We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

/pdf/branch-flow-model-relaxations-and-convexification-part-ii-3yqj9g9dqx.pdf

Branch Flow Model: Relaxations and Convexification—Part II

Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems.

https://ieeexplore.ieee.org/ielaam/5165411/8075182/7990560-aam.pdf

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relationships among them. Part II presents sufficient conditions under which the convex relaxations are exact.

/pdf/convex-relaxation-of-optimal-power-flow-part-i-formulations-52chepgnj0.pdf

Convex Relaxation of Optimal Power Flow—Part I: Formulations and Equivalence

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.

Convex Relaxation of Optimal Power Flow, Part II: Exactness

The electricity infrastructure is a critical lifeline system and of utmost importance to our daily lives. Power system resilience characterizes the ability to resist, adapt to, and timely recover from disruptions. The resilient power system is intended to cope with low probability, high risk extreme events including extreme natural disasters and man-made attacks. With an increasing awareness of such threats, the resilience of power systems has become a top priority for many countries. Facing the pressing urgency for resilience studies, the objective of this paper is to investigate the resilience of power systems. It summarizes practices taken by governments, utilities, and researchers to increase power system resilience. Based on a thorough review on the existing metrics system and evaluation methodologies, we present the concept, metrics, and a quantitative framework for power system resilience evaluation. Then, system hardening strategies and smart grid technologies as means to increase system resilience are discussed, with an emphasis on the new technologies such as topology reconfiguration, microgrids, and distribution automation; to illustrate how to increase system resilience against extreme events, we propose a load restoration framework based on smart distribution technology. The proposed method is applied on two test systems to validify its effectiveness. In the end, challenges to the power system resilience are discussed, including extreme event modeling, practical barriers, interdependence with other critical infrastructures, etc.

Battling the Extreme: A Study on the Power System Resilience

The intent of the study detailed in this paper is to demonstrate the benefits of inverter var control on a fast timescale to mitigate rapid and large voltage fluctuations due to the high penetration of photovoltaic generation and the resulting reverse power flow. Our approach is to formulate the volt/var control as a radial optimal power flow (OPF) problem to minimize line losses and energy consumption, subject to constraints on voltage magnitudes. An efficient solution to the radial OPF problem is presented and used to study the structure of optimal inverter var injection and the net benefits, taking into account the additional cost of inverter losses when operating at non-unity power factor. This paper will illustrate how, depending on the circuit topology and its loading condition, the inverter's optimal reactive power injection is not necessarily monotone with respect to their real power output. The results are demonstrated on a distribution feeder on the Southern California Edison system that has a very light load and a 5 MW photovoltaic (PV) system installed away from the substation.

Optimal inverter VAR control in distribution systems with high PV penetration

Motivated by the need to cope with rapid and random fluctuations of renewable generation, we presents a model that augments the traditional Volt/VAR control through switched controllers on a slow timescale with inverter control on a fast timescale. The optimization problem is generally nonconvex and therefore hard to solve. We propose a simple convex relaxation and prove that it is exact provided over-satisfaction of load is allowed. Hence Volt/VAR control over radial networks is efficiently solvable. Simulations of a real-world distribution circuit illustrates that the proposed inverter control achieves significant improvement over the IEEE 1547 standard in terms of power quality and power savings.

/pdf/inverter-var-control-for-distribution-systems-with-1kbwj9ylou.pdf

Inverter VAR control for distribution systems with renewables

We consider a class of local volt/var control schemes where the control decision on the reactive power at a bus depends only on the local bus voltage. These local algorithms form a feedback dynamical system and collectively determine the bus voltages of a power network. We show that the dynamical system has a unique equilibrium by interpreting the dynamics as a distributed algorithm for solving a certain convex optimization problem whose unique optimal point is the system equilibrium. Moreover, the objective function serves as a Lyapunov function implying global asymptotic stability of the equilibrium. The optimization based model does not only provide a way to characterize the equilibrium, but also suggests a principled way to engineer the control. We apply the results to study the parameter setting for the inverter-based volt/var control in the proposed IEEE 1547.8 standard.

/pdf/equilibrium-and-dynamics-of-local-voltage-control-in-1qn7m6bxn0.pdf

Masoud Farivar

Papers

Branch Flow Model: Relaxations and Convexification—Part II

Optimal inverter VAR control in distribution systems with high PV penetration

Inverter VAR control for distribution systems with renewables

Equilibrium and dynamics of local voltage control in distribution systems

Equilibrium and Dynamics of Local Voltage Control in Distribution