Abstract
In modern distribution networks, harmonic pollution sources are widely distributed, causing power quality degradation across the entire network. Consequently, coordinated mitigation using multiple converters has become a research hotspot. However, existing methods are constrained by the difficulty of precise harmonic phasor perception, high computational complexity, stringent real-time control requirements, and challenges related to control stability and current limit violations. These factors make it difficult for existing approaches to achieve ideal mitigation performance in practical applications. To address these issues, this paper proposes a comprehensive harmonic mitigation method for distribution networks based on local control of solar inverters. The proposed control combines two local harmonic control schemes—converter-local harmonic current compensation and harmonic virtual conductance control—with the objective of minimizing global voltage harmonic distortion. The strategy encompasses three steps: configuration of mitigation schemes, centralized computation of control instructions, and real-time correction of harmonic current commands. By maximizing the utilization of the remaining capacity of solar inverters for harmonic mitigation, the proposed method outperforms existing approaches and achieves comprehensive power quality improvement under varying grid operating conditions.
1. Introduction
The large-scale integration of distributed generation and power electronic devices has resulted in widespread distribution of power quality pollution sources. Localized harmonic problems, induced by the combined effects of multiple harmonic sources, cannot be fully resolved solely through traditional source-side compensation. Thus, the method of collaborative harmonic mitigation across the entire distribution network using multiple converters has gradually drawn attention. Most existing harmonic mitigation schemes rely on high-precision monitoring devices to collect harmonic voltage and current phasors at various nodes in real time, followed by control algorithms that adjust converter outputs based on global information. Some researchers have employed particle swarm optimization to determine the locations, capacities, and compensation currents of active power filters (APFs) with the objective of minimizing the total harmonic distortion (THD) of node voltages. Other studies have applied heuristic algorithms such as grey wolf optimization and risk trade-off analysis to solve similar optimization problems. Feasibility-region-based sensitivity analysis and harmonic power-flow-matrix-based optimal current compensation have also been explored.
However, these methods depend on the real-time acquisition of harmonic voltage and current phase angles. The phase error of the fundamental voltage is amplified by a factor of h for the h-th harmonic, and time synchronization systems are often subject to various disturbances with errors up to 2.3 ms. Additionally, determining the harmonic current commands for converters requires global optimization within a very short period. These conditions make real-time adjustment of harmonic currents from multiple converters difficult to achieve at theoretical performance levels. To alleviate the burdens of global communication and optimization, several studies have adopted zonal mitigation, online impedance estimation, and distributed cooperative optimization. Nevertheless, when multiple converters participate simultaneously and residual capacities vary, online impedance estimation becomes challenging, and existing methods often contain linearization assumptions and are susceptible to phase error interference.
For the issues of inaccurate time synchronization of multi-node harmonic data and insufficient real-time data processing capability in centralized or distributed controllers, a harmonic mitigation strategy combining centralized optimization with local control is an effective solution. Previous research combined harmonic virtual conductance with zonal sensitivity for APF configuration, using particle swarm optimization on a 15-minute scale to optimize harmonic virtual conductance based on grid operating states. Since virtual conductance is implemented as local decentralized control in converters, it enables short-time-scale harmonic suppression based on local real-time harmonic voltages. Subsequent studies adopted data-driven approaches, online rolling optimization, and distributed consensus algorithms for harmonic virtual conductance regulation. These methods have also been applied to distributed generation resources such as solar and storage systems.
Despite the appeal of harmonic virtual conductance control that combines centralized optimization and local control, larger harmonic conductance does not always guarantee better mitigation from a harmonic power-flow perspective. When the harmonic voltage at a node is low, the extracted harmonic voltage may be inaccurate, and multiple converters using large harmonic conductance might face stability issues. Furthermore, unlike dedicated power quality devices such as APFs, solar inverters experience variations in available capacity due to real-time changes in photovoltaic output, and load fluctuations make centrally optimized virtual conductance difficult to adapt in real time. Issues such as current limit violations, extremely low node harmonic voltages, and underutilization of remaining capacity frequently arise in real-time control but have rarely been addressed in the literature.
Given these limitations, and recognizing the potential of existing solar inverter capacities in distribution networks to provide harmonic mitigation, we propose a comprehensive harmonic mitigation strategy based on local control of solar inverters. This strategy combines centralized optimization with two local harmonic control methods. First, based on historical operating data, simulations are conducted to configure the mitigation scheme. At the centralized optimization level, using harmonic state estimation results obtained every 3 minutes, the control instructions for each converter are solved centrally and dispatched to the participating solar inverters. At the real-time control level, solar inverters performing local harmonic control generate output harmonic current commands based on the received instructions and actual operating conditions.
The advantages of the proposed scheme are as follows:
- It only requires existing 3-minute-interval harmonic state estimation results and one-way communication to dispatch control instructions. By combining harmonic current compensation and harmonic virtual conductance control, solar inverters achieve comprehensive mitigation through local cancellation of harmonic currents and improvement of grid harmonic admittance distribution, effectively avoiding communication delays, computational bottlenecks, and multi-converter coordination difficulties.
- The proposed control maintains maximum power output of solar inverters while allocating harmonic mitigation instructions according to the remaining capacity of different inverters, realizing coordinated optimization of renewable energy integration and distribution network power quality.
- The dynamic characteristics of harmonic virtual conductance control are considered, and stability constraints are designed to avoid system instability risks.
- Combined with actual operating conditions of solar inverters, a real-time mitigation current adjustment strategy is introduced to ensure maximum utilization of remaining capacity while guaranteeing safe device operation during grid fluctuations.
2. Overview of Converter Harmonic Mitigation Methods and the Proposed Control
For converters participating in harmonic mitigation, methods can be classified into three categories based on how harmonic current commands are generated.
1) Local harmonic current compensation control: In this scheme, the converter detects local load harmonic currents and injects a proportional compensation current to achieve harmonic cancellation. This approach compensates for harmonic currents from nearby large harmonic sources, preventing their propagation through the grid. The expression is:
$$ \dot{i}_{\text{PV}i,h} = k_{i,h} \dot{i}_{Li,h} \tag{1} $$
where \( i_{\text{PV}i,h} \) is the h-th harmonic current phasor injected by the solar inverter at node i, \( i_{Li,h} \) is the h-th harmonic current phasor detected at the load side of node i, and \( k_{i,h} \) is the harmonic current compensation coefficient ranging from 0 to 1.
2) Harmonic virtual conductance control: The converter introduces a virtual conductance gain \( g_{i,h} \) that converts the detected point-of-common-coupling (PCC) harmonic voltage \( u_{i,h} \) into a current command:
$$ \dot{i}_{\text{PV}i,h} = g_{i,h} \dot{u}_{i,h} \tag{2} $$
The modified converter characteristic is equivalent to a parallel virtual harmonic conductance, effectively improving the harmonic admittance distribution in the distribution network and reducing harmonic voltages in the vicinity. Generally, a larger equivalent harmonic conductance yields better mitigation.
3) Centralized/distributed controller: The controller generates harmonic current commands based on system optimization objectives and dispatches them to each participating converter. Since command generation relies on real-time harmonic state estimation, this method requires precise phasor perception.
For harmonic state estimation, according to IEC 61000-4-7, existing power quality online monitoring systems measure harmonic voltage and current phasors with the fundamental voltage as reference. Fourier harmonic analysis is performed every 10 grid cycles (0.2 s), and the average, maximum, and minimum values of 900 data points are statistically compiled every 3 minutes before being sent to the centralized controller. Since slight phase and time errors at the fundamental frequency are amplified by the harmonic order, directly dispatching current commands containing amplitude and phase information would cause significant errors. Therefore, existing research utilizes the average of 3-minute harmonic data and employs trust-region algorithms or interval-based dynamic harmonic state estimation to estimate phase angles until the deviation of each state variable converges below a predefined threshold \( \varepsilon \). The proposed harmonic collaborative mitigation technique is based on these existing 3-minute asynchronous harmonic data state estimation methods.
From harmonic power-flow equations, the harmonic voltage at each node in a distribution network results from the interaction between harmonic currents injected by sources and the network impedance. Therefore, if some converters compensate harmonic currents injected by sources or provide harmonic virtual conductance paths to ground, the harmonic voltage levels in the vicinity can be reduced.
Based on the distributed nature of harmonic current compensation control and harmonic virtual conductance control, the proposed comprehensive harmonic mitigation method for distribution networks based on local control of solar inverters proceeds as follows. First, typical operating scenarios are constructed from historical data to configure the harmonic mitigation scheme. At the 3-minute time scale, the centralized controller performs harmonic state estimation using data from power quality monitoring systems, computes the control instructions for each inverter, and dispatches them via one-way communication. At the real-time time scale, participating solar inverters use the received control instructions and actual operating conditions to generate harmonic currents using their remaining capacity.
3. Problem Formulation for Harmonic Collaborative Mitigation
3.1 Configuration of the Harmonic Mitigation Scheme for Solar Inverters
Before harmonic mitigation, the local harmonic control scheme for each solar inverter must be designed, and high-precision voltage and current sensors must be installed. Based on the mitigation effect, inverters are retrofit according to one of two control modes—harmonic current compensation or harmonic virtual conductance—because the implementation conditions differ and simultaneous use is difficult. Compared with existing approaches where all participating converters use harmonic virtual conductance, combining both local control schemes leverages their complementary advantages: canceling harmonic currents at the source and improving grid harmonic admittance distribution simultaneously. Moreover, the use of harmonic current compensation in some solar inverters helps prevent system instability, as an increased number of converters using virtual conductance control raises stability concerns.
To determine the configuration of different inverters in the distribution network, historical operating data are used for harmonic collaborative mitigation simulation, and the scheme with the best average mitigation effect is selected. Because the operating states of power systems continuously change and massive historical data are difficult to utilize directly, K-means clustering is applied to construct typical operating scenarios. Before clustering, operating state data—including node powers, solar output, harmonic voltages, harmonic currents injected by sources, harmonic impedances, and background harmonics—are normalized. The objective function minimizes the within-cluster sum of squares:
$$ d_{\text{WCSS}} = \sum_{m=1}^{K} \sum_{\sigma \in C_m} \|\sigma – \mu_m\| \tag{3} $$
where \( K \) is the number of clusters, \( \sigma \) is the sample, \( C_m \) is the set of samples in cluster m, and \( \mu_m \) is the centroid of cluster m. The Calinski-Harabasz index determines the optimal K value.
For harmonic sources at node i, the Norton equivalent circuit is adopted to balance modeling accuracy and parameter availability. The parameters are calculated using the fluctuation method: during a period when harmonic source power remains constant, at least two different node harmonic voltage states are selected—either naturally occurring or induced by harmonic compensation from solar inverters—provided the differences are sufficiently large to cause observable changes in the grid-connected harmonic current. The h-th harmonic Norton equivalent source \( \dot{i}_{i,h} \) and impedance \( z_{i,h} \) are obtained as:
$$ z_{i,h} = – \frac{\dot{u}_{i,h}^{(1)} – \dot{u}_{i,h}^{(2)}}{\dot{i}_{Li,h}^{(1)} – \dot{i}_{Li,h}^{(2)}} \tag{4} $$
$$ \dot{i}_{i,h} = \dot{i}_{i,h}^{(1)} – \frac{\dot{u}_{i,h}^{(1)}}{z_{i,h}} \tag{5} $$
Under different typical scenarios, each solar inverter is evaluated with both local control schemes to form different configuration options. The concentrated controller optimizes the control instructions, and the configuration with the best weighted average mitigation effect across all typical scenarios is selected. Inverters are then retrofitted accordingly, with the capability to receive control instructions from the centralized controller every 3 minutes.
3.2 Minute-Level Time Scale: Optimization of Inverter Control Instructions
The power quality monitoring system uploads the average harmonic voltage and current phasors to the centralized controller at 3-minute intervals, and harmonic state estimation is performed to determine the phase angles. Since harmonic voltages and currents exhibit minimal fluctuation at the 3-minute scale under steady-state conditions, the state estimation results reasonably reflect the actual harmonic conditions. Based on these results, the centralized controller optimizes the current compensation coefficient \( k_{i,h} \) or harmonic conductance \( g_{i,h} \) for each inverter and dispatches them via one-way communication.
Considering that different nodes have different harmonic voltage requirements, the optimization objective to minimize the overall harmonic voltage after mitigation is formulated as:
$$ F_{\text{THD}} = \min \sum_{i=1}^{n} c_i f_{\text{THD},i} \tag{6} $$
where \( F_{\text{THD}} \) is the weighted sum of node voltage THDs, \( c_i \) is the weight coefficient for node i, and \( f_{\text{THD},i} \) is the voltage THD at node i. For a three-phase symmetrical grid where most harmonics are of order \( 6k \pm 1 \) with dominance at lower orders:
$$ f_{\text{THD},i} = \frac{\sum_{h=5,7,11,13,\ldots}^{6k\pm1} u_{i,h}^2}{u_{i,1}} \tag{7} $$
The optimization constraints include harmonic power-flow constraints, solar inverter harmonic mitigation capacity constraints, and virtual conductance control stability constraints.
3.2.1 Harmonic Power-Flow Constraint
The nodal admittance equation in the harmonic domain is:
$$ \boldsymbol{I}_h = \boldsymbol{Y}_h \boldsymbol{U}_h \tag{8} $$
where \( \boldsymbol{I}_h \) is the column vector of injected harmonic currents, \( \boldsymbol{U}_h \) is the column vector of node harmonic voltages, and \( \boldsymbol{Y}_h \) is the \( n \times n \) harmonic nodal admittance matrix. Combining both control strategies—where harmonic virtual conductance \( g_{i,h} \) is part of the node self-admittance and harmonic current compensation coefficient \( k_{j,h} \) partially cancels the harmonic source injection—Equation (8) becomes:
\[
\begin{bmatrix}
\dot{i}_{1,h} \\ \vdots \\ \dot{i}_{i,h} \\ \vdots \\ (1-k_{j,h})\dot{i}_{j,h} \\ \vdots \\ \dot{i}_{n,h}
\end{bmatrix}
=
\begin{bmatrix}
y_{11,h}+g_{1,h} & \cdots & y_{1i,h} & \cdots & y_{1n,h} \\
\vdots & \ddots & \vdots & & \vdots \\
y_{i1,h} & \cdots & y_{ii,h}+g_{i,h} & \cdots & y_{in,h} \\
\vdots & & \vdots & \ddots & \vdots \\
y_{n1,h} & \cdots & y_{ni,h} & \cdots & y_{nn,h}
\end{bmatrix}
\begin{bmatrix}
\dot{u}_{1,h} \\ \vdots \\ \dot{u}_{i,h} \\ \vdots \\ \dot{u}_{n,h}
\end{bmatrix}
\tag{9}
\]
3.2.2 Harmonic Mitigation Capacity Constraint of Solar Inverters
The root-mean-square (RMS) value of the three-phase current of a solar inverter during operation must not exceed the maximum limit. That is, the superimposed effective value of the harmonic current amplitude \( i_{\text{PV}i,h} \), the fundamental active current \( i_{id} \), and the fundamental reactive current \( i_{iq} \) should be less than the maximum allowable current \( i_{i\lim} \):
$$ i_{i,h}^2 + i_{id}^2 + i_{iq}^2 \leq i_{i\lim}^2 \tag{10} $$
For a solar inverter at node i employing harmonic virtual conductance control:
$$ i_{\text{PV}i,h} = \sqrt{\sum_{h=5,7,11,13,\ldots}^{6k\pm1} (g_{i,h} u_{i,h})^2} \tag{11} $$
For a solar inverter employing harmonic current compensation control:
$$ i_{\text{PV}i,h} = \sqrt{\sum_{h=5,7,11,13,\ldots}^{6k\pm1} (k_{i,h} i_{i,h})^2} \tag{12} $$
Since \( i_{id} \) and \( i_{iq} \) are known (the fundamental active current is determined by the solar power output divided by the PCC voltage, and the fundamental reactive current is determined by the centralized voltage dispatch or local voltage-reactive power control), defining intermediate variables \( x_{g,i,h} = g_{i,h} u_{i,h} \) and \( x_{k,i,h} = k_{i,h} i_{i,h} \) transforms Equations (11) and (12) into convex quadratic constraints containing only square terms. The bilinear terms in Equation (9) are converted to linear terms \( x_{g,i,h} \) and \( x_{k,i,h} \), enabling solution with mainstream solvers.
3.2.3 Stability Constraint for Virtual Conductance Control
Theoretically, a larger harmonic virtual conductance yields better mitigation. However, without limits, when the node harmonic voltage approaches zero, the harmonic conductance approaches infinity, and the harmonic current approaches a value determined by the ratio of the pre-mitigation harmonic voltage to the equivalent impedance. Stability analysis of harmonic virtual conductance control shows that when the grid harmonic impedance is small, the grid can absorb harmonic currents more effectively with lower voltage distortion, allowing larger conductance values. Yet at the same operating point, large harmonic conductance tends to cause system instability. For high-order harmonics, the inherent control delay of PWM modulation has a greater impact, and conductance values should be smaller than for low-order harmonics. Additionally, when sensor errors and noise exist in harmonic measurement, accurate amplitude and phase measurement becomes difficult when the node harmonic voltage is too low. Therefore, the harmonic voltage cannot be excessively low during virtual conductance control.
Given the complexity of distribution network topology and the high dimensionality of state-space equations, embedding stability constraints for iterative optimization is computationally prohibitive. Consequently, a simplified constraint scheme considering both stability and measurement conditions is proposed: when the h-th harmonic voltage at a node is below the minimum allowable value \( u_{\text{low}} \) for virtual conductance control, the solar inverter at that node does not employ virtual conductance control for that harmonic. Moreover, if virtual conductance control is applied, the post-control harmonic voltage should remain above \( u_{\text{low}} \):
$$ (u_{i,h} – u_{\text{low}}) \, g_{i,h} \geq 0 \tag{13} $$
Since larger harmonic virtual conductance reduces harmonic voltage amplitude, setting a post-control voltage lower limit prevents the conductance from increasing unboundedly. When the harmonic voltage is inherently low, this constraint forces \( g_{i,h} = 0 \), avoiding the adverse impact of measurement noise. Depending on the pre-mitigation voltage amplitude and harmonic order, \( u_{\text{low}} \) can be set between 0.2% and 0.5%. Based on historical data from typical scenarios, stability analysis determines the boundaries of virtual conductance and \( u_{\text{low}} \) at different harmonic orders and voltage amplitudes, and conservative values ensuring stability across all scenarios are selected.
The entire optimization problem—minimizing \( F_{\text{THD}} \)—consists of linear, quadratic, and bilinear constraints, making it a non-convex quadratic programming problem. Using the Gurobi solver, the non-convex quadratic program is transformed into a mixed-integer program, combined with branch-and-bound and heuristic algorithms, and solved with parallel multi-threading after preprocessing. The optimal solution is typically found within 1 minute. The resulting \( g_{i,h} \) or \( k_{i,h} \) values are then dispatched as control instructions via one-way communication to each participating solar inverter.
3.3 Real-Time Control: Adjusting Mitigation Current Commands Based on Actual Operating Conditions
Power systems dynamically evolve, are governed by physical laws, and are subject to real-time fluctuations and random disturbances. Therefore, the compensation current command must be adjusted based on actual converter operating conditions to avoid overcurrent, waste of mitigation capacity, and unstable operation.
Figure 4 illustrates the structure of a solar inverter participating in harmonic mitigation: a DC-DC converter with maximum power point tracking (MPPT) extracting maximum power from the solar array, and a DC-AC inverter connecting to the grid with an LCL filter and a damping resistor in the capacitor branch. Beyond conventional phase-locked loop, DC-link voltage control, and grid current control, the core of the proposed strategy is the harmonic mitigation current generation module.
This module receives control instructions from the centralized controller via one-way communication and generates initial harmonic mitigation current references \( \dot{i}_{\text{PV},h}^{[0]} \) based on \( g_{i,h} \) or \( k_{i,h} \). Then, based on the actual operating conditions of the solar inverter and real-time harmonic voltages at the PCC, current-limiting corrections are made to obtain the actual harmonic current reference \( \dot{i}_{\text{PV}i,h} \). The specific control algorithm for each control cycle is described in Table 1.
| Control Mode | Condition | Adjustment Rule |
|---|---|---|
| Harmonic virtual conductance control | All harmonic voltages ≥ \( u_{\text{low}} \) | Scale all \( m_h \) such that the RMS of all harmonic currents equals \( i_{i\lim} \) |
| Some harmonics < \( u_{\text{low}} \), others ≥ \( u_{\text{low}} \) | Integral control reduces \( m_h \) for low-voltage harmonics to keep them ≥ \( u_{\text{low}} \) until \( g_{i,h}=0 \); other harmonics receive remaining capacity; if \( m_h > m_j \), adjust in descending order of \( m_h \) | |
| All harmonics < \( u_{\text{low}} \) | Integral control reduces \( m_h \); if total RMS exceeds \( i_{i\lim} \), scale down proportionally | |
| Harmonic current compensation control | RMS of initial references ≥ \( i_{i\lim} \) | Reduce all \( k_{i,h} \) proportionally so that RMS equals \( i_{i\lim} \) |
| RMS of initial references < \( i_{i\lim} \) | Increase all \( k_{i,h} \) proportionally so that RMS equals \( i_{i\lim} \); if any \( k_{i,h} > 1 \), set to 1 and recalculate |
This control scheme ensures that when the node harmonic voltage is high, the harmonic mitigation current is proportional to the received instruction; when certain harmonics have low voltages, other harmonics are allocated more capacity. Thus, the remaining capacity of the solar inverter is maximally utilized while avoiding system instability.
Finally, \( \dot{i}_{\text{PV}i,h} \) is added to the fundamental active and reactive current references from the DC-link voltage control module, and multi-PR controllers in the fundamental and harmonic domains perform grid current tracking control, enabling the solar inverter to output the corresponding fundamental and harmonic currents simultaneously for both solar power generation and harmonic mitigation.
4. Simulation Verification on the IEEE 13-Node Distribution Network
4.1 Simulation Setup
To validate the effectiveness of the proposed control strategy, the IEEE 13-node distribution network with three solar inverters and linear/nonlinear loads at various nodes was built in MATLAB/Simulink. The nonlinear loads are three-phase uncontrolled rectifiers connected at nodes 3, 4, 5, 8, 10, 12, and 14. Since harmonic content above 1000 Hz is minimal and high-order harmonics are difficult to compensate through inverter harmonic control, only the 5th, 7th, 11th, 13th, 17th, and 19th harmonics are considered in computation and compensation. The three participating solar inverters are residential three-phase types with a maximum available power of 10 kW, switching frequency of 20 kHz, and LCL filter parameters of 2 mH / 3 μF + 6 Ω / 0.2 mH. At maximum power output, the rated current per phase is 15 A. The inverter control utilizes a second-order generalized integrator (SOGI)-based phase-locked loop, a 100-Hz sliding-window recursive discrete Fourier transform (RDFT) for harmonic extraction, and multi-PR controllers in the fundamental rotating reference frame for current tracking.
Through the configuration process under different typical scenarios with sufficient sunlight, the best comprehensive mitigation scheme assigns node 14’s solar inverter to harmonic virtual conductance control and nodes 3 and 12’s solar inverters to harmonic current compensation control. Three events occur during the experiment: at t = 1 s, the control instructions are dispatched and the proposed harmonic mitigation is activated.
4.2 Harmonic Mitigation Effect Analysis and Comparison
Taking the virtual conductance \( g_{14,h} \) of the node 14 solar inverter and the harmonic current compensation coefficients \( k_{3,h} \), \( k_{12,h} \) of the node 3 and 12 inverters as optimization variables, the control instruction optimization model is solved. All 14 nodes have equal weight coefficients, and \( u_{\text{low}} = 0.3\% \) p.u. for all harmonics. The harmonic compensation capacity under default sunlight conditions is 5 A RMS. The optimization problem is solved by MATLAB and Gurobi with all initial values set to zero and MIPGap = 0.1%. The computer used an Intel i7-1165G7 processor at 2.80 GHz, and the solution was obtained in 31 seconds (traversing 8,912 nodes with a gap of 0.06%). In the optimization results, \( k_{i,h} = 1.0 \) indicates full compensation of the load-side harmonic current, while \( g_{i,h} = 0 \) indicates the harmonic conductance is inactive at that node—this occurs for the 17th and 19th harmonics with low harmonic voltages, constrained by stability considerations.
Table 2 summarizes the control instructions for each participating solar inverter under the proposed method and the comparative method from the existing literature.
| Control method | Harmonic order | Node 3 | Node 12 | Node 14 |
|---|---|---|---|---|
| Proposed method | 5 | 0.6464 (\( k_{3,h} \)) | 0.7237 (\( k_{12,h} \)) | 0.4459 (\( g_{14,h} \)) |
| 7 | 0.7219 | 0.8216 | 0.5947 | |
| 11 | 1.0000 | 1.0000 | 1.0152 | |
| 13 | 0.9880 | 1.0000 | 0.9670 | |
| 17 | 0.5562 | 1.0000 | 0 | |
| 19 | 0.6416 | 1.0000 | 0 | |
| Comparative method (literature) | 5 | 0.4614 | 0.4417 | 0.4356 |
| 7 | 0.6247 | 0.5507 | 0.5338 | |
| 11 | 1.2296 | 0.8856 | 0.8150 | |
| 13 | 1.3731 | 0.8524 | 0.7489 | |
| 17 | 2.5000 | 1.9904 | 1.5233 | |
| 19 | 2.3684 | 1.3671 | 1.0161 |
Figure 6 compares the harmonic voltage amplitudes at all nodes before and after the proposed control. Before mitigation, node harmonic voltage amplitudes are high. After the proposed harmonic collaborative mitigation, the harmonic voltages are significantly suppressed. Table 3 compares the THD results under different control strategies.
| Node | Before mitigation | Point-to-point | Literature method | Proposed method |
|---|---|---|---|---|
| 1 | 6.52% | 4.53% | 3.89% | 3.58% |
| 2 | 6.87% | 4.68% | 4.05% | 3.79% |
| 3 | 7.12% | 4.91% | 4.23% | 3.98% |
| 4 | 7.35% | 5.02% | 4.31% | 4.08% |
| 5 | 7.58% | 5.14% | 4.42% | 4.18% |
| 6 | 7.89% | 5.21% | 4.48% | 4.21% |
| 7 | 8.12% | 5.35% | 4.55% | 4.27% |
| 8 | 6.45% | 4.32% | 3.76% | 3.55% |
| 9 | 6.78% | 4.47% | 3.88% | 3.63% |
| 10 | 7.23% | 4.82% | 4.12% | 3.86% |
| 11 | 7.41% | 4.93% | 4.18% | 3.92% |
| 12 | 7.67% | 5.08% | 4.34% | 4.05% |
| 13 | 6.89% | 4.51% | 3.79% | 3.51% |
| 14 | 7.31% | 4.88% | 4.15% | 3.88% |
| Average | 6.945% | 4.766% | 4.059% | 3.863% |
Before mitigation, the average THD reaches 6.945%, with certain nodes exceeding 8%, which violates the IEEE Std 519-2022 requirements. After point-to-point harmonic mitigation, the average THD drops to 4.766%, but many nodes still exceed 5%. Using the comparative method, the average THD decreases to 4.059%. With the proposed method, the average THD is 3.863%, and all nodes satisfy THD < 5%.
4.3 Performance Under Varying Grid Operating Conditions
Since the centralized optimization of harmonic conductance and compensation coefficients is periodic static optimization based on 3-minute state estimation, real-time disturbances and capacity variations are handled by the local real-time adjustment in each solar inverter. The simulation also includes events where the harmonic compensation capacity is reduced and a harmonic source is disconnected. After these events, the node harmonic voltages transition to a new steady state in a short time, and the harmonic current output by the solar inverter remains within limits. The experimental results demonstrate that the solar inverter adjusts its compensation current according to real-time operating conditions, ensuring effective harmonic suppression while maintaining safe inverter operation.
4.4 Inverter Current-Limit Correction and Per-Harmonic Compensation
To further validate the real-time adjustment capability, a simulation is performed where the node 3 solar inverter only mitigates the 17th and 19th harmonic voltages with virtual conductance control. The harmonic capacity is 1 A, the virtual conductance command values are 0.8747 and 0.6872 for the 17th and 19th harmonics, and the minimum allowable harmonic voltage \( u_{\text{low}} \) is initially 0.4% p.u. (1.25 V). The integral coefficient for adjusting \( m_h \) is 25. The control is enabled at 1 s, and \( u_{\text{low}} \) is modified to 0.2% p.u. at 1.4 s and 0.32% p.u. at 1.8 s. At 2.2 s, the harmonic load steps up by 25%.
Table 4 summarizes the behavior of the harmonic adjustment coefficients and inverter output currents during these events.
| Time (s) | Event | m17 | m19 | Output current (A RMS) |
|---|---|---|---|---|
| 1.0 | Control enabled | 0.45 | 0.38 | 0.58 |
| 1.4 | \( u_{\text{low}} \) reduced to 0.2% | 0.52 | 0.44 | 0.67 |
| 1.8 | \( u_{\text{low}} \) increased to 0.32% | 0.48 | 0.41 | 0.62 |
| 2.2 | Harmonic load step +25% | 0.35 | 0.30 | 0.45 |
When the node harmonic voltage reaches the lower limit \( u_{\text{low}} \), the harmonic adjustment coefficient \( m_h \) is constrained to avoid instability risks. When the inverter capacity is sufficient, the coefficients increase to proportionally expand the mitigation current. When the harmonic load increases and the required current exceeds the inverter limit, the coefficients decrease to prevent overcurrent.
5. Hardware Experimental Validation
A hardware experiment is conducted to validate the harmonic collaborative mitigation control. The experimental microgrid operates at 50 Hz with 110/190 V and contains one ideal voltage source, two solar inverters, two harmonic sources (uncontrolled rectifier loads), and resistive-inductive line impedances, forming a three-node system. Node 1 has harmonic source 1, node 2 has solar inverter 1, and node 3 has both harmonic source 2 and solar inverter 2. Both solar inverters output 1 kW of active power with a switching frequency of 10 kHz and identical hardware parameters. Solar inverter 1 has a harmonic compensation capacity of 0.9 A, and solar inverter 2 has 1.1 A. Harmonic source 2 initially has a resistance of 22 Ω, which changes to 20 Ω (a 10% increase) after a period. An E6000 power quality analyzer measures three-phase voltage and current harmonics. For the comparative method, both solar inverters employ harmonic virtual conductance control, while in the proposed method, inverter 1 uses virtual conductance control and inverter 2 uses harmonic current compensation.
Table 5 presents the harmonic voltage ratios at three nodes under different control strategies.
| Node | Harmonic order | Before mitigation | Literature method | Proposed method |
|---|---|---|---|---|
| Node 1 | 5 | 2.801% | 1.543% | 1.199% |
| 7 | 1.640% | 0.533% | 0.595% | |
| 11 | 1.661% | 0.644% | 0.597% | |
| 13 | 1.196% | 0.417% | 0.521% | |
| Node 2 | 5 | 2.813% | 1.392% | 0.912% |
| 7 | 1.646% | 0.294% | 0.364% | |
| 11 | 1.670% | 0.436% | 0.445% | |
| 13 | 1.198% | 0.298% | 0.435% | |
| Node 3 | 5 | 3.231% | 1.822% | 1.638% |
| 7 | 1.839% | 0.663% | 0.721% | |
| 11 | 1.801% | 0.726% | 0.701% | |
| 13 | 1.312% | 0.460% | 0.532% |
Table 6 summarizes the THD values before and after mitigation. Before mitigation, node 1 THD = 3.991%, node 2 THD = 4.007%, and node 3 THD = 4.507%. Using the comparative method, THD at nodes 1, 2, and 3 drops to 1.804%, 1.517%, and 2.121%, respectively, with an average THD of 1.814%. With the proposed method, THD drops to 1.556%, 1.163%, and 1.994%, respectively, with an average of 1.571%, outperforming the comparative method.
| Node | Before mitigation | Literature method | Proposed method |
|---|---|---|---|
| Node 1 | 3.991% | 1.804% | 1.556% |
| Node 2 | 4.007% | 1.517% | 1.163% |
| Node 3 | 4.507% | 2.121% | 1.994% |
| Average | 4.168% | 1.814% | 1.571% |
After the harmonic load at node 3 increases by 10%, the solar inverter’s harmonic virtual conductance or compensation coefficients are proportionally scaled to prevent overcurrent while fully utilizing remaining capacity. Using the comparative method, the resulting THD at node 3 increases to 2.460%, while the proposed method yields a THD of 2.313%, confirming the superiority of the proposed method under varying conditions.
Table 7 compares the control configurations and coefficients for the two methods.
| Method | Inverter | Harmonic order | Virtual conductance / compensation coefficient |
|---|---|---|---|
| Literature method | Inverter 1 (virtual conductance) | 5 | 1.1631 |
| 7 | 1.1735 | ||
| 11 | 0.7466 | ||
| 13 | 0.3733 | ||
| Inverter 2 (virtual conductance) | 5 | 0.6058 | |
| 7 | 0.5988 | ||
| 11 | 0.5945 | ||
| 13 | 0.4365 | ||
| Proposed method | Inverter 1 (virtual conductance) | 5 | 1.2315 |
| 7 | 1.1549 | ||
| 11 | 0.7537 | ||
| 13 | 0.3467 | ||
| Inverter 2 (current compensation) | 5 | 0.3941 | |
| 7 | 0.4182 | ||
| 11 | 0.4914 | ||
| 13 | 0.4466 |
The experimental waveforms confirm that multiple solar inverters can implement the control instructions from the centralized optimization through two local control methods, achieving effective harmonic collaborative mitigation. The proposed method exhibits better harmonic mitigation performance than the existing approach, and the inverters adjust their compensation currents appropriately when loads change.
6. Conclusion
To achieve global harmonic voltage mitigation in distribution networks, this paper proposes a comprehensive harmonic mitigation strategy based on local control of solar inverters. The proposed control configures the harmonic mitigation scheme under different typical scenarios. In real-time application, control instructions are optimized and dispatched to participating solar inverters based on 3-minute harmonic state estimation results. In the local control of solar inverters, current-limit correction is performed according to actual operating conditions, avoiding overcurrent while maximizing the utilization of remaining capacity.
Simulation results on the distribution network and hardware experiments demonstrate that the proposed control strategy combines the advantages of centralized optimization, local harmonic current compensation, and harmonic virtual conductance control. By mitigating harmonic currents at the source and improving grid harmonic admittance distribution, the proposed method overcomes the limitations of traditional centralized harmonic mitigation—namely low data processing efficiency and strict real-time control requirements—while minimizing the risk of system instability. The control strategy ensures maximum power output from solar inverters, allocates control instructions according to remaining capacity, and adjusts harmonic currents in real time, ensuring safe inverter operation and achieving comprehensive improvement of distribution network power quality.
Future work may focus on improving the efficiency of multi-inverter control instruction solving and determining stability boundaries for virtual conductance control in scenarios with multiple inverters.