Probability and Mathematical Statistics in power systems

In this work, we introduce a new generalization of the Inverted Weibull distribution called the alpha power Extended Inverted Weibull distribution using the alpha power transformation method. This approach adds an extra parameter to the baseline distribution. The statistical properties of this distribution including the mean, variance, coefficient of variation, quantile function, median, ordinary and incomplete moments, skewness, kurtosis, moment and moment generating functions, reliability analysis, Lorenz and Bonferroni and curves, Ré nyi of entropy and order statistics are studied. We consider the method of maximum likelihood for estimating the model parameters and the observed information matrix is derived. Simulation method and three real life data sets are presented to demonstrate the effectiveness of the new model.

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.

A strategy for adaptive control and energetic optimization of aerobic fermentors was implemented, with both air flow and agitation speed as manipulated variables. This strategy is separable in its components: control, optimization, estimation. We optimized parameter’s estimation (from the usual KLa correlation) using sinusoidal excitation of air flow and agitation speed. We have implemented parameter’s estimation trough recursive least squares algorithm with forgetting factor. We carried separate essays on control, optimization and estimation algorithms. We carried our essays using an original computational simulation environment, with noise and delay generating facilities for data sampling and filtering.

Our results show the convergence and robustness of the estimation algorithm used, improved with use of both forgetting factor and KLa dead-band facilities. Control algorithm used in our work compares favorably with PID using the integrated area criteria for deviation between oxygen molarity and critical molarity (set point). Optimization algorithm clearly reduces energetic consumption, respecting critical molarity. Integration of control, optimization and adaptive algorithms was implemented, but future work is needed for stability. Methods were defined and implemented for stability improvement. We have implemented data acquisition and computer manipulation of air flow and agitation speed for actual fermentors.

In this work, we introduce a new generalization of the Inverted Weibull distribution called the alpha power Extended Inverted Weibull distribution using the alpha power transformation method. This approach adds an extra parameter to the baseline distribution. The statistical properties of this distribution including the mean, variance, coefficient of variation, quantile function, median, ordinary and incomplete moments, skewness, kurtosis, moment and moment generating functions, reliability analysis, Lorenz and Bonferroni and curves, Ré nyi of entropy and order statistics are studied. We consider the method of maximum likelihood for estimating the model parameters and the observed information matrix is derived. Simulation method and three real life data sets are presented to demonstrate the effectiveness of the new model.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

RESUMO A perspectiva de uma maior participação das fontes eólicas no Sistema Interligado Nacional aponta para a necessidade de incluir a previsão de curto prazo da geração eólica nos procedimentos da operação em tempo real e da programação diária da operação. No presente trabalho apresentam-se duas abordagens para previsão probabilística da geração eólica: regressão quantílica e uma metodologia híbrida que congrega processos gaussianos, estimador de densidade de probabilidade e simulação de Monte Carlo. Uma avaliação comparativa das duas abordagens propostas foi conduzida por meio de um experimento computacional com dados reais de um parque eólico localizado na Galícia, Espanha. PALAVRAS-CHAVE Energia eólica, Estimação de densidade, Previsão probabilística, Processo gaussiano, Regressão quantílica 1.0-INTRODUÇÃO O comportamento intermitente da velocidade do vento impõe desafios para a integração segura e econômica dos aproveitamentos eólicos aos sistemas de potência. Para viabilizar a integração da geração eólica os centros de operação demandam novas ferramentas de gerenciamento de energia elétrica, entre as quais se destaca a previsão da geração eólica [1]. Em geral, os métodos de previsão privilegiam a predição do valor esperado da geração eólica sem, contudo, quantificar e informar apropriadamente a incerteza associada à previsão. Em virtude do comportamento aleatório e intermitente do vento, a previsão do valor esperado da geração eólica sem uma quantificação adequada da incerteza não atende completamente as necessidades da operação. Por exemplo, em sistemas com grande participação de fontes eólicas a reserva operativa deve ser dimensionada de maneira a cobrir as incertezas nas previsões de carga e de geração eólica [2]. A densidade de probabilidade contém toda a informação de uma variável aleatória, portanto, para lidar com a incerteza deve-se considerar as previsões de densidade de probabilidade da geração eólica fornecidas por métodos de previsão probabilística [3]. O presente trabalho tem por objetivo descrever e comparar duas propostas para previsão probabilística da geração eólica: a regressão quantílica [4] e uma metodologia híbrida que congrega processos gaussianos [5], estimador de densidade de probabilidade [6] e simulação de Monte Carlo [7]. A implementação e análise comparativa das duas abordagens foi conduzida por meio de um programa em linguagem R [8] aplicado em um experimento computacional que contou com dados reais provenientes de um parque eólico localizado na Galícia, Espanha. A seguir, na seção 2 tem-se uma descrição da metodologia híbrida e na seção 3 uma breve introdução à regressão quantílica. Na sequência, na seção 4 apresentam-se algumas métricas para avaliação da qualidade de previsões probabilísticas. Os principais resultados são descritos na seção 5, em especial, as previsões probabilísticas das médias horárias da geração eólica até 48 horas à frente, ao longo de uma semana. Por fim, na seção 6 tem-se as conclusões do trabalho.

A fixed company of players observes a person selected from a fixed queue. After each observation, players are asked to bet the dollar secret from others, either on the fact that person is bald, or on what is not. A denite formula of the gain is suggested, such that every time after bets the gain of each player from a given company are completely determined by this formula. However, before bets player's gain is an uncertain value. Is it possible for a given company of players and a given queue of people before bets to build a correct mathematical model of uncertaint gain of each player within the framework of Kolmogorov's probability theory? If not, what else do you need to add to the foundations of probability theory so that before bets to be able to use this model for decision making? The paper answers these questions within the framework of the new theory of experience and of chance (the certainty theory) [1] that consists of two dual halves: the believability theory and the probability theory, and that is intended for the mathematical description of experienced-random experiments, the uncertainty in outcomes of which is generated by the observer. A xed company of players observes a person selected from a xed queue. After each observation, players are asked to bet the dollar secret from others, either on the fact that the person is bald, or on what is not. If, as a result of the bets made, the player's choice is in the minority, the player loses the bet. If, on the other hand, in the majority, the player is returned his bet plus an equal share of the loser players' bets. With a draw, the equality of the number of betting on the bald and the number of betting on the not bald, all players remain at their own: they are returned their bets 1. Every time after the bets, the gain of each player from this company is fully certain. But before the bets, the gain is an uncertain value. Is it possible for a given company of players and a given queue of persons before bets to build a correct mathematical model of uncertain gain of each player within the framework of Kolmogorov's probability theory?
c ⃝ 2016 O.Yu.Vorobyev Oleg Vorobyev (ed.), Proc. XV FAMEMS'2016, Krasnoyarsk: SFU

Imagine that you are conducting an experiment, as a result of which the event occurs or does not occur. You must determine whether this event occurred or not. Your experience tells you that many times in the past you, being exactly in the same circumstances, only in half of the cases came to the conclusion this event happened. In this situation I would say your believability of this event happened is equaled to the half. But you can't answer in this way. You are required to answer only either ``Yes'' or ``No''. What would you do in this situation? You must not think such situations are big rare. Refresh your memory what are you doing every day under uncertainty? Every day you are doing only it and no more. The question arises how you do this action?
If I say you ``toss a coin in your mind, or in your thoughts'' I'll do the mistake. The point is every one of us has an ability to make the believabilistic choice. You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.
Therefore, to denote this action, I will choose the new expression ``tossing an experienced coin'' in an experienced experiment that becomes dual to the usual expression ``tossing a coin'' in a random experiment.

We introduce the set-theoretic language for the element-set labelling a Cartesian product by measurable binary relations intended for the
labelling, or for the naming of parts and details of the construction that we are going to propose in the theory of experience and of chance as a
mathematical model of an event as a dual pair.

Summary form only given. This work presents a methodology for calculating the optimal generation capacity reserve and siting in power systems planning. The proposed methodology is based on a least-cost planning approach which considers along the planning horizon, the capacity reserve investment costs and the operational expected costs, comprising thermal generation and interruption costs. This problem is solved by using the Benders decomposition technique in which investment and operation subproblems solutions are obtained iteratively and separately thus allowing the use of tailor-made algorithms for each subproblem. The operation subproblem corresponds to a probabilistic evaluation of the minimum operation cost in composite generation/transmission systems, in which the system states are selected through Monte Carlo simulation or state enumeration and the system performance is analysed by a nonlinear optimal power flow solved by an interior point algorithm. The proposed methodology is illustrated in a case study with a modification of the IEEE Reliability Test System (MRTS).