The proliferation of solar photovoltaic (PV) generation has fundamentally altered the fault response characteristics of modern power systems. Unlike synchronous generators, which are characterized by a fixed internal voltage behind a reactance, a grid-connected solar inverter is a fully controllable power electronic interface. Its behavior during grid disturbances is not governed by inherent electromagnetic principles but is dictated by its programmed Fault Ride-Through (FRT) control strategy. This shift introduces a paradigm where fault characteristics are strong functions of control objectives, presenting both challenges and opportunities for system protection and stability analysis.
A critical scenario arises during asymmetrical faults, such as single-line-to-ground or phase-to-phase faults. During such events, the grid voltage contains both positive and negative sequence components. Traditional generation sources contribute only a positive sequence internal voltage; the negative sequence network is passive except at the fault location. However, a modern solar inverter, with its independent control capabilities in the positive and negative sequence rotational frames, can actively regulate its negative sequence current output. This ability to inject controlled negative sequence current transforms the inverter from a passive element in the negative sequence network into an active, controlled negative sequence source. The specific control objective for this negative sequence current—whether to suppress it, eliminate power oscillations, or achieve another goal—profoundly impacts the resulting fault current waveform and the equivalent impedance “seen” by the network. This article delves into the analytical derivation of steady-state fault current for a solar inverter, comprehensively examines the implications of different negative sequence control strategies, and characterizes the resulting fault signatures and equivalent negative sequence impedance.
Control Foundation of a Grid-Connected Solar Inverter
The typical control architecture for a grid-following solar inverter employs a cascaded structure with an outer power (or DC-link voltage) control loop and an inner current control loop. For fault analysis, the fast dynamics of the inner current regulator are most critical. With well-designed Proportional-Integral (PI) controllers in the synchronous reference frame, the output current can rapidly track its reference value. Therefore, for steady-state fault analysis, the inverter’s output sequence currents can be assumed to equal their reference values dictated by the FRT strategy.
Let us define the positive sequence current reference in the positive (d+q+) synchronous rotating frame as $$i^{+*}=i^{+*}_d + j i^{+*}_q$$, and the negative sequence current reference in the negative (d–q–) synchronous rotating frame as $$i^{-*}=i^{-*}_d + j i^{-*}_q$$. The steady-state output sequence currents satisfy:
$$ i^{+} = i^{+*} $$
$$ i^{-} = i^{-*} $$
where $$ i^{+} = i^{+}_d + j i^{+}_q $$ and $$ i^{-} = i^{-}_d + j i^{-}_q $$.
Assuming the d+-axis is aligned with the positive sequence grid voltage vector at time t=0, the phase currents can be derived by performing the inverse Park transformation on both sequence components and summing them:
$$ i_a = |i^{+}| \cos(\omega_1 t + \phi^{+}) + |i^{-}| \cos(\omega_1 t – \phi^{-}) $$
$$ i_b = |i^{+}| \cos(\omega_1 t – 120^{\circ} + \phi^{+}) + |i^{-}| \cos(\omega_1 t + 120^{\circ} – \phi^{-}) $$
$$ i_c = |i^{+}| \cos(\omega_1 t + 120^{\circ} + \phi^{+}) + |i^{-}| \cos(\omega_1 t – 120^{\circ} – \phi^{-}) $$
where $$\phi^{+} = \arctan(i^{+}_q / i^{+}_d)$$ and $$\phi^{-} = \arctan(i^{-}_q / i^{-}_d)$$, and $$\omega_1$$ is the fundamental angular frequency.
This can be consolidated to express the amplitude of each phase current:
$$ I_{a} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 + 2|i^{+}||i^{-}|\cos(\phi^{+} + \phi^{-}) } $$
$$ I_{b} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 + 2|i^{+}||i^{-}|\cos(\phi^{+} + \phi^{-} + 120^{\circ}) } $$
$$ I_{c} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 + 2|i^{+}||i^{-}|\cos(\phi^{+} + \phi^{-} – 120^{\circ}) } $$
The key observation is that the relative magnitude of the three phase currents depends not only on the magnitudes of the positive and negative sequence currents $$|i^{+}|$$ and $$|i^{-}|$$, but critically on the phase angle sum $$(\phi^{+} + \phi^{-})$$. This sum is directly controlled by the negative sequence control objective.
Fault Ride-Through Control Strategies: Positive and Negative Sequence Objectives
Positive Sequence FRT Control
Grid codes mandate that a solar inverter must support the grid voltage during faults by injecting reactive current. Typically, the q-axis positive sequence current reference $$i^{+*}_q$$ is set according to the depth of the voltage sag at the point of common coupling (PCC). A common specification is:
$$ i^{+*}_q = \begin{cases}
0, & U_T > 0.9 \\
K \cdot (0.9 – U_T) \cdot I_N, & 0.2 \le U_T \le 0.9 \\
1.05 \cdot I_N, & U_T < 0.2
\end{cases} $$
where $$U_T$$ is the per-unit PCC voltage, $$I_N$$ is the per-unit rated current, and $$K$$ is a gain (often 1.5 or 2.0). The d-axis positive sequence current reference $$i^{+*}_d$$, governing active power output, is typically managed to respect the total current capability of the solar inverter. A standard approach prioritizes reactive current support: after allocating current for the mandated $$i^{+*}_q$$ and any controlled $$i^{-*}$$, the remaining capacity within the maximum current limit (e.g., 1.2 pu) is used for active current $$i^{+*}_d$$.
Negative Sequence Control Strategies
During unbalanced voltage sags, the instantaneous power output of the solar inverter contains a double-frequency oscillating component. Different control objectives can be defined for the negative sequence current to mitigate specific effects. Let $$e^{+}=e^{+}_d + j e^{+}_q$$ and $$e^{-}=e^{-}_d + j e^{-}_q$$ be the positive and negative sequence grid voltages in their respective rotating frames. With d-axis voltage orientation ($$e^{+}_q = 0$$), the power oscillation terms are functions of both sequence voltages and currents.
Three primary negative sequence control objectives are prevalent:
- Target I: Suppress Negative Sequence Current. The goal is to inject balanced currents. The reference is: $$ i^{-*} = 0 $$.
- Target II: Suppress Reactive Power Oscillations. The goal is to eliminate the 2ω ripple in the reactive power Q. This leads to the reference:
$$ i^{-*}_d = \frac{ e^{-}_d i^{+*}_d + e^{-}_q i^{+*}_q }{ e^{+}_d }, \quad i^{-*}_q = \frac{ e^{-}_q i^{+*}_d – e^{-}_d i^{+*}_q }{ e^{+}_d } $$ - Target III: Suppress Active Power Oscillations. The goal is to eliminate the 2ω ripple in the active power P. This leads to the reference:
$$ i^{-*}_d = -\frac{ e^{-}_d i^{+*}_d + e^{-}_q i^{+*}_q }{ e^{+}_d }, \quad i^{-*}_q = -\frac{ e^{-}_q i^{+*}_d – e^{-}_d i^{+*}_q }{ e^{+}_d } $$
These three targets can be expressed by a unified formula using a parameter ρ:
$$ i^{-*}_d = \rho \frac{ e^{-}_d i^{+*}_d + e^{-}_q i^{+*}_q }{ e^{+}_d }, \quad i^{-*}_q = \rho \frac{ e^{-}_q i^{+*}_d – e^{-}_d i^{+*}_q }{ e^{+}_d } $$
where ρ = 0 for Target I, ρ = +1 for Target II, and ρ = -1 for Target III. In phasor form, this simplifies to a crucial relationship:
$$ i^{-} = \rho \frac{e^{-}}{e^{+}_d} (i^{+}_d – j i^{+}_q) = \rho \frac{e^{-}}{e^{+}_d} \cdot (i^{+})^{*} $$
This implies that the phase relationship between the output positive and negative sequence currents is fixed by the control target and the negative sequence voltage: $$ \phi^{+} + \phi^{-} = \arg(\rho \cdot e^{-}) $$.
Analysis of Fault Current Characteristics Under Different Strategies
The unified current relationship allows for a systematic analysis of fault current signatures. The characteristics depend heavily on the fault type and the chosen control target ρ.
Single-Line-to-Ground Fault (Phase-A)
Consider a solid Phase-A to ground fault. The pre-fault voltage is assumed nominal. Post-fault, the Phase-A voltage collapses to a fraction λ (0 ≤ λ < 1), while phases B and C remain near their pre-fault magnitude. The resulting negative sequence voltage is primarily a real quantity in the d-axis: $$e^{-}_d = (\lambda – 1)/3$$, $$e^{-}_q ≈ 0$$. Therefore, $$\arg(e^{-}) ≈ 0^{\circ}$$ or $$180^{\circ}$$ depending on the sign of $$e^{-}_d$$ (which is negative).
- Target I (ρ=0): $$i^{-}=0$$. The solar inverter outputs only positive sequence current. All three phase currents have equal magnitude: $$I_a = I_b = I_c = |i^{+}|$$.
- Target II (ρ=+1): $$\phi^{+} + \phi^{-} ≈ 180^{\circ}$$. Substituting into the phase current amplitude formulas shows that the faulted phase (A) current magnitude is the minimum of the three: $$I_a = |\,|i^{+}| – |i^{-}|\,|$$, while the two healthy phases (B, C) have equal and larger magnitude: $$I_b = I_c = \sqrt{ |i^{+}|^2 + |i^{-}|^2 + |i^{+}||i^{-}| }$$.
- Target III (ρ=-1): $$\phi^{+} + \phi^{-} ≈ 0^{\circ}$$. Here, the opposite occurs. The faulted phase (A) current magnitude is the maximum: $$I_a = |i^{+}| + |i^{-}|$$, and the healthy phases have equal but smaller magnitude: $$I_b = I_c = \sqrt{ |i^{+}|^2 + |i^{-}|^2 – |i^{+}||i^{-}| }$$.
Phase-to-Phase Fault (B-C)
For a solid Phase-B to Phase-C fault, analysis of the sequence network shows the negative sequence voltage $$\arg(e^{-})$$ is shifted by approximately $$180^{\circ}$$ compared to the Phase-A ground fault case.
- Target I (ρ=0): Again, currents remain balanced: $$I_a = I_b = I_c = |i^{+}|$$.
- Target II (ρ=+1): Now $$\phi^{+} + \phi^{-} ≈ 0^{\circ}$$. This leads to the healthy phase (A) having the maximum current, and the faulted phases (B, C) having equal, smaller currents. This signature is identical to that of a Phase-A ground fault under Target III.
- Target III (ρ=-1): Here $$\phi^{+} + \phi^{-} ≈ 180^{\circ}$$. This leads to the healthy phase (A) having the minimum current, and the faulted phases (B, C) having equal, larger currents. This signature is identical to that of a Phase-A ground fault under Target II.
These results are summarized comprehensively in the table below, which highlights the symmetry and cross-over of fault signatures between different fault types and control strategies.
| Negative Sequence Control Target | Fault Type | Faulted Phase(s) Current Magnitude | Non-Faulted Phase(s) Current Magnitude |
|---|---|---|---|
| Target I (Suppress I⁻) | Single-Phase-to-Ground | $$I_{f} = |i^{+}|$$ | $$I_{nf} = |i^{+}|$$ |
| Phase-Phase or Phase-Phase-Ground | $$I_{f} = |i^{+}|$$ | $$I_{nf} = |i^{+}|$$ | |
| Target II (Suppress Q Osc.) | Single-Phase-to-Ground | $$I_{f} = |\,|i^{+}| – |i^{-}|\,|$$ | $$I_{nf} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 + |i^{+}||i^{-}| }$$ |
| Phase-Phase or Phase-Phase-Ground | $$I_{f} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 – |i^{+}||i^{-}| }$$ | $$I_{nf} = |i^{+}| + |i^{-}|$$ | |
| Target III (Suppress P Osc.) | Single-Phase-to-Ground | $$I_{f} = |i^{+}| + |i^{-}|$$ | $$I_{nf} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 – |i^{+}||i^{-}| }$$ |
| Phase-Phase or Phase-Phase-Ground | $$I_{f} = \sqrt{ |i^{+}|^2 + |i^{-}|^2 + |i^{+}||i^{-}| }$$ | $$I_{nf} = |\,|i^{+}| – |i^{-}|\,|$$ |
Critical Implications: The controlled nature of the solar inverter fault current is starkly evident. A given fault type can manifest vastly different current magnitude patterns depending on the software-defined control target. Conversely, completely different fault types can produce identical current magnitude patterns if paired with specific control strategies. This behavior is fundamentally different from synchronous machines and poses significant challenges for conventional fault identification, classification (e.g., phase selection), and overcurrent protection setting algorithms that rely on predictable current magnitude ratios.
Equivalent Negative Sequence Impedance Characteristic
The concept of equivalent impedance is central to many protection and power flow analysis techniques. For a solar inverter, the equivalent negative sequence impedance $$Z^{-}$$ is defined as the ratio of the negative sequence voltage at its terminals to the negative sequence current it injects:
$$ Z^{-} = -\frac{e^{-}}{i^{-}} $$
The negative sign convention is used to represent the impedance as a sink for negative sequence current. Substituting the unified control law $$i^{-} = \rho \frac{e^{-}}{e^{+}_d} (i^{+}_d – j i^{+}_q)$$ yields:
$$ Z^{-} = -\frac{e^{+}_d}{\rho (i^{+}_d – j i^{+}_q)} $$
This is a profound result. It shows that the equivalent negative sequence impedance of the solar inverter is not a passive RL element but a complex quantity directly controlled by its positive sequence current output and the parameter ρ.
- Target I (ρ=0): Since $$i^{-}=0$$, the impedance is theoretically infinite $$(|Z^{-}| \to \infty)$$. The inverter presents an open circuit to the negative sequence network.
- Target II (ρ=+1): $$ Z^{-} = -\frac{e^{+}_d}{i^{+}_d – j i^{+}_q} $$. Given that during voltage sags $$i^{+}_q \ge 0$$ (reactive support) and $$i^{+}_d \ge 0$$ (active power export), the denominator lies in the fourth quadrant of the complex plane. Therefore, the impedance $$Z^{-}$$ lies in the second quadrant: its angle $$\angle Z^{-}$$ is between $$90^{\circ}$$ and $$180^{\circ}$$. This resembles an inductive impedance.
- Target III (ρ=-1): $$ Z^{-} = \frac{e^{+}_d}{i^{+}_d – j i^{+}_q} $$. In this case, the impedance lies in the fourth quadrant: its angle $$\angle Z^{-}$$ is between $$-90^{\circ}$$ and $$0^{\circ}$$. This resembles a capacitive impedance.
The magnitude in both active control cases is $$|Z^{-}| = e^{+}_d / \sqrt{ (i^{+}_d)^2 + (i^{+}_q)^2 } = e^{+}_d / |i^{+}|$$, which is simply the ratio of the positive sequence d-axis voltage to the positive sequence current magnitude.
Implications for Protection: The highly variable and controlled nature of $$Z^{-}$$ significantly impacts protection schemes based on negative sequence quantities. Directional elements that rely on a fixed angular relationship between negative sequence voltage and current (e.g., expecting a fault source behind the relay to have a characteristic negative sequence impedance angle) may maloperate. The solar inverter can appear as either an inductive or capacitive source in the negative sequence network depending solely on its software configuration, invalidating traditional assumptions used in fault location and system unbalance analysis.
Synthesis and Conclusion
The integration of solar inverter-based generation necessitates a fundamental re-evaluation of power system fault response models. This analysis has demonstrated that the fault characteristics of a solar inverter during asymmetrical disturbances are not intrinsic properties but are explicitly programmed via its sequence control strategies. The key findings are synthesized as follows:
- Controlled Fault Signatures: The steady-state phase current magnitudes during asymmetrical faults are dictated by the combination of the positive sequence current reference (set by active/reactive power support requirements) and the negative sequence current reference (set by objectives like oscillation suppression). The resulting patterns can defy expectations based on synchronous machine behavior, with the faulted phase not necessarily carrying the highest current.
- Strategy-Dependent Equivalence: The solar inverter‘s equivalent negative sequence impedance is a dynamic, controlled quantity. Its angle can swing from highly inductive to highly capacitive based on the chosen negative sequence control target (ρ), while its magnitude is inversely proportional to the positive sequence current output.
- Cross-Mapping of Fault Phenomena: A specific fault signature (pattern of phase current magnitudes) is not uniquely tied to a fault type. For example, the signature of a single-phase ground fault under a “suppress reactive power oscillation” strategy is identical to that of a phase-phase fault under a “suppress active power oscillation” strategy. This ambiguity challenges traditional fault analysis and protection logic.
The behavior of the modern solar inverter underscores that in the future power grid, “fault characteristics” are increasingly a system design choice rather than a physical inevitability. This has far-reaching consequences. Protection engineers must account for the possible control modes of inverter-based resources in their designs. System planners and operators need to consider the implications of these control strategies on fault levels, voltage unbalance propagation, and the stability of other grid-connected devices. Future research must focus on developing protection algorithms that are adaptive or robust to these controlled fault signatures, and on defining grid code requirements that balance the inverter’s support functions with the need for predictable and manageable fault responses. Understanding the detailed interplay between control strategy and fault response, as laid out here, is the essential first step in ensuring the reliable and secure operation of power systems dominated by power electronic interfaces like the solar inverter.