Utilization of a Time Domain Simulator in the Technical and Economic Analysis of a Wind Turbine Electric Drive Train

The amount of installed wind power has been growing exponentially during the past ten years. As wind turbines have become a significant source of electrical energy, the interactions between the turbines and the electric power network need to be studied more thoroughly than before. Especially, the behavior of the turbines in fault situations is of prime importance; simply disconnecting all wind turbines from the network during a voltage drop is no longer acceptable, since this would contribute to a total network collapse. These requirements have been a contributor to the increased role of simulations in the study and design of the electric drive train of a wind turbine. When planning a wind power investment, the selection of the site and the turbine are crucial for the economic feasibility of the installation. Economic feasibility, on the other hand, is the factor that determines whether or not investment in wind power will continue, contributing to green electricity production and reduction of emissions. In the selection of the installation site and the turbine (siting and site matching), the properties of the electric drive train of the planned turbine have so far been generally not been taken into account. Additionally, although the loss minimization of some of the individual components of the drive train has been studied, the drive train as a whole has received less attention. Furthermore, as a wind turbine will typically operate at a power level lower than the nominal most of the time, efficiency analysis in the nominal operating point is not sufficient. This doctoral dissertation attempts to combine the two aforementioned areas of interest by studying the applicability of time domain simulations in the analysis of the economicfeasibility of a wind turbine. The utilization of a general-purpose time domain simulator, otherwise applied to the study of network interactions and control systems, in the economic analysis of the wind energy conversion system is studied. The main benefits of the simulation-based method over traditional methods based on analytic calculation of losses include the ability to reuse and recombine existing models, the ability to analyze interactions between the components and subsystems in the electric drive train (something which is impossible when considering different subsystems as independent blocks, as is commonly done in theanalytical calculation of efficiencies), the ability to analyze in a rather straightforward manner the effect of selections other than physical components, for example control algorithms, and the ability to verify assumptions of the effects of a particular design change on the efficiency of the whole system. Based on the work, it can be concluded that differences between two configurations can be seen in the economic performance with only minor modifications to the simulation models used in the network interaction and control method study. This eliminates the need ofdeveloping analytic expressions for losses and enables the study of the system as a whole instead of modeling it as series connection of independent blocks with no lossinterdependencies. Three example cases (site matching, component selection, control principle selection) are provided to illustrate the usage of the approach and analyze its performance.

Genesis and Emplacement of Carbonatites and Lamprophyres in the Svecofennian Domain

A small carbonatite dyke swarm has been identified at Naantali, southwest Finland. Several swarms of shoshonitic lamprophyres are also known along the Archean-Proterozoic boundary in eastern Finland and northwest Russia. These intrusions, along with the carbonatite intrusion at Halpanen, eastern Finland, represent a stage of widespread low-volume mantle-sourced alkaline magmatism in the Svecofennian Domain. Using trace element and isotope geochemistry coupled with precise geochronology from these rocks, a model is presented for the Proterozoic metasomatic evolution of the Fennoscandian subcontinental lithospheric mantle. At ~2.2-2.06 Ga, increased biological production in shallow seas linked to continental rifting, resulted in increased burial rates of organic carbon. Subduction between ~1.93-1.88 Ga returned organic carbon-enriched sediments of mixed Archean and Proterozoic provenance to the mantle. Dehydration reactions supplied water to the mantle wedge, driving arc volcanism, while mica, amphibole and carbonate were brought deeper into the mantle with the subducting slab. The cold subducted slab was heated conductively from the surrounding warm mantle, while pressures continued to gradually increase as a result of crustal thickening. The sediments began to melt in a two stage process, first producing a hydrous alkaline silicate melt, which infiltrated the mantle wedge and crystallised as metasomatic veins. At higher temperatures, carbonatite melt was produced, which preferentially infiltrated the pre-existing metasomatic vein network. At the onset of post-collisional extension, deep fault structures formed, providing conduits for mantle melts to reach the upper crust. Low-volume partial melting of the enriched mantle at depths of at least 110 km led to the formation of first carbonatitic magma and subsequently lamprophyric magma. Carbonatite was emplaced in the upper crust at Naantali at 1795.7 ± 6.8 Ma; lamprophyres along the Archean-Proterozoic boundary were emplaced between 1790.1 ± 3.3 Ma and 1781 ± 20 Ma.

Boundary identification in the domain of a parabolic partial differential equation

Bakgrunden och inspirationen till föreliggande studie är tidigare forskning i tillämpningar på randidentifiering i metallindustrin. Effektiv randidentifiering möjliggör mindre säkerhetsmarginaler och längre serviceintervall för apparaturen i industriella högtemperaturprocesser, utan ökad risk för materielhaverier. I idealfallet vore en metod för randidentifiering baserad på uppföljning av någon indirekt variabel som kan mätas rutinmässigt eller till en ringa kostnad. En dylik variabel för smältugnar är temperaturen i olika positioner i väggen. Denna kan utnyttjas som insignal till en randidentifieringsmetod för att övervaka ugnens väggtjocklek. Vi ger en bakgrund och motivering till valet av den geometriskt endimensionella dynamiska modellen för randidentifiering, som diskuteras i arbetets senare del, framom en flerdimensionell geometrisk beskrivning. I de aktuella industriella tillämpningarna är dynamiken samt fördelarna med en enkel modellstruktur viktigare än exakt geometrisk beskrivning. Lösningsmetoder för den s.k. sidledes värmeledningsekvationen har många saker gemensamt med randidentifiering. Därför studerar vi egenskaper hos lösningarna till denna ekvation, inverkan av mätfel och något som brukar kallas förorening av mätbrus, regularisering och allmännare följder av icke-välställdheten hos sidledes värmeledningsekvationen. Vi studerar en uppsättning av tre olika metoder för randidentifiering, av vilka de två första är utvecklade från en strikt matematisk och den tredje från en mera tillämpad utgångspunkt. Metoderna har olika egenskaper med specifika fördelar och nackdelar. De rent matematiskt baserade metoderna karakteriseras av god noggrannhet och låg numerisk kostnad, dock till priset av låg flexibilitet i formuleringen av den modellbeskrivande partiella differentialekvationen. Den tredje, mera tillämpade, metoden kännetecknas av en sämre noggrannhet förorsakad av en högre grad av icke-välställdhet hos den mera flexibla modellen. För denna gjordes även en ansats till feluppskattning, som senare kunde observeras överensstämma med praktiska beräkningar med metoden. Studien kan anses vara en god startpunkt och matematisk bas för utveckling av industriella tillämpningar av randidentifiering, speciellt mot hantering av olinjära och diskontinuerliga materialegenskaper och plötsliga förändringar orsakade av “nedfallande” väggmaterial. Med de behandlade metoderna förefaller det möjligt att uppnå en robust, snabb och tillräckligt noggrann metod av begränsad komplexitet för randidentifiering.

Towards input/output-free modelling of linear infinite-dimensional systems in continuous time

This dissertation describes a networking approach to infinite-dimensional systems theory, where there is a minimal distinction between inputs and outputs. We introduce and study two closely related classes of systems, namely the state/signal systems and the port-Hamiltonian systems, and describe how they relate to each other. Some basic theory for these two classes of systems and the interconnections of such systems is provided. The main emphasis lies on passive and conservative systems, and the theoretical concepts are illustrated using the example of a lossless transfer line. Much remains to be done in this field and we point to some directions for future studies as well.

Applications of graph transformation in tools for domain-specific modeling languages

The use of domain-specific languages (DSLs) has been proposed as an approach to cost-e ectively develop families of software systems in a restricted application domain. Domain-specific languages in combination with the accumulated knowledge and experience of previous implementations, can in turn be used to generate new applications with unique sets of requirements. For this reason, DSLs are considered to be an important approach for software reuse. However, the toolset supporting a particular domain-specific language is also domain-specific and is per definition not reusable. Therefore, creating and maintaining a DSL requires additional resources that could be even larger than the savings associated with using them. As a solution, di erent tool frameworks have been proposed to simplify and reduce the cost of developments of DSLs. Developers of tool support for DSLs need to instantiate, customize or configure the framework for a particular DSL. There are di erent approaches for this. An approach is to use an application programming interface (API) and to extend the basic framework using an imperative programming language. An example of a tools which is based on this approach is Eclipse GEF. Another approach is to configure the framework using declarative languages that are independent of the underlying framework implementation. We believe this second approach can bring important benefits as this brings focus to specifying what should the tool be like instead of writing a program specifying how the tool achieves this functionality. In this thesis we explore this second approach. We use graph transformation as the basic approach to customize a domain-specific modeling (DSM) tool framework. The contributions of this thesis includes a comparison of di erent approaches for defining, representing and interchanging software modeling languages and models and a tool architecture for an open domain-specific modeling framework that e ciently integrates several model transformation components and visual editors. We also present several specific algorithms and tool components for DSM framework. These include an approach for graph query based on region operators and the star operator and an approach for reconciling models and diagrams after executing model transformation programs. We exemplify our approach with two case studies MICAS and EFCO. In these studies we show how our experimental modeling tool framework has been used to define tool environments for domain-specific languages.

Frequency-Domain and Wide-Pulse Time-Domain Measurements of Lanthanide Luminescence and Lanthanide-Based Resonance Energy Transfer

Resonance energy transfer (RET) is a non-radiative transfer of the excitation energy from the initially excited luminescent donor to an acceptor. The requirements for the resonance energy transfer are: i) the spectral overlap between the donor emission spectrum and the acceptor absorption spectrum, ii) the close proximity of the donor and the acceptor, and iii) the suitable relative orientations of the donor emission and the acceptor absorption transition dipoles. As a result of the RET process the donor luminescence intensity and the donor lifetime are decreased. If the acceptor is luminescent, a sensitized acceptor emission appears. The rate of RET depends strongly on the donor–acceptor distance (r) and is inversely proportional to r6. The distance dependence of RET is utilized in binding assays. The proximity requirement and the selective detection of the RET-modified emission signal allow homogeneous separation free assays. The term lanthanide-based RET is used when luminescent lanthanide compounds are used as donors. The long luminescence lifetimes, the large Stokes’ shifts and the intense, sharply-spiked emission spectra of the lanthanide donors offer advantages over the conventional organic donor molecules. Both the organic lanthanide chelates and the inorganic up-converting phosphor (UCP) particles have been used as donor labels in the RET based binding assays. In the present work lanthanide luminescence and lanthanide-based resonance energy transfer phenomena were studied. Luminescence lifetime measurements had an essential role in the research. Modular frequency-domain and time-domain luminometers were assembled and used successfully in the lifetime measurements. The frequency-domain luminometer operated in the low frequency domain ( 100 kHz) and utilized a novel dual-phase lock-in detection of the luminescence. One of the studied phenomena was the recently discovered non-overlapping fluorescence resonance energy transfer (nFRET). The studied properties were the distance and temperature dependences of nFRET. The distance dependence was found to deviate from the Förster theory and a clear temperature dependence was observed whereas conventional RET was completely independent of the temperature. Based on the experimental results two thermally activated mechanisms were proposed for the nFRET process. The work with the UCP particles involved the measurement of the luminescence properties of the UCP particles synthesized in our laboratory. The goal of the UCP particle research is to develop UCP donor labels for binding assays. In the present work the effect of the dopant concentrations and the core–shell structure on the total up-conversion luminescence intensity, the red–green emission ratio, and the luminescence lifetime was studied. Also the non-radiative nature of the energy transfer from the UCP particle donors to organic acceptors was demonstrated for the first time in aqueous environment and with a controlled donor–acceptor distance.

Professionalization of controllers in Finland

Professions are a special category of occupations that possesses exclusive rights over its domain of expertise. Professions apply expert knowledge in their work by using professional discretion and judgment to solve their clients’ problems. With control over their expert knowledge base, professions are able to control the supply of practitioners in their field and regulate the practice in their market. Professionalization is the process during which occupations attempt to gain the status of a profession. The benefits of becoming a profession are extensive – professional autonomy, social and financial rewards, prestige, status, and an exclusive community are only a few of the privileges that established professions possess. Many aspiring occupations have tried and failed to gain the status of a profession and one of these groups is the occupation of controllers in Finland. The objective of this study to uncover, why controllers have not professionalized, which properties of the occupation correspond with the elements generally regarded to pertain to professions, and which aspects of the occupational group may hinder the professionalization project. The professionalization project of controllers is analyzed using a multi-actor model of professionalization, in which practitioners, clients, the state, training institutions, and employing organizations are considered to affect the project. The properties of the occupation of controllers are compared to features generally associated with professions. The research methodology for this thesis is qualitative, and the study is conducted as an exploratory research. The data is primarily gathered using semi-structured interviews, which were conducted between March and May 2013 lasting from 40 minutes to an hour. In total, four controllers were interviewed, who worked for different companies operating in different industries, and whose experience of working as a controller varied between a few years to nearly 15 years. The data in this study indicates that although controllers possess qualities that distinguish professions from other occupational groups, the professionalization of controllers may not be plausible. Controllers enjoy considerable autonomy in organizations, and they possess a strong orientation towards serving their clients. The more profound problem with the occupation is its non-exclusive, indistinct knowledge base that does not rely solely on a single knowledge base. Controllers’ expertise is relatively organization-specific and built on several different fields of knowledge and not just management accounting, which could be considered as their primary knowledge base. In addition, controllers have not organized themselves, which is a quintessential, but by no means a sufficient prerequisite for professionalization.

Target audience analysis in cyber domain

This thesis aims to find an effective way of conducting a target audience analysis (TAA) in cyber domain. There are two main focal points that are addressed; the nature of the cyber domain and the method of the TAA. Of the cyber domain the object is to find the opportunities, restrictions and caveats that result from its digital and temporal nature. This is the environment in which the TAA method is examined in this study. As the TAA is an important step of any psychological operation and critical to its success, the method used must cover all the main aspects affecting the choice of a proper target audience. The first part of the research was done by sending an open-ended questionnaire to operators in the field of information warfare both in Finland and abroad. As the results were inconclusive, the research was completed by assessing the applicability of United States Army Joint Publication FM 3-05.301 in the cyber domain via a theory-based content analysis. FM 3- 05.301 was chosen because it presents a complete method of the TAA process. The findings were tested against the results of the questionnaire and new scientific research in the field of psychology. The cyber domain was found to be “fast and vast”, volatile and uncontrollable. Although governed by laws to some extent, the cyber domain is unpredictable by nature and not controllable to reasonable amount. The anonymity and lack of verification often present in the digital channels mean that anyone can have an opinion, and any message sent may change or even be counterproductive to the original purpose. The TAA method of the FM 3-05.301 is applicable in the cyber domain, although some parts of the method are outdated and thus suggested to be updated if used in that environment. The target audience categories of step two of the process were replaced by new groups that exist in the digital environment. The accessibility assessment (step eight) was also redefined, as in the digital media the mere existence of a written text is typically not enough to convey the intended message to the target audience. The scientific studies made in computer sciences and both in psychology and sociology about the behavior of people in social media (and overall in cyber domain) call for a more extensive remake of the TAA process. This falls, however, out of the scope of this work. It is thus suggested that further research should be carried out in search of computer-assisted methods and a more thorough TAA process, utilizing the latest discoveries of human behavior. ---------------------------------------------------------------------------------------------------------------------------------- Tämän opinnäytetyön tavoitteena on löytää tehokas tapa kohdeyleisöanalyysin tekemiseksi kybertoimintaympäristössä. Työssä keskitytään kahteen ilmiöön: kybertoimintaympäristön luonteeseen ja kohdeyleisöanalyysin metodiin. Kybertoimintaympäristön osalta tavoitteena on löytää sen digitaalisesta ja ajallisesta luonteesta juontuvat mahdollisuudet, rajoitteet ja sudenkuopat. Tämä on se ympäristö jossa kohdeyleisöanalyysiä tarkastellaan tässä työssä. Koska kohdeyleisöanalyysi kuuluu olennaisena osana jokaiseen psykologiseen operaatioon ja on onnistumisen kannalta kriittinen tekijä, käytettävän metodin tulee pitää sisällään kaikki oikean kohdeyleisön valinnan kannalta merkittävät osa-alueet. Tutkimuksen ensimmäisessä vaiheessa lähetettiin avoin kysely informaatiosodankäynnin ammattilaisille Suomessa ja ulkomailla. Koska kyselyn tulokset eivät olleet riittäviä johtopäätösten tekemiseksi, tutkimusta jatkettiin tarkastelemalla Yhdysvaltojen armeijan kenttäohjesäännön FM 3-05.301 soveltuvuutta kybertoimintaympäristössä käytettäväksi teorialähtöisen sisällönanalyysin avulla. FM 3-05.301 valittiin koska se sisältää kokonaisvaltaisen kohdeyleisöanalyysiprosessin. Havaintoja verrattiin kyselytutkimuksen tuloksiin ja psykologian uusiin tutkimuksiin. Kybertoimintaympäristö on tulosten perusteella nopea ja valtava, jatkuvasti muuttuva ja kontrolloimaton. Vaikkakin lait hallitsevat kybertoimintaympäristöä jossakin määrin, on se silti luonteeltaan ennakoimaton eikä sitä voida luotettavasti hallita. Digitaalisilla kanavilla usein läsnäoleva nimettömyys ja tiedon tarkastamisen mahdottomuus tarkoittavat että kenellä tahansa voi olla mielipide asioista, ja mikä tahansa viesti voi muuttua, jopa alkuperäiseen tarkoitukseen nähden vastakkaiseksi. FM 3-05.301:n metodi toimii kybertoimintaympäristössä, vaikkakin jotkin osa-alueet ovat vanhentuneita ja siksi ne esitetään päivitettäväksi mikäli metodia käytetään kyseisessä ympäristössä. Kohdan kaksi kohdeyleisökategoriat korvattiin uusilla, digitaalisessa ympäristössä esiintyvillä ryhmillä. Lähestyttävyyden arviointi (kohta 8) muotoiltiin myös uudestaan, koska digitaalisessa mediassa pelkkä tekstin läsnäolo ei sellaisenaan tyypillisesti vielä riitä halutun viestin välittämiseen kohdeyleisölle. Tietotekniikan edistyminen ja psykologian sekä sosiologian aloilla tehty tieteellinen tutkimus ihmisten käyttäytymisestä sosiaalisessa mediassa (ja yleensä kybertoimintaympäristössä) mahdollistavat koko kohdeyleisöanalyysiprosessin uudelleenrakentamisen. Tässä työssä sitä kuitenkaan ei voida tehdä. Siksi esitetäänkin että lisätutkimusta tulisi tehdä sekä tietokoneavusteisten prosessien että vielä syvällisempien kohdeyleisöanalyysien osalta, käyttäen hyväksi viimeisimpiä ihmisen käyttäytymiseen liittyviä tutkimustuloksia.

Stiffness based trajectory planning and feedforward based vibration suppression control of parallel robot machines

The dissertation proposes two control strategies, which include the trajectory planning and vibration suppression, for a kinematic redundant serial-parallel robot machine, with the aim of attaining the satisfactory machining performance. For a given prescribed trajectory of the robot's end-effector in the Cartesian space, a set of trajectories in the robot's joint space are generated based on the best stiffness performance of the robot along the prescribed trajectory. To construct the required system-wide analytical stiffness model for the serial-parallel robot machine, a variant of the virtual joint method (VJM) is proposed in the dissertation. The modified method is an evolution of Gosselin's lumped model that can account for the deformations of a flexible link in more directions. The effectiveness of this VJM variant is validated by comparing the computed stiffness results of a flexible link with the those of a matrix structural analysis (MSA) method. The comparison shows that the numerical results from both methods on an individual flexible beam are almost identical, which, in some sense, provides mutual validation. The most prominent advantage of the presented VJM variant compared with the MSA method is that it can be applied in a flexible structure system with complicated kinematics formed in terms of flexible serial links and joints. Moreover, by combining the VJM variant and the virtual work principle, a systemwide analytical stiffness model can be easily obtained for mechanisms with both serial kinematics and parallel kinematics. In the dissertation, a system-wide stiffness model of a kinematic redundant serial-parallel robot machine is constructed based on integration of the VJM variant and the virtual work principle. Numerical results of its stiffness performance are reported. For a kinematic redundant robot, to generate a set of feasible joints' trajectories for a prescribed trajectory of its end-effector, its system-wide stiffness performance is taken as the constraint in the joints trajectory planning in the dissertation. For a prescribed location of the end-effector, the robot permits an infinite number of inverse solutions, which consequently yields infinite kinds of stiffness performance. Therefore, a differential evolution (DE) algorithm in which the positions of redundant joints in the kinematics are taken as input variables was employed to search for the best stiffness performance of the robot. Numerical results of the generated joint trajectories are given for a kinematic redundant serial-parallel robot machine, IWR (Intersector Welding/Cutting Robot), when a particular trajectory of its end-effector has been prescribed. The numerical results show that the joint trajectories generated based on the stiffness optimization are feasible for realization in the control system since they are acceptably smooth. The results imply that the stiffness performance of the robot machine deviates smoothly with respect to the kinematic configuration in the adjacent domain of its best stiffness performance. To suppress the vibration of the robot machine due to varying cutting force during the machining process, this dissertation proposed a feedforward control strategy, which is constructed based on the derived inverse dynamics model of target system. The effectiveness of applying such a feedforward control in the vibration suppression has been validated in a parallel manipulator in the software environment. The experimental study of such a feedforward control has also been included in the dissertation. The difficulties of modelling the actual system due to the unknown components in its dynamics is noticed. As a solution, a back propagation (BP) neural network is proposed for identification of the unknown components of the dynamics model of the target system. To train such a BP neural network, a modified Levenberg-Marquardt algorithm that can utilize an experimental input-output data set of the entire dynamic system is introduced in the dissertation. Validation of the BP neural network and the modified Levenberg- Marquardt algorithm is done, respectively, by a sinusoidal output approximation, a second order system parameters estimation, and a friction model estimation of a parallel manipulator, which represent three different application aspects of this method.

Transition management in action within the context of nutrient economy

Tämä työ tutkii ja tarkastelee transitio-kokeilua ravinnetaloudessa. Transitio-kokeilu on toimintatutkimusprojekti, joka toteutetaan systeemisen muutoksen ajattelun mukaisesti alhaalta ylöspäin. Ravinnetalous määritetään tarkemmin työn kautta sekä analysoidaan monitaso-perspektiivin näkökulmasta. Ravinnetalous on terminä varsin tuntematon ja tarvitsee enemmän tunnettavuutta laajemman yleisön edessä. Transitio-areenan ja transitio-visioiden kehittäminen ovat työn keskipisteessä, koska ne ovat tärkeimpiä vaiheita transition alkuvaiheessa. Joukko sidosryhmätoimijoita osallistuu transitio areenaan sekä visioiden jatkokehittelyyn. Visio(t) luodaan ensisijaisesti backcasting-menetelmällä, jota myös täydennetään tavanomaisella ennustamisella. Backcasting- menetelmä on osin osallistava ja siinä käytetään ravinteiden planeettarajoja kvantitatiivisina pääperiaatteina, minkä tuloksena myös visiot ovat osin kvantitatiivisia. Transitio areenan kokoaminen ja fasilitointi aiheuttavat hankalia kysymyksiä, jotka tarvitsevat jatko-tutkimusta. Alhaalta-ylöspäin organisoitu transitio-arena houkuttelee niche-toimijoita, mutta epäonnistuu sitouttamaan julkisen vallan toimijoita. Toimintamallin voimasuhteet, politiikka ja transition vakiinnuttaminen tulisivat olla jatko-toimenpiteinä niin tutkimuksessa kuin toiminnassakin.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.