In my analysis, I treat the solar inverter not as a simple ideal source but as a fast, current-limited power-electronic interface whose transient behavior is shaped by its DC-link voltage controller, positive-sequence current controller, negative-sequence current controller, low-voltage ride-through logic, and hard current limiter. This viewpoint is important because a solar inverter does not respond to an asymmetric fault in the same way as a synchronous machine. When an asymmetric fault occurs, negative-sequence voltage and current components appear, and those components enter the solar inverter control loops. The result is a second-order oscillation at twice the fundamental frequency in the DC-link voltage, and that oscillation later produces a third-harmonic component in the solar inverter short-circuit current. My purpose here is to explain the mechanism, derive the analytical expression, summarize the controlling factors, and evaluate the consequences for transformer differential protection, especially current-transformer saturation blocking logic.
I use the term solar inverter repeatedly because every harmonic mechanism discussed below is tied to its controllers and limits. A solar inverter operating under balanced conditions may appear fairly clean from a harmonic perspective. However, during an asymmetric fault, the solar inverter can become a significant source of third-harmonic current. This third-harmonic current is not merely a power-quality nuisance. It can satisfy the harmonic restraint or saturation blocking criterion of a transformer differential relay, block the low-slope differential element, and create a risk that an internal transformer fault is not cleared quickly. I therefore organize my discussion around four questions: how the solar inverter is modeled, how the third harmonic is generated, how the third harmonic depends on control and limiting, and how protection can remain dependable.
The solar inverter circuit that I consider consists of a PV array, a DC-link capacitor, a three-phase voltage-source converter, and an RL filter connected to the grid. The PV array delivers current and power to the DC link. The solar inverter transfers power from the DC side to the AC side. I denote the array current and power by \(i_{array}\) and \(P_{array}\), the DC-link input current and power by \(i_{dc}\) and \(P_{in}\), the DC-link voltage by \(u_{dc}\), the DC-link capacitance by \(C\), the solar inverter output voltages by \(u_a,u_b,u_c\), the output currents by \(i_a,i_b,i_c\), the grid voltages by \(e_a,e_b,e_c\), and the filter resistance and inductance by \(R\) and \(L\). I apply the voltage and current reference directions consistently so that the DC-link energy balance can be written directly.
For the AC side, I use the synchronous rotating \(dq\) frame. The solar inverter current controller is usually implemented as a proportional-integral controller with feedforward decoupling and grid-voltage feedforward. In my notation, the positive-sequence current references are \(i_d^{+*}\) and \(i_q^{+*}\), the measured positive-sequence currents are \(i_d^+\) and \(i_q^+\), and the grid voltages are \(e_d^+\) and \(e_q^+\). The current PI gains are \(k_{ip}\) and \(k_{ii}\). The synchronous angular frequency is \(\omega_1\). The resulting solar inverter voltage references are as follows:
$$
u_d^{+*} = \left(k_{ip}+\frac{k_{ii}}{s}\right)\left(i_d^{+*}-i_d^+\right)-\omega_1 L i_q^+ + R i_d^+ + e_d^+
$$
$$
u_q^{+*} = \left(k_{ip}+\frac{k_{ii}}{s}\right)\left(i_q^{+*}-i_q^+\right)+\omega_1 L i_d^+ + R i_q^+ + e_q^+
$$
This model is simple, but it contains the essential path by which a DC-link ripple becomes an AC current ripple. The DC-link voltage error is processed by the outer voltage controller, which produces the active-current reference \(i_d^{+*}\). Therefore, any ripple in \(u_{dc}\) appears as a ripple in \(i_d^{+*}\). If the negative-sequence current reference is calculated from \(i_d^{+*}\), then the same ripple can also appear in the negative-sequence current command. This is the start of the third-harmonic mechanism.
I summarize the main variables and gains in Table 1. The table is not a substitute for the equations, but it helps me keep track of which controller parameters amplify the harmonic path.
| Symbol | Meaning in my solar inverter model | Role in third-harmonic generation |
|---|---|---|
| \(u_{dc}\) | DC-link voltage of the solar inverter | Carries the second-order oscillation that drives the third harmonic |
| \(i_d^+,i_q^+\) | Positive-sequence \(d\)- and \(q\)-axis currents | \(i_d^+\) controls active power and is most strongly affected by DC-link ripple |
| \(i_d^-,i_q^-\) | Negative-sequence \(d\)- and \(q\)-axis currents | Negative-sequence commands may inherit ripple from positive-sequence commands |
| \(k_{up},k_{ui}\) | DC-link voltage PI gains | Higher gains can increase the transfer of DC ripple into current commands |
| \(k_{ip},k_{ii}\) | Current PI gains of the solar inverter | Set the dynamic response and magnitude of voltage ripple components |
| \(I_{lim}\) | Current limit of the solar inverter | Determines whether active-current ripple is passed or clipped |
| \(\rho\) | Negative-sequence control objective coefficient | Selects which power oscillation is minimized or maximized |
The positive-sequence control of the solar inverter has two main tasks. The first is to regulate the DC-link voltage, which indirectly controls active power. The second is to inject reactive current according to the grid code. In normal operation, the reactive-current reference is often zero. During a voltage dip, the solar inverter must prioritize reactive current. I use the following low-voltage ride-through logic for the positive-sequence reactive-current reference:
$$
i_q^{+*} = 0,\qquad U_T > 0.9\ \text{p.u.}
$$
$$
i_q^{+*} \ge 1.5(0.9-U_T)I_N,\qquad 0.2\ \text{p.u.} \le U_T \le 0.9\ \text{p.u.}
$$
$$
i_q^{+*} \ge 1.05 I_N,\qquad U_T < 0.2\ \text{p.u.}
$$
Here \(U_T\) is the per-unit terminal voltage and \(I_N\) is the rated current of the solar inverter. This logic is important because it determines how much current margin remains for active current and negative-sequence current. If the solar inverter uses most of its current capability for reactive current, then the active-current ripple may be clipped by the limiter, and the third harmonic may decrease. If the voltage dip is shallow enough that the solar inverter can still deliver active current, then the active-current ripple can pass through and the third harmonic can become large. This is one of the central conclusions of my analysis.
When the grid voltage is unbalanced, the solar inverter sees positive-sequence and negative-sequence components. The instantaneous active and reactive powers delivered by the solar inverter contain DC terms plus second-order oscillatory terms. I write the active and reactive powers as follows:
$$
P = P_0 + P_{c2}\cos(2\omega_1 t) + P_{s2}\sin(2\omega_1 t)
$$
$$
Q = Q_0 + Q_{c2}\cos(2\omega_1 t) + Q_{s2}\sin(2\omega_1 t)
$$
The coefficients are functions of positive- and negative-sequence voltages and currents. In my notation, the positive-sequence voltages are \(e_d^+,e_q^+\), the negative-sequence voltages are \(e_d^-,e_q^-\), the positive-sequence currents are \(i_d^+,i_q^+\), and the negative-sequence currents are \(i_d^-,i_q^-\). The expanded coefficients are summarized in Table 2.
| Power coefficient | Expression in my solar inverter model |
|---|---|
| \(P_0\) | \(e_d^+ i_d^+ + e_q^+ i_q^+ + e_d^- i_d^- + e_q^- i_q^-\) |
| \(P_{c2}\) | \(e_d^- i_d^+ + e_q^- i_q^+ + e_d^+ i_d^- + e_q^+ i_q^-\) |
| \(P_{s2}\) | \(e_q^- i_d^+ – e_d^- i_q^+ – e_q^+ i_d^- + e_d^+ i_q^-\) |
| \(Q_0\) | \(e_q^+ i_d^+ – e_d^+ i_q^+ + e_q^- i_d^- – e_d^- i_q^-\) |
| \(Q_{c2}\) | \(e_q^- i_d^+ – e_d^- i_q^+ + e_q^+ i_d^- – e_d^+ i_q^-\) |
| \(Q_{s2}\) | \(-e_d^- i_d^+ – e_q^- i_q^+ + e_d^+ i_d^- + e_q^+ i_q^-\) |
I now consider the negative-sequence control objectives. A solar inverter can be controlled to pursue different goals during unbalanced voltage. A common set of objectives is: suppress negative-sequence current, suppress the second-order reactive-power oscillation, or suppress the second-order active-power oscillation. I encode these objectives by a coefficient \(\rho\). When \(\rho=0\), the solar inverter suppresses negative-sequence current. When \(\rho=1\), it suppresses the second-order reactive-power oscillation. When \(\rho=-1\), it suppresses the second-order active-power oscillation. The negative-sequence current references I use are:
$$
i_d^{-*} = \rho \frac{e_d^- i_d^{+*} + e_q^- i_q^{+*}}{e_d^+}
$$
$$
i_q^{-*} = \rho \frac{e_q^- i_d^{+*} – e_d^- i_q^{+*}}{e_d^+}
$$
This is a very compact expression, but it has a deep consequence. The negative-sequence current reference is a function of the positive-sequence active-current reference \(i_d^{+*}\). If \(i_d^{+*}\) contains a second-order ripple because of DC-link voltage ripple, then \(i_d^{-*}\) may contain a second-order ripple as well. That ripple then flows through the negative-sequence current controller and contributes to a third harmonic in the solar inverter output current. The negative-sequence control objective therefore changes both the power oscillation and the harmonic content. I summarize this in Table 3.
| \(\rho\) | Negative-sequence objective | Effect on active-power second-order ripple | Typical effect on third harmonic from solar inverter |
|---|---|---|---|
| 0 | Suppress negative-sequence current | Depends on grid asymmetry and current limit | Can be moderate if active-current ripple is not clipped |
| 1 | Suppress reactive-power second-order ripple | Often increases active-power second-order ripple | Usually the largest third harmonic among the three objectives |
| -1 | Suppress active-power second-order ripple | Directly reduces active-power second-order ripple | Usually the smallest third harmonic among the three objectives |
Current limiting is unavoidable in a solar inverter. The semiconductor devices and filter components have thermal and instantaneous current limits. I define the current-limit magnitude as \(I_{lim}=1.2I_N\). The positive-sequence and negative-sequence current commands may each be reasonable individually, but their sum may exceed the limit. Therefore, the solar inverter controller must choose a priority order. I use the following priority logic. When the terminal voltage is above \(0.9\) p.u., the solar inverter does not need to inject reactive current for low-voltage ride-through, so active current has priority and the remaining current margin can be used for negative-sequence current. When the terminal voltage is at or below \(0.9\) p.u., reactive current has priority because the grid code requires voltage support. After that, negative-sequence current may be used to limit power oscillation. Active current is considered last. If the DC-link voltage rises too high, a chopper or unloading circuit may dissipate excess energy. This priority logic is summarized in Table 4.
| Condition | First priority | Second priority | Third priority | Consequence for third harmonic |
|---|---|---|---|---|
| \(U_T>0.9\) p.u. | Active current | Negative-sequence current | Reactive current usually zero | Active-current ripple can pass, so third harmonic may appear if asymmetry exists |
| \(0.2\le U_T\le0.9\) p.u. | Reactive current | Negative-sequence current | Active current if margin remains | Third harmonic depends strongly on remaining active-current margin |
| \(U_T<0.2\) p.u. | Reactive current | Negative-sequence current | Active current usually minimal | Active-current ripple is clipped, so third harmonic may be reduced |
I next derive the DC-link voltage oscillation. If I neglect losses in the solar inverter, the DC-link voltage equation is:
$$
C\frac{du_{dc}}{dt} = i_{array}-i_{dc} = \frac{P_{array}-P}{u_{dc}}
$$
I assume that the average active power \(P_0\) is approximately equal to the array power \(P_{array}\). The oscillatory part of \(P\) then charges and discharges the DC-link capacitor. Solving for the DC-link voltage gives an expression of the following form:
$$
u_{dc} = \frac{1}{C}\sqrt{-\frac{P_{c2}\sin(2\omega_1 t)}{\omega_1} + \frac{P_{s2}\cos(2\omega_1 t)}{\omega_1} + u_{dc0}^2}
$$
where \(u_{dc0}\) is the DC-link voltage before the fault. This expression shows that the DC-link voltage contains a second-order oscillation. The amplitude of that oscillation is proportional to the amplitude of the active-power second-order ripple. Therefore, the negative-sequence control objective that maximizes active-power ripple will also maximize the DC-link voltage ripple. I often approximate the DC-link voltage as a DC term plus a single second-order cosine term:
$$
u_{dc} \approx U_{dc0} + U_{dc2}\cos(2\omega_1 t + \phi_{dc2})
$$
This approximation is useful for deriving the third-harmonic current. I now let the second-order DC-link component be:
$$
u_{dc2} = U_{dc2}\cos(2\omega_1 t + \phi_{dc2})
$$
The outer voltage controller in the solar inverter processes this ripple. The voltage error is multiplied by the voltage PI controller and then used as the active-current reference. The current controller then processes the resulting current reference. For the \(d\)-axis path, the voltage reference component caused by \(u_{dc2}\) can be written in the Laplace domain as:
$$
U_{d2}^{+*}(s) = U_{dc2}(s)\left(k_{up}+\frac{k_{ui}}{s}\right)\left(k_{ip}^+ + \frac{k_{ii}^+}{s}\right)
$$
I perform an inverse Laplace transformation. The result contains a cosine term at \(2\omega_1\), a sine term at \(2\omega_1\), and a DC offset. I write it as:
$$
u_{d2}^{+*} = U’_{dc2}\cos(2\omega_1 t + \phi_{dc2}) + U”_{dc2}\sin(2\omega_1 t + \phi_{dc2}) – U”’_{dc2}
$$
where
$$
U’_{dc2} = U_{dc2}\left(k_{up}k_{ip}^+ – \frac{k_{ui}k_{ii}^+}{4\omega_1^2}\right)
$$
$$
U”_{dc2} = U_{dc2}\left(\frac{k_{up}k_{ii}^+}{2\omega_1} + \frac{k_{ui}k_{ip}^+}{2\omega_1}\right)
$$
The DC offset \(U”’_{dc2}\) does not contribute to the third harmonic, so I do not carry it further. The important point is that the second-order ripple in the DC-link voltage becomes a second-order ripple in the \(d\)-axis voltage reference. Because the \(q\)-axis path does not see the same DC-link voltage ripple in this derivation, I set \(u_{q2}^{+*}=0\). I then transform the \(d\)-axis second-order component back to the stationary \(abc\) frame. The transformation produces two frequencies: a fundamental-frequency component and a third-harmonic component. The three-phase voltage references become:
$$
u_{A2}^{+*} = M_{dc2}\left[\cos(\omega_1 t + \phi’_{dc2}) + \cos(3\omega_1 t + \phi’_{dc2})\right]
$$
$$
u_{B2}^{+*} = M_{dc2}\left[\cos\left(\omega_1 t + \phi’_{dc2} + \frac{2\pi}{3}\right) + \cos\left(3\omega_1 t + \phi’_{dc2} – \frac{2\pi}{3}\right)\right]
$$
$$
u_{C2}^{+*} = M_{dc2}\left[\cos\left(\omega_1 t + \phi’_{dc2} – \frac{2\pi}{3}\right) + \cos\left(3\omega_1 t + \phi’_{dc2} + \frac{2\pi}{3}\right)\right]
$$
The amplitude \(M_{dc2}\) and phase \(\phi’_{dc2}\) are given by:
$$
M_{dc2} = \frac{1}{2}\sqrt{(U’_{dc2})^2 + (U”_{dc2})^2}
$$
$$
\phi’_{dc2} = \phi_{dc2} – \phi_1
$$
$$
\phi_1 = \arctan\left(\frac{U”_{dc2}}{U’_{dc2}}\right)
$$
Substituting the expressions for \(U’_{dc2}\) and \(U”_{dc2}\) gives an explicit dependence of the third-harmonic amplitude on the DC-link ripple and the PI gains:
$$
M_{dc2} = \frac{U_{dc2}}{2}\sqrt{
k_{up}^2(k_{ip}^+)^2
+ \frac{k_{ui}^2(k_{ii}^+)^2}{16\omega_1^4}
+ \frac{k_{up}^2(k_{ii}^+)^2}{4\omega_1^2}
+ \frac{k_{ui}^2(k_{ip}^+)^2}{4\omega_1^2}
}
$$
This formula is one of the central results of my study. It tells me that the third harmonic in the solar inverter output current is not a fixed property of the device. It is a function of the DC-link voltage ripple \(U_{dc2}\) and the voltage and current PI gains. If the solar inverter controller is tuned aggressively, the third harmonic can be amplified. If the control objective reduces active-power ripple, then \(U_{dc2}\) is small, and the third harmonic is small. If the control objective increases active-power ripple, then \(U_{dc2}\) is large, and the third harmonic can be very large.
The same logic applies to the negative-sequence path. The negative-sequence current reference is a function of the positive-sequence current references. When the positive-sequence active-current reference contains a second-order ripple, the negative-sequence current reference also contains a second-order ripple. That ripple passes through the negative-sequence current controller and produces additional third-harmonic voltage and current components. Therefore, I consider both the positive-sequence and negative-sequence control paths when I estimate the total third harmonic of the solar inverter.
I now summarize the main characteristics of the third harmonic from a solar inverter under asymmetric faults. The third harmonic is not caused by magnetic saturation in the traditional sense. It is caused by the control response to negative-sequence components. The third harmonic exists as long as the asymmetric fault remains. It is not a transient harmonic that disappears after a few cycles. Its magnitude depends on the depth and type of asymmetry, the active-power level before the fault, the reactive-current requirement during the fault, the negative-sequence control objective, and the current limiter. I list these dependencies in Table 5.
| Factor | How it affects third harmonic in the solar inverter | Practical interpretation |
|---|---|---|
| Voltage dip depth | Very shallow dip gives small negative-sequence disturbance and small third harmonic. Very deep dip may force reactive-current priority and clip active-current ripple, reducing third harmonic. | The largest third harmonic often occurs at intermediate voltage dip or with moderate fault resistance. |
| Pre-fault active power | Low pre-fault active power leaves more current margin for active-current ripple. High pre-fault active power may saturate the limiter and clip the ripple. | The same fault can produce different harmonic levels depending on solar irradiance. |
| Negative-sequence objective | Objective that minimizes active-power ripple reduces DC-link ripple and third harmonic. Objective that minimizes reactive-power ripple often increases active-power ripple and third harmonic. | The control choice for the solar inverter is also a protection-relevant choice. |
| Reactive-current priority | Strong reactive-current priority reduces available active-current margin and can clip the ripple. | Low-voltage ride-through may reduce third harmonic in severe dips. |
| Current limit | Hard limit clips second-order current oscillations. | The solar inverter third harmonic is not simply proportional to fault severity. |
| PI gains | Higher gains can increase the transfer from DC-link ripple to voltage and current ripple. | Controller tuning by the solar inverter manufacturer matters. |
From my derivations and simulations, the most severe condition for third-harmonic current from a solar inverter is usually the following: the voltage dip is not so deep that active current disappears, the pre-fault active power is modest, the negative-sequence control uses the objective that minimizes reactive-power ripple, and the solar inverter still has current margin to output the full active-current command. In that condition, the active-power second-order ripple is large, the DC-link voltage ripple is large, the active-current command contains a large second-order component, and the solar inverter output current contains a large third harmonic. I have observed third-harmonic ratios approaching \(40\%\) to \(45\%\) of the fundamental current in such cases.
I now turn to the protection problem. Transformer differential protection is widely used because it can isolate internal faults with good sensitivity. However, differential protection must deal with current-transformer saturation. When a current transformer saturates, its secondary current is distorted, and a differential current can appear even for an external fault. To prevent maloperation, differential relays commonly use harmonic blocking. A typical third-harmonic blocking criterion is:
$$
I_{\phi3} > K_3 I_{\phi1}
$$
where \(I_{\phi3}\) is the third-harmonic current in phase \(\phi\), \(I_{\phi1}\) is the fundamental current in the same phase, and \(K_3\) is the third-harmonic restraint coefficient. A second-harmonic criterion is also often used:
$$
I_{\phi2} > K_2 I_{\phi1}
$$
The relay blocks or restrains the low-slope differential element when these harmonic conditions are satisfied. The intention is reasonable for conventional systems: during external faults with current-transformer saturation, the current waveform becomes distorted and contains harmonics. Blocking the low-slope element prevents a false trip. The relay then relies on a high-slope differential element with a higher pickup for internal faults. In a conventional system, a current-transformer saturation condition is associated with a large fault current. Therefore, if the fault is internal, the high-slope element sees a large differential current and can still trip.
In a solar inverter system, the situation is different. The third harmonic can be generated by the solar inverter itself during an internal asymmetric fault, without any current-transformer saturation. If the differential relay sees this third harmonic, it may assert the saturation blocking criterion and block the low-slope differential element. If the system is weak or the connection line is long, the system-side contribution may also be limited. Then the differential current may not be large enough to reach the high-slope element. The result is a blocked low-slope element, an unreached high-slope element, and an uncleared internal transformer fault. This is the protection risk that I investigate.
I summarize the differential protection logic in Table 6. The exact settings vary by manufacturer and application, but the structure is similar. The low-slope element provides sensitivity for internal faults. The high-slope element provides security for heavy through-faults and current-transformer saturation. Harmonic blocking is applied to the low-slope element.
| Element | Typical characteristic | Purpose | Effect of solar inverter third harmonic |
|---|---|---|---|
| Low-slope differential | Low pickup, one or more shallow slopes | Detect internal faults with limited current | May be blocked by third-harmonic criterion caused by the solar inverter |
| High-slope differential | Higher pickup, steeper slope | Remain secure during CT saturation and heavy external faults | May be insensitive if the solar inverter and system both provide limited current |
| Second-harmonic blocking | \(I_{\phi2}>K_2I_{\phi1}\) | Detect CT saturation during energization or fault | May or may not be asserted depending on the solar inverter control and fault type |
| Third-harmonic blocking | \(I_{\phi3}>K_3I_{\phi1}\) | Detect CT saturation and waveform distortion | Highly relevant because the solar inverter can produce strong third harmonic without CT saturation |
To represent the differential characteristics, I use the following definitions. The differential current is the magnitude of the phasor sum of the currents from all sides, after ratio matching:
$$
I_d = \left|\sum_k \bar{I}_k\right|
$$
The restraint current is often defined as half the sum of the current magnitudes:
$$
I_r = \frac{1}{2}\sum_k |\bar{I}_k|
$$
A low-slope percentage differential element may operate when:
$$
I_d > I_{qd}
$$
$$
I_d > S_{l1} I_r
$$
A high-slope element may operate when:
$$
I_d > I_{qh}
$$
$$
I_d > S_h I_r
$$
where \(I_{qd}\) and \(I_{qh}\) are pickup thresholds and \(S_{l1}\) and \(S_h\) are slopes. When the third-harmonic blocking condition is true, the low-slope element is blocked. The high-slope element may remain enabled. If the internal fault current is small, the high-slope element may not operate. I therefore see a protection gap.
The key question is whether the third harmonic is caused by current-transformer saturation or by the solar inverter. I can distinguish these causes by examining the fundamental current magnitude and the waveform interval. When a current transformer is lightly saturated, the secondary current may still have a significant fundamental component. When it is heavily saturated, the secondary current often exhibits a noticeable interval angle or dead angle in each half-cycle. A solar inverter, in contrast, is current-limited. Its fundamental short-circuit current is usually not far above its rated current. Moreover, the solar inverter current waveform does not exhibit the same saturation interval angle because there is no magnetic core saturation in the same sense. Therefore, I propose an auxiliary criterion for third-harmonic blocking. If the third harmonic exceeds the threshold, but the fundamental current is below a specified multiple of rated current and the interval angle is below a threshold, then the third harmonic is not likely to be caused by current-transformer saturation, and the low-slope differential element should not be blocked. I write this concept as:
$$
\left(I_{\phi3} > K_3 I_{\phi1}\right) \quad \text{and} \quad \left(I_{\phi1} < K_{fund} I_N\right) \quad \text{and} \quad \left(\theta_{int} < \theta_{set}\right)
$$
$$
\Rightarrow \text{do not block the low-slope differential element}
$$
Here \(K_{fund}\) is a fundamental-current threshold, \(I_N\) is the rated current referred to the relay, and \(\theta_{int}\) is the interval angle of the current waveform. In my studies, a value of \(K_{fund}=2\) is a reasonable starting point because a solar inverter is unlikely to provide a fundamental short-circuit current far above twice its rated current. The interval-angle threshold should be set based on the relay sampling and the current-transformer characteristics. This auxiliary logic preserves the original CT saturation blocking when actual saturation occurs, but prevents a solar inverter-generated third harmonic from blocking a genuine internal fault.
There is also a control-side countermeasure. The solar inverter can add a compensation term to the current or voltage control path to cancel the second-order DC-link voltage ripple before it becomes a third harmonic. For example, if the DC-link ripple is measured or estimated as:
$$
u_{dc2} = U_{dc2}\cos(2\omega_1 t + \phi_{dc2})
$$
then a feedforward compensation can be added to the active-current reference:
$$
i_d^{+*} = i_{d,ref}^{+*} – G_{dc}(s) u_{dc2}
$$
where \(G_{dc}(s)\) is a compensation transfer function tuned to cancel the effect of the DC-link ripple on the current command. Alternatively, the solar inverter can explicitly calculate the second-order active-power ripple and adjust the negative-sequence current command to cancel it. A properly designed compensation can reduce the third harmonic from the solar inverter without compromising low-voltage ride-through. However, this countermeasure requires coordination between the solar inverter controller and the protection system. From a protection perspective, it is safer not to rely solely on compensation because not all solar inverters may implement it, and the compensation may be disabled during certain grid conditions.
I now present the simulation cases that I use to verify the mechanism. The system I model consists of a solar inverter-based PV plant connected to a collector transformer, then to a step-up transformer, and then to an equivalent grid through a line. I vary irradiance, fault type, fault resistance, system impedance, and negative-sequence control objective. I record the DC-link voltage, active and reactive power, positive-sequence active-current reference, negative-sequence current reference, phase currents, and third-harmonic ratios. The simulation cases are summarized in Table 7.
| Case | Irradiance | Fault condition | Negative-sequence objective | Purpose |
|---|---|---|---|---|
| 1A | 400 W/m² | Phase B to ground through 50 Ω at high-voltage side | Minimize reactive-power ripple | Show severe third harmonic when active-current margin exists |
| 1B | 400 W/m² | Same as 1A | Minimize active-power ripple | Show reduction of third harmonic |
| 2A | 400 W/m² | BC two-phase short circuit with 0 Ω | Minimize reactive-power ripple | Show clipping of active-current ripple in severe dip |
| 2B | 400 W/m² | BC two-phase short circuit with 5 Ω | Minimize reactive-power ripple | Show large third harmonic with moderate dip |
| 3 | 1000 W/m² | BC two-phase short circuit with 5 Ω | Minimize reactive-power ripple | Show that high pre-fault active power limits third harmonic |
In Case 1A, the solar inverter uses the objective that minimizes reactive-power ripple. The active-power second-order ripple becomes large. The DC-link voltage develops a strong second-order oscillation. The positive-sequence active-current reference and the negative-sequence current reference both show second-order oscillations. The phase currents exhibit a pronounced third harmonic. In my simulation, the largest phase third-harmonic ratio reaches about \(45\%\). In Case 1B, the solar inverter uses the objective that minimizes active-power ripple. The DC-link voltage ripple is much smaller, and the third-harmonic ratio falls to about \(5\%\). This comparison confirms that the negative-sequence control objective has a first-order effect on the third harmonic from the solar inverter.
In Case 2A, the fault is a solid two-phase short circuit. The voltage dip is severe. The solar inverter prioritizes reactive current, and the active-current reference is pushed to its limit or to zero. The second-order active-current ripple is clipped by the limiter. Even though the DC-link voltage may still show some oscillation, the current command does not pass the ripple. The third-harmonic ratio in the phase currents is only about \(5\%\). In Case 2B, the fault has a \(5\,\Omega\) resistance. The voltage dip is less severe. The solar inverter has enough current margin to output active current, and the active-current ripple is not clipped. The third-harmonic ratio rises to nearly \(40\%\). This pair of cases shows that a more severe fault does not necessarily mean a larger third harmonic. The limiting behavior of the solar inverter creates a non-monotonic relationship between fault severity and harmonic content.
In Case 3, the irradiance is high, so the pre-fault active power is high. During the fault, the solar inverter is already close to its current limit. The active-current ripple is clipped by the limiter. The third-harmonic ratio is much smaller than in Case 2B, with a maximum of about \(12\%\). This case shows that the operating point of the solar inverter matters. A solar inverter that is already producing near rated power has less flexibility to reproduce the second-order active-current ripple, so the third harmonic is reduced.
I also examine the differential protection behavior. In Case 1A and Case 2B, the third-harmonic content at the low-voltage side of the transformer is large enough to satisfy the CT saturation blocking criterion. The low-slope differential element is blocked. When the system impedance is small, the differential current trajectory may still enter the high-slope operating region, and the relay can trip. When the system impedance is large, the differential current trajectory may not reach the high-slope region. The relay remains blocked, and the internal fault is not cleared. This is the dangerous scenario. I summarize the protection outcomes in Table 8.
| Case | Third-harmonic ratio | Low-slope element | High-slope element with weak system | Protection outcome |
|---|---|---|---|---|
| 1A, target that minimizes reactive-power ripple | Up to about 45% | Blocked | May not operate | Risk of uncleared internal fault |
| 1B, target that minimizes active-power ripple | About 5% | Not blocked | Not needed | Correct low-slope operation |
| 2A, solid two-phase fault | About 5% | Not blocked | Not needed | Correct operation |
| 2B, two-phase fault with 5 Ω | Up to about 40% | Blocked | May not operate | Risk of uncleared internal fault |
| 3, high irradiance | Up to about 12% | Usually not blocked | Not needed | Correct operation in most settings |
These results confirm my theoretical derivation. The third harmonic from the solar inverter is not an incidental artifact. It is a predictable consequence of the control architecture. The most severe harmonic condition occurs when the solar inverter has enough current margin to reproduce the second-order active-current ripple but chooses a negative-sequence objective that maximizes active-power ripple. The protection risk occurs when the harmonic blocking criterion is satisfied and the high-slope element cannot compensate because the total fault current is limited.
I can summarize the analytical chain in a compact form. The chain begins with an asymmetric fault:
$$
\text{asymmetric fault}
\Rightarrow \text{negative-sequence voltage and current}
\Rightarrow \text{second-order power oscillation}
$$
The second-order power oscillation then creates a DC-link voltage ripple:
$$
\text{second-order power oscillation}
\Rightarrow u_{dc2}
$$
The DC-link ripple passes through the voltage controller and current controller:
$$
u_{dc2} \Rightarrow i_d^{+*} \Rightarrow u_d^{+*}
$$
The negative-sequence reference may also inherit the ripple:
$$
i_d^{+*} \Rightarrow i_d^{-*}
$$
The \(abc\) transformation turns the second-order \(dq\) ripple into a third harmonic:
$$
2\omega_1 \text{ in } dq \Rightarrow \omega_1 \text{ and } 3\omega_1 \text{ in } abc
$$
The third harmonic then affects the transformer differential relay:
$$
I_{\phi3} > K_3 I_{\phi1}
\Rightarrow \text{low-slope differential element blocked}
$$
If the high-slope element cannot operate, the final outcome is:
$$
\text{internal fault not cleared}
$$
This chain is the core of my analysis. It also shows where countermeasures can be applied. The solar inverter manufacturer can reduce \(u_{dc2}\) through control. The protection engineer can modify the harmonic blocking logic to distinguish solar inverter-generated harmonics from CT saturation harmonics. Both approaches are useful, and together they provide defense in depth.
I also want to emphasize the difference between harmonic generation by a solar inverter and harmonic generation by a saturated current transformer. In a saturated current transformer, the harmonic content is tied to the magnetic core behavior and the secondary burden. The fundamental current is usually large because saturation requires a large primary current. The waveform has a characteristic interval angle. In a solar inverter, the harmonic content is tied to the control loops and the current limiter. The fundamental current may be only slightly above rated current. The waveform may be smooth and continuous because the solar inverter uses high-frequency switching and current control. Therefore, a protection criterion that only checks third-harmonic magnitude can be fooled by the solar inverter. A protection criterion that additionally checks fundamental magnitude and waveform interval is more robust.
I propose the following practical settings for the auxiliary logic. The third-harmonic restraint coefficient \(K_3\) should remain as designed for CT saturation. The fundamental-current threshold \(K_{fund}\) can be set to about \(2.0\) per unit of relay-rated current, but it should be coordinated with the maximum solar inverter fault current. The interval-angle threshold \(\theta_{set}\) can be set to a value that is above the normal interval angle of a solar inverter current waveform but below the interval angle of a saturated CT. The exact values depend on the relay sampling rate, the CT performance, and the solar inverter current-control bandwidth. In my simulations, the solar inverter current waveform did not exhibit a large interval angle even when the third-harmonic ratio was high. Therefore, the auxiliary logic correctly identified that the third harmonic was not caused by CT saturation.
There is another important consideration. The third harmonic from the solar inverter may be phase-dependent. In an asymmetric fault, one phase may have a much larger third-harmonic ratio than the other phases. In my Case 1A, the C-phase current had the highest ratio. In Case 2B, the C phase again had the highest ratio. This phase dependence occurs because the negative-sequence voltage and the positive-sequence current references combine differently in each phase. A differential relay that uses per-phase harmonic blocking may block one phase while allowing another phase to operate. However, transformer differential protection often uses some combination of per-phase and common blocking. If the harmonic blocking is common to all phases, a high third harmonic in one phase can block the entire low-slope element. This makes the protection problem more severe. I therefore recommend that the auxiliary logic be applied per phase where possible, and that the relay logic be reviewed for the solar inverter interconnection.
I also note that the third-harmonic current from the solar inverter can interact with transformer connection compensation. A transformer differential relay performs phase compensation and zero-sequence compensation according to the transformer vector group. The third-harmonic components may be affected by this compensation. In some vector groups, third-harmonic currents may circulate in a delta winding and not appear at the terminals. In other configurations, the third harmonic may appear in the relay currents. Therefore, the actual third-harmonic content seen by the relay depends on the transformer winding connection and the location of the solar inverter. My analysis focuses on the solar inverter output current, but the relay sees the compensated currents. A full protection study should include the transformer vector group and the CT connections.
I further summarize the influence of fault type in Table 9. Different asymmetric faults produce different negative-sequence components and different power oscillations. A single-phase-to-ground fault can produce a significant negative-sequence voltage. A phase-to-phase fault also produces negative-sequence voltage but with different phase relationships. The third-harmonic content may be highest in different phases depending on the fault type. A balanced three-phase fault does not produce negative-sequence components, so the mechanism described here is not active in the same way. A solar inverter may still produce harmonics during a balanced fault because of other nonlinearities, but the second-order-to-third-harmonic path requires asymmetry.
| Fault type | Negative-sequence production | Expected third harmonic from solar inverter | Protection concern |
|---|---|---|---|
| Single-phase-to-ground | High | Can be high, depending on control objective and current limit | Per-phase blocking may be triggered |
| Phase-to-phase | High | Can be high, often with one phase dominant | Common blocking may block all phases |
| Two-phase-to-ground | High | Can be high | Depends on transformer compensation |
| Three-phase balanced | Low or zero | The described second-order path is weak | CT saturation logic may still be relevant if fault current is high |
I now consider the effect of system strength. In a strong system, the system-side fault current is large. Even if the low-slope differential element is blocked by third harmonic, the high-slope element may see enough differential current to operate. In a weak system, the system-side fault current is limited. The solar inverter itself is also current-limited. Therefore, the total differential current may not reach the high-slope threshold. This is why the protection risk is greatest in weak grids or in systems with long collector lines. The solar inverter third harmonic does not need to be the only source of fault current; it only needs to be large enough to block the sensitive element while the total fault current is too small for the insensitive element. I summarize this in Table 10.
| System condition | System-side fault current | Solar inverter fault current | High-slope element | Overall risk |
|---|---|---|---|---|
| Strong grid | High | Limited by solar inverter | Likely to operate | Lower |
| Weak grid | Low | Limited by solar inverter | May not operate | High |
| Long collector line | Reduced by impedance | Limited by solar inverter | May not operate | High |
| High pre-fault solar inverter output | Varies | Less active-current ripple due to limiter | Harmonic blocking less likely | Moderate to low |
| Low pre-fault solar inverter output | Varies | More active-current ripple can pass | Harmonic blocking more likely | High |
From the perspective of the solar inverter control designer, the third harmonic can be reduced by selecting a negative-sequence control objective that does not amplify active-power ripple. However, that choice may increase reactive-power ripple or negative-sequence current. There is a trade-off between power quality, current quality, and protection behavior. A grid code may require one objective over another during fault ride-through. Therefore, the protection engineer cannot assume that the solar inverter will always choose the least harmful objective. The protection system should be robust to the worst-case objective.
I also consider the effect of the DC-link voltage control bandwidth. If the DC-link voltage controller is slow, the second-order ripple may not be fully transferred to the active-current command. If the controller is fast, the ripple may be transferred more effectively. However, the current controller also has a bandwidth. The third-harmonic amplitude is largest when the control gains and the ripple frequency interact favorably. The expression for \(M_{dc2}\) shows that the third-harmonic amplitude depends on the product of voltage and current PI gains. This means that two solar inverters with the same DC-link ripple can produce different third-harmonic currents. Protection settings should therefore be tested with the actual solar inverter controller or with a validated model.
I further note that the third-harmonic current can affect other protection functions beyond transformer differential protection. Distance protection may be affected if the harmonic content influences the extracted fundamental phasor. Overcurrent protection may see a distorted waveform. Harmonic-based islanding detection may be affected. However, in this analysis I focus on transformer differential protection because the CT saturation blocking logic is directly sensitive to third-harmonic magnitude. The same mechanism could be relevant to any protection function that uses third-harmonic restraint.
I can summarize my recommended countermeasures in Table 11. The table groups them into solar inverter control measures and protection measures. The control measures reduce the source of the third harmonic. The protection measures improve the ability of the relay to distinguish the source of the third harmonic. A combined approach is best.
| Category | Measure | Benefit | Limitation |
|---|---|---|---|
| Solar inverter control | Compensate DC-link second-order voltage ripple | Reduces third-harmonic generation at the source | Requires controller access and may be disabled |
| Solar inverter control | Select negative-sequence objective that limits active-power ripple | Reduces DC-link ripple and third harmonic | May increase other power oscillations |
| Solar inverter control | Coordinate current limiter with harmonic compensation | Prevents uncontrolled clipping | Reduces available active current during faults |
| Protection | Add fundamental-current supervision to third-harmonic blocking | Distinguishes solar inverter third harmonic from CT saturation | Requires setting coordination |
| Protection | Add waveform interval-angle supervision | Detects actual CT saturation more reliably | Depends on relay capability and sampling |
| Protection | Use per-phase blocking rather than common blocking | Prevents one phase from blocking all phases | May require relay logic changes |
| Protection | Review high-slope element sensitivity for weak systems | Ensures internal faults can still be cleared | May reduce security for external faults |
I want to close with a concise restatement of the main findings. First, an asymmetric fault creates negative-sequence components at the solar inverter terminals. Second, those negative-sequence components interact with the positive-sequence control and produce second-order active-power and reactive-power oscillations. Third, the second-order active-power oscillation creates a second-order DC-link voltage ripple. Fourth, the DC-link voltage ripple passes through the voltage and current controllers and appears in the positive-sequence and negative-sequence current references. Fifth, the transformation from the \(dq\) frame to the \(abc\) frame converts the second-order ripple into a third-harmonic component in the solar inverter output current. Sixth, the magnitude of that third harmonic depends strongly on the negative-sequence control objective, the current limiter, the pre-fault active power, and the fault severity. Seventh, the third harmonic can satisfy the CT saturation blocking criterion of a transformer differential relay even when no CT saturation exists. Eighth, in a weak system, the high-slope differential element may not operate, so an internal transformer fault may remain uncleared. Ninth, auxiliary supervision based on fundamental current magnitude and waveform interval angle can prevent this maloperation. Tenth, solar inverter control compensation can reduce the harmonic at the source.
In my view, the most important engineering lesson is that the solar inverter should not be treated as a passive harmonic source. It is an active controlled source whose harmonic signature depends on its control objectives and limits. A protection study for a solar inverter interconnection should therefore include asymmetric faults, multiple negative-sequence control objectives, multiple irradiance levels, and weak-system conditions. The third-harmonic current from the solar inverter is not always large, but when it is large, it can have a direct and adverse effect on transformer differential protection. Understanding the mechanism allows the protection engineer to design blocking logic that is secure for CT saturation but dependable for internal faults.
I have presented the analytical derivation, the influencing factors, the protection risk, and the countermeasures in a unified framework. The equations for the DC-link voltage ripple, the current-reference ripple, and the third-harmonic amplitude provide a basis for quantifying the phenomenon. The tables summarize the qualitative relationships and the simulation cases. The overall conclusion is that third-harmonic generation by a solar inverter during asymmetric faults is a real and predictable phenomenon, and that transformer differential protection must be engineered with this phenomenon in mind. When the solar inverter control and the protection logic are coordinated, the risk of an uncleared internal fault can be greatly reduced, and the benefits of solar inverter integration can be realized without compromising protection dependability.