Geometry of the Feasible Output Regions of Grid-Interfacing Inverters with Current Limits

Abstract

Many resources in the grid connect to power grids via programmable grid-interfacing inverters that can provide grid services and offer greater control flexibility and faster response times compared to synchronous generators. However, the current through the inverter needs to be limited to protect the semiconductor components. Existing controllers are designed using somewhat ad hoc methods, for example, by adding current limiters to preexisting control loops, which can lead to stability issues or overly conservative operations. In this paper, we study the geometry of the feasible output region of a current-limited inverter. We show that under a commonly used model, the feasible region is convex. We provide an explicit characterization of this region, which allows us to efficiently find the optimal operating points of the inverter. We demonstrate how knowing the feasible set and its convexity allows us to improve upon existing grid-forming inverters such that their steady-state currents always remain within the current magnitude limit, whereas standard grid-forming controllers can lead to instabilities and violations.

Figures (5)

• Figure 1: We model the inverter as a controllable voltage source in the $\mathrm{dq}$-frame connected via an $RL$ filter to an infinite bus.
• Figure 2: The set of feasible inverter currents ($\mathcal{I})$ (top left) always forms a circle of radius $I_\mathrm{max}$ due to the magnitude constraint on $\mathbf{I}_\mathrm{dq}$. The shapes of feasible $(P,Q) (\hbox{top right}), (P,V_\mathrm{dq}^2) (\hbox{bottom left})$, and $(Q,V_\mathrm{dq}^2) (\hbox{bottom right})$ regions depend on specific network parameters, namely $RL$ filter impedance values and grid-side voltage magnitude $||\mathbf E_\mathrm{dq}||_2$.
• Figure 3: Graphical representation of the quadratic equation $f_1(\mu)$ with its roots and quadratic inequality $f_2(\mu)\leq0$ that corresponds to the interval $\mu \in [\underline{\mu},\overline{\mu}]$. Convexity of set $\mathcal{C}$ depends on at least one of $\mu^-, \mu ^+$ falling within this interval.
• Figure 4: The two plots compare the current magnitude response (top) and $(P,V^{2})$ region trajectory (bottom) for the same droop controller tracking two different setpoints: the original infeasible setpoint (dashed blue line) and the best feasible setpoint output from the optimization problem (solid blue line). Providing an updated, optimized setpoint to the droop controller instead of the original reference setpoint ensures it achieves a safe steady-state value regardless of whether the original setpoint was within the feasible operating region. The transient response violates the current constraints for a short time, which might be tolerable sorensen2013thermalfirouz2014efficiency or can be mitigated with other techniques qoria2020currentfan2022review.
• Figure 5: The top plot is the response of the optimal controller current magnitude to a feasible $(P^*,Q^*)$ setpoint and the bottom plot is the response of the optimal controller to an infeasible $(P^*,V^{2*})$ setpoint. In both simulations, the inverter current magnitude remains within the safe operating region. The infeasible setpoint in the bottom plot causes the inverter current to settle at the boundary of the safe operating region.

Theorems & Definitions (5)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof