Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.

Stochastic Approach to Hosting Limit of Transmission System and Improving Method Utilizing HVDC

According to the global de-carbonization trends, renewable energy integration has become an increasingly important issue in power systems. To achieve 100% renewable energy integration and operate a system with these resources, it is necessary to appropriately evaluate the system hosting capability and prepare appropriate planning and operation strategies using the evaluation result. So far, these interests have focused particularly on distribution-level systems. However, although the hosting limit in transmission-level systems requires further consideration, previous study is limited. This study introduces the constraints on the transmission-level hosting limit. In addition, a stochastic estimation of the hosting limit methodology in the transmission system and the use of a high voltage direct current system to improve hosting capacity are proposed and evaluated. Moreover, these methodology-based simulations are conducted using possible scenarios on the IEEE 39 bus system with some constraints, and the simulation results are presented herein. The results showed that the HVDC location selection and operation using the proposed method and optimization technique is appropriate. The strategy can be used to integrate more renewable energy. Furthermore, the proposed methodology can be applied to renewable energy integration scenario establishing a plan.

Effective Optimisation of the Patient Circuits of an Oncology Day Hospital: Mathematical Programming Models and Case Study

In this paper, we first use the information we have on the patients of an oncology day hospital to distribute the treatment schedules they have in each of the visits to this centre. To do this, we propose a deterministic mathematical programming model in such a way that we minimise the duration of the waiting room stays of the total set of patients and taking into account the restrictions of the circuit. Secondly, we will look for a solution to the same problem under a stochastic approach. This model will explicitly consider the existing uncertainty in terms of the different times involved in the circuit, and this model also allows the reorganisation of the schedules of medical appointments with oncologists. The models are complemented by a tool that solves the problem of assigning nurses to patients. The work is motivated by the particular characteristics of a real hospital and the models are used and compared with data from this case.

Stochastic estimation of the distribution of soil water repellency on the soil surface in a humid-temperate forest

Soil water repellency (SWR) increases surface runoff and preferential flows. Thus, quantitative evaluation of SWR distribution is necessary to understand water movements. Because the variability of SWR distribution makes it difficult to measure directly, we developed a method for estimating an SWR distribution index, defined as the areal fraction of surface soil showing SWR (SWRarea). The theoretical basis of the method is as follows: (1) SWRarea is equivalent to the probability that a position on the soil surface is drier than the critical water content (CWC); SWR is present (droplets absorbed in >10 s) when the soil surface is drier than the CWC and absent when it is wetter. (2) CWC and soil moisture content (θ) are normally distributed independent variables. (3) Thus, based on probability theory, the cumulative normal distribution of θ – CWC (f(x)) can be obtained from the distributions of CWC and θ, and f(0), the cumulative probability that θ – CWC < 0, gives the SWRarea. To investigate whether the method gives reasonable results, we repeatedly measured θ at 0–5 cm depth and determined the water repellency of the soil surface at multiple points in fixed plots with different soils and topography in a humid-temperate forest. We then calculated the CWC from the observed θ–SWR relationship at each point. We tested the normality of the CWC and θ distributions and the correlation between CWC and θ. Then, we determined f(x) from the CWC and θ distributions and estimated the SWRarea on each measurement day. Although CWC and θ were both normally distributed, in many cases they were correlated. Nevertheless, the CWC–θ dependency had little effect on the estimation error, and f(x) explained 69% of the SWRarea variability. Our findings show that a stochastic approach is useful for estimating SWRarea.