Abstract
Today's power grid is in a transitional stage to cater to the needs of energy efficiency, climate change, and environmental targets. In the process of designing the future power grid, one of the most fundamental models to be utilized is AC optimal power flow (AC-OPF). Since the feasible space of AC-OPF is non-convex, the optimization models developed using it often result in multiple local minima. To avoid such computational challenges in solving optimization models, various relaxation methods have been developed in the past. In the literature, these relaxation methods are mainly tested on specific networks. However, the scalability of relaxation techniques on branch-flow-based AC-OPF is yet to be explored. In this context, this paper compares the performance of different relaxation methods with the well-established MATPOWER AC-OPF solver in terms of the mean square error (MSE), maximum squared error, minimum and maximum values of voltage magnitude, and the average simulation time. In addition, the scalability of these models is tested on various radial and mesh networks with nodes ranging from 33 to 6655 nodes and 9 to 6515 nodes, respectively. In this manner, the trade-off between computational complexity and solution accuracy is demonstrated and analyzed in depth. This provides an enhanced understanding of the suitability and efficiency of the compared relaxation methods, helping, in turn, the efficiency of optimization models for varying sizes and types (i.e., radial or meshed) of networks.

Abstract
Nowadays, decentralized microgrids (DC-MGs) have become a popular topic due to the effectiveness and the less complexity. In fact, DC-MGs resist to share their internal information with the distribution system operator (DSO) to protect their privacy and compete in the electricity market. Further, lack of information sharing among MGs in normal operation conditions leads to form a competitive market. However, in emergency operation conditions, it results numerous challenges in managing network outages. Therefore, this paper presents a hierarchical model consisting of three stages to enhance the resilience of DC-MGs. In all stages, the network outage management is performed considering the reported data of MGs. In the first stage, proactive actions are performed with the aim of increasing the network readiness against the upcoming windstorm. In the second stage, generation scheduling, allocation of mobile units and distribution feeder reconfiguration (DFR) are operated by DSO to minimize operating costs. In the final stage, the repair crew is allocated to minimize the energy not served (ENS). Un-certainties of load demand, wind speed and solar radiation are considered, and the effectiveness of the proposed model is investigated by integrating to the 118-bus distribution network. Finally, the results of the simulation indicate that DFR and proactive actions decrease the ENS by 19,124 kWh and 4101 kWh, respectively. Further, the sharing of information among MGs leads to a 48.16% growth in the supply service level to critical loads, and consequently a 3.47% increase in the resilience index.

Abstract
This paper presents a mathematical problem formulation for energy management systems for smart homes. The flexibility can be provided by a home energy management system (HEMS) in a local energy community. The main concept is to model the flexibility provision and flexibility procurement within the energy community that can be provided to the aggregator from active consumers. The integrated energy management model is coded as a standard mixed-integer linear programming (MILP) model which can be solved by open-source tools like the PuLP package developed in Python. The simulation results confirm the performance of the proposed model in terms of flexibility provided by the centralized integrated energy system introduced in this paper.

Abstract
This paper evaluates lines repair and maintenance impacts on generation-transmission expansion planning (GTEP), considering the transmission and generation reliability. The objective is to form a balance between the transmission and generation expansion and operational costs and reliability, as well as lines repair and main-tenance costs. For this purpose, the transmission system reliability is represented by the value of loss of load (LOL) and load shedding owing to line outages, and generation reliability is formulated by the LOL and load shedding indices because of transmission congestion and outage of generating units. The implementation results of the model on the IEEE RTS show that including line repair and maintenance as well as line loading in GTEP leads to optimal generation and transmission plans and significant savings in expansion and operational costs.

Abstract
Virtual Power Plants (VPPs) are one of the concepts introduced in modern power systems to handle the increasing number of the distributed generation (DG) units. Technical VPPs (TVPPs) consider both financial and technical perspectives of using DGs in the system. Besides, secure and reliable operation of the system is a priority. In this paper, optimal operation of technical virtual power plants in a reconfigurable network is formulated as an optimization problem to resolve the probable contingency problem in the lines of the system. The VPP is assumed to be a multi-carrier energy system including combined heat and power (CHP), renewable DGs and dispatchable DGs beside thermal and electrical storage systems and loads. The uncertainties of renewable based DGs and demand levels are handled using chance constrained programming (CCP). By using CCP in presence of uncertain parameters, the security of the system can be guaranteed in predefined level of probability. Finally, to evaluate the effectiveness, quality and applicability of the proposed methodology, the problem is structured as a mixed-integer nonlinear programming (MINLP) problem which is solved using General Algebraic Modeling System (GAMS) software via Baron solver.

Abstract
This paper presents a planning and operational strategy to improve the recoverability of distribution systems (DSs) to deal with a set of possible line fault scenarios. The strategy simultaneously optimizes the allocation of dispatchable distributed generation (DG) units while coordinating a dynamic restoration process based on a radial topology reconfiguration, an islanding operation, a demand response program, and the pre-positioning and dispatch of mobile emergency storage units. The uncertainty and variability associated with solar irradiation and demand are captured via a multi-period formulation based on a stochastic mixed-integer linear programming model. The objective function of this model minimizes the investment cost of new dispatchable DG units and the amount of energy shedding within the system. Simulations are performed on adapted 33-node and 53-node test systems to validate the proposed strategy under four different test conditions, numerical results reveal the advantages of simultaneously solving the planning and operational stages to improve the recoverability of the system.