The increasing integration of renewable energy sources, particularly utility-scale photovoltaic (PV) plants, has fundamentally altered the characteristics of power systems. A critical aspect of this transformation lies in the fault response of these inverter-based resources, which differs significantly from traditional synchronous generators. While much research has focused on the fundamental frequency magnitude and phase of fault currents, the harmonic content, especially during unbalanced conditions, presents a less explored but significant challenge for system protection. This analysis delves into the mechanism behind the generation of pronounced third harmonic currents in grid-connected solar inverter outputs during asymmetrical faults, models its behavior under various control schemes, and investigates its consequential impact on critical protection functions like transformer differential protection. We will also propose and discuss effective mitigation strategies.
The core of a grid-tied PV system is the power electronic solar inverter, responsible for converting DC power from the PV array into AC power synchronized with the grid. A typical two-level voltage source converter (VSC) structure forms the basis, as shown in the system context below. The dynamic behavior of this solar inverter during faults is governed by its control system, typically implemented in the synchronous rotating (dq) reference frame for decoupled control of active and reactive power.
The standard positive-sequence control structure for a solar inverter aims to regulate the DC-link voltage (which dictates active power transfer) and the reactive current injection. The voltage equations in the dq-frame, incorporating PI controllers and feedforward terms, are:
$$u^{*}_{d} = (k_{ip} + \frac{k_{ii}}{s})(i^{*}_{d} – i_{d}) – \omega_{1} L i_{q} + R i_{d} + e_{d}$$
$$u^{*}_{q} = (k_{ip} + \frac{k_{ii}}{s})(i^{*}_{q} – i_{q}) + \omega_{1} L i_{d} + R i_{q} + e_{q}$$
where \(u^{*}_{d}, u^{*}_{q}\) are the d- and q-axis voltage references, \(i^{*}_{d}, i^{*}_{q}\) are the current references, \(i_{d}, i_{q}\) are the measured currents, \(e_{d}, e_{q}\) are the grid voltages, \(k_{ip}, k_{ii}\) are the PI gains for the current controller, and \(\omega_{1}\) is the grid angular frequency.
During grid voltage imbalances caused by asymmetrical faults, negative-sequence components appear. To manage the resulting double-frequency oscillations in power, advanced solar inverter controllers implement negative-sequence control strategies. These strategies define specific objectives: eliminating negative-sequence current (Objective I), suppressing reactive power oscillation (Objective II), or suppressing active power oscillation (Objective III). The negative-sequence current references \(i^{-*}_{d}, i^{-*}_{q}\) are derived from the positive-sequence references and the measured negative-sequence grid voltages. Furthermore, due to the limited overcurrent capability of power semiconductors (typically 1.2 pu), a current priority scheme is enforced during low-voltage ride-through (LVRT) events, often prioritizing reactive current support over active power transfer or negative-sequence compensation.
The genesis of the third harmonic lies in the interaction between these control loops and the inherent power pulsations during imbalance. An asymmetrical fault causes oscillating power injection at twice the fundamental frequency (2ω₁). The instantaneous active power \(P\) and reactive power \(Q\) can be expressed as:
$$P = P_{0} + P_{c2}\cos(2\omega_{1}t) + P_{s2}\sin(2\omega_{1}t)$$
$$Q = Q_{0} + Q_{c2}\cos(2\omega_{1}t) + Q_{s2}\sin(2\omega_{1}t)$$
The coefficients \(P_{c2}, P_{s2}, Q_{c2}, Q_{s2}\) are functions of both positive- and negative-sequence voltages and currents. This double-frequency power ripple directly affects the DC-link dynamics. Ignoring losses, the DC-link capacitor equation is:
$$C \frac{du_{dc}}{dt} = \frac{P_{array} – P}{u_{dc}}$$
Assuming the average power \(P_0\) matches the PV array output \(P_{array}\), the solution shows that the double-frequency power components \(P_{c2}, P_{s2}\) induce a double-frequency ripple on the DC-link voltage \(u_{dc}\):
$$u_{dc} \approx \sqrt{ \frac{1}{C} \left[ -\frac{P_{c2} \sin(2\omega_{1}t)}{\omega_{1}} + \frac{P_{s2} \cos(2\omega_{1}t)}{\omega_{1}} \right] + u^{2}_{dc0} }$$
For small ripples, this simplifies to a sinusoidal component: \(u_{dc\_2} = U_{dc\_2} \cos(2\omega_{1}t + \phi_{dc\_2})\). This voltage ripple is fed into the DC-link voltage controller, which generates the positive-sequence d-axis current reference \(i^{+*}_{d}\). Consequently, \(i^{+*}_{d}\) itself acquires a 2ω₁ component. This component propagates through the current controller. The transfer function from the DC voltage error to the d-axis voltage command involves two PI controllers in series (voltage outer loop and current inner loop). When a sinusoidal disturbance at 2ω₁ passes through these controllers, the output in the dq-domain contains components at the same frequency. Transforming this double-frequency component from the positive-sequence dq-frame (rotating at ω₁) back to the three-phase stationary (abc) frame yields:
$$u^{+*}_{abc\_2} \propto \cos(2\omega_{1}t) \cdot
\begin{bmatrix}
\cos(\omega_{1}t)\\
\cos(\omega_{1}t – 2\pi/3)\\
\cos(\omega_{1}t + 2\pi/3)
\end{bmatrix} = \frac{1}{2}
\begin{bmatrix}
\cos(\omega_{1}t) + \cos(3\omega_{1}t)\\
\cos(\omega_{1}t – 2\pi/3) + \cos(3\omega_{1}t + 2\pi/3)\\
\cos(\omega_{1}t + 2π/3) + \cos(3\omega_{1}t – 2\pi/3)
\end{bmatrix}$$
The result clearly shows the production of both a fundamental-frequency component (which manifests as a negative-sequence component in the balanced fundamental frame) and a third harmonic (3ω₁) component in the voltage reference. This third harmonic voltage then causes a third harmonic current to flow from the solar inverter into the grid. A similar process occurs in the negative-sequence control loop, where the 2ω₁ component in the negative-sequence current references can also generate third harmonic currents.
The magnitude of this third harmonic current is not constant but depends heavily on the operating point and control strategy of the solar inverter. Our analysis identifies the following key influencing factors:
| Influencing Factor | Effect on Third Harmonic Magnitude | Rationale |
|---|---|---|
| Fault Severity (Voltage Dip) | Non-linear, peaks at moderate dips | Small dips cause small power ripple. Very deep dips trigger LVRT reactive priority, limiting active current ripple and thus DC voltage ripple. |
| Pre-fault Active Power Output | Inversely proportional | High pre-fault current leaves little room for increase during fault due to current limiter, suppressing the 2ω₁ modulation of the active current. |
| Negative-Sequence Control Objective | Objective II > I > III | Objective II minimizes reactive power oscillation but maximizes active power oscillation (Pc2, Ps2), leading to larger DC ripple. |
| Current Limiter & Priority | Critical determining factor | The sequence of saturating positive-sequence reactive, negative-sequence, and positive-sequence active current references directly limits the 2ω₁ components in the control loops. |
| Controller Gains (kup, kui, kip, kii) | Proportional | Higher gains amplify the processing of the DC-link ripple through the control path, increasing the generated 3rd harmonic voltage. |
The analytical expression for the amplitude of the third harmonic voltage component \(M_{dc\_2}\) generated from the DC-link ripple confirms these dependencies:
$$M_{dc\_2} = \frac{U_{dc\_2}}{2} \sqrt{ \frac{k_{up}^2 (k_{ip}^2) + k_{ui}^2 (k_{ii}^2)}{16\omega_{1}^4} + \frac{k_{up}^2 (k_{ii}^2)}{4\omega_{1}^2} + \frac{k_{ui}^2 (k_{ip}^2)}{4\omega_{1}^2} }$$
where \(U_{dc\_2}\) is the magnitude of the DC-link 2ω₁ ripple. This shows a direct proportionality to the DC ripple and the controller gains.
The presence of significant third harmonic currents during faults has profound implications for protection systems. A primary concern is its interaction with transformer differential protection. Traditionally, transformer differential relays incorporate harmonic restraint/blocking criteria, notably second and third harmonic blocking, to prevent maloperation during magnetizing inrush or current transformer (CT) saturation. The logic is that high harmonic content indicates a non-fault condition like inrush or saturation during an external fault.
The third harmonic blocking criterion is typically: \(I_{\phi3} > K_3 \cdot I_{\phi1}\), where \(I_{\phi3}\) and \(I_{\phi1}\) are the third harmonic and fundamental phase currents, and \(K_3\) is a restraint coefficient (e.g., 0.15-0.2). If met, the sensitive low-set percentage differential characteristic is blocked, leaving a higher-set, less sensitive characteristic to operate for internal faults.
The critical issue arises when an internal transformer fault occurs in a system fed by a solar inverter. Under the specific conditions outlined earlier, the inverter may feed fault current containing >20% third harmonic. This can satisfy the blocking criterion. Unlike traditional faults where CT saturation provides large fundamental current, inverter-fed fault currents are limited (~1.2 pu). If the grid source is also weak, the total differential current may be insufficient to operate the high-set characteristic. This creates a protection blind spot: the low-set element is blocked by the harmonic, and the high-set element fails to pick up, leaving the internal fault uncleared.
To address this challenge, mitigation strategies can be applied both at the inverter control level and the protection relay level.
1. Inverter-Level Mitigation: The most direct approach is to compensate for the double-frequency ripple within the solar inverter control loops. A feedforward or feedback compensation term can be added to the current references to cancel out the effect of the \(u_{dc\_2}\) ripple. For example, an estimate of the 2ω₁ component in \(i^{+*}_{d}\) can be calculated based on measured power oscillations and subtracted from the reference before it enters the current controller, effectively breaking the harmonic generation chain.
2. Protection-Level Enhancement: The relay logic can be improved to better discriminate between third harmonic from CT saturation and from the solar inverter. An enhanced blocking logic can be proposed:
$$ \text{Block Low-Set Diff} = \left[ I_{\phi3} > K_3 \cdot I_{\phi1} \right] \ \textbf{AND} \ \left[ I_{\phi1} > 2.0 I_n \ \textbf{OR} \ \text{CT\_Saturation\_Detected} \right] $$
where \(I_n\) is the rated current. CT saturation detection can use well-known criteria like the presence of waveform “flat spots” or high levels of even harmonics. This logic ensures that the third harmonic block only applies when the fundamental current is high (indicative of a strong source and potential CT saturation) or direct saturation is detected. For inverter-fed faults with low fundamental current and no saturation signature, the blocking is overridden, allowing the sensitive low-set differential to operate correctly.
To validate the theoretical analysis, detailed time-domain simulations were performed for a 100 MW PV plant connected through a step-up transformer. Various asymmetric fault scenarios (single-phase, phase-to-phase) at different locations, with different pre-fault power levels and negative-sequence control objectives, were tested. The results consistently demonstrated high third harmonic content under the predicted conditions. For instance, during a phase-to-phase fault with 5Ω fault resistance, with the solar inverter using control Objective II and operating at low pre-fault power, the phase current third harmonic ratio reached nearly 40%. In such a case, for a system with high impedance (weak grid), the transformer differential protection scenario described previously was observed, confirming the risk. Conversely, scenarios with deep voltage dips (triggering reactive priority) or high pre-fault power showed significantly reduced third harmonic levels, aligning perfectly with the theoretical model.
In conclusion, the asymmetric fault response of modern grid-following solar inverters is complex and can lead to the injection of substantial third harmonic currents into the network. This phenomenon is intrinsically linked to the double-frequency power oscillations and their interaction with the DC-link dynamics and dual-sequence control loops. The harmonic level is highly dependent on the operating point and control strategy of the solar inverter. This non-characteristic harmonic poses a tangible risk to transformer differential protection schemes that rely on harmonic blocking principles, potentially causing a failure to trip for internal faults under specific, but plausible, system conditions. A comprehensive solution requires attention from both sides: inverter manufacturers can implement ripple compensation techniques to suppress the harmonic at its source, while protection engineers should consider adopting enhanced relay logic that can differentiate between harmonics originating from inverter control and those from traditional CT saturation. As power systems evolve with higher penetrations of inverter-based resources, such detailed understanding and co-engineering of source and protection behavior become paramount for maintaining system reliability and security.