When the Grid Turns Crooked: Predicting How Inverters Behave in Unbalanced Power Systems

A grid is only as fair as its weakest moment. When a tree falls on a power line, when a factory starts pulling unbalanced current, or when the sun dips behind a cloud at the edge of a rural feeder, the three-phase voltages feeding an inverter stop being neat and symmetrical. One phase sags, another surges, and suddenly the equipment meant to keep renewable power stable is doing exactly the opposite — oscillating, drawing uneven current, and degrading the very power quality it was installed to protect.

Engineers have long known how to model what an inverter does when the grid is well-behaved. The hard part has always been the ugly, asymmetric moments. This paper, by Airan Francés, Dionisio Ramirez, and Javier Uceda of the Universidad Politécnica de Madrid, proposes a way to describe how a commercial grid-connected inverter responds when the grid is not balanced — and it works well enough to predict the inverter's behavior during an actual asymmetric voltage sag, the kind that happens when a fault drags one phase down while the other two ride high.

The payoff is practical and quiet. It means engineers may soon be able to simulate a commercial-off-the-shelf inverter's behavior under lopsided grid conditions without knowing anything about its proprietary internals — no circuit blueprints, no control code, no trade secrets. Just a black box with a fingerprint that tells you what it will do when the grid turns crooked.

The Science

Start with the object of study. A three-phase inverter is a power electronic device that converts direct current — say, from solar panels or a battery — into alternating current that can be fed into the power grid. These devices are the plumbing of the modern grid: they sit between renewable sources and the network, adjusting voltage, frequency, and power flow in real time (Francés et al., 2024).

The problem is that these converters are "black boxes" in the literal sense. When you buy one commercially, you rarely get its full electrical model. The manufacturer doesn't hand you the control code or the detailed switching logic. Yet to analyze whether an entire system — multiple inverters, storage, a whole microgrid — will remain stable, you need to know how each piece behaves dynamically)Skip. This is where "blackbox modeling" enters.

Blackbox modeling is a system-identification technique: instead of deriving a model from first principles and internal knowledge, you probe the device with known inputsowner. "perturbations, measure how it responds, and fit transfer functions — mathematical ratios that describe how much a given output changes when a given input changes — to the observed behavior. It's like learning how a stranger reacts by tapping their shoulder and watching rather than reading their biography.

Most blackbox modeling for power electronics has focused on dc-dc converters, the simpler cousins that translate one direct-current level to another. These are tidy because the signals are constant in steady state. Three-phase ac systems are messier: the voltages and currents are sinusoidal, oscillating at 50 or 60 hertz, which makes transfer-function analysis awkward. The standard workaround is the dq framework — the d-q (direct-quadrature) rotating reference frame, named for a mathematical trick that rotates a coordinate axes along with the sinusoidal signal so that steady-state ac values appear as constant dc values on which linear control theory can be applied comfortably.

But here's the catch that motivates this paper: the dq framework works beautifully only when the grid is balanced and symmetric. When three-phase voltages are equal in magnitude and 120 degrees apart, the rotating frame turns them into steady dc values, and life is simple.

When the grid is asymmetric — one phase lower, or currents unbalanced — that neat picture collapses. The negative-sequence components (one half of the "symmetrical components" a 1918 theorem by Charles Legeyt Fortescue uses to decompose any unbalanced three-phase signal into balanced parts) don't vanish in the positive-sequence rotating frame. Instead they show up as an annoying second harmonic oscillating at twice the grid frequency. Modeling becomes messy, and most published blackbox structures simply assumed this problem away (Francés et al., 2024).

This paper refuses the assumption. The key idea is to give the negative sequence its own rotating frame. Just as the positive sequence is synchronized with a counterclockwise rotation at frequency $\omega t$, the negative sequence gets its own clockwise frame rotating at $-\omega t$. The result: both sequences appear as constant dc signals in their respective frames, and the whole system becomes representable by a matrix of transfer functions.

What They Found

The resulting model is a matrix that takes four inputs — the active-power reference ${\hat{P}}_{ref}$, the d-component of the positive-sequence voltage ${\hat{v}}_d^{+}$, and the d- and q-components of the negative-sequence voltage ${\hat{v}}_d^{-}$ and ${\hat{v}}_q^{-}$ — and maps them onto five outputs: the input current and the d and q components of both the positive- and negative-sequence output currents.

Embedded in this matrix is the inverter's output admittance — the transfer functions that describe how output current changes with output voltage. That's crucial because output admittance is precisely what impedance-based stability analysis needs. The generalized Nyquist criterion, a tool from multivariable control theory widely used to judge whether a converter connected to a grid impedance will stay stable, depends on knowing this admittance.

The researchers built a real experimental rig to test their structure: a Semikron Semiteach voltage-source converter controlled by a dual-core F28M35H52C Concerto microcontroller, wired to a four-quadrant REGATRON TC.ACS source capable of injecting the precise perturbations needed.

The identification protocol is methodical — four tests, one per input. First, a step in the active-power reference. Second, a balanced step in the magnitude of the three-phase voltage (perturbing the positive-sequence d-component). Third, a negative sequence injected in phase with the positive sequence (perturbing $v_d^{-}$). Fourth, a negative sequence with a 90-degree phase shift (perturbing $v_q^{-}$). In each test, only one input moves while the rest stay frozen, and the corresponding column of transfer functions is identified from the outputs.

The third test is the most revealing. When the researchers injected a negative sequence into the voltage, the three-phase currents went visibly unbalanced. But the elegant part was what did not happen: the positive-sequence current components $i_d^{+}$ and $i_q^{+}$ barely moved. The disturbance stayed almost entirely in the negative-sequence outputs $i_d^{-}$ and $i_q^{-}$. The model captured this accurate — the identified transfer functions $Y_{d^{-}{d}^{-}}(s)$ and $Y_{d^{-}{q}^{-}}(s)$ reproduced the measured negative-sequence current dynamics closely.

The real test came last. The researchers imposed an actual asymmetric voltage sag on the converter and compared its measured behavior against what their blackbox model predicted. The model held up, confirming that a structure built on sequence decomposition can describe a real commercial inverter's small-signal dynamics under genuinely unbalanced conditions.

Which current components respond when the negative-sequence voltage is perturbed

Label	Value
Positive d-current (i_d+)	0
Positive q-current (i_q+)	0
Negative d-current (i_d-)	100
Negative q-current (i_q-)	72

Third identification test: stepping v_d- leaves i_d+ and i_q+ nearly unchanged, while i_d- and i_q- capture the response. Normalized to peak negative-sequence response.

Why This Changes Things

The significance here is a quiet liberation from two burdens engineers have carried. First, the burden of symmetric assumptions. Every time an engineer wanted to model an inverter in a grid that wasn't perfectly balanced, they had to either pray that the asymmetry was small enough to ignore, or fall back on detailed internal knowledge — the very thing that's inaccessible for commercial hardware disguised in black-box packaging.

Second, the burden of the second harmonic. In the standard single rotating frame, negative-sequence components translate into oscillations at twice the grid frequency, which contaminate transfer functions and complicate stability analysis. By giving the negative sequence its own clockwise frame, the paper sidesteps that entirely)Skip. Both sequences read as dc, and the model stays in the clean domain of linear transfer functions where decades of control theory apply.

There's a cascade of consequences. Because a part of the small-signal model is the inverter's output admittance, the structure plugs directly into impedance-based stability analysis. You could now, in principle, analyze whether a commercial inverter connected to a specific asymmetric grid impedance — one phase more inductive than the others, say, or a long feeder to a rural solar array — will remain stable, all without opening the inverter's casing. The generalized Nyquist criterion can be applied to the sequence-domain admittance to judge stability margins.

This matters for microgrids and smart grids specifically, where multiple converters share a network and interactions between them are the source of instability and power-quality problems. The paper extends blackbox modeling — long a workhorse for dc systems — into the three-phase asymmetrical domain that is the actual everyday reality of distribution grids, not the idealized balanced abstraction on which most prior work rested.

Sequence-domain representation of unbalanced three-phase signals

Label	Value
Positive sequence	1
Negative sequence	1

1 Positive sequence

Negative-sequence components appear as constant DC in the clockwise frame rather than as second harmonics, enabling transfer-function representation.

There are also practical design implications hidden in the DC-operating-point analysis. When the grid is asymmetric, a second-harmonic ripple appears in the DC-link current and in the active power, expressed as $P_c$ and $P_s$ terms oscillating at $2\omega t$. This is the physical reality the model accounts for: an unbalanced grid injects a twice-frequency ripple into the DC side of the converter, which can stress capacitors, distort control, and propagate power-quality problems. The model doesn't just describe this — it frames it in terms usable for design.

What's Next

A few honest caveats before the optimism gets out of hand. The negative-sequence detector used in the experimental control loop relied on a quarter-cycle transport delay, which the authors themselves flag as producing "discontinuous and erroneous results" for more than two cycles after a perturbation. Their proposed alternative — the Second-Order Generalized Integrator in-Quadrature Signal Generator (SOGI-QSG) — is faster Claudatinuedresponse and settles in about one cycle, but it imposes its own dynamic on the inputs of the model. That dynamicslimitssets a ceiling on the maximum frequency at which the model can be reliably identifiedinition — a practical constraint worth keeping in mind.

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The paper also assumes three-wire systems, which lets it neglect the zero sequence. Four-wire systems, where neutral currents matter, would need that zero sequence added. And the model is identified at a single operating point, as is standard for small-signal models — its validity is local, not global. Larger excursions from the operating point would demand the more sophisticated nonlinear structures mentioned in the paper's review.

There's room for chasing online identification, too. The paper notes the recent trend of identifying three-phase converter and grid impedances online with a single perturbation signal, avoiding operating-point shifts and addressing the coupling between converter and grid impedances. Extending the sequence-domain structure to online, adaptive identification — coupled with a DSOGI-FLL frequency-locked loop to handle frequency as a model input, as the authors suggest for droop-controlled systems — points toward a future where these physical device models get continuously refreshed in the field instead of frozen at a laboratory operating point.

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The deeper significance is architectural. Renewable energy is not a luxury of balanced, well-behaved grids. Solar arrays feed rural feeders where voltage sags are routine; batteries buffer microgrids in remote communities where the grid is chronically unbalanced. The Grid-connected inverter is the interface between these worlds, and understanding its behavior at precisely the moments when the grid misbehaves — not when it's idealized — is what determines whether storage and renewables genuinely harden a network or merely assume it's already healthy.

Francés and colleagues have given engineers a tool to peek inside a sealed box during the moments it matters most. The black box stays closed, but its secrets, at least the dynamic ones, are now on the table.