1. Introduction

2. Typical Control and Disturbance Analyses of Grid-Forming Inverters

2.2. Disturbance Analysis after Considering Line Impedance Parameter Perturbation

In order to ensure a suitable voltage-following capability for output voltage control, the bandwidth of the voltage-current dual-loop closure is set to be much larger than that of the external power control loop. As a result, the VSG-controlled GFM can be simplified as a controllable voltage source [27]. After introducing the virtual impedance control, the equivalent circuit diagram of its access to the AC grid is shown in Figure 2. Where E is the virtual internal potential amplitude of the VSG; δ is the phase difference between the virtual internal potential of the VSG and the grid voltage, which is the power angle; U_o and δ_o are the voltage amplitude at the output of the GFM and the phase difference between the GFM output and the grid voltage, respectively, after virtual impedance compensation is performed; $\tilde{S}$, P and Q are the actual complex power, active power and reactive power output from the virtual synchronous machine, respectively.

Conventional phase quantities are built on the assumption that the power electronic system is a quasi-steady state, which presupposes that the rate of change of the amplitude and phase of the system voltages, currents, and other physical quantities is much less than the system's base frequency angular velocity ω₀. In a quasi-steady state, the amplitude within the frequency bandwidth (-ω₀, ω₀) is considered to be constant. Therefore, the electrical quantities can be approximated as a sinusoidal waveform, which can be expressed in terms of the phase quantity method. However, in the dynamic phase volume concept, the phase volume change rate is close to the angular velocity of the fundamental frequency, although it is still less than that. At the same time, its frequency bandwidth is also close to, but less than, the fundamental frequency, and within this bandwidth, the amplitude is no longer constant but varies over a range. Thus, the dynamic phase can more accurately describe the system's behavior during unsteady or transient processes.

For inductive elements, the dynamic phase model [20] is:

$${\dot{U}}_{\mathrm{L}}=L\frac{d{\dot{I}}_{\mathrm{L}}}{dt}+j{\omega }_{0}L{\dot{I}}_{\mathrm{L}}$$

(4)

where ${\dot{U}}_{\mathrm{L}}$ is denoted as the dynamic phase form of the variable u_L , which is the inductive voltage; ${\dot{I}}_{\mathrm{L}}$ is denoted as the dynamic phase form of the variable i_L , which is the inductive current.

When building the quasi-static model, it will be ignored that the differential terms of the dynamic phasors on the right side of the equal sign with respect to time are often neglected.

According to Figure 2 combined with Equation (4), the relationship between the GFM output voltage and output current I can be obtained as:

$$\left\{\begin{array}{c}\dot{E}-{\dot{U}}_{\mathrm{o}}=(R_{\mathsf{\nu }}+\mathrm{j}{X}_{\mathsf{\nu }})\dot{I}\\ {\dot{U}}_{\mathrm{o}}-{\dot{U}}_{\mathrm{g}}=(R_{\mathrm{g}}+\mathrm{j}{X}_{\mathrm{g}})\dot{I}+L_{\mathrm{g}}\frac{d\dot{I}}{dt}\end{array}\right.$$

(5)

Since the virtual impedance link is designed for steady-state synchronous impedance control, the differential term is not considered in the virtual impedance modeling.

Laplace transformation of Equation (5) and collation gives the current flowing into the grid as:

$$\begin{array}{c}\dot{I}=\frac{E\angle \delta -U_{\mathrm{o}}\angle {\delta }_{\mathrm{o}}}{R_{\mathrm{v}}+\mathrm{j}{X}_{\mathrm{v}}}=\frac{U_{\mathrm{o}}\angle {\delta }_{\mathrm{o}}-U_{\mathrm{g}}\angle 0}{R_{\mathrm{g}}+j{X}_{\mathrm{g}}+s{L}_{\mathrm{g}}}=\frac{E\angle \delta -U_{\mathrm{g}}\angle 0}{R+jX+s{L}_{\mathrm{g}}}\end{array}$$

(6)

where R=R_v+R_g and X=X_v+X_g. Then, the inverter output complex power can be found as:

$$\begin{array}{l}\begin{array}{ll}\tilde{S}& =3{\dot{U}}_{\mathrm{o}}\cdot {\dot{I}}^{*}\\ & =P+jQ\\ & =3\left[E\angle \delta -\dot{I}\left(R_{\mathrm{v}}+j\omega L_{\mathrm{v}}\right)\right]\cdot {\dot{I}}^{*}\\ & =3\left[E\angle \delta {\dot{I}}^{*}-\dot{I}{\dot{I}}^{*}\left(R_{\mathrm{v}}+j\omega L_{\mathrm{v}}\right)\right]\end{array}\end{array}$$

(7)

The actual output active power P and reactive power Q of the GFM considering the dynamic process can be obtained as:

$$\begin{array}{c}P=\frac{3\left[R_{\mathrm{v}}(E{U}_{\mathrm{g}}\cos \delta -{U_{\mathrm{g}}}^2)+(R_{\mathrm{g}}+s{L}_{\mathrm{g}})(E^2-E{U}_{\mathrm{g}}\cos \delta )+XE{U}_{\mathrm{g}}\sin \delta \right]}{{(R+s{L}_{\mathrm{g}})}^2+X^2}\\ Q=\frac{3\left[X_{\mathrm{v}}(E{U}_{\mathrm{g}}\cos \delta -{U_{\mathrm{g}}}^2)+X_{\mathrm{g}}(E^2-E{U}_{\mathrm{g}}\cos \delta )-(R+s{L}_{\mathrm{g}})E{U}_{\mathrm{g}}\sin \delta \right]}{{(R+s{L}_{\mathrm{g}})}^2+X^2}\end{array}$$

(8)

The output power expression of the GFM at (δ₀, E₀), linearized and small-signal modeled, can be obtained by taking the partial derivatives for P and Q, respectively:

$$\left[\begin{array}{c}\Delta P\\ \Delta Q\end{array}\right]={\left.\left[\begin{array}{cc}\frac{\partial P}{\partial \delta }& \frac{\partial P}{\partial E}\\ \frac{\partial Q}{\partial \delta }& \frac{\partial Q}{\partial E}\end{array}\right]\right|}_{({\delta }_0,E_0)}\left[\begin{array}{c}\Delta \delta \\ \Delta E\end{array}\right]=G_{\mathrm{v}}\left[\begin{array}{c}\Delta \delta \\ \Delta E\end{array}\right]$$

(9)

Where

$$\begin{array}{ll}{G}_{\mathrm{v}}\left(s\right)& =\left[\begin{array}{cc}{G}_{\left(\mathrm{v}\right)\mathrm{P}-\mathsf{\delta }}(s)& G_{\left(\mathrm{v}\right)\mathrm{P}-\mathrm{E}}(s)\\ G_{\left(\mathrm{v}\right)\mathrm{Q}-\mathsf{\delta }}(s)& G_{\left(\mathrm{v}\right)\mathrm{Q}-\mathrm{E}}(s)\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{k_{11}s+k_{12}}{g_1{s}^2+g_{2}s+g_3}& \frac{k_{21}s+k_{22}}{g_1{s}^2+g_{2}s+g_3}\\ \frac{k_{31}s+k_{32}}{g_1{s}^2+g_{2}s+g_3}& \frac{k_{41}s+k_{42}}{g_1{s}^2+g_{2}s+g_3}\end{array}\right]\end{array}$$

(10)

where the individual numerator denominator coefficients of the transfer function are:

(11)

From Equation (11), the power dynamic small signal model of GFM is shown in Figure 3.

Rewriting the small-signal model in the time-domain progression can be obtained:

$$\left\{\begin{array}{c}\Delta \ddot{P}+\frac{g_2}{g_1}\Delta \dot{P}+\frac{g_3}{g_1}\Delta P=\frac{k_{12}}{g_1}\Delta \delta +\frac{k_{11}}{g_1}\Delta \dot{\delta }+\frac{k_{21}}{g_1}\Delta \dot{E}+\frac{k_{22}}{g_1}\Delta E\\ \Delta \ddot{Q}+\frac{g_2}{g_1}\Delta \dot{Q}+\frac{g_3}{g_1}\Delta Q=\frac{k_{42}}{g_1}\Delta E+\frac{k_{41}}{g_1}\Delta \dot{E}+\frac{k_{31}}{g_1}\Delta \dot{\delta }+\frac{k_{32}}{g_1}\Delta \delta \end{array}\right.$$

(12)

In actual working conditions, there is a parameter perturbation in the line impedance. Not only is there an accuracy error between the actual and nominal parameters of the components, but also, if the grid is affected by factors such as environment, electromagnetic interference, mechanical stress changes, etc., the line impedance parameters also deviate from the nominal value, and the effect of power decoupling will become worse. Therefore, the system can be separated according to Equation (13) as: (13)

(13)

Where the coefficients with the label n indicate the use of nominal line impedance parameters, and the coefficients without the label n indicate the use of actual line impedance parameters. As the control structure between active and reactive power is similar, the active power ring control ring is analyzed as an example. Let:

$$\left\{\begin{array}{l}{f}_{\mathrm{p}}=\left(\frac{g_2^n}{g_1^n}-\frac{g_2}{g_1}\right)\Delta \dot{P}+\left(\frac{g_3^n}{g_1^n}-\frac{g_3}{g_1}\right)\Delta P+\left(\frac{k_{12}^n}{g_1^n}-\frac{k_{12}}{g_1}\right)\Delta \delta +\frac{k_{11}}{g_1}\Delta \dot{\delta }+\frac{k_{21}}{g_1}\Delta \dot{E}+\frac{k_{22}}{g_1}\Delta E\\ f=-\frac{g_2^n}{g_1^n}\Delta \dot{P}-\frac{g_3^n}{g_1^n}\Delta P+f_{\mathrm{p}}\end{array}\right.$$

(14)

where f is the total disturbance and f_p is the disturbance without model information. The total disturbance f has several components, including: $\frac{k_{22}}{g_1}\Delta E$ and $\frac{k_{21}}{g_1}\Delta \dot{E}$ are the disturbances coupled between active and reactive power after being compensated by virtual impedance; $\left(\frac{g_2^n}{g_1^n}-\frac{g_2}{g_1}\right)\Delta \dot{P}$, $\left(\frac{g_3^n}{g_1^n}-\frac{g_3}{g_1}\right)\Delta P$ and $\left(\frac{k_{12}^n}{g_1^n}-\frac{k_{12}}{g_1}\right)\Delta \delta$ are the disturbances due to model deviation; $\frac{g_2^n}{g_1^n}\Delta \dot{P}$ and $\frac{g_3^n}{g_1^n}\Delta P$ are the model information of the system, which is a part of the model different from the integral series-type standard type, and are estimated together in the estimation of the total disturbances, but when power decoupling is performed, in order not to change the model of the original system, only the disturbances are compensated when compensates the disturbance f_p that does not contain the model information.

3. Decoupling Control Strategy of Grid-Forming Inverter Based on Reduced-Order ESO

4. Hardware-in-the-Loop Experimental Verification

5. Conclusions

References

1. Zhao, G.; Yang, H. Parallel Control of Converters with Energy Storage Equipment in a Microgrid. Electronics 2019, 8, 1110. [Google Scholar] [CrossRef]
2. Matevosyan, J.; Badrzadeh, B.; Prevost, T.; Quitmann, E.; Ramasubramanian, D.; Urdal, H.; Achilles, S.; MacDowell, J.; Hsien Huang, S.; Vital, V.; et al. Grid-Forming Inverters: Are They the Key for High Renewable Penetration? IEEE Power and Energy Magazine 2023, 21, 77–86. [Google Scholar] [CrossRef]
3. Chen, Y.; Luo, J.; Zhang, X.; Li, X.; Wang, W.; Ding, S. A Virtual Synchronous Generator Secondary Frequency Modulation Control Method Based on Active Disturbance Rejection Controller. Electronics 2023, 12, 4587. [Google Scholar] [CrossRef]
4. Wang, L.; Zhou, H.; Hu, X.; Hou, X.; Su, C.; Sun, K. Adaptive Inertia and Damping Coordination (AIDC) Control for Grid-Forming VSG to Improve Transient Stability. Electronics 2023, 12, 2060. [Google Scholar] [CrossRef]
5. Rosso, R.; Wang, X.; Liserre, M.; Lu, X.; Engelken, S. Grid-Forming Converters: Control Approaches, Grid-Synchronization, and Future Trends—A Review. IEEE Open Journal of Industry Applications 2021, 2, 93–109. [Google Scholar] [CrossRef]
6. Arghir, C.; Jouini, T.; Dörfler, F. Grid-Forming Control for Power Converters Based on Matching of Synchronous Machines. Automatica 2018, 95, 273–282. [Google Scholar] [CrossRef]
7. Garzon, O.D.; Nassif, A.B.; Rahmatian, M. Grid Forming Technologies to Improve Rate of Change in Frequency and Frequency Nadir: Analysis-Based Replicated Load Shedding Events. Electronics 2024, 13, 1120. [Google Scholar] [CrossRef]
8. Johnson, B.B.; Sinha, M.; Ainsworth, N.G.; Dörfler, F.; Dhople, S.V. Synthesizing Virtual Oscillators to Control Islanded Inverters. IEEE Transactions on Power Electronics 2016, 31, 6002–6015. [Google Scholar] [CrossRef]
9. Unruh, P.; Nuschke, M.; Strauß, P.; Welck, F. Overview on Grid-Forming Inverter Control Methods. Energies 2020, 13, 2589. [Google Scholar] [CrossRef]
10. Hu, P.; Jiang, K.; Ji, X.; Cai, Y.; Wang, B.; Liu, D.; Cao, K.; Wang, W. A Novel Grid-Forming Strategy for Self-Synchronous PMSG under Nearly 100% Renewable Electricity. Energies 2023, 16, 6648. [Google Scholar] [CrossRef]
11. Sun, Z.; Zhu, F.; Cao, X. Study on a Frequency Fluctuation Attenuation Method for the Parallel Multi-VSG System. Front. Energy Res. 2021, 9. [Google Scholar] [CrossRef]
12. Shang, L.; Hu, J.; Yuan, X.; Chi, Y.; Tang, H. Modeling and Improved Control of Virtual Synchronous Generators Under Symmetrical Faults of Grid. Zhongguo Dianji Gongcheng Xuebao/Proceedings of the Chinese Society of Electrical Engineering 2017, 37, 403–411. [Google Scholar] [CrossRef]
13. Jiang, K.; Liu, D.; Cao, K.; Xiong, P.; Ji, X. Small-Signal Modeling and Configuration Analysis of Grid-Forming Converter under 100% Renewable Electricity Systems. Electronics 2023, 12, 4078. [Google Scholar] [CrossRef]
14. Song, Z.; Zhang, J.; Tang, F.; Wu, M.; Lv, Z.; Sun, L.; Zhao, T. Small Signal Modeling and Parameter Design of Virtual Synchronous Generator to Weak Grid. In Proceedings of the 2018 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), May 2018; pp. 2618–2624. [Google Scholar]
15. Wu, H.; Ruan, X.; Yang, D.; Chen, X.; Zhao, W.; Lv, Z.; Zhong, Q.-C. Small-Signal Modeling and Parameters Design for Virtual Synchronous Generators. IEEE Transactions on Industrial Electronics 2016, 63, 4292–4303. [Google Scholar] [CrossRef]
16. Sarojini, R.K.; Palanisamy, K. Small Signal Modelling and Determination of Critical Value of Inertia for Virtual Synchronous Generator. In Proceedings of the 2019 Innovations in Power and Advanced Computing Technologies (i-PACT), March 2019; Vol. 1, pp. 1–6. [Google Scholar]
17. Yu, J.; Qi, Y.; Deng, H.; Liu, X.; Tang, Y. Evaluating Small-Signal Synchronization Stability of Grid-Forming Converter: A Geometrical Approach. IEEE Transactions on Industrial Electronics 2022, 69, 9087–9098. [Google Scholar] [CrossRef]
18. Zhang, L.; Harnefors, L.; Nee, H.-P. Power-Synchronization Control of Grid-Connected Voltage-Source Converters. IEEE Transactions on Power Systems 2010, 25, 809–820. [Google Scholar] [CrossRef]
19. D’Arco, S.; Suul, J.A.; Fosso, O.B. Small-Signal Modelling and Parametric Sensitivity of a Virtual Synchronous Machine. In Proceedings of the 2014 Power Systems Computation Conference, August 2014; pp. 1–9. [Google Scholar]
20. Stankovic, A.M.; Sanders, S.R.; Aydin, T. Dynamic Phasors in Modeling and Analysis of Unbalanced Polyphase AC Machines. IEEE Transactions on Energy Conversion 2002, 17, 107–113. [Google Scholar] [CrossRef]
21. Wu, T.; Liu, Z.; Liu, J.; Wang, S.; You, Z. A Unified Virtual Power Decoupling Method for Droop-Controlled Parallel Inverters in Microgrids. IEEE Transactions on Power Electronics 2016, 31, 5587–5603. [Google Scholar] [CrossRef]
22. De Brabandere, K.; Bolsens, B.; Van den Keybus, J.; Woyte, A.; Driesen, J.; Belmans, R. A Voltage and Frequency Droop Control Method for Parallel Inverters. IEEE Transactions on Power Electronics 2007, 22, 1107–1115. [Google Scholar] [CrossRef]
23. Li, Y.; Li, Y.W. Power Management of Inverter Interfaced Autonomous Microgrid Based on Virtual Frequency-Voltage Frame. IEEE Transactions on Smart Grid 2011, 2, 30–40. [Google Scholar] [CrossRef]
24. Peng, Z.; Wang, J.; Bi, D.; Wen, Y.; Dai, Y.; Yin, X.; Shen, Z.J. Droop Control Strategy Incorporating Coupling Compensation and Virtual Impedance for Microgrid Application. IEEE Transactions on Energy Conversion 2019, 34, 277–291. [Google Scholar] [CrossRef]
25. Peng, Z.; Wang, J.; Wen, Y.; Bi, D.; Dai, Y.; Ning, Y. Virtual Synchronous Generator Control Strategy Incorporating Improved Governor Control and Coupling Compensation for AC Microgrid. IET Power Electronics 2019, 12. [Google Scholar] [CrossRef]
26. Li, C.; Yang, Y.; Mijatovic, N.; Dragicevic, T. Frequency Stability Assessment of Grid-Forming VSG in Framework of MPME With Feedforward Decoupling Control Strategy. IEEE Transactions on Industrial Electronics 2022, 69, 6903–6913. [Google Scholar] [CrossRef]
27. Rathnayake, D.B.; Bahrani, B. Multivariable Control Design for Grid-Forming Inverters With Decoupled Active and Reactive Power Loops. IEEE Transactions on Power Electronics 2023, 38, 1635–1649. [Google Scholar] [CrossRef]
28. Guerrero, J.M.; Vicuna, L.G. de; Matas, J.; Castilla, M.; Miret, J. Output Impedance Design of Parallel-Connected UPS Inverters with Wireless Load-Sharing Control. IEEE Transactions on Industrial Electronics 2005, 52, 1126–1135. [Google Scholar] [CrossRef]
29. Rodríguez-Cabero, A.; Roldán-Pérez, J.; Prodanovic, M. Virtual Impedance Design Considerations for Virtual Synchronous Machines in Weak Grids. IEEE Journal of Emerging and Selected Topics in Power Electronics 2020, 8, 1477–1489. [Google Scholar] [CrossRef]
30. Wen, T.; Zhu, D.; Zou, X.; Jiang, B.; Peng, L.; Kang, Y. Power Coupling Mechanism Analysis and Improved Decoupling Control for Virtual Synchronous Generator. IEEE Transactions on Power Electronics 2021, 36, 3028–3041. [Google Scholar] [CrossRef]
31. He, J.; Li, Y.W. Analysis, Design, and Implementation of Virtual Impedance for Power Electronics Interfaced Distributed Generation. IEEE Transactions on Industry Applications 2011, 47, 2525–2538. [Google Scholar] [CrossRef]
32. Wang, Y.; Wai, R.-J. Newly-Designed Power Decoupling Strategy for Virtual Synchronous Generator in Micro-Grid. In Proceedings of the 2021 IEEE International Future Energy Electronics Conference (IFEEC), November 2021; pp. 1–6. [Google Scholar]
33. Wang, Y.; Wai, R.-J. Adaptive Fuzzy-Neural-Network Power Decoupling Strategy for Virtual Synchronous Generator in Micro-Grid. IEEE Transactions on Power Electronics 2022, 37, 3878–3891. [Google Scholar] [CrossRef]
34. Dong, N.; Li, M.; Chang, X.; Zhang, W.; Yang, H.; Zhao, R. Robust Power Decoupling Based on Feedforward Decoupling and Extended State Observers for Virtual Synchronous Generator in Weak Grid. IEEE Journal of Emerging and Selected Topics in Power Electronics 2023, 11, 576–587. [Google Scholar] [CrossRef]
35. Han, J. From PID to Active Disturbance Rejection Control. IEEE Transactions on Industrial Electronics 2009, 56, 900–906. [Google Scholar] [CrossRef]
36. Fu, C.; Tan, W. Parameters Tuning of Reduced-Order Active Disturbance Rejection Control. IEEE Access 2020, 8, 72528–72536. [Google Scholar] [CrossRef]
37. Engler, A.; Soultanis, N. Droop Control in LV-Grids. In Proceedings of the 2005 International Conference on Future Power Systems, November 2005; p. 6. [Google Scholar]