The proliferation of power electronic converters, driven by the integration of active loads such as distributed generation, distributed energy storage, electric vehicles, and DC loads, is fundamentally reshaping the distribution network. This trend towards an increasingly power-electronics-dominated grid introduces significant power quality challenges, with harmonic pollution being a primary concern. Conventional converters, due to the non-linear characteristics of their switching devices and modulation algorithms, are inherent sources of harmonic currents. While traditional mitigation devices like active power filters offer a solution, they represent a passive, add-on approach that does not address the problem at its source. The typical grid-connected solar inverter, primarily focused on maximizing power extraction and ensuring high-quality output current, has limited capability to improve broader network power quality due to constraints imposed by its controller bandwidth, switching frequency, and topology. However, the inherent controllability of the modern solar inverter presents a compelling opportunity. By implementing advanced control strategies, the solar inverter can be transformed from a mere generator into an active participant in power quality management, specifically in harmonic suppression, thereby enhancing grid power quality and potentially reducing the overall cost of harmonic mitigation.
Traditional methods for enabling harmonic compensation in inverters often rely on detecting the harmonic current drawn by local non-linear loads using current transformers. This approach, while effective for the specific load being measured, has inherent limitations. It requires physical modification of the installation for sensor placement, and its compensation capability is restricted to that particular load circuit. When the local non-linear load is inactive, the compensation capacity sits idle, leading to underutilization of the inverter’s capabilities. This is particularly uneconomical when the solar generation power is low, leaving a significant portion of the inverter’s capacity available. A more holistic and efficient approach is to target the voltage quality at the Point of Common Coupling (PCC). Since grid impedance is largely resistive at harmonic frequencies, harmonic current distortions manifest as voltage distortions at the PCC. Therefore, a solar inverter can be controlled to suppress these voltage harmonics, benefiting all connected equipment downstream of the PCC, regardless of the number or location of individual harmonic-producing loads. This paper delves into a control strategy centered on the concept of a Virtual Resistance, enabling the solar inverter to perform simultaneous maximum power point tracking (MPPT) and active harmonic damping.
Virtual Resistance-Based Harmonic Extraction and Control
The core of the proposed strategy is a harmonic extraction method that does not require load current measurement. Instead, it utilizes the PCC voltage. The fundamental principle is to make the solar inverter emulate a programmable resistance specifically for harmonic frequencies. By detecting the harmonic voltage components at the PCC and dividing them by a virtual resistance value, a harmonic current reference signal is generated. This signal is then used by the inverter’s current control loop to inject compensating currents.
The process begins with the accurate extraction of harmonic voltage components from the measured PCC voltage. While methods like the p-q theory exist, they can be sensitive to voltage waveform distortion. A robust alternative for three-phase systems is the $i_p$-$i_q$ method based on instantaneous reactive power theory. This method is less susceptible to voltage distortion. For a set of three-phase voltages $v_a$, $v_b$, $v_c$ and currents $i_a$, $i_b$, $i_c$, the transformation into the $\alpha\beta$ stationary frame is given by:
$$
\begin{bmatrix}
v_{\alpha} \\ v_{\beta}
\end{bmatrix}
= \sqrt{\frac{2}{3}}
\begin{bmatrix}
1 & -\frac{1}{2} & -\frac{1}{2} \\
0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}
\end{bmatrix}
\begin{bmatrix}
v_a \\ v_b \\ v_c
\end{bmatrix}
$$
A similar transformation is applied to currents to get $i_\alpha$ and $i_\beta$. These are then transformed into synchronous reference frame coordinates rotating at the fundamental frequency $\omega$:
$$
\begin{bmatrix}
i_p \\ i_q
\end{bmatrix}
=
\begin{bmatrix}
\sin(\omega t) & -\cos(\omega t) \\
-\cos(\omega t) & -\sin(\omega t)
\end{bmatrix}
\begin{bmatrix}
i_{\alpha} \\ i_{\beta}
\end{bmatrix}
$$
Under ideal sinusoidal conditions, $i_p$ and $i_q$ are DC quantities corresponding to the active and reactive components. In the presence of harmonics, they contain AC oscillatory components. Low-Pass Filters (LPFs) are used to extract the DC components $\overline{i_p}$ and $\overline{i_q}$. The inverse transformation then yields the fundamental frequency components of the current. The harmonic components are found by subtracting these fundamental components from the original measured signals.
In the virtual resistance method, we apply this detection principle not to current, but to the PCC voltage. The three-phase PCC voltages are measured and processed using the $i_p$-$i_q$ (or a similar) detection algorithm to isolate their harmonic components, denoted as $v_{ah}$, $v_{bh}$, $v_{ch}$. The harmonic current reference signal $i_{h, ref}$ for the solar inverter is then computed as:
$$
\begin{bmatrix}
i_{ah,ref} \\ i_{bh,ref} \\ i_{ch,ref}
\end{bmatrix}
= -\frac{1}{R_{v}}
\begin{bmatrix}
v_{ah} \\ v_{bh} \\ v_{ch}
\end{bmatrix}
$$
where $R_v$ is the virtual resistance value. The negative sign indicates that the inverter should inject current to oppose the harmonic voltage (sink harmonic power). This harmonic current reference is superimposed onto the fundamental current reference generated by the power/voltage outer control loop (e.g., from MPPT and DC-link voltage regulation). The combined reference $i^*_{o,abc}$ is then fed to the fast inner current control loop, typically a PI or PR controller in the synchronous frame, which drives the Pulse Width Modulation (PWM) stage to generate the switching signals for the inverter bridge. The overall control block diagram is structured as follows:
The implementation of this strategy transforms the solar inverter’s role. Beyond its primary function of converting DC from PV panels into grid-synchronized AC power, it now actively monitors the grid conditions at its connection point. By dynamically injecting compensating harmonic currents, it acts as a damping element, reducing the harmonic voltage distortion at the PCC. This functionality is always active as long as the inverter is connected and has available capacity, making it a persistent and proactive asset for power quality improvement, unlike load-specific solutions.
Theoretical Foundation: Norton Equivalent and Optimal Virtual Resistance
The effectiveness of the virtual resistance method can be analyzed using a simplified single-phase equivalent circuit for a specific harmonic frequency. The network as seen from the PCC can be represented by a Norton equivalent circuit, combining the effects of the grid background harmonic voltage, grid impedance, linear loads, and non-linear loads.
Let the grid background harmonic voltage be $U_{sh} = \alpha_u + j\beta_u$ and the grid impedance be $Z_s = R_s + jX_s$. The local linear load impedance is $Z_L = R_L + jX_L$. The non-linear load is modeled as a harmonic current source $I_{NL} = \alpha_i + j\beta_i$. The solar inverter is represented by its virtual resistance $R_v$ at the harmonic frequency under consideration. Using Norton’s theorem, the entire network upstream and parallel to the PCC (including $U_{sh}$, $Z_s$, $Z_L$, and $I_{NL}$) can be simplified to a single equivalent current source $I_{eq}$ in parallel with an equivalent impedance $Z_{eq}$.
The equivalent current source $I_{eq}$ and impedance $Z_{eq}$ are derived as:
$$
I_{eq} = \left( \frac{\alpha_u R_s + \beta_u X_s}{R_s^2 + X_s^2} + \alpha_i \right) + j\left( \frac{\beta_u R_s – \alpha_u X_s}{R_s^2 + X_s^2} + \beta_i \right) = \alpha_{ieq} + j\beta_{ieq}
$$
$$
Z_{eq} = \frac{R_L(R_s^2+X_s^2) + R_s(R_L^2+X_L^2)}{(R_s+R_L)^2+(X_s+X_L)^2} + j\frac{X_L(R_s^2+X_s^2) + X_s(R_L^2+X_L^2)}{(R_s+R_L)^2+(X_s+X_L)^2} = R_{eq} + jX_{eq}
$$
The harmonic voltage at the PCC, $U_{eq}$, and the current drawn by the virtual resistor, $I_v$, are then:
$$
U_{eq} = \frac{R_v (R_{eq} + jX_{eq})}{R_v + R_{eq} + jX_{eq}} I_{eq}
$$
$$
I_v = \frac{U_{eq}}{R_v}
$$
The harmonic power absorbed by the solar inverter (modeled as $R_v$) is $P_v = \Re\{ U_{eq} \cdot I_v^* \}$. To maximize the harmonic suppression effect, we aim to maximize $P_v$, meaning the inverter sinks the maximum possible harmonic power from the PCC. Finding the maximum of $P_v$ with respect to $R_v$ yields the optimal condition:
$$
\frac{dP_v}{dR_v} = 0 \quad \Rightarrow \quad R_{v,opt} = |Z_{eq}| = \sqrt{R_{eq}^2 + X_{eq}^2}
$$
Substituting the expression for $Z_{eq}$, the optimal virtual resistance is:
$$
R_{v,opt} = \sqrt{ \frac{(R_s^2+X_s^2) \cdot (R_L^2+X_L^2)}{(R_s+R_L)^2+(X_s+X_L)^2} }
$$
This result confirms the existence of an optimal virtual resistance value that maximizes the absorbed harmonic power. At this point, the virtual resistance matches the magnitude of the equivalent system impedance as seen from the PCC. Operating at this point leads to the greatest reduction in PCC harmonic voltage. However, in practice, the parameters $R_s, X_s, R_L, X_L$ are unknown and time-varying. Therefore, a real-time search algorithm is necessary to track the optimal $R_v$.
Online Search for Maximum Harmonic Power Point (MHPP)
Given that the function $P_v = f(R_v)$ is unimodal (has a single maximum) for a given operating condition, a one-dimensional search algorithm can be employed. The Golden-Section Search method is a suitable, efficient choice for this purpose. The algorithm iteratively narrows down an interval $[a, b]$ known to contain the optimum $R_{v,opt}$.
- Initialization: Define an initial search interval $[a, b]$ for $R_v$. This interval can be estimated based on typical ranges for grid and load impedance (e.g., from short-circuit ratio analysis or typical values).
- Evaluation: Choose two test points $R_{v1}$ and $R_{v2}$ within the interval using the golden ratio ($\phi \approx 0.618$).
- $R_{v1} = b – \phi(b-a)$
- $R_{v2} = a + \phi(b-a)$
- Measurement: Apply each $R_v$ value briefly to the control algorithm. Measure or estimate the resulting harmonic power $P_v(R_{v1})$ and $P_v(R_{v2})$. This can be done by calculating $P_v \approx V_{h, rms} \cdot I_{v, rms}$ for the dominant harmonic, where $V_{h}$ and $I_{v}$ are extracted via the detection algorithm.
- Interval Update:
- If $P_v(R_{v1}) > P_v(R_{v2})$, then the optimum lies in $[a, R_{v2}]$. Set $b = R_{v2}$.
- If $P_v(R_{v2}) > P_v(R_{v1})$, then the optimum lies in $[R_{v1}, b]$. Set $a = R_{v1}$.
- Convergence Check: If the interval width $|b-a|$ is below a specified tolerance, stop and set $R_{v,opt} = (a+b)/2$. Otherwise, go back to Step 2.
This search process runs continuously but slowly relative to the fundamental control loops, adapting the virtual resistance to changing grid and load conditions to maintain near-optimal harmonic suppression. The following table contrasts the proposed method with traditional approaches.
| Aspect | Traditional Load-Current Feedback APF | Virtual-Resistance Based Solar Inverter |
|---|---|---|
| Compensation Target | Specific, measured non-linear load current. | PCC voltage harmonic distortion (system-level). |
| Installation Requirement | Requires CTs on load feeders, more invasive. | Only requires voltage measurement at inverter terminals. |
| Scope of Impact | Local to the compensated load. | Global for all loads downstream of the PCC. |
| Inverter Utilization | Compensation capacity idle when local load is off. | Always active when grid-connected, utilizes available capacity. |
| Economic Efficiency | Lower, requires dedicated hardware or underutilizes inverter. | Higher, adds functionality to existing solar inverter asset. |
| Adaptability | Fixed to pre-defined load. | Self-tuning to system impedance via MHPP search. |
Simulation Analysis and Experimental Verification
To validate the proposed control strategy, comprehensive simulations were conducted under various grid scenarios. The performance metric was the Total Harmonic Distortion (THD) of the PCC voltage. The solar inverter control included both the MPPT/power control outer loop and the inner virtual-resistance-based harmonic current loop.
Scenario 1: Grid with Background Voltage Harmonics. The grid voltage contained 5th (10% of fundamental) and 7th (5% of fundamental) harmonics. Without harmonic compensation, the PCC voltage THD was 10.35%. After activating the virtual resistance control in the solar inverter, the THD was reduced to 6.02%.
Scenario 2: Ideal Grid with Local Non-linear Load. A three-phase diode rectifier with an RL load was connected. Without compensation, the PCC voltage THD was 4.58%. With the solar inverter’s harmonic suppression active, the THD dropped to 3.02%.
Scenario 3: Combined Scenario (Background Harmonics + Non-linear Load). This represents a realistic worst-case. The initial PCC voltage THD was 15.27%. The virtual-resistance-controlled solar inverter successfully reduced it to 10.42%.
The following table summarizes the simulation results for the 5th and 7th harmonics, demonstrating the consistent effectiveness of the method across different distortion sources.
| Scenario | Condition | Voltage THD | 5th Harmonic (%) | 7th Harmonic (%) |
|---|---|---|---|---|
| 1: Background Harmonics | Without Compensation | 10.35% | 9.38 | 4.38 |
| With Solar Inverter Compensation | 6.02% | 5.40 | 2.62 | |
| 2: Non-linear Load | Without Compensation | 4.58% | 3.38 | 2.35 |
| With Solar Inverter Compensation | 3.02% | 2.03 | 1.46 | |
| 3: Combined | Without Compensation | 15.27% | 12.81 | 7.70 |
| With Solar Inverter Compensation | 10.42% | 8.52 | 5.58 |
Experimental validation was performed using a solar inverter prototype controlled by a digital signal processor (DSP). The harmonic extraction algorithm targeted the 5th, 7th, 11th, and 13th harmonics. The experiment clearly showed a significant and simultaneous drop in the positive and negative sequence magnitudes of these harmonic voltages upon enabling the virtual-resistance-based compensation function, confirming the system-level suppression capability.
A separate experiment was conducted to validate the Maximum Harmonic Power Point (MHPP) search algorithm. Starting with an initial virtual conductance ($G_v = 1/R_v$) of 0.5, the harmonic suppression was activated. Upon initiating the golden-section search, the algorithm successfully adjusted $G_v$. The measured harmonic power increased until $G_v$ reached approximately 2.3 (i.e., $R_v \approx 0.43$), at which point the power stabilized at its maximum value. The PCC harmonic voltage was minimized at this operating point. This experiment confirmed the unimodal nature of $P_v = f(R_v)$ and the practicality of the online search algorithm in a real solar inverter.
Conclusion
The integration of a virtual-resistance-based harmonic suppression strategy into the control architecture of a grid-connected solar inverter presents a significant advancement in the functionality of distributed energy resources. This approach moves beyond the traditional paradigm of the solar inverter as a simple power conversion device, repurposing it into an active grid-supporting unit that autonomously improves local power quality. The method offers several key advantages: it requires only local voltage measurement, making installation simple; it suppresses voltage harmonics at the PCC, benefiting all downstream loads; and it maximizes the utilization of the inverter’s available capacity. The theoretical analysis, grounded in Norton equivalent circuit theory, proves the existence of an optimal virtual resistance that maximizes harmonic power absorption and minimizes voltage distortion. The accompanying online search algorithm enables the solar inverter to adapt to changing grid conditions and maintain near-optimal performance. Simulation and experimental results consistently demonstrate the strategy’s effectiveness in reducing PCC voltage THD under various harmonic pollution scenarios. Therefore, equipping solar inverters with this dual functionality of energy generation and active harmonic filtering is a technically sound and economically efficient pathway towards building more resilient and power-quality-conscious power-electronics-dominated distribution networks.