There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Numerical Initial Value Problems in Ordinary Differential Equations

Solar and Terrestrial Radiation

Energy is an essential ingredient of socio-economic development and economic growth. Renewable energy sources like wind energy is indigenous and can help in reducing the dependency on fossil fuels. Wind is the indirect form of solar energy and is always being replenished by the sun. Wind is caused by differential heating of the earth's surface by the sun. It has been estimated that roughly 10 million MW of energy are continuously available in the earth's wind. Wind energy provides a variable and environmental friendly option and national energy security at a time when decreasing global reserves of fossil fuels threatens the long-term sustainability of global economy. This paper reviews the wind resources assessment models, site selection models and aerodynamic models including wake effect. The different existing performance and reliability evaluation models, various problems related to wind turbine components (blade, gearbox, generator and transformer) and grid for wind energy system have been discussed. This paper also reviews different techniques and loads for design, control systems and economics of wind energy conversion system.

A review of wind energy technologies

The probability density function (PDF) of wind speed is important in numerous wind energy applications. A large number of studies have been published in scientific literature related to renewable energies that propose the use of a variety of PDFs to describe wind speed frequency distributions. In this paper a review of these PDFs is carried out. The flexibility and usefulness of the PDFs in the description of different wind regimes (high frequencies of null winds, unimodal, bimodal, bitangential regimes, etc.) is analysed for a wide collection of models. Likewise, the methods that have been used to estimate the parameters on which these models depend are reviewed and the degree of complexity of the estimation is analysed in function of the model selected: these are the method of moments (MM), the maximum likelihood method (MLM) and the least squares method (LSM). In addition, a review is conducted of the statistical tests employed to see whether a sample of wind data comes from a population with a particular probability distribution. With the purpose of cataloguing the various PDFs, a comparison is made between them and the two parameter Weibull distribution (W.pdf), which has been the most widely used and accepted distribution in the specialised literature on wind energy and other renewable energy sources. This comparison is based on: (a) an analysis of the degree of fit of the continuous cumulative distribution functions (CDFs) for wind speed to the cumulative relative frequency histograms of hourly mean wind speeds recorded at weather stations located in the Canarian Archipelago; (b) an analysis of the degree of fit of the CDFs for wind power density to the cumulative relative frequency histograms of the cube of hourly mean wind speeds recorded at the aforementioned weather stations. The suitability of the distributions is judged from the coefficient of determination R2. Amongst the various conclusions obtained, it can be stated that the W.pdf presents a series of advantages with respect to the other PDFs analysed. However, the W.pdf cannot represent all the wind regimes encountered in nature such as, for example, those with high percentages of null wind speeds, bimodal distributions, etc. Therefore, its generalised use is not justified and it will be necessary to select the appropriate PDF for each wind regime in order to minimise errors in the estimation of the energy produced by a WECS (wind energy conversion system). In this sense, the extensive collection of PDFs proposed in this paper comprises a valuable catalogue.

A review of wind speed probability distributions used in wind energy analysis: Case studies in the Canary Islands

Recently differential transform method (DTM) has been used to solve various partial differential equations. In this paper, an alternative approach called the reduced differential transform method (RDTM) is presented to overcome the demerit of complex calculation of differential transform method. Comparing the methodology with some known techniques shows that the present approach is effective and powerful. In addition, three test problems of mathematical physics are discussed to illustrate the effectiveness and the performance of the reduced differential transform method.

Reduced Differential Transform Method for Partial Differential Equations

The main objective of the present study is to estimate wind power potential using the two Weibull parameters of the wind speed distribution function, the shape parameter k (dimensionless) and the scale parameter c (m/s). In this regard, a methodology that uses three various techniques (maximum likelihood, least squares, and method of moments) for estimating the Weibull parameters was given first. The methodology was then applied to a region in Turkey. Finally, the parameter techniques were compared to Monte-Carlo simulation in different sample sizes, and the best parameter estimation techniques belonging to the sample sizes were also determined.

Estimation of Wind Power Potential Using Weibull Distribution

This paper presents the generalization of the differential transformation method to n-dimensional case in order to solve partial differential equations (PDEs). A distinctive practical feature of this method is its ability to solve especially nonlinear differential equations efficiently. We apply our results to a few initial boundary-value problems to illustrate the proposed method.

n-Dimensional differential transformation method for solving PDEs

Delay differential equations (DDEs) are a large and important class of dynamical systems. Many phenomena in applied sciences can be successfully modelled by these equations [1-4]. For example, in natural or control systems, a controller monitors the state of the system and make adjustments to the system based on its observations. Therefore a delay occurs between the observation and the control action. In particular, they become powerful models in explaining physical phenomena when ODEbased models fail. There are different kinds of DDEs studied by various methods [5-12] In this paper, we deal with the generalization of the pantograph equation which is studied in [5,6] given by

Approximate Solutions of Generalized Pantograph Equations by the Differential Transform Method

In this paper, a general framework of the reduced differential transform method is presented for solving the generalized Korteweg–de Vries equations. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology with some known techniques shows that the present approach is effective and powerful. In addition, three test problems of mathematical physics are discussed to illustrate the effectiveness and the performance of the reduced differential transform method.

https://www.mdpi.com/2297-8747/15/3/382/pdf

Galip Oturanc

Papers

Reduced Differential Transform Method for Partial Differential Equations

Estimation of Wind Power Potential Using Weibull Distribution

n-Dimensional differential transformation method for solving PDEs

Approximate Solutions of Generalized Pantograph Equations by the Differential Transform Method

Reduced Differential Transform Method for Generalized KdV Equations