Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments,

This paper continues the investigation of Du and Lou (J. European Math Soc, to appear), where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)$ for $x$ over a varying interval $(g(t), h(t))$ was examined. Here $x=g(t)$ and $x=h(t)$ are free boundaries evolving according to $g'(t)=-\mu u_x(t, g(t))$, $h'(t)=-\mu u_x(t,h(t))$, and $u(t, g(t))=u(t,h(t))=0$. We answer several intriguing questions left open in the paper of Du and Lou.First we prove the conjectured convergence result in the paper of Du and Lou for the general case that $f$ is $C^1$ and $f(0)=0$. Second, for bistable and combustion types of $f$, we determine the asymptotic propagation speed of $h(t)$ and $g(t)$ in the transition case. More presicely, we show that when the transition case happens, for bistable type of $f$ there exists a uniquely determined $c_1>0$ such that $\lim_{t\to\infty} h(t)/\ln t=\lim_{t\to\infty} -g(t)/\ln t=c_1$, and for combustion type of $f$, there exists a uniquely determined $c_2>0$ such that $\lim_{t\to\infty} h(t)/\sqrt t=\lim_{t\to\infty} -g(t)/\sqrt t=c_2$. Our approach is based on the zero number arguments of Matano and Angenent, and on the construction of delicate upper and lower solutions

The Fisher-KPP equation over simple graphs: Varied persistence states in river networks

In this article, we study the growth and spread of a new species in a river network with two or three branches via the Fisher-KPP advection-diffusion equation over some simple graphs with every edge a half infinite line. We obtain a rather complete description of the long-time dynamical behavior for every case under consideration, which can be loosely described by a trichotomy (see Remark 1.7), including two different kinds of persistence states as parameters vary. The phenomenon of "persistence below carrying capacity" revealed here appears new, which does not occur in related models of the existing literature where the river network is represented by graphs with finite-lengthed edges, or the river network is simplified to a single infinite line

LSTM-based energy management for electric vehicle charging in commercial-building prosumers

As typical prosumers, commercial buildings equipped with electric vehicle (EV) charging piles and solar photovoltaic panels require an effective energy management method. However, the conventional optimization-model-based building energy management system faces significant challenges regarding prediction and calculation in online execution. To address this issue, a long short-term memory (LSTM) recurrent neural network (RNN) based machine learning algorithm is proposed in this paper to schedule the charging and discharging of numerous EVs in commercial-building prosumers. Under the proposed system control structure, the LSTM algorithm can be separated into offline and online stages. At the offline stage, the LSTM is used to map states (inputs) to decisions (outputs) based on the network training. At the online stage, once the current state is input, the LSTM can quickly generate a solution without any additional prediction. A preliminary data processing rule and an additional output filtering procedure are designed to improve the decision performance of LSTM network. The simulation results demonstrate that the LSTM algorithm can generate near-optimal solutions in milliseconds and significantly reduce the prediction and calculation pressures compared with the conventional optimization algorithm

Aggregated feasible active power region for distributed energy resources with a distributionally robust joint probabilistic guarantee

Distributed Energy Resources (DERs) have valuable flexibility to provide grid services. The Aggregated Feasible Active Power Region (AFAPR) is useful for aggregating DERs and reducing the computational burden in system-wide DER scheduling. However, the uncertainty of DERs calls for a reliable AFAPR. This paper proposes a novel surrogate polytope method for deriving the inner approximation of the AFAPR that is jointly reliable for all DER constraints and linear network constraints across the scheduling period. Instead of directly applying the chance constraints to the low-level DER constraints and network constraints, the proposed method applies the Wasserstein Distributionally Robust Joint Chance Constraint (WDRJCC) to the surrogate polytope approximation of the AFAPR, which is reformulated into a tractable set of Mixed Integer Linear Programming (MILP) constraints. Our derived inner approximation to the reliable AFAPR is less conservative while still being reliable, as demonstrated by comparisons with four benchmarks in extensive case studies, and with the nonlinear Z-Bus power flow simulation applied to validate the satisfaction of network constraints. The historical data size required is small, making the proposed method easier to deploy. The scale of MILP constraints is small and does not increase with the network size nor with the number of DERs