With the rapid integration of power electronic devices into distribution networks, harmonic pollution has become a critical issue, leading to voltage distortions and potential equipment failures. Solar inverters, which share topological similarities with active power filters, offer a promising solution for harmonic mitigation due to their controllable harmonic emission capabilities. However, leveraging solar inverters for harmonic management involves multiple stakeholders, including solar users who have significant autonomy in pricing their services. This autonomy often results in uncoordinated pricing strategies, undermining the efficiency of harmonic mitigation efforts. To address this, we propose a novel harmonic trading model that incorporates a harmonic management aggregator as a third party to standardize pricing and coordinate harmonic power transactions. Our approach employs a two-layer master-slave game framework, where the upper layer involves a price game between the aggregator and harmonic source users, and the lower layer focuses on price negotiations between the aggregator and solar users. This model aims to balance the interests of all participants while ensuring effective harmonic suppression. We validate our model using the IEEE-13 node system, demonstrating significant improvements in harmonic mitigation and economic benefits for all parties involved.
The increasing penetration of renewable energy sources, particularly solar photovoltaic systems, has transformed distribution networks into active systems with bidirectional power flows. Solar inverters play a pivotal role in converting DC power from solar panels to AC power for grid integration. Beyond their primary function, solar inverters can be repurposed for harmonic compensation, akin to active power filters, by injecting counter-harmonic currents. This dual functionality not only enhances grid stability but also provides economic opportunities for solar users. However, the lack of a structured market mechanism for harmonic trading hinders the widespread adoption of this approach. In this paper, we introduce the concept of a harmonic management aggregator to facilitate a coordinated market for harmonic power, addressing the pricing disparities among solar users. Our two-layer game-theoretic model ensures that all participants—solar users, the aggregator, and harmonic source users—achieve optimal outcomes through strategic interactions.
The core of our model lies in the hierarchical decision-making process. In the upper layer, harmonic source users, as leaders, determine their harmonic power demand based on the price set by the aggregator. The aggregator, as a follower, then allocates this demand among solar users and adjusts the harmonic power price accordingly. In the lower layer, the aggregator acts as the leader, proposing harmonic power supply quotas to solar users, who respond by setting their prices based on their operational constraints and solar generation profiles. This bidirectional博弈 ensures that prices reflect both demand and supply conditions, leading to a stable equilibrium. We formulate the utility functions for each participant, considering costs, revenues, and technical constraints such as harmonic distortion limits and power flow equations.
To illustrate the mathematical foundation, we define the harmonic power demand at time t as $$D_h(t) = \sum_{H=2}^{\infty} U_1(t) I_H(t)$$, where $$U_1(t)$$ is the fundamental voltage RMS and $$I_H(t)$$ is the H-th harmonic current consumed by the harmonic source user. The utility function for harmonic source users is given by $$I_{HS} = \sum_{t=1}^{24} \lambda_h D_h(t)$$, where $$\lambda_h$$ is the price per kVA of harmonic power. Constraints include cost-effectiveness compared to traditional APF installation: $$d V_d \leq V_{APF}$$, where $$d$$ is the number of typical production days, $$V_d$$ is the daily cost through solar inverters, and $$V_{APF}$$ is the annual cost of APF. Additionally, voltage total harmonic distortion (THD) must not exceed the maximum allowable limit: $$\text{THD}_U^i(t) \leq \text{THD}_U^{\text{max}}$$ for each node i.
For the harmonic management aggregator, the utility function is $$I_{HA} = F_{HA} – C_{HA}$$, where $$F_{HA} = \sum_{t=1}^{24} \lambda_h D_h(t)$$ is the revenue from selling harmonic power, and $$C_{HA} = \sum_{i=1}^{n} \sum_{t=1}^{24} \lambda_i D_i(t)$$ is the cost of purchasing harmonic power from solar users. Here, $$\lambda_i$$ is the price per kVA from solar user i, and $$D_i(t)$$ is the harmonic power supplied by solar user i at time t. The aggregator must ensure that revenue exceeds costs to maintain profitability.
Solar users maximize their utility $$I_{PV,i} = F_{PV,i} = \sum_{t=1}^{24} \lambda_i D_i(t)$$, subject to constraints including fundamental power flow, harmonic power flow, and inverter capacity limits. The power flow constraints can be expressed using the admittance matrix for fundamental and harmonic frequencies. For instance, the fundamental power flow at node i is given by $$P_i + jQ_i = V_i \sum_{j=1}^{N} Y_{ij} V_j^*$$, where $$Y_{ij}$$ is the admittance matrix element. Similarly, for the H-th harmonic, the current injection from solar inverters must satisfy harmonic power flow equations.
We prove the existence of a Stackelberg equilibrium (SE) for our two-layer game model. The equilibrium strategies $$(D_h^*, D_i^*, \lambda_i^*)$$ satisfy the conditions: $$U_{HS}(D_h^*, D_i^*, \lambda_i^*) \leq U_{HS}(D_h, D_i^*, \lambda_i^*)$$ for harmonic source users, $$U_{HA}(D_h^*, D_i^*, \lambda_i^*) \geq U_{HA}(D_h^*, D_i, \lambda_i)$$ for the aggregator, and $$U_{PV,i}(D_h^*, D_i^*, \lambda_i^*) \geq U_{PV,i}(D_h^*, D_i^*, \lambda_i)$$ for solar users. This ensures that no participant can unilaterally improve their utility by changing their strategy.
In our simulation, we use the IEEE-13 node system to validate the model. The system includes one harmonic source user at node 633 and three solar users at nodes 646, 684, and 675, each with a rated capacity of 500 kW. The harmonic source user has a load of 155 + j110 kVA, with harmonic currents dominated by the 3rd, 5th, and 7th orders. The voltage THD must remain below 5%. We compare three scenarios: Scenario 1 without the aggregator, Scenario 2 with the aggregator, and Scenario 3 with the aggregator considering communication delays.
The harmonic current profile and voltage THD for the harmonic source node are analyzed. For instance, at 03:00 and 09:00, the THD exceeds 5%, necessitating harmonic mitigation. The results demonstrate that solar inverters effectively reduce THD when coordinated through the aggregator. In Scenario 1, without the aggregator, solar users set prices independently, leading to significant disparities. For example, at 03:00, the harmonic power prices are 0.221, 0.129, and 0.131 USD/kVA for solar users 1, 2, and 3, respectively, with corresponding supplies of 36.43, 17.08, and 9.23 kVA. The daily cost for the harmonic source user is 16.978 USD, resulting in an annual cost of 2,546.7 USD.
In Scenario 2, with the aggregator, prices are more uniform, with maximum differences of only 0.045 USD/kVA. The harmonic power supplies increase to 46.16, 28.51, and 10.37 kVA at 03:00, and 37.54, 17.67, and 5.10 kVA at 09:00 for solar users 1, 2, and 3, respectively. The daily revenues for solar users rise to 13.031, 7.150, and 2.389 USD, while the aggregator’s revenue is 24.388 USD, exceeding its costs. The harmonic source user’s cost increases but remains below the APF alternative.
Scenario 3 introduces communication delays, where solar user 2 at 03:00 and solar user 3 at 09:00 are unavailable. Despite this, the model maintains functionality, with the aggregator achieving revenues of 14.004 USD at 03:00 and 9.559 USD, covering costs of 9.817 USD and 8.741 USD, respectively. This highlights the robustness of our approach in real-world conditions with intermittent communications.
To further analyze the economic and technical impacts, we present a comparison of key metrics across the scenarios in the following tables. These tables summarize the harmonic power prices, supplies, and participant revenues, emphasizing the role of solar inverters in enhancing system performance.
| Time | Scenario | Solar User 1 | Solar User 2 | Solar User 3 |
|---|---|---|---|---|
| 03:00 | Scenario 1 | 0.221 | 0.129 | 0.131 |
| 03:00 | Scenario 2 | 0.166 | 0.138 | 0.163 |
| 03:00 | Scenario 3 | 0.153 | — | 0.159 |
| 09:00 | Scenario 1 | 0.207 | 0.116 | 0.124 |
| 09:00 | Scenario 2 | 0.143 | 0.182 | 0.137 |
| 09:00 | Scenario 3 | 0.139 | 0.178 | — |
| Time | Solar User | Supply (kVA) | Revenue (USD) |
|---|---|---|---|
| 03:00 | 1 | 46.16 | 7.66 |
| 2 | 28.51 | 3.93 | |
| 3 | 10.37 | 1.69 | |
| 09:00 | 1 | 37.54 | 5.37 |
| 2 | 17.67 | 3.22 | |
| 3 | 5.10 | 0.70 |
The mathematical formulation of the harmonic power flow is critical for understanding the constraints. For each harmonic order H, the nodal current injections relate to voltages through the harmonic admittance matrix $$Y_H$$: $$I_H = Y_H V_H$$, where $$I_H$$ is the vector of harmonic currents and $$V_H$$ is the vector of harmonic voltages. Solar inverters contribute to $$I_H$$ by injecting compensatory currents, which are controlled to minimize THD. The total harmonic distortion at node i is calculated as $$\text{THD}_U^i = \frac{\sqrt{\sum_{H=2}^{\infty} |V_H^i|^2}}{|V_1^i|} \times 100\%$$, where $$V_H^i$$ is the H-th harmonic voltage at node i.
In our optimization, we use an improved particle swarm algorithm to solve the game-theoretic model, ensuring convergence to the Stackelberg equilibrium. The algorithm iteratively updates the strategies of leaders and followers until no participant can improve their utility. This process accounts for the dynamic interactions in the harmonic market, with solar inverters adjusting their outputs based on real-time pricing signals.
The results indicate that the inclusion of a harmonic management aggregator not only standardizes prices but also incentivizes higher participation from solar users. The increased harmonic power supply from solar inverters leads to better THD control, with violations reduced to within acceptable limits. Economically, all parties benefit: solar users gain additional revenue, the aggregator profits from the price spread, and harmonic source users achieve cost-effective harmonic mitigation compared to APF installation.
In conclusion, our two-layer master-slave game model provides a robust framework for harmonic trading in distribution networks with high solar inverter penetration. By introducing a harmonic management aggregator, we address the challenges of uncoordinated pricing and enhance the economic viability of using solar inverters for harmonic compensation. Future work will focus on long-term trading mechanisms, such as quarterly or annual contracts, to ensure sustainability. Additionally, integrating advanced communication technologies and machine learning for predictive pricing could further optimize the market operations. The versatility of solar inverters in providing grid services underscores their potential beyond mere energy conversion, paving the way for smarter and more resilient distribution systems.