Examensarbetet har utforts pa uppdrag av enheten for Data- och Elektroteknik pa skolan for Teknik och Halsa, KTH. Denna studie om transformatorer omfattar deras funktioner och anvandningsomraden, hur de ar uppbyggda, hur karnan och lindningarna som ar transformatorns viktigaste delar ar uppbyggda, vilka materiel som anvands vid konstruktion av dessa samt vilka typer det finns av dem beskrivs i rapporten. Vidare beskrivs flera olika transformatortypers funktioner och anvandningsomraden. Eftersom en viktig del av studien varit pa vilka satt transformatorer kan utvecklas for okad miljovanlighet och livslangd gar rapporten igenom olika vegetabiliska oljor, vilka fordelar de har gentemot mineraloljor och vilka egenskaper de har samt pa vilket satt de ar skonsammare mot miljon. HVDC-enheter transformerar hoga spanningar fran vaxelspanning till likspanning for mindre resistiva forluster och darmed lagre energiforluster vilket i sin tur leder till en renare miljo. Teorigenomgangen av transformatorn ligger till grund for utformningen av en laboration for undervisningsbruk, bl.a. i en kurs om elkraftteknik som ges i hogskoleingenjorsprogrammet Elektroteknik pa KTH. / This degree project was carried out on behalf of the unit for Computer and Electrical engineering at the School of Technology and Health, KTH. This study about transformers, their function and uses, the way they’re built, how the core and windings, which are the most important parts of a transformer, are made and what kind of materials are used to make them. Several types of transformers and their functions are described. A crucial part of the study has been to search for different ways to develop transformers to make them environmentally safer and to increase their life-span. As such this work describes different transformer oils, what advantages and disadvantages they have compared to mineral oils, what properties they have and in what way they’re less harmful for the environment. HVDC units transform high voltages from Alternating Current to Direct Current to lessen the resistive losses and energy losses which lead to a cleaner environment. The theory examination of the transformer is the basis for the formation of a lab for educational use, among others in a course about electrical power that’s included in Bachelor of Science in Electrical Engineering at KTH.

Physical asset management in the electric power sector encompasses the scheduling of the maintenance and replacement of grid components, as well as decisions about investments in new components. Data plays a crucial role in these decisions. The importance of data is increasing with the transformation of the power system and its evolution toward smart grids. This thesis deals with questions related to data management as a way to improve the performance of asset management decisions. Data management is defined as the collection, processing, and storage of data. Here, the focus is on the collection and processing of data. First, the influence of data on the decisions related to assets is explored. In particular, the impacts of data quality on the replacement time of a generic component (a line for example) are quantified using a scenario approach, and failure modeling. In fact, decisions based on data of poor quality are most likely not optimal. In this case, faulty data related to the age of the component leads to a non-optimal scheduling of component replacement. The corresponding costs are calculated for different levels of data quality. A framework has been developed to evaluate the amount of investment needed into data quality improvement, and its profitability. Then, the ways to use available data efficiently are investigated. Especially, the possibility to use machine learning algorithms on real-world datasets is examined. New approaches are developed to use only available data for component ranking and failure prediction, which are two important concepts often used to prioritize components and schedule maintenance and replacement. A large part of the scientific literature assumes that the future of smart grids lies in big data collection, and in developing algorithms to process huge amounts of data. On the contrary, this work contributes to show how automatization and machine learning techniques can actually be used to reduce the need to collect huge amount of data, by using the available data more efficiently. One major challenge is the trade-offs needed between precision of modeling results, and costs of data management. / <p>QC 20210330</p>

The aim of the study is to analyze the possibility of how and where wind farms should be integrated on the electrical grid. The challenges mainly concern grid capacity and transmission losses. Economic factors will be regarded as well. To fulfill the aim, the Skellefteälven river in Sweden is selected as study object. A regional grid along the river is thereupon simulated with regards to five existing hydro power plants, four electrical consumption points, and the national grid. Additionally, four wind farms are placed on probable sites around the grid. Considering the large amount of data to be calculated in this study, a grid model assembled through numerical analysis in MATLAB is henceforth deemed optimal. Through load flow simulation, the voltage variations and power losses are calculated. Hence, the costs of the losses is found. The investment costs for upgrading the grid are also determined. As the results show, an upgrade of the electrical grid certainly requires a relatively large investment sum. Nevertheless, the return of the project will eventually surpass the initial costs. Accordingly, there are economic benefits of investing in upgrading the grid capacity. / Syftet med studien är att analysera möjligheten till hur och var vindkraftsparker borde integreras i elnätet. Utmaningarna rör främst nätkapacitet och ledningsförluster. Ekonomiska faktorer kommer även att undersökas. För att uppnå syftet bedrivs en fallstudie, där Skellefteälven väljs som studieområde. Ett regionalnät är utformat längs älven med hänsyn till fem existerande vattenkraftverk, fyra valda konsumtionsnoder och stamnätet. Utöver dessa placeras även fyra vindkraftsparker ut på lämpliga ställen. Med tanke på hur mycket data som behandlas vid beräkningarna simuleras därför nätet med hjälp av numerisk analys i MATLAB. Genom att köra effektflödesberäkningar räknas spänningsvariationer och effektförluster fram. Därifrån kan kostnader för ledningsförluster tas fram. Vidare framtas även investeringskostnader för uppgradering av nätet. Resultaten visar att en uppgradering kräver en relativt stor investeringssumma. Däremot kommer inkomsten efter en genomförd uppgradering tillslut att överstiga initialkostnaden. Därav finns det ekonomiska fördelar med att investera i en ökad nätkapacitet.

Due to the result of the sudden fossil fuels over-night price rises of 1973/1974, coupled with the depletion of the traditional energy resources, many initiatives globally have addressed the efficient use of these resources. Since then, several renewable energy sources have been introduced as alternatives to traditional resources to protect environmental resources and to improve quality of life. Globally, there are more than a quarter of the human population experiencing an energy crisis, particularly those living in the rural areas of developing countries. One typical example of this is Nigeria. This is a country with approximately 80% of her population consistently relying on combustible biomass from wood and its charcoal derivative. Nigeria has an abundant amount of both renewable and fossil fuel resources, but due to the lack of a reasonable energy policy (until recently), it has concentrated on traditional fossil fuels alone. Renewable energy is now Globally considered as a solution for mitigating climate change and environmental pollution. To assess the sustainability of renewable energy systems, the use of sustainability indicators is often necessary. These indicators are not only able to evaluate all the sustainability criteria of the renewable energy sources,1 but also can provide numerical results of sustainability assessment for different objective systems.

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[ES] La naturaleza aleatoria que caracteriza algunos fenómenos en sistemas físicos reales (e.g., ingeniería, biología, economía, finanzas, epidemiología y otros) nos ha planteado el desafío de un cambio de paradigma del modelado matemático y el análisis de sistemas dinámicos, y a tratar los fenómenos aleatorios como variables aleatorias o procesos estocásticos. Este enfoque novedoso ha traído como consecuencia nuevas especificidades que la teoría clásica del modelado y análisis de sistemas dinámicos deterministas no ha podido cubrir. Afortunadamente, maravillosas contribuciones, realizadas sobre todo en el último siglo, desde el campo de las matemáticas por científicos como Kolmogorov, Langevin, Lévy, Itô, Stratonovich, sólo por nombrar algunos; han abierto las puertas para un estudio bien fundamentado de la dinámica de sistemas físicos perturbados por ruido. En la presente tesis se discute el uso de ecuaciones diferenciales algebraicas estocásticas (EDAEs) para el modelado de sistemas multifísicos en red afectados por perturbaciones estocásticas, así como la evaluación de su estabilidad asintótica a través de exponentes de Lyapunov (ELs). El estudio está enfocado en EDAEs d-index-1 y su reformulación como ecuaciones diferenciales estocásticas ordinarias (EDEs). Fundamentados en la teoría ergódica, es factible analizar los ELs a través de sistemas dinámicos aleatorios (SDAs) generados por EDEs subyacentes. Una vez garantizada la existencia de ELs bien definidas, hemos procedido al uso de técnicas de simulación numérica para determinar los ELs numéricamente. Hemos implementado métodos numéricos basados en descomposición QR discreta y continua para el cómputo de la matriz de solución fundamental y su uso en el cálculo de los ELs. Las características numéricas y computacionales más relevantes de ambos métodos se ilustran mediante pruebas numéricas. Toda esta investigación sobre el modelado de sistemas con EDAEs y evaluación de su estabilidad a través de ELs calculados numéricamente, tiene una interesante aplicación en ingeniería. Esta es la evaluación de la estabilidad dinámica de sistemas eléctricos de potencia. En el presente trabajo de investigación, implementamos nuestros métodos numéricos basados en descomposición QR para el test de estabilidad dinámica en dos modelos de sistemas eléctricos de potencia de una-máquina bus-infinito (OMBI) afectados por diferentes perturbaciones ruidosas. El análisis en pequeña-señal evidencia el potencial de las técnicas propuestas en aplicaciones de ingeniería. / [CA] La naturalesa aleatòria que caracteritza alguns fenòmens en sistemes físics reals (e.g., enginyeria, biologia, economia, finances, epidemiologia i uns altres) ens ha plantejat el desafiament d'un canvi de paradigma del modelatge matemàtic i l'anàlisi de sistemes dinàmics, i a tractar els fenòmens aleatoris com a variables aleatòries o processos estocàstics. Aquest enfocament nou ha portat com a conseqüència noves especificitats que la teoria clàssica del modelatge i anàlisi de sistemes dinàmics deterministes no ha pogut cobrir. Afortunadament, meravelloses contribucions, realitzades sobretot en l'últim segle, des del camp de les matemàtiques per científics com Kolmogorov, Langevin, Lévy, Itô, Stratonovich, només per nomenar alguns; han obert les portes per a un estudi ben fonamentat de la dinàmica de sistemes físics pertorbats per soroll. En la present tesi es discuteix l'ús d'equacions diferencials algebraiques estocàstiques (EDAEs) per al modelatge de sistemes multifísicos en xarxa afectats per pertorbacions estocàstiques, així com l'avaluació de la seua estabilitat asimptòtica a través d'exponents de Lyapunov (ELs). L'estudi està enfocat en EDAEs d-index-1 i la seua reformulació com a equacions diferencials estocàstiques ordinàries (EDEs). Fonamentats en la teoria ergòdica, és factible analitzar els ELs a través de sistemes dinàmics aleatoris (SDAs) generats per EDEs subjacents. Una vegada garantida l'existència d'ELs ben definides, hem procedit a l'ús de tècniques de simulació numèrica per a determinar els ELs numèricament. Hem implementat mètodes numèrics basats en descomposició QR discreta i contínua per al còmput de la matriu de solució fonamental i el seu ús en el càlcul dels ELs. Les característiques numèriques i computacionals més rellevants de tots dos mètodes s'illustren mitjançant proves numèriques. Tota aquesta investigació sobre el modelatge de sistemes amb EDAEs i avaluació de la seua estabilitat a través d'ELs calculats numèricament, té una interessant aplicació en enginyeria. Aquesta és l'avaluació de l'estabilitat dinàmica de sistemes elèctrics de potència. En el present treball de recerca, implementem els nostres mètodes numèrics basats en descomposició QR per al test d'estabilitat dinàmica en dos models de sistemes elèctrics de potència d'una-màquina bus-infinit (OMBI) afectats per diferents pertorbacions sorolloses. L'anàlisi en xicotet-senyal evidencia el potencial de les tècniques proposades en aplicacions d'enginyeria. / [EN] The random nature that characterizes some phenomena in the real-world physical systems (e.g., engineering, biology, economics, finance, epidemiology, and others) has posed the challenge of changing the modeling and analysis paradigm and treat these phenomena as random variables or stochastic processes. Consequently, this novel approach has brought new specificities that the classical theory of modeling and analysis for deterministic dynamical systems cannot cover. Fortunately, stunning contributions made overall in the last century from the mathematics field by scientists such as Kolmogorov, Langevin, Lévy, Itô, Stratonovich, to name a few; have opened avenues for a well-founded study of the dynamics in physical systems perturbed by noise. In the present thesis, we discuss stochastic differential-algebraic equations (SDAEs) for modeling multi-physical network systems under stochastic disturbances, and their asymptotic stability assessment via Lyapunov exponents (LEs). We focus on d-index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Supported by the ergodic theory, it is feasible to analyze the LEs via the random dynamical system (RDSs) generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important numerical and computational features of both methods are illustrated through numerical tests. All this investigation concerning systems modeling through SDAEs and their stability assessment via computed LEs finds an appealing engineering application in the dynamic stability assessment of power systems. In this research work, we implement our QR-based numerical methods for testing the dynamic stability in two types of single-machine infinite-bus (SMIB) power system models perturbed by different noisy disturbances. The analysis in small-signal evidences the potential of the proposed techniques in engineering applications. / Mi agradecimiento al estado ecuatoriano que, a través del Programa de Becas para el Fortalecimiento y Desarrollo del Talento Humano en Ciencia y Tecnología 2012 de la Secretaría Nacional de Educación Superior, Ciencia y Tecnología (SENESCYT), han financiado mis estudios de doctorado. / González Zumba, JA. (2020). Dynamic Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Electric Power Systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/158558

In der Intralogistik führen technologieabhängig gekoppelte, drehzahlvariable Mehrmotorensysteme zu hohen Schwankungen im elektrischen Leistungsbedarf, der u.a. von der transportierten Gütermenge abhängt. Aufgrund der Überlappung der Lastanforderungen einzelner Verbraucher entstehen beträchtliche Lastmaxima, die das Versorgungsnetz belasten und eine Überdimensionierung der Infrastruktur erfordern. Ein räumlich ausgedehnter Gleichspannungsverbund in Kombination mit einem Energiespeichersystem sowie die Anpassung der Steuerung bieten Lösungen zur Verringerung der Netzanschlussleistung und des Energiebedarfs. Darüber hinaus ermöglichen solche Systeme eine effektive Rückgewinnung von potentieller und Bremsenergie. Am Beispiel eines Lagersystems mit Regalbediengeräten wird ermittelt, welche Reduktion des Netzleistungsbedarfs und Energieverbrauchs erreicht werden kann. Dabei werden ebenfalls die Auswirkungen auf die Durchsatzleistung des Systems untersucht. Dazu kommen Modelle des Bewegungsablaufes, der Komponenten der Antriebsstränge mit Abbildung von Mechanik, Getriebe, Motoren, Umrichter und der Speicher in einem Materialflussmodell zum Einsatz. Die Materialflusssimulationsumgebung wird dazu um die kontinuierlichen Modelle zu einem kombinierten Systemmodell erweitert und leistungsbegrenzende Steuerungsansätze im Zusammenwirken mit dem erweiterten Gleichspannungsverbund sowie den elektrischen Speichersystemen analysiert. Abschließend erfolgt die Ermittlung der Wirtschaftlichkeit der einzelnen Lösungen. Das Ergebnis ist ein Verfahren zur effizienten Betriebsweise von Intralogistiksystemen sowie eine Methode zur Dimensionierung eines räumlich ausgedehnten Gleichspannungsverbunds mit elektrischem Speicher. Zusammenfassend werden wirtschaftliche Maßnahmen identifiziert, die elektrische Lastmaxima reduzieren, die Schwankungen in der externen Energieversorgung verringern und die Energieeffizienz von Intralogistiksystemen verbessern. / In intralogistics, technologically linked, variable-speed multi-motor systems cause high fluctuations in electrical power demand due to variations in the amount of transported goods. These fluctuations often result in overlapping load demands by various consumers, which strain the supply network and require overdimensioning of infrastructure. To address this issue, a physically extended DC link, in combination with an energy storage system as well as adapted control, can reduce both grid power demand and energy consumption while allowing for the recovery of potential and braking energy. To evaluate the achievable reduction of grid power and energy demand, models were developed for individual components such as drive trains with motor, inverter, storage, axis coordination and material flow simulation with sequence control. An example of a warehouse system with storage and retrieval machines was used to determine the effects on throughput performance. To achieve this, the material flow simulation environment was extended to a combined system model, and performance-limiting control approaches were analyzed in interaction with the extended DC link and storage. Subsequently, economic feasibility of the individual solutions was determined. The result is a procedure for efficient operation of intralogistics systems and a methodology for dimensioning a physically extended DC link with electrical storage. The overall objective was to identify economic solutions that reduce electrical load peaks, balance external energy supply, and make intralogistics systems more energy-efficient. In conclusion, this study identifies economically viable solutions that can reduce electrical load peaks, balance external energy supply, and improve energy efficiency of intralogistics systems.

Eine angemessene Qualität der Elektroenergie ist Grundvoraussetzung für den störungsfreien Betrieb aller angeschlossenen Geräte und Anlagen und spielt in den Verteilungsnetzen moderner Industriegesellschaften wie Deutschland eine zentrale Rolle. Die Elektroenergiequalität (EEQ) wird in Strom- und Spannungsqualität unterteilt. Während die Stromqualität maßgeblich im Verantwortungsbereich der Hersteller von Geräten und Anlagen liegt, sind für die Sicherung einer angemessenen Spannungsqualität im Wesentlichen die Netzbetreiber verantwortlich. Durch die technische Weiterentwicklung bspw. neuer Gerätetechnologien und die zunehmende Integration dezentraler Erzeugungsanlagen wie Photovoltaikanlagen ist zu erwarten, dass die EEQ auch künftig weiter an Bedeutung gewinnt. Die EEQ im Niederspannungsverteilungsnetz ist abhängig von Ort und Zeit und wird durch verschiedene Qualitätskenngrößen beschrieben. Die örtliche und zeitliche Abhängigkeit resultieren aus einer Vielzahl verschiedener Einflussfaktoren, welche sich entweder der elektrischen oder der nicht-elektrischen Umgebung des betrachteten Verteilungsnetzes zuordnen lassen. Die elektrische Umgebung wird durch die Art und Anzahl angeschlossener Verbraucher bzw. Erzeuger (Abnehmer- bzw. Erzeugerstruktur) sowie Struktur und technische Parameter des Verteilungsnetzes (Netzstruktur) bestimmt. Die nicht-elektrische Umgebung umfasst u.a. Einflüsse der klimatischen Umgebung wie bspw. Temperatur oder Globalstrahlung. Ziel dieser Arbeit ist die systematische Identifikation korrelativer Zusammenhänge zwischen den genannten Umgebungseinflüssen und der EEQ sowie deren Quantifizierung auf Basis geeigneter Indizes und Kenngrößen. Die Ergebnisse der Arbeit helfen grundlegende Prinzipien der Ausprägung der Elektroenergiequalität im öffentlichen Verteilungsnetz besser zu verstehen sowie die Verteilungsnetze im Hinblick auf die Elektroenergiequalität zu charakterisieren und zu klassifizieren. Analog zu den Standard-Lastprofilen erfolgt die Definition von Standard-Qualitätsprofilen. / Power quality levels in public low voltage grids are influenced by many factors which can either be assigned to the electrical environment (connected consumers, connected genera-tion, network characteristics) or to the non-electrical environment (e.g. climatic conditions) at the measurement site. Type and amount of connected consumers (consumer topology) are expected to have a very high impact on power quality (PQ) levels. The generation topology is characterized by number and kind of equipment and generating installations like photovoltaic systems which are connected to the LV grid. The electrical parameters of the grid define the network topology. The parameters which are most suitable to describe each of the three topologies and the climatic environment will be identified. Voltage and current quality in public low voltage (LV) grids vary depending on location and time. They are quantified by a set of different parameters which either belong to events (e.g. dips) or to variations (e.g. harmonics). This thesis exclusively addresses continuous parameters describing variations. Continuous phenomena like harmonics are closely linked to an one-day-cycle which implies a more or less periodic behavior of the continuous power quality parameters. Consumer topologies such as office buildings or residential areas differ in their use of equipment. Time series analysis is used to distinguish between different consumer topologies and to identify characteristic weeks. The clustering of one-day time series is applied to identify characteristic days within the weeks of certain topologies. Based on the results, emission profiles for certain current quality parameters of different consumer topologies will be defined. Due to the characteristic harmonic current emission of certain consumer topologies which represents the typical user behaviour a classification system is developed. It is used to automatically classify the emission profiles of harmonic currents for unknown measurements and to estimate a likely consumer topology. A classification measure is introduced in order to identify unusual or false classified emission profiles. The usage behaviour of equipment by customers usually varies over the year. Subsequently, the levels of PQ parameters like harmonics may show seasonal variations which are identified by using newly defined parameters. The introduction of new device technologies on a large scale like the transition from incandescent to LED lamps might result in long-term changes to the levels of PQ parameters (e.g. harmonics). The analysis of the long-term behavior (trend) will be applied in order to quantify global trends (looking on the measurement duration as a whole) and local trends (looking on individual segments of the whole time series).

Esta dissertação apresenta uma contribuição aos estudos de planejamento do sistema elétrico do Estado de São Paulo, ressaltando os benefícios que se obteria, com uma maior integração das fontes renováveis à matriz elétrica desse estado (notadamente as fontes de energia solar fotovoltaica, eólica, aproveitamento dos resíduos sólidos urbano RSU e vinhaça), visando o aumento da segurança e independência energética, redução de perdas, e benefícios ambientais. Para tanto, apresenta-se o conceito das usinas virtuais, cujo propósito é obter um melhor e maior aproveitamento dos recursos energéticos normalmente dispersos espacialmente - incluindo nesse rol a microgeração distribuída, a autoprodução e cogeração, e mesmo as fontes não renováveis que se encontram perto dos centros de consumo - com o objetivo de coordenar o seu funcionamento conjunto para satisfazer os requisitos da demanda de energia, por meio do uso da Tecnologia da Informação e Telecomunicações ou das Redes Inteligentes (Smart Grid), e, além disso, configurando a usina virtual de maneira que ela atue , no que diz respeito ao sistema interconectado de energia elétrica, rede de transmissão ou distribuição, como se fosse uma planta de geração de grande porte. Apresenta-se uma análise e projeções da operação e do suprimento do subsistema Sudeste/Centro Oeste (SE/CO), com foco especial no Sistema Elétrico do Estado de São Paulo, com suas interações com o SIN. / This Thesis presents contributions for the planning of the electrical system of São Paulo State Brazil, emphasizing benefits obtained with a better integration of renewables (mainly solar photovoltaics, wind energy, from municipal solid waste and sugarcane vinasse) aiming energy security and independence, loss reduction and environmental benefits. For this, the concept of VPP Virtual Power Plant is introduced, in which a better and broader use of energetic resources, geographically dispersed, including distributed microgeneration, autoproduction, cogeneration, and even nonrenewable sources, is facilitated. Those disperse energy resources, in general located near the consumer, are grouped and jointly managed, in order of to satisfy the demand requirements, using Smart Grid and Information and Telecommunication Technologies, and making the VPP to act, for the transmission or distribution system point of view, as a conventional large electrical power plant. An analysis and forecasts of the operation of the Brazilian Southeast/Center West (SE/CO) subsystem, with special focus on São Paulo State electrical system, and its interactions with the Brazilian Interconnected Electrical System (SIN), is presented.

Neste trabalho é apresentada uma abordagem fuzzy para a estabilização de uma classe de sistemas não-lineares com parâmetros descritos por saltos Markovianos. Uma nova modelagem fuzzy de sistemas é formulada para representar esta classe de sistemas na vizinhança de pontos de operação escolhidos. A estrutura deste sistema fuzzy é composta de dois níveis, um para descrição dos saltos Markovianos e outro para descrição das não-linearidades no estado do sistema. Condições suficientes para a estabilização estocástica do sistema fuzzy considerado são derivadas usando uma função de Lyapunov acoplada. O projeto de controle fuzzy é então formulado a partir de um conjunto de desigualdades matriciais lineares. Em adição, um exemplo de aplicação, envolvendo a representação da operação de um sistema elétrico de potência em esquema de co-geração por um sistema com saltos Markovianos, é construído para validação dos resultados. / This work deals with the fuzzy-model-based control design for a class of Markovian jump nonlinear systems. A new fuzzy system modeling is proposed to approximate the dynamics of this class of systems. The structure of the new fuzzy system is composed of two levels, a crisp level which describes the Markovian jumps and a fuzzy level which describes the system nonlinearities. A sufficient condition on the existence of a stochastically stabilizing controller using a Lyapunov function approach is presented. The fuzzy-model-based control design is formulated in terms of a set of linear matrix inequalities. In addition, simulation results for a single-machine infinite-bus power system in cogeneration scheme, whose operation is modeled as an Markovian jump nonlinear system, are presented to illustrate the applicability of the technique.