Hyperbolic type metrics and distortion of quasiconformal map pings

This Ph.D. thesis consists of four original papers. The papers cover several topics from geometric function theory, more specifically, hyperbolic type metrics, conformal invariants, and the distortion properties of quasiconformal mappings. The first paper deals mostly with the quasihyperbolic metric. The main result gives the optimal bilipschitz constant with respect to the quasihyperbolic metric for the M¨obius self-mappings of the unit ball. A quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is also proved. The second paper studies some distortion estimates for the class of quasiconformal self-mappings fixing the boundary values of the unit ball or convex domains. The distortion is measured by the hyperbolic metric or hyperbolic type metrics. The results provide explicit, asymptotically sharp inequalities when the maximal dilatation of quasiconformal mappings tends to 1. These explicit estimates involve special functions which have a crucial role in this study. In the third paper, we investigate the notion of the quasihyperbolic volume and find the growth estimates for the quasihyperbolic volume of balls in a domain in terms of the radius. It turns out that in the case of domains with Ahlfors regular boundaries, the rate of growth depends not merely on the radius but also on the metric structure of the boundary. The topic of the fourth paper is complete elliptic integrals and inequalities. We derive some functional inequalities and elementary estimates for these special functions. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.

Price spike forecasting in a competitive day-ahead energy market

Electricity price forecasting has become an important area of research in the aftermath of the worldwide deregulation of the power industry that launched competitive electricity markets now embracing all market participants including generation and retail companies, transmission network providers, and market managers. Based on the needs of the market, a variety of approaches forecasting day-ahead electricity prices have been proposed over the last decades. However, most of the existing approaches are reasonably effective for normal range prices but disregard price spike events, which are caused by a number of complex factors and occur during periods of market stress. In the early research, price spikes were truncated before application of the forecasting model to reduce the influence of such observations on the estimation of the model parameters; otherwise, a very large forecast error would be generated on price spike occasions. Electricity price spikes, however, are significant for energy market participants to stay competitive in a market. Accurate price spike forecasting is important for generation companies to strategically bid into the market and to optimally manage their assets; for retailer companies, since they cannot pass the spikes onto final customers, and finally, for market managers to provide better management and planning for the energy market. This doctoral thesis aims at deriving a methodology able to accurately predict not only the day-ahead electricity prices within the normal range but also the price spikes. The Finnish day-ahead energy market of Nord Pool Spot is selected as the case market, and its structure is studied in detail. It is almost universally agreed in the forecasting literature that no single method is best in every situation. Since the real-world problems are often complex in nature, no single model is able to capture different patterns equally well. Therefore, a hybrid methodology that enhances the modeling capabilities appears to be a possibly productive strategy for practical use when electricity prices are predicted. The price forecasting methodology is proposed through a hybrid model applied to the price forecasting in the Finnish day-ahead energy market. The iterative search procedure employed within the methodology is developed to tune the model parameters and select the optimal input set of the explanatory variables. The numerical studies show that the proposed methodology has more accurate behavior than all other examined methods most recently applied to case studies of energy markets in different countries. The obtained results can be considered as providing extensive and useful information for participants of the day-ahead energy market, who have limited and uncertain information for price prediction to set up an optimal short-term operation portfolio. Although the focus of this work is primarily on the Finnish price area of Nord Pool Spot, given the result of this work, it is very likely that the same methodology will give good results when forecasting the prices on energy markets of other countries.

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