The rapid proliferation of distributed energy resources (DERs) and electric vehicle charging infrastructure marks a pivotal shift towards a more decentralized and sustainable power system. This transformation, however, is predominantly mediated by power electronic converters, the ubiquitous on-grid inverter being the primary interface. While enabling the integration of renewables, this widespread power electronics penetration introduces significant power quality challenges, with harmonic pollution emerging as a critical and systemic issue. Unlike traditional, centralized harmonic sources, modern distribution networks contend with a high-density, dispersed, and dynamic array of harmonic injections. Conventional point-of-use mitigation strategies, such as passive or dedicated active power filters, are increasingly inadequate—both economically and technically—for addressing this new paradigm of network-wide harmonic distortion.
This context underscores the need for innovative, resource-efficient harmonic management strategies. A promising avenue lies in leveraging the existing fleet of on-grid inverters not just for power conversion, but as active power quality service providers. By modifying their control algorithms, these grid-tied inverters can be transformed into Multifunctional Grid-Tied Inverters (MFGTIs), capable of providing ancillary services like reactive power support, harmonic current compensation, and resonance damping alongside their primary power injection function. This approach maximizes asset utilization, defers dedicated filter investments, and turns potential polluters into part of the solution. The core challenge then evolves from simply adding filters to intelligently coordinating the harmonic compensation efforts of multiple, distributed on-grid inverters to achieve optimal system-wide performance.
This article presents a systematic, model-based methodology for the coordinated harmonic compensation of multiple MFGTIs in a distribution network. The proposed Harmonic Compensation Management System (HCMS) addresses the fundamental question: given a network with several candidate on-grid inverters, which ones should be activated for harmonic compensation, what should their compensation currents be, and where should we monitor to effectively assess system-wide harmonic distortion? We introduce the Relative Gain Array (RGA), a tool from multi-variable control theory, to analyze the coupling and effectiveness of potential compensation sources. This analysis guides the optimal selection and configuration of MFGTIs. Furthermore, a pragmatic measurement point selection strategy is developed to render the scheme feasible for large networks without requiring exhaustive monitoring. Finally, the optimal compensation currents for the selected MFGTIs are calculated using a Least-Squares algorithm, respecting individual device capacity constraints. The efficacy of the proposed HCMS is validated through detailed case studies on a representative low-voltage distribution network.
System Modeling and MFGTI Control Framework
The foundation of any coordinated control strategy is a robust model of the individual actuators—in this case, the MFGTIs. A typical three-phase voltage-source on-grid inverter with an LCL output filter is considered. For harmonic compensation, precise tracking of non-fundamental frequency currents is required. While dq-frame control is common for fundamental power flow, its inherent cross-coupling and need for multiple coordinate transformations for harmonics complicate the controller design. The stationary αβ-frame offers a simpler alternative, as it eliminates phase coupling and allows for the direct implementation of resonant controllers for selective harmonic frequencies.
Therefore, the control structure for the MFGTI is implemented in the αβ-frame. The core current controller is a Quasi-Proportional-Resonant (QPR) regulator. The transfer function of the QPR controller, designed to regulate the fundamental and key harmonic currents (e.g., 5th, 7th, 11th, 13th), is given by:
$$G_{\text{QPR}}(s) = k_p + \frac{2k_{r1}\omega_c s}{s^2 + 2\omega_c s + \omega_1^2} + \sum_{h \in H} \frac{2k_{rh}\omega_c s}{s^2 + 2\omega_c s + \omega_h^2}$$
where $k_p$ is the proportional gain, $k_{r1}$ and $k_{rh}$ are the resonant gains for the fundamental and the $h$-th harmonic, respectively, $\omega_c$ is the cutoff frequency, $\omega_1$ is the fundamental frequency, and $\omega_h$ is the frequency of harmonic $h$ belonging to the set $H$ of compensated orders. Active damping via capacitor current feedback is typically employed to stabilize the LCL filter resonance. When the MFGTI operates in its multifunctional mode, the current reference $i_{\alpha\beta}^*$ is composed of two parts: the fundamental current $i_{1,\alpha\beta}^*$ for active/reactive power delivery and the harmonic compensation current $i_{h,\alpha\beta}^*$ dictated by the higher-level HCMS: $i_{\alpha\beta}^* = i_{1,\alpha\beta}^* + i_{h,\alpha\beta}^*$. The power stage of the on-grid inverter, including the DC source, switching bridge, and LCL filter, can be linearized as an equivalent gain $k_{\text{PWM}}$ for harmonic frequency analysis. The resulting closed-loop model approximates the MFGTI as a controlled harmonic current source with a specific output impedance. For coordination analysis, the key relationship is the effective current gain from the harmonic reference $i_{h}^*$ to the injected current $i_{h}$. Under high-gain control and within its bandwidth, this can be considered as unity, i.e., $i_{h} \approx i_{h}^*$.
The Harmonic Compensation Management System (HCMS) Algorithm
The HCMS is the central intelligence that determines the optimal harmonic compensation strategy. It operates in three sequential stages: 1) Optimal Configuration of Compensation Sources using RGA, 2) Strategic Selection of Harmonic Measurement Points, and 3) Calculation of Optimal Compensation Currents via Least-Squares.
Stage 1: Optimal Source Configuration via Relative Gain Array (RGA) Analysis
In a network with multiple candidate MFGTIs, activating all of them is neither necessary nor optimal. Compensation sources can interfere with each other due to network coupling. The RGA provides a quantitative measure of this interaction and the effectiveness of a given on-grid inverter in controlling voltages at specific nodes.
For a given harmonic order $h$, the network’s nodal admittance matrix $\mathbf{Y}_h$ is constructed. The impedance matrix is $\mathbf{Z}_h = \mathbf{Y}_h^{-1}$. Consider $n_c$ candidate compensation sources located at nodes indexed by vector $\mathbf{c}$. The transfer relationship between compensation currents $\mathbf{I}_{c,h}$ and node voltages $\mathbf{V}_h$ is $\mathbf{V}_h = \mathbf{Z}_{c,h} \mathbf{I}_{c,h} + \mathbf{V}_{h0}$, where $\mathbf{Z}_{c,h}$ is the submatrix of $\mathbf{Z}_h$ corresponding to the compensation nodes, and $\mathbf{V}_{h0}$ is the background harmonic voltage.
The RGA matrix $\mathbf{\Lambda}_h$ for this input-output pair is defined as:
$$\mathbf{\Lambda}_h = \mathbf{Z}_{c,h} \odot (\mathbf{G}_h^{-1})^T$$
where $\mathbf{G}_h = \mathbf{Y}_h(\mathbf{c}, \mathbf{c})$ is the mutual impedance matrix among compensation nodes, and $\odot$ denotes the element-wise (Hadamard) product. The element $\lambda_{ij,h}$ relates the effect of the $j$-th compensation source on the $i$-th node voltage.
Traditional RGA analysis focuses on diagonal elements. We propose a comprehensive set of metrics to evaluate a candidate on-grid inverter at node $i$ for harmonic $h$:
- Coupling Degree ($\gamma_{i,h}$): Measures the alignment between the source and its intended local impact. A value of 1 indicates perfect pairing.
$$\gamma_{i,h} = \min(\lambda_{ii,h}, \frac{1}{\lambda_{ii,h}})$$ - Independence Index ($\eta_{i,h}$): Assesses the decoupling from other compensation sources. A higher value indicates less interference.
$$\eta_{i,h} = 1 – \frac{\max_{j \neq i}(|\lambda_{ij,h}|)}{|\lambda_{ii,h}|}$$ - Comprehensive Efficacy ($\xi_{i,h}$): A unified score combining coupling and independence.
$$\xi_{i,h} = \gamma_{i,h} \cdot \eta_{i,h}$$
For comparing sources across different harmonic orders, a normalized efficacy is used: $\xi_{i,h}^* = \xi_{i,h} / \max_j(\xi_{j,h})$. The optimal set of MFGTIs is selected by choosing those with the highest aggregated (e.g., sum or average) $\xi_{i,h}^*$ scores across all harmonics of concern. This ensures the selected on-grid inverters are both effective and minimally interacting.
Stage 2: Strategic Selection of Harmonic Measurement Points
Full network harmonic monitoring is impractical. We propose a method to select a strategic subset of measurement nodes (typically 30-50% of total nodes) that best represent the system’s harmonic state. Two criteria are combined:
- Electrical Distance ($D_i$): Reflects coupling to all selected compensation sources. A smaller aggregate impedance implies stronger coupling.
$$D_i = \sum_{h \in H} \sum_{j \in \mathbf{c}} \frac{1}{|z_{ij,h}|}$$
where $z_{ij,h}$ is the $h$-th harmonic impedance between node $i$ and compensation source $j$. - Harmonic Sensitivity ($S_i$): Reflects a node’s overall sensitivity to harmonic currents anywhere in the network.
$$S_i = \sum_{h \in H} \sum_{j=1}^{n} |z_{ij,h}|$$
The normalized metrics $D_i^*$ and $S_i^*$ are combined into a single merit index $M_i$:
$$M_i = \alpha D_i^* + (1-\alpha) S_i^*$$
where $\alpha$ is a weighting factor (e.g., 0.5 for equal balance). The final measurement set $\mathcal{M}$ is formed by including: a) the main source/PCC node, b) the nodes hosting the selected MFGTIs, and c) the top-ranked nodes according to $M_i$ until the desired number of measurement points is reached. This ensures critical points related to sources, actuators, and sensitive/propagating locations are monitored.
Stage 3: Least-Squares Optimization of Compensation Currents
With the optimal MFGTIs ($\mathbf{c}^*$) and measurement points ($\mathcal{M}$) determined, the final step is to compute their optimal harmonic injection currents. Let $\mathbf{V}_{m0,h}$ be the vector of measured $h$-th harmonic voltages before compensation. Let $\mathbf{Z}_{m,c^*,h}$ be the impedance submatrix from compensation nodes $\mathbf{c}^*$ to measurement nodes $\mathcal{M}$. The goal is to find the compensation current vector $\mathbf{I}_{c,h}$ that minimizes the squared Euclidean norm of the compensated voltages at the measurement points:
$$\min_{\mathbf{I}_{c,h}} J_h = || \mathbf{V}_{m0,h} + \mathbf{Z}_{m,c^*,h} \mathbf{I}_{c,h} ||^2$$
This is a standard linear least-squares problem. The optimal solution, subject to no constraints, is given by the Moore-Penrose pseudoinverse:
$$\mathbf{I}_{c,h}^{opt} = – (\mathbf{Z}_{m,c^*,h}^H \mathbf{Z}_{m,c^*,h})^{-1} \mathbf{Z}_{m,c^*,h}^H \mathbf{V}_{m0,h}$$
where $^H$ denotes the Hermitian transpose. If the computed current for any on-grid inverter exceeds its available capacity $I_{\text{max}}$, the current is clipped at that limit. The resulting currents, after scaling by the on-grid inverter’s closed-loop gain, form the harmonic reference commands $i_{h,\alpha\beta}^*$ sent to each MFGTI.
Case Study: Implementation and Results
The proposed HCMS was tested on a detailed 20-node, 400V low-voltage distribution network model. The system included a grid connection at Node 1 (with background voltage harmonics), two three-phase diode rectifier nonlinear loads at Nodes 14 and 18, and four identical 500 kVA MFGTIs at Nodes 10, 17, 19, and 20, each operating with a 300 kW fundamental power setpoint. The compensation targeted the 5th, 7th, 11th, and 13th harmonics, with a per-inverter harmonic current limit $I_{\text{max}} = 60$ A.
Optimal Source Configuration Results
The RGA-based analysis was performed for the four candidate MFGTIs. The normalized comprehensive efficacy $\xi_{i,h}^*$ for each harmonic order was calculated. The aggregate scores consistently identified the MFGTIs at Nodes 17 and 19 as the most effective and least coupled pair. The ranking was: Node 17 > Node 19 > Node 10 > Node 20. Consequently, the HCMS configured the MFGTIs at Nodes 17 and 19 as the active compensation sources.
Measurement Point Selection Results
Applying the proposed selection criteria ($\alpha=0.5$ and targeting ~40% coverage), the HCMS selected the following measurement nodes: $\mathcal{M} = \{1, 10, 13, 14, 17, 18, 19, 20\}$. This set includes the source (1), all MFGTI locations (10,17,19,20), the main harmonic load buses (14,18), and one additional sensitive node (13) identified by the merit index $M_i$.
Compensation Performance Under Various Scenarios
Scenario A: Without Background Harmonics (Only Nonlinear Loads). The HCMS calculated the optimal compensation currents for the MFGTIs at Nodes 17 and 19. The compensation currents for the fundamental harmonic components are summarized below:
| Harmonic Order | MFGTI @ Node 17 | MFGTI @ Node 19 | ||
|---|---|---|---|---|
| Magnitude (A) | Phase (°) | Magnitude (A) | Phase (°) | |
| 5th | 44.8 | 174.4 | 44.5 | 162.2 |
| 7th | 25.7 | -177.4 | 26.6 | 166.4 |
| 11th | 14.7 | 9.8 | 13.8 | -15.8 |
| 13th | 11.3 | 12.5 | 10.8 | -16.6 |
The performance was compared against several alternative MFGTI activation schemes (e.g., using only Node 17, using Nodes 10 & 17, using all four). The figure of merit was the average reduction in Total Harmonic Distortion (THD) across all 20 nodes per unit of total compensation current injected. The HCMS-selected pair (Nodes 17 & 19) achieved the highest efficiency. Simulation results showed the system voltage THD was reduced from an uncontrolled maximum of over 6% to below 2.5% at all nodes, with an average THD of approximately 1.5%.
Scenario B: With Background Voltage Harmonics. 5th and 7th harmonic voltages were added at the grid source (Node 1). The HCMS algorithm automatically recomputed the optimal currents. The compensation strategy effectively mitigated both the background and load-generated harmonics. The post-compensation voltage THD profile was significantly improved, with all node voltages complying with standard limits (e.g., IEC 61000-3-6). This demonstrates the adaptability of the coordinated on-grid inverter control to varying network harmonic conditions.
Scenario C: Comparison with Full Measurement Set. To validate the measurement selection strategy, the optimal currents were also computed using harmonic data from all 20 nodes (full observability). The resulting compensation currents and the final network-wide THD were nearly identical to those obtained using only the 8 strategically selected measurement points. This confirms that the proposed selection method captures the essential harmonic state of the network, making the HCMS practical for large-scale implementation.
Scenario D: Algorithm Efficiency Comparison. The Least-Squares solution was compared against a heuristic search method (Particle Swarm Optimization – PSO). While PSO could find a similar solution, it required significantly more computation time (over 50x longer for a comparable accuracy setup). The deterministic Least-Squares approach provides a fast, reliable, and repeatable solution essential for real-time or near-real-time control of multiple on-grid inverters.
Conclusion
The increasing harmonic pollution in power-electronics-dominated distribution networks demands a shift from localized to system-wide, resource-cooperative mitigation strategies. This article has presented a comprehensive Harmonic Compensation Management System (HCMS) that enables the optimal coordination of multiple Multifunctional Grid-Tied Inverters. The three-pillar approach—RGA-based source configuration, criteria-driven measurement point selection, and Least-Squares current optimization—provides a rigorous yet practical framework.
The core insight is to treat the fleet of on-grid inverters not merely as independent units but as a coordinated system whose harmonic injection capabilities can be optimally dispatched. The RGA analysis ensures the selected inverters work effectively together with minimal counterproductive interaction. The measurement strategy ensures the control is based on sufficient, strategically relevant data without requiring unrealistic monitoring infrastructure. Finally, the Least-Squares solver efficiently computes the precise compensation currents needed to minimize distortion at critical network points.
Case study results validate that the HCMS can significantly reduce system-wide voltage THD under various operating conditions, utilizing the existing on-grid inverter infrastructure. This method enhances power quality, improves asset utilization, and provides a scalable, cost-effective pathway for managing harmonics in the future grid. Future work will integrate dynamic constraints, consider unbalanced conditions, and explore distributed versions of the algorithm for fully decentralized execution among the on-grid inverters.