An econometric model for intraday electricity trading

This paper develops an econometric price model with fundamental impacts for intraday electricity markets of 15-min contracts. A unique dataset of intradaily updated forecasts of renewable power generation is analysed. We use a threshold regression model to examine how 15-min intraday trading depends on the slope of the merit order curve. Our estimation results reveal strong evidence of mean reversion in the price formation mechanism of 15-min contracts. Additionally, prices of neighbouring contracts exhibit strong explanatory power and a positive impact on prices of a given contract. We observe an asymmetric effect of renewable forecast changes on intraday prices depending on the merit-order-curve slope. In general, renewable forecasts have a higher explanatory power at noon than in the morning and evening, but price information is the main driver of 15-min intraday trading. This article is part of the theme issue ‘The mathematics of energy systems’.

APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS

In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system. In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system. For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values ​​which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.

Some Properties of the Exeter Transformation

The Exeter point of a given triangle ABC is the center of perspective of the tangential triangle and the circummedial triangle of the given triangle. The process of the Exeter point from the centroid serves as a base for defining the Exeter transformation with respect to the triangle ABC, which maps all points of the plane. We show that a point, its image, the symmedian, and three exsymmedian points of the triangle are on the same conic. The Exeter transformation of a general line is a fourth-order curve passing through the exsymmedian points. We show that each image point can be the Exeter transformation of four different points. We aim to determine the invariant lines and points and some other properties of the transformation.

MODELING OF THE 2ND ORDER CURVES AND SURFACES OF ENGINEERING STRUCTURES SHELLS BASED ON THEIR BASIS

The paper describes an example of modeling an arc of a 2nd order curve using an engineering discriminant and its analytical description based on a graphical algorithm for constructing a curve in point calculus. Examples of modeling the surfaces of engineering structures shells on an elliptical and rectangular plan are given. Research methods include geometric algorithms: modeling of 2nd order curves passing through 3 predetermined points in advance and having tangents at the start and end points, and shell surfaces based on them; analytical definition of curves arcs and sections of surfaces using the mathematical apparatus point calculation in a given parametrization and taking into account all predetermined geometric conditions. This approach can be widely used in the practice of modeling the shells of engineering structures for various technical purposes. It allows the designer to choose the best curvature of the shell surface, which will have the necessary strength characteristics, technical aesthetics and artistic expressiveness. The possibility of dividing the surface of the shell into finite elements of a given amount is also provided for studying the stress-strain state of the shell under the action of various loads in the systems of finite element analysis.

Intraday Electricity Pricing of Night Contracts

This paper investigates the intraday electricity pricing of 15-min. contracts in night hours. We tailor a recently introduced econometric model with fundamental impacts, which is successful in describing the pricing of day contracts. Our estimation results show that the mean reversion and the positive price impact of neighboring contracts are generic features of the price formation process on the intraday market, independent of the time of day. Intraday auction prices have higher explanatory power for the pricing of night than day contracts, particularly, for the first and last 15-min. contract in a night hour. Intradaily updated forecasts of wind power infeed are the only significant fundamental factors for intraday electricity prices at night. Neither expected conventional capacities nor the slope of the merit order curve contribute to explaining price dynamics. Overall, we conclude that fundamentals lose in importance in night hours and the 15-min. intraday market is rather driven by price information.

Design method of a novel variable speed ratio line gear mechanism

Based on the space curve meshing theory, a novel noncircular line gear mechanism was advanced, namely, this paper presented a design method of the variable speed ratio noncircular line gear with coplanar axes. Firstly, the universal contact curve equations of the constant speed ratio and variable speed ratio line teeth were established. After the constraint equations of the rotating angle of the driving and driven variable speed ratio noncircular line gears were analyzed and established, the relationship between the rotating angle of the driven variable speed ratio noncircular line gear and the parameter t in the VSR area was assumed to be a piecewise fourth-order curve. Then, the contact curve equations of the variable speed ratio noncircular line gears were derived, and the entity models of variable speed ratio noncircular line gears were built. The prototypes of the parallel axis and intersecting axis variable speed ratio noncircular line gears were manufactured by Stereo Lithography Apparatus, and the speed ratios were measured on the kinematic test rig. The kinematic and finite element analysis results demonstrate that the relationship between the rotating angle of the driven variable speed ratio noncircular line gear and the parameter t conforms to the designed function and the noncircular line teeth smoothly achieve the preset VSR transmission. The proposed design method is helpful to design the variable speed ratio noncircular line gears with lower theoretical sliding rate and wider variation range of the speed ratio; consequently, the designed variable speed ratio noncircular line gears have better applicability in specific variable speed ratio applications.