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This paper uses convex inner approximations (CIA) of the AC power flow to tackle the optimization problem of quantifying a 3-phase distribution feeder’s capacity to host distributed energy resources (DERs). This is often connoted hosting capacity (HC), but herein we consider separative bounds for each node on positive and negative DER injections, which ensures that injections within these nodal limits satisfy feeder voltage and current limits and across nodes sum up to the feeder HC. The methodology decomposes a 3-phase feeder into separate phases and applies CIA-based techniques to each phase. An analysis is developed to determine the technical condition under which this per-phase approach can still satisfy network constraints. New approaches are then presented that modify the per-phase optimization problems to overcome conservativeness inherent to CIA methods and increase overall HC, including selectively modifying the per-phase impedances and iteratively relaxing per-phase voltage bounds. Discussion is included on trade-offs and feasibility. To validate the methodology, simulation-based analysis is conducted with the IEEE 37-node test feeder and a real 534-node unbalanced radial distribution feeder. 

Scenario reduction (SR) aims to identify a small yet representative scenario set to depict the underlying uncertainty, which is critical to scenario-based stochastic optimization (SBSO) of power systems. Existing SR techniques commonly aim to achieve statistical approximation to the original scenario set. However, SR and SBSO are commonly considered as two distinct and decoupled processes, which cannot guarantee a superior approximation of the original optimality. Instead, this paper incorporates the SBSO problem structure into the SR process and introduces a novel problem-driven scenario reduction (PDSR) framework. Specifically, we project the original scenario set in distribution space onto the mutual decision applicability between scenarios in problem space. Subsequently, the SR process, embedded by a distinctive problem-driven distance metric, is rendered as a mixed-integer linear programming formulation to obtain the representative scenario set while minimizing the optimality gap. Furthermore, ex-ante and ex-post problem-driven evaluation indices are proposed to evaluate the SR performance. Numerical experiments on two two-stage stochastic economic dispatch problems validate the effectiveness of PDSR, and demonstrate that PDSR significantly outperforms existing SR methods by identifying salient (e.g., worst-case) scenarios, and achieving an optimality gap of less than 0.1% within acceptable computation time. 

For fast timescales or long prediction horizons, the AC optimal power flow (OPF) problem becomes a computational challenge for large-scale, realistic AC networks. To overcome this challenge, this paper presents a novel network reduction methodology that leverages an efficient mixed-integer linear programming (MILP) formulation of a Kron-based reduction that is optimal in the sense that it balances the degree of the reduction with resulting modeling errors in the reduced network. The method takes as inputs the full AC network and a pre-computed library of AC load flow data and uses the graph Laplacian to constraint nodal reductions to only be feasible for neighbors of non-reduced nodes. This results in a highly effective MILP formulation which is embedded within an iterative scheme to successively improve the Kron-based network reduction until convergence. The resulting optimal network reduction is, thus, grounded in the physics of the full network. The accuracy of the network reduction methodology is then explored for a 100+ node medium-voltage radial distribution feeder example across a wide range of operating conditions. It is finally shown that a network reduction of 25-85% can be achieved within seconds and with worst-case voltage magnitude deviation errors within any super node cluster of less than 0.01pu. These results illustrate that the proposed optimization-based approach to Kron reduction of networks is viable for larger networks and suitable for use within various power system applications. 

This paper presents a market-based optimization framework wherein Aggregators can compete for nodal capacity across a distribution feeder and guarantee that allocated flexible capacity cannot cause overloads or congestion. This mechanism, thus, allows Aggregators with allocated capacity to pursue a number of services at the whole-sale market level to maximize revenue of flexible resources. Based on Aggregator bids of capacity (MW) and network access price ($/MW), the distribution system operator (DSO) formulates an optimization problem that prioritizes capacity to the different Aggregators across the network while implicitly considering AC network constraints. This grid-aware allocation is obtained by incorporating a con- vex inner approximation into the optimization framework that prioritizes hosting capacity to different Aggregators. We adapt concepts from transmission-level capacity market clearing, utility demand charges, and Internet-like bandwidth allocation rules to distribution system operations by incorporating nodal voltage and transformer constraints into the optimization framework. Simulation based results on IEEE distribution networks showcase the effectiveness of the approach.