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The non-linearity and non-convexity of the AC power flow equations may induce convergence problems to the Newton-Raphson (NR) algorithm. Indeed, as shown by Thorp and Naqavi, the NR algorithm may exhibit a fractal behavior. Furthermore, under heavy loading conditions or if some of the line reactances are relatively large compared to the others, the Jacobian matrix becomes ill-conditioned, which may cause the divergence of this algorithm. To address the aforementioned problems for radial power distribution systems, we propose in this paper to apply a sinusoidal transform to map the AC power flow equations into a convex quadratic form, which includes nodebased and Pythagorean equations. The good performance of the proposed approach is demonstrated via simulations carried out on several power distribution systems. 

Facing stochastic variations of the loads due to an increasing penetration of renewable energy generation, online decision making under uncertainty in modern power systems is capturing power researchers' attention in recent years. To address this issue while achieving a good balance between system security and economic objectives, we propose a surrogate-enhanced scheme under a joint chance-constrained (JCC) optimal power-flow (OPF) framework. Starting from a stochastic-sampling procedure, we first utilize the copula theory to simulate the dependence among multivariate uncertain inputs. Then, to reduce the prohibitive computational time required in the traditional Monte-Carlo (MC) method, we propose to use a polynomial-chaos-based surrogate that allows us to efficiently evaluate the power-system model at non-Gaussian distributed sampled values with a negligible computing cost. Learning from the MC simulated samples, we further proposed a hybrid adaptive approach to overcome the conservativeness of the JCC-OPF by utilizing correlation of the system states, which is ignored in the traditional Boole's inequality. The simulations conducted on the modified Illinois test system demonstrate the excellent performance of the proposed method. 

Power systems serve social communities that consist of residential, commercial, and industrial customers. As a result, the disaster resilience of a power system should account for social community resilience. The social behavior and psychological features of all stakeholders involved in a disaster influence the level of power system preparedness, mitigation, recovery, adaptability, and resilience. Hence, there is a need to consider the social community's effect on the power system and the dependence between them in determining a power system's resilient to human-made and natural hazards. The social community, such as a county, city, or state, consists of various stakeholders, e.g., social consumers, social prosumers, and utilities. In this paper, we develop a multi-dimensional output-oriented method to measure resilience. The three key ideas for measuring power system resilience are the multi-dimensionality, output-oriented, and degraded functionality aspects of the power system. To this end, we develop an artificial society based on neuroscience, social science, and psychological theories to model the behavior of consumers and prosumers and the interdependence between power system resilience, comsumer and prosumer well-being, and community capital. Both mental health and physical health are used as metrics of well-being, while the level of cooperation is used to measure community capital resilience.