I detta examensarbete har en tillförlitlighetsmetod tagits fram som väger samman kostnadspåverkande faktorer för investeringar och underhåll i eldistributionssystem med kostnader förknippade med avbrott. Syftet med metoden är att möjliggöra för nätplanerare och projektörer att jämföra olika investeringsalternativ för att kunna kostnadseffektivisera investeringarna i eldistributionssystem. Med tillförlitlighetsanalys kan den förväntade avbrottstiden i systemets uttagspunkter beräknas. Kopplas en kostnad till den förväntade avbrottstiden i uttagspunkterna kan den förväntade avbrottskostnaden för distributionssystemet beräknas.   Syftet med simuleringarna har varit att undersöka om det går att minimera den totala kostnaden, investeringskostnader samt förväntade avbrottskostnad, genom slumpvisa investeringar i systemets ledningssektioner. Simuleringarna har genomförts på ett fiktivt mindre system. Systemet möjliggör analys av resultaten, men är ändå tillräckligt komplext för att påvisa problematik som kan uppstå. Simuleringarna visar att den framtagna metoden är robust och fungerar på önskat vis. Det visade sig också att simuleringstiden, även med ett stort antal slumpvisa investeringstillfällen, är kort vilket gör att modellen även kan användas för mer komplexa nät. / This MSc project develops a method that combines factors that affect the investment and maintenance costs in electrical power systems with costs associated with outage. This gives the system planner an opportunity to compare different alternatives for investments to receive a low total cost. Using system reliability analysis the outage time for the load points in the system can be decided. If the expected outage time is associated with a cost, it is possible to calculate the expected outage cost for the system.   The purpose of the simulations has been to examine if it is possible to minimize the total cost by doing randomized investments on power lines in the system. The simulations are carried out on a fictional, small system. Because of the specific design, it is easy to analyze results but also complex enough to show problems that can occur in the system. The simulations show that the model is robust and the expected results were accomplished. The simulations also showed that, even for a large amount of events, the time for the simulation is short. This shows that the model is suitable also for larger systems.

Stochastic analysis provides a rationale for the treatment of uncertainties, founded on the principles of probability theory and statistics, and is concerned with a quantifiable measure of the confidence or the reliability associated with any design process. In this thesis, a stochastic approach is employed in the design of flexible pavement structures, to facilitate the development of safe and reliable pavement structures. The important aspects that have been explored in sufficient detail include the system reliability and global sensitivity analysis, and the spatial and temporal uncertainties that pervade the life of pavements. Chapter 1 of the thesis provides an introduction to the stochastic modelling of flexible pavements and its significance in the present day. Highlighting the need for this study, this chapter also enumerates its objectives and presents an overview of the organization of the thesis. Chapter 2 provides a review of the existing literature of the design of flexible pavements and the approaches adopted to deal with the various sources of uncertainties in a probabilistic setting. The estimation of the uncertainties in fundamental pavement design inputs and their integration into the general performance prediction procedures has become a required component of the modern Mechanistic-Empirical pavement design methodology, which has been described in detail. This chapter also provides the scope of the thesis by identifying the areas of stochastic analysis that have received little attention in the flexible pavement design, which include the effect of spatial variability on the pavement structural responses and the techniques of global sensitivity analysis. Chapter 3 provides a detailed overview of the various methodologies adopted in this thesis to carry out the stochastic modelling of flexible pavements. The fundamental technique adopted for the analysis of reliability is the Monte Carlo Simulation (MCS), which relies upon a numerical/analytical model of the physical system, i.e. the pavement model and a probabilistic description of the design parameters represented by random variables or random fields. The high computational expense associated with the MCS, particularly in the case of random fields, is tackled by the use of meta-models based on the stochastic response surface methodology. The chapter outlines the steps followed to develop the meta-models in the form of Polynomial Chaos Equations (PCEs) and its extension to the Sparse PCE that can conveniently represent the spatial variability of the pavement fields. Chapter 4 deals with the probabilistic modelling of flexible pavements, where the design parameter and model uncertainties are quantified based on the available literature studies. The global sensitivity analysis, which aims to study the impact of the input uncertainty on the variation of a model output (critical pavement responses) through uncertainty propagation, is achieved by the construction of the Polynomial Chaos Equations (PCEs). To implement the global sensitivity analysis in a system reliability framework, a generalized approach based on Bayes’ theorem and the concept of entropy as a sensitivity measure, has been proposed in this chapter. Chapter 5 deals with the characterization of the spatial variability inherent in the pavement layer by employing random fields and analyzing the effect on the pavement responses. The discretization of the random field into a vector of random variables is achieved through the simple Midpoint Discretization and the efficient Expansion Optimal Linear Estimation method. Since the computational effort in stochastic problems is proportional to the number of random variables involved, it is desirable to use a small number of random variables to represent the random field. To achieve this, the principle of transformation of the original random variables into a set of uncorrelated random variables through an eigenvalue orthogonalization procedure is adopted. To further increase the computational efficiency of generating random fields for Monte Carlo Simulation, the variance reduction technique of Latin Hypercube Sampling and the meta-modelling technique using Sparse Polynomial Chaos Equations (SPCEs) are implemented. The primary focus of this chapter is to analyze the influence of the spatial variability of the pavement layer moduli, including its anisotropic characteristics on the pavement structural responses. Chapter 6 focuses on the time-dependent reliability of the pavement structures as they age in service, with due consideration given to degradation of strength with traffic loading. The study is concerned with the fatigue reliability and thereby only the decrease in the asphalt modulus with time is considered as a function of the accumulated damage due to repeated loading, whose uncertainty is determined by the uncertainties of material parameters and the traffic loading. The time-dependent model adopted in this chapter can be quite effortlessly embedded in the Mechanistic-Empirical design framework, and provides a tool to effectively schedule the maintenance of the pavement structure and ensure that the reliability level remains at the desired level for the entire design life of the structure. Chapter 7 summarizes the various studies reported in this thesis and highlights the important conclusions.

In most of the urban cities of developing countries piped water supply is intermittent and they receive water on alternate days for about few hours. The Unaccounted For Water (UFW) in these cities is very high due to aged infrastructure, poor management and operation of the system. In the cities of developing countries, supplied water is not able to meet the demand and there is huge gap between supply and demand of water. To meet the water demand people are depending on other sources of water like groundwater, rain water harvesting, waste water treatment, desalination etc. Huge quantity of groundwater is extracted without any account for the quantity of water used. The main challenge for water authorities is to meet the consumer demands at varying loading conditions. However, the present execution of decisions in the operational management of WDS is through manual control. The manual control of valve throttling and control of pump speed, reduces the efficiency and operation of WDS. In such cases, system modeling coupled with automated control can play a significant role in the appropriate execution and operation of the system. In the past few decades, there has been a major development in the field of modeling and analysing water distribution systems. Most of the people in Indian mega cities are facing water problems as they are not able to receive safe reliable drinking water. In rapidly growing cities, the water resources management has been a major concern for the Government. There is always a need to optimize the available water resources when the rate of demand constantly beats the rate of replenishments. Mathematical modeling of WDS has become an indispensible tool since the ages to model any type of WDS. Development of mathematical models of WDS is necessary to analyse the system behavior for a wide range of operating conditions. Using models, problems can be anticipated in proposed or existing systems, and solutions can be evaluated before time, money, and materials are invested in a real-world project. In the present study, we have developed a model of WDS of a typical city like Bangalore, India and analysed them for several scenarios and operating conditions. Bangalore WDS is modeled using EPANET. Before a network model is used for analysis purpose, it must be ensured that the model is predicting the behavior of the system with reasonable accuracy. The process of matching the parameters of the developed model and the field observed data is known as calibration. All WDS require calibration for effective modeling and simulation of the system. Demand and roughness are the most uncertain parameters and they are adjusted repeatedly to get the required head at nodes and flow in the pipes. The calibration parameters usually include pipe roughness, valve settings, pipe diameter and demand. Pipe roughness, valve settings and pipe diameter are associated with the flow conditions and the demands relate to the boundary conditions. For Bangalore WDS, the values of roughness coefficient and demand are available; and the values of valve settings are not available. Hence, this value is estimated during calibration process. Dynamic Inversion (DI) nonlinear controller with Proportional Integral Derivative (PID) features (DI-PID) is used for calibrating WDS for valve settings on the basis of observed flow and roughness coefficient. From the obtained results it is observed that, controllers are capable of achieving the target flow to all the GLRs with acceptable difference between the flow meter readings and the simulated flow. After calibrating any real WDS to the field observed data, it will be useful for water authorities if the consumer demands are met up to certain extent. This can be achieved by using the concept of equitable distribution of water to different consumers. In the urban cities of developing countries, often large quantities of water are supplied to only a few consumers, leading to inequitable water supply. It is a well known fact that quantity of water supplied from the source is not distributed equitably among the consumers. Aged pipelines pump failures, improper management of water resources are some of the main reasons for it. Equitable water to different consumers can be provided by operating the system in an efficient manner. Most of the urban cities receive water from the source to intermediate reservoirs and from these reservoirs water is supplied to consumers. Therefore, to achieve equitable water supply, these two supply levels have to be controlled using different concepts/ techniques. The water requirement of each of the reservoirs has to be calculated, which may depend on the number of consumers and consumer category. Each reservoir should receive its share of water to satisfy its consumer demand and also there must be provision to accommodate shortages, if any. The calibrated model of Bangalore WDS is used to achieve equitable water supply quantity to different zones of Bangalore city. The city has large undulating terrain among different zones which leads to unequal distribution of water. Dynamic Inversion (DI) nonlinear controller with Proportional Integral Derivative (PID) features (DI-PID) is used for valve throttling to achieve the target flows to different zones/reservoirs of the city at different levels. Equitable water distribution to different reservoirs, when a part of the source fails to supply water is also discussed in this thesis. From the obtained results it is observed that, controllers were responding in all the cases in different levels of targets for such a huge network. When there is change in supply pattern to achieve the equitable supply of water to different zones, the hydraulics of the WDS will change. Therefore, it is necessary to understand whether the system is able to handle these changes. The concept of reliability can be used to analyse the performance of WDS for wide range of operating conditions. Reliability analysis of a WDS for both normal and likely to occur situations will give a better quality of service to its consumers. Calculating both hydraulic and mechanical reliability is important as the chances of occurrence of both the failure scenarios are equal in a WDS. In the present study, a methodology is presented to model the nodal, system and total reliability for water supply networks by considering the hydraulic and mechanical failure scenarios. These two reliability measures together give the total reliability of the system. Analysing a real and complex WDS for the probable chances of occurrence of the failure scenarios; and then to anlyse the total reliability of the system is not reported in the literature and this analysis is carried out in the present study for Bangalore city WDS. The hydraulics of the system for all the operating conditions is analysed using EPANET. Hydraulic reliability is calculated by varying the uncertain independent parameters (demand, roughness and source water) and mechanical reliability is calculated by assuming system component failures. The system is analysed for both the reliability scenarios by considering different chances of failure that may occur in a real WDS; and hence the total reliability is calculated by making different combinations of hydraulic and mechanical failure scenarios. Sensitivity analysis for all the zones is also carried out to understand the behavior of different demand points for large fluctuation in hydraulics of the system. From the study, it is observed that, Hydraulic reliability decreases as the demand variation increases. But, as the roughness variation increases, there is no much change in the nodal or system reliability. Consumer demand or reliability of the WDS can be increased by saving the water lost in the system. This can be achieved by tracking the water parcel from the source till the consumer end, which will give an idea about the performance of different stages and zones in achieving the target flows. Huge quantity of water is lost in WDS and hence it is necessary to account for the water lost at different levels, hence the system can be managed in a better way. In most of the intermittent water supply systems demand is controlled by supply side; there is also a need to understand the demand variation at the consumer end which in turn affects the supply. Matching this varied supply-demand gap at various levels is challenging task. To get a better control of such problem, water balance (WB) equations need to be derived at various levels. When we derive these WB equations it should be emphasized that UFW is one of the major component of this equation. Given this back ground of the complex problem, for a typical city like Bangalore, an attempt is made to derive WB equations at various levels. In the present study, stage-wise and zone-wise WB is analysed for different months based on the flow meter readings. The conceptual model developed is calibrated, validated and also the performance of the model is analysed by giving a chance of error in the flow measurement. Based on all the above observations, stage-wise and zone-wise water supply weights are also calculated. From the study it is found that, there is no much loss of water in all the four stages of supply. Water loss is minimal of about 3 % till water reaches from source to GLRs. Water is transferred between the stages during some days of the month, may be due to shortage of water or due to unexpected demand. Huge quantity of water is lost in the distribution main which is of about 40 to 45% for all the moths which is analysed. This type of model will be extremely useful for water supply managers to manage their resources more efficiently and this study is discussed in detail as a part of this thesis. As mentioned above, huge quantity of groundwater is used in urban cities and the quantity of water extracted is not accounted. In the present study, zone wise and sub zone-wise piped water and ground water used in different parts of the cities is analysed with the help of available data. From the study it is observed that, the quantity of piped water supply and UFW is consistent for the time period analysed and the quantity of water withdrawn from the borewells are varying considerably depending on the yield of the borewlls in different zones. The main components of urban water supply are piped water, ground water, rainfall and runoff generated, UFW, waste water produced and other water quantities which may be minute. In future, to manage the water resources properly, integrated water management is necessary in city scale which will give an idea about the total water produced and the water utilized at the consumer end. Therefore, integrated water management concept is carried out in Hebbal region, (a small part of Bangalore) using the available data. From the analysis we noticed that, domestic water supplied to North sub zones are better when comparing to East sub zones. This type of total water balance can be studied in other parts of Bangalore, to understand the behavior of different water components and to make better decisions. The developed model, analysis and operating conditions of this study can be applied to other similar cities like Bangalore. This type of study may be useful to water authorities for better control of the resources, or in making better decisions and these types of models will act as decision support systems.