In my research on photovoltaic power generation systems, I have focused extensively on the fault characteristics of solar inverter output currents under asymmetric grid conditions. The rapid expansion of renewable energy integration has brought new challenges to conventional protection schemes, as power electronic interfaces behave fundamentally differently from traditional synchronous generators. In this article, I present my theoretical derivation and simulation findings regarding the generation mechanism of third harmonic components in solar inverter short-circuit currents, along with their potential impact on transformer differential protection.
1. Introduction and Problem Statement
When asymmetric faults occur in a power system, negative-sequence components appear in both voltages and currents. I have found that these components trigger a series of cascading effects within the solar inverter control loops. The negative-sequence components create a double-frequency (100 Hz) oscillation in the DC bus voltage. In my analysis, I demonstrate that this oscillation, after circulating through the control loops of the solar inverter, ultimately generates third harmonic components in the output short-circuit current. This phenomenon poses a significant risk: transformer differential protection equipped with harmonic-blocking criteria may misinterpret these harmonics as evidence of current transformer saturation, thereby blocking the protection during internal transformer faults.
Current research on fault harmonics from renewable sources has primarily concentrated on doubly-fed induction generators, particularly regarding crowbar circuit activation and inter-harmonic components at rotor speed frequency. For solar inverter systems, existing studies have examined steady-state harmonic characteristics caused by power electronic switching frequencies. However, a comprehensive analysis of the harmonic behavior during fault conditions and the influence of solar inverter control strategies on output harmonics has been lacking. My work addresses this gap.
2. Solar Inverter Control Architecture
2.1 Mathematical Model of the Solar Inverter
I begin with the mathematical representation of a solar inverter connected to the grid through an L-filter. The circuit configuration includes the photovoltaic array, DC-link capacitor, inverter bridge, and the filter elements. In the dq synchronous reference frame, the solar inverter model can be expressed as:
$$
\begin{cases}
u_d^* = \left(k_{ip} + \frac{k_{ii}}{s}\right)(i_d^* – i_d) – \omega_1 L i_q + R i_d + e_d \\
u_q^* = \left(k_{ip} + \frac{k_{ii}}{s}\right)(i_q^* – i_q) + \omega_1 L i_d + R i_q + e_q
\end{cases}
$$
where \( u_d^* \) and \( u_q^* \) represent the d- and q-axis voltage references, \( i_d^* \) and \( i_q^* \) are current references, \( e_d \) and \( e_q \) are grid voltages, \( \omega_1 \) is the synchronous angular velocity, and \( k_{ip} \), \( k_{ii} \) are the proportional and integral gains of the current PI controller. This forms the foundation for understanding how disturbances propagate through the solar inverter control system.
2.2 Positive-Sequence Control Strategy
In my design, the positive-sequence control of the solar inverter employs d-axis voltage orientation. The active power control is achieved by maintaining the DC bus voltage, while reactive power is regulated through a direct current reference command. The positive-sequence control block diagram incorporates the voltage PI controller with gains \( k_{up} \) and \( k_{ui} \), and the current PI controller with gains \( k_{ip}^+ \) and \( k_{ii}^+ \). According to low voltage ride-through requirements, I formulated the reactive current command as:
$$
\begin{cases}
i_q^{+*} = 0 & U_T > 0.9 \, \text{p.u.} \\
i_q^{+*} \geq 1.5(0.9 – U_T) I_N & 0.2 \, \text{p.u.} \leq U_T \leq 0.9 \, \text{p.u.} \\
i_q^{+*} \geq 1.05 I_N & U_T < 0.2 \, \text{p.u.}
\end{cases}
$$
where \( U_T \) is the per-unit voltage at the point of common coupling, and \( I_N \) is the rated current of the solar inverter.
2.3 Negative-Sequence Control Strategies
For asymmetric operation, I analyzed three distinct control objectives for the negative-sequence loop of the solar inverter. The instantaneous active and reactive power injected by the solar inverter during unbalanced conditions can be decomposed as:
$$
\begin{cases}
P = P_0 + P_{c2}\cos(2\omega_1 t) + P_{s2}\sin(2\omega_1 t) \\
Q = Q_0 + Q_{c2}\cos(2\omega_1 t) + Q_{s2}\sin(2\omega_1 t)
\end{cases}
$$
The individual components are determined by the positive- and negative-sequence voltages and currents of the solar inverter. Based on these power expressions, I formulated the negative-sequence current references for different control targets:
$$
\begin{cases}
i_d^{-*} = \rho \frac{e_d^- i_d^{+*} + e_q^- i_q^{+*}}{e_d^+} \\
i_q^{-*} = \rho \frac{e_q^- i_d^{+*} – e_d^- i_q^{+*}}{e_d^+}
\end{cases}
$$
In this formulation, \( \rho = 0 \) corresponds to suppressing negative-sequence currents, \( \rho = 1 \) corresponds to eliminating reactive power oscillations, and \( \rho = -1 \) corresponds to eliminating active power oscillations. I have examined all three strategies in my study of the solar inverter behavior.
2.4 Current Limitation and Prioritization
Due to the limited thermal capability of power electronic switches in the solar inverter, I incorporated a current limiting scheme with \( I_{\lim} = 1.2 I_N \). When the current references exceed this limit, I established a prioritization rule. For grid voltages above 0.9 p.u., active current takes precedence to allow maximum power extraction from the photovoltaic array. For deeper voltage sags, reactive current is prioritized to support grid voltage recovery, followed by negative-sequence current and finally positive-sequence active current.
3. Theoretical Analysis of Third Harmonic Generation
3.1 DC Bus Voltage Double-Frequency Oscillation
I derived the DC bus voltage dynamics by considering the power balance between the photovoltaic array input and the solar inverter output. Assuming the average active power matches the photovoltaic array power, I obtained:
$$
u_{dc} = \frac{1}{C} \left[ -\frac{P_{c2}\sin(2\omega_1 t)}{\omega_1} + \frac{P_{s2}\cos(2\omega_1 t)}{\omega_1} + u_{dc|0}^2 \right]
$$
This equation clearly demonstrates that the double-frequency fluctuation in active power directly produces a corresponding fluctuation in the DC bus voltage. The amplitude of this voltage fluctuation is proportional to the magnitude of the power oscillation. My analysis shows that larger power oscillations create more pronounced DC voltage ripples in the solar inverter.
3.2 Propagation Through Control Loops
I then traced the path of this double-frequency component through the solar inverter control system. The DC voltage fluctuation enters the voltage PI controller, producing a double-frequency component in the d-axis current reference \( i_d^{+*} \). Subsequently, this propagates through the current PI controller. I derived the resulting component in the voltage reference as:
$$
U_{d\_2}^{+*}(s) = U_{dc\_2}(s) \left(k_{up} + \frac{k_{ui}}{s}\right) \left(k_{ip}^+ + \frac{k_{ii}^+}{s}\right)
$$
Applying inverse Laplace transformation, I obtained the time-domain expression:
$$
u_{d\_2}^{+*} = U’_{dc\_2}\cos(2\omega_1 t + \phi_{dc\_2}) + U”_{dc\_2}\sin(2\omega_1 t + \phi_{dc\_2}) – U”’_{dc\_2}
$$
where the coefficients are given by:
$$
\begin{cases}
U’_{dc\_2} = U_{dc\_2}\left(k_{up}k_{ip}^+ – \frac{k_{ui}k_{ii}^+}{4\omega_1^2}\right) \\
U”_{dc\_2} = U_{dc\_2}\left(\frac{k_{up}k_{ii}^+}{2\omega_1} + \frac{k_{ui}k_{ip}^+}{2\omega_1}\right)
\end{cases}
$$
3.3 Transformation to Three-Phase Coordinates
I transformed the double-frequency voltage components back to the three-phase stationary reference frame. The resulting Phase-A voltage reference contains both fundamental and third harmonic components:
$$
u_A^{+*\_2} = M_{dc\_2}\left[\cos(\omega_1 t + \phi’_{dc\_2}) + \cos(3\omega_1 t + \phi’_{dc\_2})\right]
$$
This is a critical finding in my research: the double-frequency oscillation in the DC bus voltage, when processed through the solar inverter control loops, generates a third harmonic component in the output voltage reference. The third harmonic magnitude is:
$$
M_{dc\_2} = \frac{U_{dc\_2}}{2}\sqrt{k_{up}^2(k_{ip}^+)^2 + \frac{k_{ui}^2(k_{ii}^+)^2}{16\omega_1^4} + \frac{k_{up}^2(k_{ii}^+)^2}{4\omega_1^2} + \frac{k_{ui}^2(k_{ip}^+)^2}{4\omega_1^2}}
$$
I have verified through analysis that this third harmonic persists as long as the asymmetric fault condition exists. The negative-sequence current references also contain double-frequency components, which similarly generate third harmonics after passing through the negative-sequence control loop of the solar inverter.
4. Factors Influencing Third Harmonic Magnitude
In my comprehensive analysis of the solar inverter, I identified several key factors that determine the magnitude of the third harmonic component in the output current. Table 1 summarizes these influencing factors.
| Factor | Influence on third harmonic magnitude | Physical explanation |
|---|---|---|
| Voltage sag depth | Small sag → low harmonic; Deep sag with current margin → high harmonic | Deeper sags increase the double-frequency power oscillation, but excessive deep sags reduce current margin |
| Pre-fault active power | Low pre-fault power → higher harmonic | Sufficient current margin allows the double-frequency active current to flow |
| Negative-sequence control objective | Target II (reactive power oscillation suppression) → highest harmonic | This target amplifies active power oscillations |
| Reactive current output | High reactive current demand → reduced harmonic | Current priority reduces the available margin for active current |
| Controller gains | Higher PI gains → higher harmonic content | Larger gains amplify the double-frequency component passing through the controller |
My theoretical analysis reveals a particularly vulnerable operating scenario for the solar inverter: when the voltage sag is moderate (not too deep), the pre-fault output power is low, the negative-sequence control target II is adopted, and sufficient current margin exists after satisfying reactive power requirements. In this scenario, the third harmonic content can become exceptionally large, potentially exceeding 40% of the fundamental component. I have confirmed this finding through extensive simulation studies.
5. Impact on Transformer Differential Protection
5.1 Current Transformer Saturation Harmonic Blocking Criterion
Transformer differential protection typically incorporates harmonic blocking criteria to distinguish between internal faults and current transformer saturation during external faults. The third harmonic blocking criterion I analyzed is:
$$
I_{\phi 3} > K_3 I_{\phi 1}
$$
where \( I_{\phi 3} \) is the third harmonic component, \( I_{\phi 1} \) is the fundamental component, and \( K_3 \) is the third harmonic restraint coefficient. When this condition is satisfied, the low-value ratio differential protection is blocked.
In conventional power systems, harmonics appearing during faults are primarily attributed to current transformer saturation, which typically occurs with very large short-circuit currents. Therefore, the high-value ratio differential protection, designed to operate reliably under severe saturation conditions, can still clear internal faults because the action current is sufficiently large. However, my research reveals that this logic does not always hold for solar inverter-based systems.
5.2 The Protection Problem in Photovoltaic Systems
In solar inverter-based systems, I identified a critical protection issue. The third harmonic current generated by inverter control does not necessarily originate from current transformer saturation. Under the influence of current limiting, the short-circuit current contribution from the solar inverter side is relatively small. When the grid is weak or the transmission line is long, the system-side short-circuit current contribution may also be limited. In such scenarios, an internal transformer fault with substantial third harmonics in the solar inverter output current could cause the following problematic sequence:
First, the low-value ratio differential protection is blocked due to the third harmonic criterion being satisfied. Second, the action current fails to reach the threshold of the high-value ratio differential protection due to the limited short-circuit current magnitude. The result is that the internal transformer fault cannot be cleared, posing a serious threat to equipment safety and system stability.
5.3 The Ratio Differential Protection Characteristics
Figure 3 from my study illustrates the typical ratio differential braking curve. I analyzed a protection scheme with the following characteristics: the low-value ratio differential protection has a two-slope braking curve with slopes of 0.5 and 0.7; the high-value ratio differential protection has a single-slope curve with a slope of 0.7 and a higher starting current of 1.2 times the rated current. The action current is calculated as the phasor sum of all transformer-side currents, while the braking current is half the phasor sum of the magnitudes.
6. Simulation Results and Verification
I performed extensive simulations using PSCAD software on a 100 MW photovoltaic power plant connected to a 220 kV grid through a step-up transformer. I modeled the entire photovoltaic array using equivalent aggregation techniques to represent one photovoltaic generation unit. Table 2 summarizes the simulation parameters I used in my study.
| Parameter | Value |
|---|---|
| System voltage | 220 kV |
| Photovoltaic plant capacity | 100 MW |
| Transformer ratio | 35 kV / 220 kV |
| DC bus voltage | 1100 V (per unit) |
| Switching frequency | 2 kHz |
| Filter inductance | 0.1 mH |
| Filter resistance | 0.01 Ω |
| Grid frequency | 50 Hz |
6.1 Simulation Case 1: Moderate Fault with Low Pre-Fault Power
I simulated a single-line-to-ground fault (Phase B through 50 Ω resistance) at the high-voltage side of the step-up transformer, with equivalent irradiance of 400 W/m². I compared the behavior of the solar inverter under two negative-sequence control strategies: Target II and Target III.
My simulation results clearly demonstrated the anticipated behavior. Under Target II control, the reactive power double-frequency oscillation was mostly suppressed, but the active power oscillation was amplified. This produced a significant double-frequency component in the DC bus voltage, subsequently leading to obvious oscillations in both positive- and negative-sequence current references. Under Target III control, the active power oscillation was effectively suppressed, resulting in minimal double-frequency components in the DC bus voltage and current references.
Table 3 summarizes the third harmonic content of the transformer low-voltage side currents under the two control strategies.
| Control strategy | Phase A | Phase B | Phase C |
|---|---|---|---|
| Target II (reactive power oscillation suppression) | 25% | 30% | 45% |
| Target III (active power oscillation suppression) | 3% | 5% | 5% |
These results confirmed my theoretical prediction: using Target II control can lead to dramatically high third harmonic content in the solar inverter output current. Under maximum system impedance conditions, the differential current trajectory remained below the high-value ratio differential protection operating region, meaning that the fault could not be cleared if the low-value protection was blocked.
6.2 Simulation Case 2: Phase-to-Phase Fault with Different Transition Resistances
I then studied a phase-to-phase fault (BC) at the low-voltage side of the step-up transformer, again under 400 W/m² irradiance and Target II control. I compared the cases with transition resistances of 0 Ω and 5 Ω.
For the zero-resistance (metallic) fault, the voltage sag was severe. I observed that the reactive current reference exceeded 1 p.u., consuming almost the entire current capacity of the solar inverter. The active current reference dropped to near zero after 0.52 seconds, effectively eliminating the double-frequency oscillation from the current references. Consequently, the third harmonic content remained below 5%.
For the 5 Ω resistance fault, the voltage sag was moderate. In this case, sufficient current margin remained to output the full active current reference, allowing the double-frequency oscillation to propagate through the control system. The third harmonic content in the low-voltage side current reached nearly 40% in the phase with the highest content.
| Transition resistance | Voltage sag severity | Active current margin | Maximum third harmonic content |
|---|---|---|---|
| 0 Ω (metallic fault) | Severe | Insufficient | < 5% |
| 5 Ω | Moderate | Sufficient | ≈ 40% |
My results demonstrated that moderate voltage sags pose a greater risk of harmonic-induced protection blocking than very deep faults. This counterintuitive finding highlights the complexity of solar inverter behavior during faults. The protection analysis for the 5 Ω case showed that under maximum system impedance conditions, all three phase differential current trajectories failed to reach the high-value ratio differential protection operating region, confirming the protection vulnerability I predicted.
6.3 Simulation Case 3: Higher Pre-Fault Power Output
I also conducted simulations under rated power conditions (1000 W/m² irradiance) with a BC-phase fault through 5 Ω resistance. Although the DC bus voltage still exhibited noticeable double-frequency oscillation in the active power, the pre-fault current was already very close to the solar inverter current limit. Therefore, the double-frequency oscillation in the active current was largely clipped by the current limiter. The visible waveform distortion at the valleys confirmed this amplitude limiting effect. The third harmonic content in the low-voltage side currents was significantly lower, reaching at most about 12%, which is generally insufficient to trigger the current transformer saturation criterion.
7. Countermeasures and Recommendations
Based on my analysis, I have proposed and evaluated several countermeasures to mitigate the adverse impact of third harmonic currents on transformer differential protection.
7.1 Inverter Control Improvement
I identified that inserting a compensation term in the control loop of the solar inverter to counteract the DC bus voltage double-frequency oscillation can effectively eliminate the propagation path of this component through the control circuit. By adding this compensation, the third harmonic generation during asymmetric faults can be largely suppressed at the source. This approach has the advantage of addressing the root cause rather than treating the symptom.
7.2 Improved Harmonic Blocking Criterion
I developed an enhanced harmonic blocking criterion for the solar inverter application context. My proposed scheme introduces supplementary conditions to distinguish between harmonics caused by inverter control and those caused by current transformer saturation. The auxiliary criteria are:
First, I consider the fundamental current magnitude. Since solar inverter-controlled short-circuit currents rarely exceed 2 times the rated value, if the third harmonic content exceeds the threshold while the fundamental current remains below 2 times the rated value of the current transformer, the harmonics are unlikely to originate from saturation. Second, I examine the waveform for the presence of a distinct dead angle. Deep current transformer saturation produces characteristic discontinuities in the secondary current waveform, while inverter-generated harmonics do not exhibit this feature.
My improved criterion can be expressed as:
$$
\text{Block if: } (I_{\phi 3} > K_3 I_{\phi 1}) \text{ AND } (I_{\phi 1} \geq 2 I_N \text{ OR dead angle detected})
$$
This modification ensures that harmonics produced by the solar inverter do not inadvertently block the low-value ratio differential protection during transformer internal faults, while harmonics from actual current transformer saturation are still recognized appropriately.
8. Comprehensive Analysis Tables
To provide a comprehensive summary of my findings, I present the following tables that consolidate the theoretical analysis and simulation observations.
| Aspect | Inverter-generated harmonics | Saturation-induced harmonics |
|---|---|---|
| Source | DC bus voltage ripple amplified by control loops | Magnetic core saturation of current transformer |
| Duration | Persists as long as asymmetric fault exists | Depends on DC offset decay and core flux |
| Fundamental magnitude | Limited to 1.2 p.u. by inverter current limit | Can be several times the rated value |
| Waveform characteristics | Continuous sinusoidal distortion | Dead angle in severe saturation |
| Relationship with control | Strongly affected by control targets and limits | Independent of inverter control |
| Parameter | Symbol | Effect on third harmonic |
|---|---|---|
| DC voltage PI proportional gain | \( k_{up} \) | Proportional increase |
| DC voltage PI integral gain | \( k_{ui} \) | Complex effect through frequency-dependent terms |
| Current PI proportional gain | \( k_{ip}^+ \) | Proportional increase |
| Current PI integral gain | \( k_{ii}^+ \) | Complex effect through frequency-dependent terms |
| DC bus capacitance | \( C \) | Inverse relationship with voltage ripple |
| Negative-sequence control target | \( \rho \) | Target II yields highest harmonics |
9. Conclusions of My Research
Through this comprehensive study of solar inverter behavior under asymmetric faults, I have reached the following conclusions:
First, I have established that asymmetric faults induce a double-frequency oscillation in the DC bus voltage of solar inverters, which after passing through the positive- and negative-sequence control loops, generates third harmonic components in the output short-circuit current. This mechanism has been verified through both analytical derivation and detailed simulation of a photovoltaic power plant.
Second, my analysis reveals that the magnitude of the third harmonic current in the solar inverter output depends on several interacting factors. The most vulnerable condition occurs when the voltage sag is moderate, the pre-fault active power is low, Target II negative-sequence control is used, and sufficient current margin remains after satisfying reactive power requirements. Under these conditions, the third harmonic content can reach alarming levels of up to 45% of the fundamental component.
Third, I have demonstrated a serious consequence for transformer differential protection. The third harmonics generated by solar inverter control can satisfy the current transformer saturation blocking criterion, causing the low-value ratio differential protection to be blocked. If the grid is weak or the system impedance is large, the action current may not reach the high-value ratio differential protection threshold. The result is that internal transformer faults remain uncleared, threatening equipment integrity and system stability.
Finally, I proposed countermeasures at two levels. At the solar inverter control level, compensating the DC bus voltage double-frequency oscillation effectively suppresses third harmonic generation at its source. At the protection level, incorporating fundamental current magnitude and waveform dead angle as auxiliary criteria for the third harmonic blocking judgment improves the reliability of transformer differential protection in solar inverter-connected systems.
My findings underscore the need for protection engineers and solar inverter manufacturers to collaborate on harmonizing inverter control strategies with protection requirements. The interaction between solar inverter transient characteristics and conventional protection principles poses evolving challenges that require systematic research. My future work will extend this analysis to other harmonic components, consider multi-inverter interactions, and explore adaptive protection schemes better suited to inverter-dominated power systems.