pressão e conteúdo molar são determinados através da solução acoplada de equações de balanço molar e consistênc ia de volumes. Concomitantemente, um cálculo de flash à pressão e à temperatura constantes, baseado na equação de estado de Peng e Robinson, é realizado em cada nó, atualizando as saturações e composições das fases. Para a validação do modelo proposto, análises de escoamento foram executadas baseadas em experimentos de escoamento em testemunho reportados na literatura, usando composição dos fluidos e condições de escoamento correspondentes, e geometria do meio poroso aproximada. Curvas de permeabilidade relativa medidas nos experimentos e previstas pelo modelo mostraram boa concordância quantitativa, para dois valores de tensão interfacial e três valores de velocidade de escoamento de gás. Após a validação, o modelo foi usado para avaliar alteração de molhabilidade e injeção de gás como possíveis métodos de recuperação avançada para reservatórios de gás retrógrado. Os resultados exibiram tendências similares àquelas observadas em experimentos de escoamento em testemunhos, e condições ótimas para melhoramento do escoamento foram identificadas. / [en] Liquid dropout and accumulation in gas-condensate reservoirs, especially in the near wellbore region, hinder gas flow and affect negatively the produced fluid composition. Yet, condensate banking forecasting is commonly inaccurate, as experiments seldom reproduce reservoir extreme conditions and complex fluid composition, while most pore-scale models oversimplify the physical phenomena associated with phase transitions between gas and condensate. To address this gap, a fully implicit isothermal compositional pore-network model for gas and condensate flow is presented. The proposed pore-networks consist of 3D structures of constricted circular capillaries. Condensation modes and flow patterns are attributed to the capillaries according to the medium s wettability, local saturations and influence of viscous and capillary forces. At the network nodes, pressure and molar contents are determined via the coupled solution of molar balance and volume consistency equations. Concomitantly, a PT-flash based on the Peng-Robinson equation of state is performed for each node, updating the local phases saturations and compositions. For the proposed model validation, flow analyses were carried out based on coreflooding experiments reported in the literature, with matching fluid composition and flow conditions, and approximated pore-space geometry. Predicted and measured relative permeability curves showed good quantitative agreement, for two values of interfacial tension and three values of gas flow velocity. Following the validation, the model was used to evaluate wettability alteration and gas injection as prospect enhanced recovery methods for gas-condensate reservoirs. Results exhibited similar trends observed in coreflooding experiments and conditions for optimal flow enhancement were identified.

[pt] As coquinas carbonáticas são rochas sedimentares constituídas principalmente por conchas e detritos. Elas apresentam diferentes características de textura que são afetadas pela sedimentação e diagênese, influenciando no arranjo de poros, e consequentemente, na permeabilidade. É necessário o entendimento do efeito da variação das tensões efetivas nessas formações rochosas e como problemas de dano mecânico irão afetar na produção e injeção de poços de sistemas de produção de petróleo e gás. Nesta tese, amostras de coquinas sintéticas foram confeccionadas e, com o auxílio de técnicas de microtomografia de raios-x (microCT), modelagem de rede de poros (PNM) e ensaios in situ, a influência da microestrutura dessas rochas na variação de suas propriedades devido a tensões de compressão confinada foi avaliada. O uso de microCT permitiu a análise da estrutura interna das amostras para a discriminação de poros. Uma célula para ensaios mecânicos, acoplável ao tomógrafo e capaz de aplicar tensões de compressão confinada, auxiliou no estudo da alteração da microestrutura das coquinas sob tensão. As amostras foram tomografadas em 3 condições: sem carregamento, com carregamento e após despressurização. PNM foi utilizada para simulação de transporte bifásico nas coquinas, além de relacionar dados como número de conectividade de poros, tortuosidade, permeabilidade, porosidade, diâmetro dos poros e capilares. Os resultados obtidos demonstram que a aplicação de tensões de compressão confinada resulta na redução da porosidade e permeabilidade das coquinas. A redução da porosidade total do estágio final em relação às condições iniciais das amostras foi de 87,5 por cento, 70,0 por cento e 85,9 por cento para CP1, CP2 e CP3, respectivamente. Já a permeabilidade absoluta em condições iniciais apresentou valores de 33,10, 11,60 e 22,90 mD para CP1, CP2 e CP3, respectivamente, reduzindo a zero com o aumento dos estágios de pressão, impossibilitando as coquinas de permear fluidos em seus interiores. Em relação aos estudos da literatura, a fabricação de coquinas sintéticas mostrou-se eficiente em termos de reprodutibilidade para coquinas carbonáticas autênticas, apresentando propriedades geomecânicas semelhantes. / [en] Carbonate coquinas are sedimentary rocks composed mainly of shells and fragments. They present different texture characteristics that are affected by sedimentation and diagenesis, influencing the pore arrangement and, consequently, the permeability. It is necessary to understand the effect of variations in effective stresses on these rock formations and how mechanical damage problems will affect the production and injection of wells in oil and gas production systems. In this thesis, samples of synthetic coquinas were made, and with the help of x-ray microtomography, pore network modeling techniques, and in situ tests, the influence of the microstructure of these rocks on the variation of their properties due to confined compression stresses was evaluated. The use of x-ray microtomography allowed the analysis of the samples internal structure for the discrimination of pores. Using a cell for mechanical tests, coupled with the tomograph and able to apply confined compressive stresses, helped study changes in the coquinas’ microstructure under tension. The samples were scanned under three conditions: without loading, with stress loading, and after depressurization. Pore network modeling was used to simulate single-phase transport in the coquinas, as well as for relating data such as pore connectivity number, tortuosity, permeability, porosity, pore, and capillary diameter. The results obtained demonstrate that the application of confined compressive stresses results in the reduction of porosity and permeability of the coquinas. The reduction of total porosity in the final stage relative to the initial conditions of the samples was 87.5 percent, 70.0 percent, and 85.9 percent for CP1, CP2, and CP3, respectively. Meanwhile, the absolute permeability under initial conditions presented values of 33.10, 11.60, and 22.90 mD for CP1, CP2, and CP3, respectively, which reduced to zero with the increase of pressure stages, making it impossible for the coquinas to permeate fluids within their interiors. Compared to literature studies, the production of synthetic coquinas proved to be efficient in terms of reproducibility for authentic carbonate coquinas, presenting similar geomechanical properties.

End-to-end delay measurement is deemed crucial for network models at all times as it acts as a pivotal metric of the model’s effectiveness, assists in delineating its performance ceiling, and stimulates further refinement and enhancement. This premise holds true for 5G Core Network (5GC) models as well. Commercial 5G models, with their intricate topological structures and requirement for reduced latencies, necessitate an effective model to anticipate each server’s current latency and load levels. Consequently, the introduction of a model for estimating the present latency and load levels of each network element server would be advantageous. The central content of this article is to record and analyze the packet data and CPU load data of network functions running at different user counts as operational data, with the data from each successful operation of a service used as model data for analyzing the relationship between latency and CPU load. Particular emphasis is placed on the end-to-end latency of the PDU session establishment scenario on two core functions - the Access and Mobility Management Function (AMF) and the Session Management Function (SMF). Through this methodology, a more accurate model has been developed to review the latency of servers and nodes when used by up to 650, 000 end users. This approach has provided new insights for network level testing, paving the way for a comprehensive understanding of network performance under various conditions. These conditions include strategies such as "sluggish start" and "delayed TCP confirmation" for flow control, or overload situations where the load of network functions exceeds 80%. It also identifies the optimal performance range. / Latensmätningar för slutanvändare anses vara viktiga för nätverksmodeller eftersom de fungerar som en måttstock för modellens effektivitet, hjälper till att definiera dess prestandatak samt bidrar till vidare förfining och förbättring. Detta antagande gäller även för 5G kärnnätverk (5GC). Kommersiella 5G-nätverk med sin komplexa topologi och krav på låg latens, kräver en effektiv modell för att prediktera varje servers aktuella last och latensbidrag. Följdaktligen behövs en modell som beskriver den aktuella latensen och dess beroende till lastnivå hos respektive nätverkselement. Arbetet består i att samla in och analysera paketdata och CPU-last för nätverksfunktioner i drift med olika antal slutanvändare. Fokus ligger på tjänster som används som modelldata för att analysera förhållandet mellan latens och CPU-last. Särskilt fokus läggs på latensen för slutanvändarna vid PDU session-etablering för två kärnfunktioner – Åtkomst- och mobilitetshanteringsfunktionen (AMF) samt Sessionshanteringsfunktionen (SMF). Genom denna metodik har en mer exakt modell tagits fram för att granska latensen för servrar och noder vid användning av upp till 650 000 slutanvändare. Detta tillvägagångssätt har givit nya insikter för nätverksnivåtestningen, vilket banar väg för en omfattande förståelse för nätverprestanda under olika förhållanden. Dessa förhållanden inkluderar strategier som ”trög start” och ”fördröjd TCP bekräftelse” för flödeskontroll, eller överlastsituationer där lasten hos nätverksfunktionerna överstiger 80%. Det identifierar också det optimala prestandaområdet.

One of the main features of the third generation (3G) mobile networks is their capability to provide different classes of services; especially multimedia and real-time services in addition to the traditional telephony and data services. These new services, however, will require higher Quality of Service (QoS) constraints on the network mainly regarding delay, delay variation and packet loss. Additionally, the overall traffic profile in both the air interface and inside the network will be rather different than used to be in today's mobile networks. Therefore, providing QoS for the new services will require more than what a call admission control algorithm can achieve at the border of the network, but also continuous buffer control in both the wireless and the fixed part of the network to ensure that higher priority traffic is treated in the proper way. This paper proposes and analytically evaluates a buffer management scheme that is based on multi-level priority and Complete Buffer Sharing (CBS) policy for all buffers at the border and inside the wireless network. The analytical model is based on the G/G/1/N censored queue with single server and R (R¿2) priority classes under the Head of Line (HoL) service rule for the CBS scheme. The traffic is modelled using the Generalised Exponential distribution. The paper presents an analytical solution based on the approximation using the Maximum Entropy (ME) principle. The numerical results show the capability of the buffer management scheme to provide higher QoS for the higher priority service classes.

Fernkälte bietet das Potenzial, wirtschaftlich und ökologisch vorteilhaft zur Deckung des stetig zunehmenden Klimakältebedarfs beizutragen. Im Rahmen dieser Arbeit wurde ein dynamisches thermohydraulisches Netzmodell „ISENA“ entwickelt, mit dem während der Planung und des Betriebs von Fernkältesystemen auftretende Fragen, beispielsweise in Bezug auf Wirtschaftlichkeit und Energieeffizienz, beantwortet werden können. Das Netzmodell setzt sich aus einem quasistationären hydraulischen Modell und einem instationären thermischen Modell zusammen, das auf der Verfolgung von Wasserpfropfen durch das gesamte Netz basiert (Lagrange-Ansatz). Mit diesem Modellierungsansatz können numerische Fehler sowie Bilanzungenauigkeiten vermieden werden, sodass sich eine höhere Ergebnisgüte im Vergleich zu bisher bekannten Netzmodellen erreichen lässt. Ebenfalls neu entwickelt wurde das Teilmodell zur Abbildung der Wärmeströme über die Wände unterirdischer Rohrpaare (Kälteverluste und -gewinne). Dieses Modell erlaubt die Bestimmung der instationären Rohrwand-Wärmeströme für wärmegedämmte unterirdische Rohrpaare, Rohrpaare mit gedämmtem Vor- und ungedämmtem Rücklauf sowie ungedämmte Rohrpaare. Anhand von Validierungs- und Verifikationsrechnungen wird gezeigt, dass ISENA verlässliche Ergebnisse liefert und für die praktische Anwendung geeignet ist. Abschließende Beispielrechnungen geben einen Einblick in die Untersuchungsmöglichkeiten, die das neue Modell bietet – unter anderem im Hinblick auf den Vergleich von Pumpenregelungsvarianten, den Einfluss von Rohrdämmung und Erdreicheigenschaften auf Kälteverluste und -gewinne sowie die Einbindung von Hochtemperatur-Kälteverbrauchern in den Netzrücklauf. / District cooling can provide economic and ecological benefits while supplying the increasing cooling demand for air conditioning. In the present thesis, a dynamic thermo-hydraulic model “ISENA” is presented which may be used in order to answer questions arising during design and operation of district cooling networks—e. g., that are related to economic and energy efficiency. The network model consists of a quasi-static hydraulic module and a transient thermal module being based on the tracking of water segments through the entire network (Lagrangian method). With this approach, numerical errors and inaccuracies in the balance of conserved quantities could be avoided, which eventually leads to a better reliability of the results as compared to that obtained from other network models. Additionally, a new sub-model has been developed for predicting the transient heat flux through the walls of buried pipes in order to model thermal gains and losses. This model covers un-insulated, insulated and combinations of insulated as well as un-insulated pipes. Calculations performed for the purpose of validation and verification are presented in order to demonstrate that ISENA provides reliable results and hence is suitable for practical applications. Finally, example simulations show the various possibilities provided by the new model—for example, concerning the comparison of different strategies for pump control, the influence of pipe insulation and soil properties on thermal gains and losses as well as the connection of buildings equipped with high temperature cooling systems to the return line of the network.

As redes secundárias de distribuição são item de extrema importância no processo de levar a energia produzida de uma fonte até o consumidor final. Dentro de uma Empresa Concessionária de Distribuição de Energia Elétrica o setor de planejamento, juntamente com o grupo técnico, deve realizar o planejamento dos investimentos necessários nos níveis de alta, média e baixa tensão, sendo que para a baixa tensão (redes secundárias de distribuição), normalmente utilizam-se os dados históricos e a média de investimentos dos últimos anos (de três a cinco últimos anos). Em geral não são encontrados nestas Empresas os cadastramentos das redes secundárias de distribuição ou então, quando estes existem, muitas vezes não são confiáveis. Neste ponto, esta dissertação de mestrado apresentará, aplicará e discutirá os resultados de uma metodologia de geração de redes secundárias de distribuição a partir dos dados das redes primárias de distribuição (média tensão) e dos ativos da Empresa, tendo como foco os investimentos necessários para estas dentro do planejamento agregado de investimentos. O trabalho proposto visa criar condições para que as Empresas enquadradas na situação anterior possam realizar o seu planejamento de baixa tensão valendo-se da aplicação de uma metodologia capaz de gerar tais redes. Uma vez criadas, tais redes serão inseridas dentro de um software de planejamento agregado de investimentos que fornecerá o valor total das aplicações necessárias para os próximos dez anos (planejamento decenal – longo prazo). Ao final do processo, poderá ser observado que os resultados obtidos são muito próximos dos valores esperados para as redes secundárias reais e, com isto, a dissertação discutirá como a metodologia foi implementada, os detalhes da aplicação e, até mesmo, todo o processo de planejamento para três Empresas Concessionárias de Distribuição de Energia Elétrica. Já de posse dos resultados, poderá afirmar-se que o processo produziu valores de muito boa aderência e, assim, os investimentos obtidos para as redes secundárias poderão ser aceitos pelo grupo de planejamento da Empresa. Como conseqüência dos valores de investimentos obtidos para os dez anos, poderão ser observados ainda, guardadas algumas considerações especiais, os valores dos custos marginais de baixa tensão, item de grande importância no planejamento de redes de distribuição. / The low voltage distribution networks are an item of extreme importance in the process of delivering the produced energy from a plant (supply) to the final consumer. In an Electric Power Distribution Company the planning group (sector), together with the technical group, must accomplish the planning of the necessary investments in the high voltage, medium voltage and low voltage levels, and for low voltage distribution networks, usually the historical data and the average of investments of the last years are used (from three to last five years). In general the databases of low voltage networks are not found in these Companies or then, when these exist, a lot of times they are not reliable. For this, this article will show, apply and discuss the results of a methodology for generating low voltage distribution networks starting from the database of the medium voltage distribution networks and of the assets of the Company, having as focus the necessary investments for these networks in planning of investments. The proposed work wishes to create conditions so that the Companies that are in the previous situation can accomplish their planning on low voltage distribution networks using a methodology capable to generate such nets. Once created, such nets will be inserted in a software of planning of investments that will supply the total amount of investments for them for the next ten years (long period planning). In the end of the process, it will be observed that the obtained results are very close to the expected values for the real low voltage distribution networks and, with this, the article will discuss how the methodology was implemented, the details of the application and, even, the whole planning process for three Electric Power Distribution Companies. With the results, it will be affirmed that the process produces values of very good adherence and, for this, the investments obtained for low voltage distribution networks can be accepted by the planning group in the Company. As a consequence of the investments values obtained for the ten years, it will be able to be observed, kept some special considerations, the values of the marginal costs of low voltage, item of great importance in the planning of distribution networks.