In this work we will demonstrate that the Cauchy problem associated with the Korteweg-de Vries equation, denoted by KdV, and Korteweg-de Vries modified equation, denoted by mKdV, with initial data in the space of Sobolev Hs(|R), is locally well-posed on Hs(|R), with s>3/4 for KdV and s≥1/4 for mKdV, where the notion of well-posedness includes existence, uniqueness, persistence property of solution and continuous dependence of solution with respect to the initial data. This result is based on the works of Kenig, Ponce and Vega. The technique used to obtain these results is based on fixed point Banach theorem combined with the regularizantes effects of the group associated with the linear part. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Neste trabalho demonstraremos que o problema de Cauchy associado as equações de Korteweg-de Vries, denotada por KdV, e de Korteweg-de Vries modificada, denotada por mKdV, com dado inicial no espaço de Sobolev Hs(|R), é bem posto localmente em Hs(|R), com s>3/4 para a KdV e s≥1/4 para a mKdV, onde a noção de boa postura inclui a existência, unicidade, a propriedade de persistência da solução e dependência contínua da solução com relação ao dado inicial. Este resultado é baseado nos trabalhos de Kenig, Ponce e Vega. A técnica utilizada para obter tais resultados se baseia no Teorema do Ponto Fixo de Banach combinada com os efeitos regularizantes do grupo associado com a parte linear.

Variable renewable energy (VRE) is becoming more present in the energy mix as society aims to shift power production from fossil fuels. Solar and wind power are VRE sources, which can result in fluctuations in the input to the network, creating supply uncertainty. The Swedish Transmission system operator (TSO), Svenska kraftnät (Svk), is obliged to ensure stability in the network, which must be considered when approving applications for subscriptions of connections to the transmission network. In the stability investigations performed before approving applications, Svk assumes that solar and wind power produces at their maximum capacity at their connection point before approving applications, to ensure safe and stable power system operation under any weather conditions. However, this might be an over-conservative assumption as the share of VRE sources increases. This is because there is an anti-correlation between wind and solar power, which results in a few colliding hours when both produce at full capacity. This study investigates the correlation between solar and wind power, evaluates the probability of high wind and solar power production coinciding at different percentages of their capacity, and analyses the potential for aggregating them in the studies performed to approve connections to the transmission network. The research method applied was an inductive approach, where a statistical analysis using Pearson's correlation and conditional probability was conducted based on reanalysed data collected from ERA5 and STRÅNG databases. The analysis showed a weak complementary correlation coefficient for all time periods, and the probability of very high production coinciding is small. If it occurs, it is more likely to happen during the first part of the year. The results show a spatial correlation in the probability. By implementing integration strategies, the conclusion is that there is a controllable risk when increasing the capacity. Even if there is a risk, it can be worth taking as more VRE sources can be connected to the transmission network. / Förnybar energi har blivit mer närvarande i dagens energimix när samhället strävar efter att förändra kraftproduktionen från fossilt till förnybara källor. Sol- och vindkraft är varierande förnybara energikällor vilket skapar fluktationer i inmatning till nätet samt ökad osäkerhet kring mängden tillgänglig produktion. Systemoperatören (TSO), Svenska kraftnät (Svk) är skyldig att säkerhetställa stabilitet i nätet, vilket måste beaktas vid ansökningar om anslutningar till transmissionsnätet. Denna studie undersöker därför korrelationen mellan sol- och vindkraft, samt utvärderar sannolikheten att hög sol- och vindkrafts produktion sammanfaller samtidigt. Forskningsmetoden som tillämpades var en induktiv metod, där en statistisk analys med Pearsons korrelationskoefficient såväl som sannolikheten mellan att deras produktion sammanföll samtidigt utfördes, baserat på återskapad väderdata insamlad från ERA5 och STRÅNG. Analysen visade svag komplementär korrelationskoefficient för alla tidsperioder och sannolikheten för att mycket hög produktion sammanfaller var liten. De tillfällen som produktionen sammanföll inträffade majoriteten under början av året. Genom implementering av de tillvalda strategierna är slutsatsen att det finns en kontrollerbar risk när man ökar den tillåtna anslutna kapaciteten sol- och vindkraft i transmissionsnätet. Det finns också en antikorrelation mellan sol- och vindkrafs produktion vilket resulterade i endast ett fåtal sammanfallande timmar när båda producerar på maximal kapacitet.