In this research effort, we study three exterior penalty function approaches for solving linear programming problems. These methods are an active set l2 penalty approach (ASL2), an inequality-equality based l2 penalty approach (IEL2), and an augmented Lagrangian approach (ALAG). Particular effective variants are presented for each method, along with comments and experience on alternative algorithmic strategies that were empirically investigated. Our motivation is to examine the relative performance of these different approaches based on the basic l2 penalty function in order to provide insights into the viability of these methods for solving linear programs. To test the performance of these algorithms, a set of randomly generated problems as well as a set of NETLIB test problems from the public domain are used. By way of providing a benchmark for comparisons, we also solve the test problems using CPLEX 6.0, an advanced simplex implementation. While a particular variant (ALAG2) of ALAG performed the best for randomly generated test problems, ASL2 performed the best for the NETLIB test problems. Moreover, for test problems having only equality constraints, IEL2, and ASL2 (which is a finer-tuned version of IEL2 in this case) were comparable and yielded a second-best performance in comparison with ALAG2. Furthermore, a set of problems with relatively higher density parameter values, as well as a set of low-density problems were used to determine the effect of density on the relative performances of these methods. This experiment revealed that for linear programs with a high density parameter, ASL2 is the best alternative among the tested algorithms; whereas, for low-density problems ALAG2 is the fastest method. Moreover, although our implementation was rudimentary in comparison with CPLEX, all of the tested methods attained a final solution faster than CPLEX for the set of large-scale low-density problems, sometimes as fast as requiring only 16-23% of the effort consumed by CPLEX. Average rank tests based on the computational results obtained are performed using two different statistics, that assess the speed of convergence and the quality or accuracy of the solution, in order to determine the relative effectiveness of the algorithms and to validate our conclusions. Overall, the results provide insights into selecting algorithmic strategies based on problem structure and indicate that while this class of methods is viable for computing near optimal solutions, more research is needed to design robust and competitive exterior point methods for solving linear programming problems. However, the use of the proposed variant of the augmented Lagrangian method to solve large-scale low-density linear programs is promising and should be explored more extensively. / Master of Science

Yes / This paper used different risk management indicators applied to the investment optimization performed by consumers in Distributed Generation (DG). The objective function is the total cost incurred by the consumer including the energy and capacity payments, the savings, and the revenues from the installation of DG, alongside the operation and maintenance (O&M) and investment costs. Probability density function (PDF) was used to model the price volatility in the long-term. The mathematical model uses a two-stage stochastic approach: investment and operational stages. The investment decisions are included in the first stage and which do not change with the scenarios of the uncertainty. The operation variables are in the second stage and, therefore, take different values with every realization. Three risk indicators were used to assess the uncertainty risk: Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and Expected Value (EV). The results showed the importance of migration from deterministic models to stochastic ones and, most importantly, the understanding of the ramifications of every risk indicator.

This study used discourse analysis (Gee, 2011; 2005; 1999) in order to explore a socio-situated view of how teachers created learning opportunities for the participation of immigrant learners when learning linear programming in a Grade 11 mathematics classroom in South Africa. The aim was to explore that which mathematics teachers do in classrooms with immigrant learners that they will not do if there were no immigrants. A discourse analysis approach was used in order to view the opportunities created through language use not as a tool for communication only but also as a tool for building reality. The study reported in this thesis was conducted in three different settings which are in; urban, township and rural environments. The urban environment focuses on immigrant learners who were born in the Democratic Republic of Congo (DRC), and started schooling there, in the township and rural environment it focuses on immigrant learners born in South Africa with parents born in the Republic of Mozambique or Angola. Three different mathematics classrooms were observed in their natural environment during lessons focusing on linear programming. Data was collected through a learner questionnaire issued before lesson observations. The aim of the learner questionnaire was to understand the language background of the learners in the mathematics classrooms selected for the study. The second method included lesson observation for at most five consecutive days at each setting. It involved observing teachers and immigrant learners during teaching sessions of linear programming activities. The activities included reading, writing, speaking and participating in mathematical activities. These activities were then analysed to understand how teachers created learning opportunities for the immigrant learners. The study contextualised the results from lesson observations by conducting clinical interviews with three immigrant learners, one from each site, to provide insights into the explanations on immigrant learners approaches when solving a linear programming task. The main conclusion in this study is that immigrant learners were successful in linear programming when teachers’ created learning opportunities by using code switching to support them. The main contribution of this study is that it focuses on multilingual mathematics classrooms of immigrant learners in South Africa – a context that has not yet been researched in South African vi mathematics education. Exploring language practices in multilingual mathematics classrooms of immigrant learners provides a different gaze into teaching and learning mathematics in multilingual classrooms in South Africa. Equally important is the extent to which immigrant learners are distinct to multilingual learners in the teaching and learning of linear programming. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education)

Neste trabalho é apresentada uma ferramenta de otimização para análise de problemas de alocação de água em bacias hidrográficas utilizando técnicas de programação linear e linear por partes, integradas a um modelo de amortecimentos de ondas em canais. A otimização é feita de forma global, com uso de softwares de programação linear baseados nos métodos de pontos interiores. A metodologia de uso do sistema consiste em se obter uma solução ?ótima? para situações de disponibilidade de água insuficiente a todos os usos conflitantes na bacia. A ferramenta está sendo acoplada e incorporada ao AcquaNet, um Sistema de Suporte a Decisões (SSD) para análise de sistemas de recursos hídricos, que utiliza um algoritmo de rede de fluxo afim de otimizar a alocação de água. A formulação utilizando programação linear permite a análise global do sistema e por isso, espera-se melhor aproveitamento da água disponível, seja no menor déficit de atendimento às demandas ou maior armazenamento nos reservatórios. A programação linear com utilização de métodos de pontos interiores é atualmente uma técnica bastante conhecida e bem desenvolvida. Existem vários pacotes computacionais gratuitos com implementações eficientes dos métodos de pontos interiores que motivaram sua utilização neste trabalho. / This work presents an optimization tool for analyzing the problems of water allocation in watersheds by utilizing techniques of linear and piecewise linear programming integrated to a pattern of stream flow routing. The optimization is done in a global way with the usage of linear programming packages based upon the Internal Point Methods. The methodology of the usage consists in the acquirement of an optimal solution for situation of insufficient water availability for all conflicting consumptions from the watershed. The tool is being attached and incorporated to AcquaNet, which is a decision support system (DSS) for analysis of water resources systems that utilizes a network flow algorithm, with the purpose of optimizing the water allocation. The formulation that uses the linear programming leads to the analysis of the system as a whole and for this reason it is expected a better usage of the available water with a lower deficit in the supply or a greater storage in the reservoirs. Linear Programming with Internal Point Methods is nowadays a well known and very well developed technique. There are several computational packages with efficient implementations of the Internal Points Methods freely available, and that, has brought great motivation in its usage in the present work.

Les travaux présentes dans cette thèse s'inscrivent dans le cadre des méthodes d'évacuation des populations. Ils visent à étudier les capacités et modéliser le problème d'évacuation (blessés, sinistrés, enfants, personnes agées, etc.) dans une situation de crise (attentats terroristes, catastrophes naturelles, etc.) et développer des méthodes d'aide à la décision tout en proposant une meilleure planification et des plans optimaux d'évacuation des populations de la zone de crise vers les centres hospitaliers.Notre travail consiste à résoudre le problème d'évacuation de blessés dans des zones de crise avec une nouvelle vision qui consiste à optimiser le temps de transport et par conséquent sauver le maximum des personnes touchées par cette crise d'une façon dynamique, efficace et rapide pour minimiser la perte humaine. / The work presented in this thesis is part of human evacuation methods. It aims to study the capacities, model the evacuation problem (wounded, victims, children, elderly, etc.) in a crisis situation (terrorist attacks, natural disasters, etc.) and to develops methods for decision making while proposing better planning and optimal evacuation plans for populations from the crisis zone to hospitals.Our job is to solve the wounded evacuation problem in crisis zone with a new vision that optimizes the transport time and thus saving the maximum of causalities in a dynamic, efficient and fast way in order to minimize human loss.

This study used discourse analysis (Gee, 2011; 2005; 1999) in order to explore a socio-situated view of how teachers created learning opportunities for the participation of immigrant learners when learning linear programming in a Grade 11 mathematics classroom in South Africa. The aim was to explore that which mathematics teachers do in classrooms with immigrant learners that they will not do if there were no immigrants. A discourse analysis approach was used in order to view the opportunities created through language use not as a tool for communication only but also as a tool for building reality. The study reported in this thesis was conducted in three different settings which are in; urban, township and rural environments. The urban environment focuses on immigrant learners who were born in the Democratic Republic of Congo (DRC), and started schooling there, in the township and rural environment it focuses on immigrant learners born in South Africa with parents born in the Republic of Mozambique or Angola. Three different mathematics classrooms were observed in their natural environment during lessons focusing on linear programming. Data was collected through a learner questionnaire issued before lesson observations. The aim of the learner questionnaire was to understand the language background of the learners in the mathematics classrooms selected for the study. The second method included lesson observation for at most five consecutive days at each setting. It involved observing teachers and immigrant learners during teaching sessions of linear programming activities. The activities included reading, writing, speaking and participating in mathematical activities. These activities were then analysed to understand how teachers created learning opportunities for the immigrant learners. The study contextualised the results from lesson observations by conducting clinical interviews with three immigrant learners, one from each site, to provide insights into the explanations on immigrant learners approaches when solving a linear programming task. The main conclusion in this study is that immigrant learners were successful in linear programming when teachers’ created learning opportunities by using code switching to support them. The main contribution of this study is that it focuses on multilingual mathematics classrooms of immigrant learners in South Africa – a context that has not yet been researched in South African vi mathematics education. Exploring language practices in multilingual mathematics classrooms of immigrant learners provides a different gaze into teaching and learning mathematics in multilingual classrooms in South Africa. Equally important is the extent to which immigrant learners are distinct to multilingual learners in the teaching and learning of linear programming. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education)

Neste trabalho é apresentada uma ferramenta de otimização para análise de problemas de alocação de água em bacias hidrográficas utilizando técnicas de programação linear e linear por partes, integradas a um modelo de amortecimentos de ondas em canais. A otimização é feita de forma global, com uso de softwares de programação linear baseados nos métodos de pontos interiores. A metodologia de uso do sistema consiste em se obter uma solução ?ótima? para situações de disponibilidade de água insuficiente a todos os usos conflitantes na bacia. A ferramenta está sendo acoplada e incorporada ao AcquaNet, um Sistema de Suporte a Decisões (SSD) para análise de sistemas de recursos hídricos, que utiliza um algoritmo de rede de fluxo afim de otimizar a alocação de água. A formulação utilizando programação linear permite a análise global do sistema e por isso, espera-se melhor aproveitamento da água disponível, seja no menor déficit de atendimento às demandas ou maior armazenamento nos reservatórios. A programação linear com utilização de métodos de pontos interiores é atualmente uma técnica bastante conhecida e bem desenvolvida. Existem vários pacotes computacionais gratuitos com implementações eficientes dos métodos de pontos interiores que motivaram sua utilização neste trabalho. / This work presents an optimization tool for analyzing the problems of water allocation in watersheds by utilizing techniques of linear and piecewise linear programming integrated to a pattern of stream flow routing. The optimization is done in a global way with the usage of linear programming packages based upon the Internal Point Methods. The methodology of the usage consists in the acquirement of an optimal solution for situation of insufficient water availability for all conflicting consumptions from the watershed. The tool is being attached and incorporated to AcquaNet, which is a decision support system (DSS) for analysis of water resources systems that utilizes a network flow algorithm, with the purpose of optimizing the water allocation. The formulation that uses the linear programming leads to the analysis of the system as a whole and for this reason it is expected a better usage of the available water with a lower deficit in the supply or a greater storage in the reservoirs. Linear Programming with Internal Point Methods is nowadays a well known and very well developed technique. There are several computational packages with efficient implementations of the Internal Points Methods freely available, and that, has brought great motivation in its usage in the present work.

The energy transition taking place in various parts of the world will have many effects on the current energy systems as an increasing amount of intermittent power supply gets installed every year. In Sweden, just as many other countries, this will cause both challenges and opportunities for today´s energy producers. Challenges that may arise along with an increasingly fluctuating electricity production include both power deficits at certain times and regions but also hours of over-production which can cause electricity prices to drop significantly. Such challenges will have to be met by both dispatchable power generation and dynamic consumption. Conversely, actors prepared to adapt to the new climate by implementing new technologies or innovative business models could benefit from the transition towards a fully renewable energy system.  This thesis evaluates the techno-economic potential of green hydrogen production at a combined heat and power plant with the objective to provide decision support to a district heat and electricity producer in Sweden. It was in the company’s interest to investigate how hydrogen production could help reduce the production cost of district heat as well as contribute to the reduction of greenhouse gases.  In the project, two separate business models: Power-to-gas and Power-to-power were evaluated on the basis of technical and economic performance and environmental impact. To do this, a mathematical model of the CHP plant and the hydrogen systems was developed in Python which optimizes the operation based on costs. The business models were then simulated for two different years with each year representing a distinctly different electricity market situation.  The main conclusions of the study show that Power-to-gas could already be profitable at a hydrogen retail price of 40 SEK per kg, which is the projected retail price for the transportation sector. The demand today is however limited but is expected to grow fast in the near future, especially within heavy transportation. Another limiting factor for hydrogen production showed to be the availability of storage space, as hydrogen gas even at pressures up to 200 bar require large volumes.  Power-to-power for frequency regulation was found to not be economically justifiable as the revenue for providing grid services could not outweigh the high investment costs for any of the simulated years. This resulted in a high levelized cost of energy at over 3000 SEK per MWh which was mostly due to the low capacity factor of the power-to-power system.  Finally, green hydrogen has the potential of replacing fossil fuels in sectors that is difficult to reach with electricity, for example long-haul road transport or the shipping industry. Therefore, green hydrogen production in large scale could help decarbonize many of society’s fossil-heavy segments. By also serving as a grid-balancer, hydrogen production in a power-to-gas process has the potential of becoming an important part of a renewable energy system. / Energiomställningen som äger rum i olika delar av världen kommer att ha många effekter på de nuvarande energisystemen eftersom en ökande mängd väderberoende kraftproduktion installeras varje år. I Sverige, precis som många andra länder, kommer detta att medföra både utmaningar och möjligheter för dagens energiproducenter. Utmaningar som kan uppstå tillsammans med en alltmer fluktuerande elproduktion inkluderar både kraftunderskott vid vissa tider och regioner men också timmar av överproduktion som kan få elpriserna att sjunka avsevärt. Sådana utmaningar måste mötas av både planerbar kraftproduktion och dynamisk konsumtion. Omvänt kan aktörer som är beredda att anpassa sig till det nya klimatet genom att implementera ny teknik eller innovativa affärsmodeller dra nytta av övergången till ett helt förnybart energisystem.  Denna rapport utvärderar den tekno-ekonomiska potentialen för produktion av grön vätgas vid ett kraftvärmeverk med målet att ge beslutsstöd till en fjärrvärme- och elproducent i Sverige. Det var i företagets intresse att undersöka hur vätgasproduktion kan bidra till att sänka produktionskostnaden för fjärrvärme samt bidra till att minska växthusgaser.  I projektet utvärderades två separata affärsmodeller: Power-to-gas och Power-to-power baserat på teknisk och ekonomisk prestanda samt miljöpåverkan. För att kunna göra detta utvecklades en matematisk modell i Python av kraftvärmeverket och vätgassystemen som optimerar driften baserat på kostnader. Affärsmodellerna simulerades sedan för två olika års elpriser för att undersöka modellens prestanda i olika typer av elmarknader.  De viktigaste slutsatserna i studien visar att Power-to-gas redan kan vara lönsamt till ett vätgaspris på 40 SEK per kg, vilket är det förväntade marknadspriset på grön vätgas for transportsektorn. Efterfrågan är idag begränsad men förväntas växa snabbt inom en snar framtid, särskilt inom tung transport. En annan begränsande faktor för vätgasproduktion visade sig vara tillgången på lagringsutrymme, eftersom vätgas även vid tryck upp till 200 bar kräver stora volymer.  Power-to-power för frekvensreglering visade sig inte vara ekonomiskt försvarbart, eftersom intäkterna för att tillhandahålla nättjänster inte kunde uppväga de höga investeringskostnaderna under några av de simulerade åren. Detta resulterade i en hög LCOE på över 3000 SEK per MWh, vilket främst berodde på Power-to-power-systemets låga utnyttjandegrad.  Slutligen kan det sägas att grön vätgas har stor potential att ersätta fossila bränslen i sektorer som är svåra att elektrifiera, exempelvis tunga vägtransporter eller sjöfart. Därför kan storskalig grön vätgasproduktion hjälpa till att dekarbonisera många av samhällets fossiltunga segment. Genom att dessutom fungera som balansering har väteproduktion i en Power-to-gas-process potential att bli en viktig del av ett system med stor andel förnybar energi.

Due to the increase in electrical demand and renewable penetration, electrical utilities need to improve and optimize the grid infrastructure. Fundamental components in this grid infrastructure are transformers, which are designed conservatively on the base of a static rated power. However, load and weather change continuously and hence, transformers are not used in the most efficient way. For this reason a new technology has been developed: Dynamic transformer rating (DTR). By applying DTR, it is possible to load transformers above the nameplate rating without affecting their life time expectancy. This project goes one step further and uses DTR for the short term and long term wind farm planning. The optimal wind farm is designed by applying DTR to the power transformer of the farm. The optimization is carried out using a Mixed-Integer Linear Programming (MILP) model. In respect of the transformer thermal analysis, the linearized top oil model of IEEE Clause 7 is selected. The model is executed for 4 different types of power transformers: 63 MVA, 100 MVA, 200 MVA and 400 MVA. As result, it is obtained that the net present value for the investment and the capacity of the wind farm increase linearly with respect to the size of the transformer. Then, a sensitivity analysis is carried out by modifying the wind speed, the electricity price, the lifetime of the transformer and the selected weather data. From this sensitivity analysis, it is possible to conclude that wind resources and electricity price are key parameters for the feasibility of the wind farm. / På grund av ökningen av efterfrågan av elektricitet och förnybara energin, elförsörgingsföretag måste förbättras och elnätets infrastruktur måste optimeras. Grundläggande komponenter i elnätet är transformatorer, som är designade konservativt efter en statisk märkeffekt. Laster och vädret ändras dock kontinuerligt, detta betyder att transformatorer inte används på de mest effektiva sätten. Av denna anledning har en ny teknik utvecklats: Dynamisk lastbarhet hos transformatorer (DTR). Genom att applicera DTR, gör det möjligt att belasta en transformator högre än märkdata utan att påverka den förväntade livslängden. Detta projekt går ett steg längre och använder DTR för kort och lång sikts vindkraftparkplaneringar. Den optimala vindkraftparken är designad genom att använda DLT på krafttransformatorn för vindkraftsparken. Optimeringen utförst med hjälp av Mixed-Integer Linear programming (MILP) modell. Gällande transformatorns termiska analys, så valdes den linjäriserade toppoljemodellen av IEEE Clause 7. Modellen var utförd för fyra olika krafttransformatorer: 63 MVA, 100 MVA, 200 MVA och 400 MVA. Resultatet blev att nettonuvärdet för investeringen och kapaciteten av vindkraftsparken ökade linjärt med avseende på storleken på transformatorn. En känslighetsanalys var utförd genom att ändra vindhastigheten, elpriset, livstiden av transformatorn och de valda väderdata. Från känslighetsanalysen så var det möjligt att dra slutsatsen att vindresurser och elpriset är nyckelparametrar för vindkraftsparkens genomförbarhet.

This thesis examines the optimal scheduling of a hydroelectric power plant with cascaded reservoirs each with multiple generating units under uncertainty after testing three linearization methods. These linearization methods are Successive Linear Programming, Piecewise Linear Approximations, and a Hybrid of the two together. There are two goals of this work. The first goal of this work aims to replace the nonconvex mixed-integer nonlinear program (MINLP) with a computationally efficient linearized mixed-integer linear program (MILP) that will be capable of finding a high quality solution, preferably the global optimum. The second goal is to implement a stochastic approach on the linearized method in a pseudo-rolling horizon method which keeps the ending time step fixed. Overall, the Hybrid method proved to be a viable replacement and performs well in the pseudo-rolling horizon tests. / Thesis / Master of Applied Science (MASc)