Lyapunov stability of smart inverters using linearized distflow approximation

Abstract

Fast-acting smart inverters that utilize preset operating conditions to determine real and reactive power injection/consumption can create voltage instabilities (over-voltage, voltage oscillations and more) in an electrical distribution network if set-points are not properly configured. In this work, linear distribution power flow equations and droop-based Volt-Var and Volt-Watt control curves are used to analytically derive a stability criterion using Lyapunov analysis that includes the network operating condition. The methodology is generally applicable for control curves that can be represented as Lipschitz functions. The derived Lipschitz constants account for smart inverter hardware limitations for reactive power generation. A local policy is derived from the stability criterion that allows inverters to adapt their control curves by monitoring only local voltage, thus avoiding centralized control or information sharing with other inverters. The criterion is independent of the internal time-delays of smart inverters. Simulation results for inverters with and without the proposed stabilization technique demonstrate how smart inverters can mitigate voltage oscillations locally and mitigate real and reactive power flow disturbances at the substation under multiple scenarios. The study concludes with illustrations of how the control policy can dampen oscillations caused by solar intermittency and cyberattacks. NOMENCLATURE: , Set of all lines;  , Set of all nodes; p c i , p g i , Real power demand, Real power generation at node i; q c i , q g i , Reactive power demand, Real power generation at node i; r i j , x i j , Line resistance, Line reactance of line between nodes i and j ; P i j , Q i j , Real power flow, Reactive power flow of line between nodes i and j ; v i , Voltage of node i; c i j , Current of line between nodes i and j ; s i , Rated apparent power of inverter at node i; p i , Maximum real power output of inverter at node i; q lim i , Hardware limit of reactive power generation of inverter at node i; f p,i , Volt-Watt control function of inverter at node i; f q,i , Volt-Var control function of inverter at node i; C p,i , Lipschitz constant for Volt-Watt control function of inverter at node i; C q,i , Lipschitz constant for Volt-Var control function of inverter at node i This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Adaptive Control of Distributed Energy Resources for Distribution Grid Voltage Stability

“…Consistent with Fig. 4, the dynamics of the inverter consist of nonlinear Volt-VAR & Volt-Watt controllers in series with first order low pass filters can be expressed as [10]:…”

“…, η 5 ], which can be used to alter the location and slopes of non-zero segments of the controllers. We make the following assumptions regarding these functions [5], [10], [21]: Assumption 1. The functions f p,i (v i ) and f q,i (v i ) are monotonically decreasing and continuously piece-wise differentiable.…”

“…In situations where the amount of active power generated is less than si , some inverter devices support the use of the excess system capacity for reactive power generation. The maximum amount of reactive power available for injection/consumption, denoted by qi (v i ), is a function of hardware limitations (q lim i ) and the available reactive power [10]:…”

“…A derivation of the Lipschitz constants for both the VV and VW control functions depicted in Figs. 2 -3 can be found in [10].…”

“…Although the instability threshold depends on the network's specifics, instability is reached when the slopes of the VV control curves become too steep. Numerous other works have also modeled the inverter/grid interaction as a first-order feedback controller and arrived at similar stability conditions [6]- [10]. Moreover, adaptive control approaches have been previously applied to improve the interaction of PV systems and the electric grid.…”