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Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof. 

With the proliferation of distributed energy resources (DERs) in the distribution grid, it is a challenge to effectively control a large number of DERs resilient to the communication and security disruptions, as well as to provide the online grid services, such as voltage regulation and virtual power plant (VPP) dispatch. To this end, a hybrid feedback-based optimization algorithm along with deep learning forecasting technique is proposed to specifically address the cyber-related issues. The online decentralized feedback-based DER optimization control requires timely, accurate voltage measurement from the grid. However, in practice such information may not be received by the control center or even be corrupted. Therefore, the long short-term memory (LSTM) deep learning algorithm is employed to forecast delayed/missed/attacked messages with high accuracy. The IEEE 37-node feeder with high penetration of PV systems is used to validate the efficiency of the proposed hybrid algorithm. The results show that 1) the LSTM-forecasted lost voltage can effectively improve the performance of the DER control algorithm in the practical cyber-physical architecture; and 2) the LSTM forecasting strategy outperforms other strategies of using previous message and skipping dual parameter update. 

Economic Dispatch aims to minimize the total cost of operation/generation of microgrids while meeting all the defined constraints. Since microgrids consist of distributed generators, it is imperative for these generators to communicate seamlessly with each other without any losses and to ensure secure operation of the microgrid. With the use of distributed generators, noise is inherent in the system. This paper focuses on including noises as a constraint in an islanded microgrid to find a better economic dispatch solution. It also introduces a STATCOM controller for reactive power management. The controller will help provide stability to the microgrid's voltage, output power and phase angle. This will enhance the microgrid's performance and make it a more resilient system. 

Economic Dispatch is an important problem in the operation of microgrids. It aims to minimize the total cost of operation/generation of microgrids while meeting all the defined constraints. Since microgrids consist of distributed generators, it is imperative for these generators to communicate seamlessly with each other without any losses and to ensure secure operation of the microgrid. With use of distributed generators, noise is inherent in the system. Most of the economic dispatch problems do not include noise in their analysis while finding a relevant solution. This paper focuses on including noises as a constraint in grid connected mode microgrids to find a better economic dispatch solution. This will enhance the microgrid's performance and make it a more resilient system.