On the Hilbert function of general fat points in P1×P1\mathbb{P}^1 \times \mathbb{P}^1

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.Comment: 25 pages; minor changes (Remark 1.7 added and Example 3.13 improved

Higher secant varieties of Pn×Pm\mathbb{P}^n \times \mathbb{P}^m embedded in bi-degree (1,d)(1,d)

Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.Comment: 8 page

Progress on the symmetric Strassen conjecture

Let F and G be homogeneous polynomials in disjoint sets of variables. We prove that the Waring rank is additive, thus proving the symmetric Strassen conjecture, when either F or G is a power, or F and G have two variables, or either F or G has small rank

Model-Based Identification of Alternative Bidding Zones: Applications of Clustering Algorithms with Topology Constraints

The definition of bidding zones is a relevant question for electricity markets. The bidding zones can be identified starting from information on the nodal prices and network topology, considering the operational conditions that may lead to congestion of the transmission lines. A well-designed bidding zone configuration is a key milestone for an efficient market design and a secure power system operation, being the basis for capacity allocation and congestion management processes, as acknowledged in the relevant European regulation. Alternative bidding zone configurations can be identified in a process assisted by the application of clustering methods, which use a predefined set of features, objectives and constraints to determine the partitioning of the network nodes into groups. These groups are then analysed and validated to become candidate bidding zones. The content of the manuscript can be summarized as follows: (1) A novel probabilistic multi-scenario methodology was adopted. The approach needs the analysis of features that are computed considering a set of scenarios defined from solutions in normal operation and in planned maintenance cases. The weights of the scenarios are indicated by TSOs on the basis of the expected frequency of occurrence; (2) The relevant features considered are the Locational Marginal Prices (LMPs) and the Power Transfer Distribution Factors (PTDFs); (3) An innovative computation procedure based on clustering algorithms was developed to group nodes of the transmission electrical network into bidding zones considering topological constraints. Several settings and clustering algorithms were tested in order to evaluate the robustness of the identified solution

Flexibility chart 2.0: An accessible visual tool to evaluate flexibility resources in power systems

ABSTRACT: Various aspects of power system flexibility are evaluated within the multi-country study framework of IEA Wind Task 25. Grid components and actions which have been adopted for enhancing flexibility in different areas, countries, regions are addressed, as well as how Transmission System Operators, Independent System Operators, Utilities intend to manage variable generation in their operating strategies. A visual assessment to evaluate the diversity of flexibility sources, called a “flexibility chart”, is further developed to illustrate several flexibility parameters (e.g., hydropower, pumped hydro, gas turbine, combined heat and power, interconnection and battery) in a polygonal radar (fan-shaped) chart. This enhanced version of the Flexibility Chart is an “at-a-glance” and “easy-to-understand” tool to show how to estimate the potential of flexibility resources in a given country or area, and is accessible for non-technical experts. The Flexibility Chart 2.0 is also a useful tool to compare the past and future flexibility of a system. Comparing the historical change of flexibility resources may not only be helpful to discuss energy policy in regions with high installed variable renewable generation, but also to contribute to the discussion in other regions where renewables have not been widely adopted yet.info:eu-repo/semantics/publishedVersio

Summary of experiences and studies for wind integration: IEA Wind Task 25

IEA WIND R&D Task 25 on “Design and Operation of Power Systems with Large Amounts of Wind Power” collects and shares information on wind generation impacts on power systems, with analyses and guidelines on methodologies. This paper summarizes the main results from the report published on January 2013 describing experience of wind integration as well as the most relevant wind power grid integration studies in the 15 participating countries. The studies build on the already significant experience in integrating wind power in power systems addressing concerns about the impact of wind power’s variability and uncertainty on power system security of supply and costs as well as grid reinforcement needs. The mitigation of wind power impacts includes more flexible operational methods, incentivising flexibility in other generating plants, increasing interconnection to neighbouring regions, and application of demand-side flexibility. Electricity storage is still not as cost effective in larger power systems as other means of flexibility, but is already seeing initial applications in places with limited transmission.

Wind and solar curtailment

High penetrations of wind and solar generation on power systems are resulting in increasing curtailment. Wind and solar integration studies predict increased curtailment as penetration levels grow. This paper examines experiences with curtailment on bulk power systems internationally. It discusses how much curtailment is occurring, how it is occurring, why it is occurring, and what is being done to reduce curtailment. This summary is produced as part of the International Energy Agency Wind Task 25 on Design and Operation of Power Systems with Large Amounts of Wind Power