Abstract
With the increased penetration of renewable energy sources, the grid-forming (GFM) energy storage (ES) has been considered to engage in primary frequency regulation (PFR), often necessitating the use of a frequency deadband (FDB) to prevent excessive battery charging cycling and mitigate frequency oscillations. Implementing the FDB is relatively straightforward in grid-following (GFL) control. However, implementing the FDB in GFM control presents a significant challenge since the inverter must abstain from providing active power at any frequency within the FDB. Therefore, in this paper, the performance of PFR control in the GFM-ES inverter is analyzed in detail first. Then, the FDB is implemented for GFM inverters with various types of synchronization methods, and the need for inertia response is also considered. Moreover, given the risk of oscillations near the FDB boundary, different FDB setting methods are proposed and examined, where an improved triangular hysteresis method is proposed to realize the fast response and enhanced stability. Finally, the simulation and experiment results are provided to verify the effectiveness of the above methods.
THE environment protection endeavor has been promoting the rapid expansion of sustainable and environment-friendly renewable energy sources (RESs) [
Currently, most ES inverters are integrated into the power grid in the grid-following (GFL) mode. However, in some regions with very high penetration of RESs, severe stability issues may arise in weak grids [
Therefore, the grid-forming (GFM) control is considered in the recent deployment of ES to deeply participate in the voltage/frequency support. The droop control is a well-known method that realizes voltage/frequency support and reasonable power allocation among different inverters [
PFR is one of the auxiliary services to realize fast power balance and frequency maintenance. In the PFR, the output power and frequency must adhere to a predefined curve, typically comprising a droop area, frequency deadband (FDB), and saturation area [
Conventionally, SGs participate in PFR with the frequency governor of the prime mover after the release of rotor kinetic energy [
When the GFM-ES inverter engages in the PFR, its output power is adjusted indirectly through the manipulation of the terminal voltage. Consequently, the current is not directly controlled, posing challenges to precise power control [

Fig. 1 Diagram of PFR with FDB.
Regarding the FDB control in GFM-ES inverters, the pivotal challenge lies in maintaining zero active power output within the specified FDB. This aspect has received limited attention in prior studies. The issue can be recast as the quest for precise and rapid control of the output current of GFM inverter, a topic that has garnered more research efforts. For example, after an overload or a grid fault, the GFM-ES inverters need current limiting control in the transient state [
In this paper, the improved FDB control of the GFM-ES inverter is proposed. Firstly, the PFR performance of the GFM-ES inverter is investigated in detail. Then, the zero power response strategies suitable for GFM-ES inverters with different synchronization methods (i.e., DVSC and PSC methods) within the FDB are proposed. Furthermore, different FDB setting methods are proposed to enhance the PFR capability of the GFM-ES inverter and reduce oscillations near the FDB boundary. Finally, the simulation and experiment results are given for verification. These methods address the critical barriers in the FDB control of the GFM-ES inverter, and the main work and contributions are summarized as follows.
1) The PFR performance of GFL-ES and GFM-ES inverters are investigated and compared to quantify the merits of GFM-ES in PFR, especially under different grid strengths.
2) Two distinct methods for implementing FDB with different inertia responses (IRs) have been proposed to cater to different application scenarios: ① the power-reference-based (PRB) method, which is further categorized into AC- and DC-side strategies, and ② the ES output point (EOP) voltage-frequency-based (EVFB) method. The two methods can be applied to GFM-ES inverters with different topologies, synchronization methods, and IR demands, where all of them can realize almost zero power response in the steady state within the FDB.
3) To enhance the PFR performance of the GFM-ES inverter, a step FDB with triangular hysteresis method is presented, which can accelerate the response speed in PFR and suppress the oscillation near the FDB boundary.
The control structure of a three-phase grid-connected ES inverter is illustrated in

Fig. 2 Control structure of a three-phase grid-connected ES inverter. (a) GFL-ES inverter. (b) GFM-ES inverter.
The control structure of the GFL-ES inverter is shown in
Currently, PFR primarily relies on the central controller applied as a fast frequency control manager, which involves command transmission from the station to the individual inverter, and this command transmission introduces the time delay . Neglecting the higher-order terms of the time delay, it can be approximated as a first-order inertial element , as shown in
The power response of GFL-ES inverter to frequency changes is given by:
(1) |
where kD,ES and are the droop coefficient and virtual inertia, respectively; is the angular frequency of ES inverter; pES is the output power of ES inverter; and the symbol represents the increment of the corresponding variable.
For the GFM-ES inverter, the frequency control is fundamentally governed by the PSC loop, as illustrated in
(2) |
where is the virtual inertia; and is the reference angular frequency.
The difference between (1) and (2) is that the GFL-ES inverter has an uncertain delay, and the relationship of and can be written as (3) if ignoring the delay of low-pass filter.
(3) |
The time delay impacts on the GFL-ES inverter connected to an SG with limited capacity are analyzed, which means the SG itself has PFR capability.

Fig. 3 Dynamic PFR of GFL-ES inverter with different Td1 and GFM-ES inverter.
The impact of the time constant is also investigated, as shown in

Fig. 4 Dynamic PFR of GFL-ES inverter with different Td2 and GFM-ES inverter.
After choosing suitable Td1 and Td2, the dynamic PFR in weak grids is examined, where the SCR serves as an indicator of grid strength, and a decreasing SCR means a weakening grid. As depicted in

Fig. 5 Dynamic PFR of GFM-ES and GFL-ES inverters with different SCRs. (a) GFM-ES inverter. (b) GFL-ES inverter.
In the GFM-ES inverter, the active power is determined by the voltage phase difference between the inverter terminal and EOP. This feature will cause continuously active power change within the FDB, potentially leading to a diminished ES service life and exacerbating frequency oscillations. Therefore, this section discusses how to implement FDB for different types of GFM-ES inverters.
As shown in

Fig. 6 Typical topology of grid-connected ES inverter.
The active power response of SG can be derived with the model shown in

Fig. 7 Typical model of grid-connected ES inverter.
The relationship between the angular frequencies of EOP and SG is written as [
(4) |
where , , , , , and are the droop coefficient, power coefficient, rotor inertia of SG, voltage of SG, voltage of EOP, and line impedance on SG side, respectively.
The control structure of DVSC method is shown in
(5) |

Fig. 8 Control structure of DVSC method.
where , , , , and are the power coefficient of ES, voltage of ES inverter terminal, line impedance on the ES side, capacitance of DC capacitor, and DC rated voltage, respectively; and the temporary variables and can be obtained as (6) by comparing (4) and (5) to imitate the characteristic of SG.
(6) |
According to (6), the physical meaning of the DVSC method is clear, and it can be used for guiding the design of virtual inertia and droop coefficient of ES .
For the GFM-ES inverter with DVSC method, the FDB is implemented by modifying the DC-side power reference , and then the DC/DC control loop is used to track with a inner current loop. As shown in

Fig. 9 Switch signal module of triangular hysteresis.
The FDB size is determined based on the current field application status of PFR and the requirements for the participation of RES in PFR. The triangular hysteresis size is determined by considering the need to suppress frequency oscillations and reduce the number of ES charging and discharging cycles.
Within the FDB, the post-stage power increment is equal to zero as the switch signal is 0. Therefore, setting the post-stage power reference , the DC-side power reference will equal zero, and the zero power response for PFR will be realized within the FDB. It should be noted that with the DC-side PRB method, the ES inverter can also provide IR through the DC capacitor instead of the ES within the FDB as the is related to the inertia , which also helps reduce unnecessary cycling of ES.
The control structure of the PSC method is illustrated in

Fig. 10 Control structure of PSC method.
Based on the quasi-static model, the dynamic relationship between the active power response and grid frequency disturbance can be expressed as:
(7) |
where is the difference between the voltage phases of ES inverter terminal and EOP .
In the time domain, the voltage phases of the ES inverter terminal and EOP can be expressed as (8). By combining (7) and (8), the relationship between and reference angle for active power control in the time domain can be formulated as (9).
(8) |
(9) |
From (9), different will influence the output active power in steady states, which can be illustrated by (10). Therefore, setting the droop coefficient of ES to zero or to will enable the implementation of FDB.
(10) |
As shown in

Fig. 11 AC-side power reference modifications. (a) Droop control. (b) VSG control.
In the VSG control, the switch signal module is incorporated into the feedback loop of the P-f droop control, while the IR remains active. Then, the power increment changes to zero when the frequency deviation is within the FDB. Herein, the IR unit inherently functions as an integral control loop to track the power reference of ES that is set to be zero in this state. This implies that the VSG control realizes zero power output within the FDB in steady state while IR is still active. It should be noted that AC-side PRB method cannot be applied to the droop control shown in
In the EVFB method, the frequency reference changes to within the FDB, using a PLL for frequency measurement.
(11) |

Fig. 12 EVFB method for FDB. (a) Switch signal selection module. (b) Quasi-static model for active power response.
The measured frequency is captured from , and then (10) is modified as:
(12) |
In the steady state, the measured frequency equals . Consequently, the actual output power is equal to the power reference . Therefore, the GFM-ES inverter with the EVFB method does not participate in the PFR within the FDB after setting .
Notably, the EVFB method can be applied to both VSG and droop control, and it also can be utilized in the DVSC method as all the GFM control methods have a similar relationship shown in (10). Compared with the PRB method, the key difference is that the EVFB method can reduce the IR within the FDB, as the terminal frequency is directly fed back into the control loop, and the IR is only provided in the transient state for the control delay.
In summary, two types of methods for the implementation of FDB have been proposed to cater to different application scenarios: ① the PRB method, which is further categorized into AC- and DC-side PRB methods, and ② the EVFB method. The PRB method activates the IR while the EVFB method does not activate IR within the FDB to satisfy different IR requirements. The following conclusions can be drawn.
1) The DC-side PRB method is applicable in two-stage topologies utilizing the DVSC method, whereas the AC-side PRB method is utilized for VSG control.
2) The EVFB method for FDB is universally applicable to GFM-ES inverters.
This section discusses the configuration of various types of FDB mechanisms and proposes an improved triangular hysteresis based FDB setting method, particularly aiming at enhancing the system dynamic response and eliminating oscillations near the FDB boundary.

Fig. 13 FDB setting methods. (a) NBD. (b) SDB. (c) RHSDB. (d) THSDB.
The RHSDB method is expressed as:
(13) |
where , , and are the boundaries of the hysteresis, FDB, and saturation areas, respectively; Pdb and Pmax are the output power values in the hysteresis and saturation areas, respectively; and are the frequency deviation and related power increment, respectively; and sgn is the signum function.
The rectangular hysteresis method needs a larger hysteresis band to prevent oscillation more effectively. However, this will deteriorate the performance both in droop and deadband states. To address the aforementioned issues, an enhanced triangular hysteresis method in step deadband, which is termed THSDB, is proposed, as shown in
(14) |
In summary, the EVFB method with triangular hysteresis is suitable for both DVSC and PSC methods. The PRB method with triangular hysteresis can be applied in the DVSC and VSG methods.
The triangular hysteresis size should not be excessively large. Currently, the mechanical FDB, the minimum FDB of RESs, and the frequency deviation experienced under disturbances are approximately 0.02 Hz [
The PFR deadbands for hydropower units and thermal power units are set to be Hz and ±0.033 Hz, respectively. To decrease the PFR times provided by conventional SGs, the FDB of the ES inverter needs to be smaller. Therefore, once is determined, the droop coefficient in the improved triangular hysteresis area can be derived as:
(15) |
To verify the effectiveness of the FDB control methods for the GFM-ES inverter, the system depicted in
Parameter | Value |
---|---|
kD,SG | 20 kW/Hz |
Damping factor of SG DSG |
0.12 kW/H |
JSG |
3 kg· |
FDB of SG fdb,SG | ±0.05 Hz |
kD,ES | 20 kW/Hz |
Cdc |
8×1 |
Udc | 750 V |
fdb,ES | ±0.03 Hz |
fhdb,ES | ±0.02 Hz |

Fig. 14 Simulation results of DVSC method. (a) Active power of ES inverter. (b) DC capacitor voltage. (c) Grid frequency.
As shown in
The simulation results for PSC method are presented as follows with a load disturbance of 0.2 kW at 10 s.
The simulation results of droop control are depicted in

Fig. 15 Simulation results of droop control. (a) Active power of ES inverter. (b) Grid frequency.
The simulation results for VSG are shown in

Fig. 16 Simulation results of VSG control. (a) Active power of ES inverter. (b) Grid frequency.
This subsection simulates different FDB setting methods when the ES inverter suffers a sudden load increase that causes the frequency deviation exceeding the FDB. As shown in

Fig. 17 Simulation results of different FDB setting methods. (a) Active power of ES inverter. (b) Grid frequency. (c) Frequency of ES inverter. (d) Power reference of ES inverter.
From
The THSDB method shown in
In summary, ① the EVFB method for zero active power response within the FDB is universally applicable for the GFM-ES inverter and operates with a smaller IR; ② the PRB methods with a larger IR are beneficial for the two-stage ES inverter with the DVSC and VSG methods, but the DC-side PRB method needs to consider the deviation of DC voltage; and ③ the THSDB method for FDB setting can enhance the PFR performance of ES inverter and prevent oscillation in more situations.
To further verify the effectiveness of the FDB control methods for the GFM-ES inverter, an islanded microgrid comprised of two single-stage GFM-ES inverters is built as shown in

Fig. 18 Experiment platform. (a) Overview of entire platform. (b) Detailed platform structure.
Symbol | Definition | Value |
---|---|---|
Ug | Rated grid voltage | 50 V |
Udc | Rated DC voltage | 150 V |
PVSG1,ref | Power reference of VSG1 | 750 W |
PVSG2,ref | Power reference of VSG2 | 0 W |
kD,VSG1 | Droop coefficient of VSG1 | 25 kW/Hz |
kD,VSG2 | Droop coefficient of VSG2 | 25 kW/Hz |
JVSG1 | Inertia of VSG1 |
1 kg· |
LVSG1 | Line impedance of VSG1 | 1.15 mH |
LVSG2 | Line impedance of VSG2 | 0.45 mH |
fdb,VSG2 | FDB of VSG2 | ±0.03 Hz |
fhdb,VSG2 | Boundary of hysteresis for VSG2 | ±0.02 Hz |
Rc | Constant load | 5 Ω |

Fig. 19 Experiment results of different implementation methods for FDB. (a) WDB. (b) PRB. (c) EFVB.
From
From
This subsection conducts experiments with various FDB setting methods in response to a sudden load increase of 0.99 kW, which causes the frequency deviation to exceed the FDB. Follows this, a load decrease of 0.74 kW is set after the system reaches the new steady state to confirm that the system can return to the FDB.
From

Fig. 20 Experiment results of different FDB setting methods. (a) WDB. (b) NBD. (c) SDB. (d) RHSDB. (e) THSDB. (e) Power reference of VSG2 with SDB. (f) Power reference of VSG2 with RHSDB.
The experimental results validate the effectiveness of the PRB and EVFB methods for the implementation of FDB. Additionally, the THSDB method for enhancing the PFR capability and suppressing oscillation in control loops is also verified.
In this paper, the methods for minimizing the participation of GFM-ES inverter in PFR within the FDB are presented. In addition, the THSDB method for the FDB setting is proposed to enhance PFR capability and suppress oscillations in the FDB border. The conclusions are drawn as follows.
1) The PRB and EVFB methods are proposed for the implementation of FDB with different IR in the GFM-ES inverter. The PRB method is applicable in both DVSC and VSG methods while maintaining the IR performance. The EVFB method is universally suitable for the GFM-ES inverter with reduced IR performance. These methods can reduce ES charging cycling, thereby extending the battery’s lifespan.
2) To enhance the PFR performance of ES inverter, the SDB and RHSDB methods are presented, while these methods may cause oscillations around the FDB boundary. Furthermore, the THSDB method is proposed to address oscillation, and it not only improves the PFR performance but also facilitates smoother transitions between different operating states, and reduces the risk of frequency oscillation in the FDB.
References
Z. Zhang, S. Liao, Y. Sun et al., “A parallel-type load damping factor controller for frequency regulation in power systems with high penetration of renewable energy sources,” Journal of Modern Power Systems and Clean Energy, vol. 12, no. 4, pp. 1019-1030, Jul. 2024. [Baidu Scholar]
Y. Xie, H. Yi, F. Zhuo et al., “Analysis and stabilization for full harmonic compensation oscillation in SAPF system with source current direct control,” IET Power Electronics, vol. 17, no. 1, pp. 107-120, Jan. 2024. [Baidu Scholar]
Z. Wang, H. Yi, Y. Jiang et al., “Voltage control and power-shortage mode switch of PV inverter in the islanded microgrid without other energy sources,” IEEE Transactions on Energy Conversion, vol. 37, no. 4, pp. 2826-2836, Dec. 2022. [Baidu Scholar]
J. Li, Y. Qiao, Z. Lu et al., “Integrated frequency-constrained scheduling considering coordination of frequency regulation capabilities from multi-source converters,” Journal of Modern Power Systems and Clean Energy, vol. 12, no. 1, pp. 261-274, Jan. 2024. [Baidu Scholar]
O. D. Adeuyi, M. Cheah-Mane, J. Liang et al., “Fast frequency response from offshore multiterminal VSC-HVDC schemes,” IEEE Transactions on Power Delivery, vol. 32, no. 6, pp. 2442-2452, Dec. 2017. [Baidu Scholar]
D. Sun, H. Liu, F. Zhao et al., “Comparison of inverter generators with different support control methods,” Power System Technology, vol. 44, no. 11, pp. 4359-4369, Aug. 2020. [Baidu Scholar]
X. Fu, J. Sun, M. Huang et al., “Large-signal stability of grid-forming and grid-following controls in voltage source converter: a comparative study,” IEEE Transactions on Power Electronics, vol. 36, no. 7, pp. 7832-7840, Jul. 2021. [Baidu Scholar]
I. Batarseh, K. Siri, and H. Lee, “Investigation of the output droop characteristics of parallel-connnected DC-DC converters,” in Proceedings of 1994 Power Electronics Specialist Conference, Taipei, China, Jun. 1994, pp. 1342-1351. [Baidu Scholar]
J. Driesen and K. Visscher, “Virtual synchronous generators,” in Proceedings of 2008 IEEE PES General Meeting – Conversion and Delivery of Electrical Energy in the 21st Century, Pittsburgh, USA, Jul. 2008, pp. 1-3. [Baidu Scholar]
L. Harnefors, J. Kukkola, M. Routimo et al., “A universal controller for grid-connected voltage-source converters,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 5, pp. 5761-5770, Oct. 2021. [Baidu Scholar]
C. Luo, X. Ma, T. Liu et al., “Adaptive-output-voltage-regulation-based solution for the DC-link undervoltage of grid-forming inverters,” IEEE Transactions on Power Electronics, vol. 38, no. 10, pp. 12559-12569, Oct. 2023. [Baidu Scholar]
J. Fang, H. Li, Y. Tang et al., “Distributed power system virtual inertia implemented by grid-connected power converters,” IEEE Transactions on Power Electronics, vol. 33, no. 10, pp. 8488-8499, Oct. 2018. [Baidu Scholar]
W. Zhang, Z. Wang, H. Yi et al., “Primary frequency regulation with improved frequency deadband control of grid-forming storage inverter for longer lifespan,” in Proceedings of 2023 IEEE 2nd International Power Electronics and Application Symposium, Guangzhou, China, Nov. 2023, pp. 1021-1026. [Baidu Scholar]
N. Lu, J. Fang, Y. Tang et al., “A frequency deadband-based virtual inertia control for grid-connected power converters,” in Proceedings of 2019 10th International Conference on Power Electronics and ECCE Asia, Busan, Korea (South), May 2019, pp. 1-6. [Baidu Scholar]
H. Ye, W. Pei, and Z. Qi, “Analytical modeling of inertial and droop responses from a wind farm for short-term frequency regulation in power systems,” IEEE Transactions on Power Systems, vol. 31, no. 5, pp. 3414-3423, Sept. 2016. [Baidu Scholar]
X. Liu, B. Wu, and L. Xiu, “A fast positive-sequence component extraction method with multiple disturbances in unbalanced conditions,” IEEE Transactions on Power Electronics, vol. 37, no. 8, pp. 8820-8824, Aug. 2022. [Baidu Scholar]
X. Liu, L. Xiong, B. Wu et al., “Phase locked-loop with decaying DC transient removal for three-phase grids,” International Journal of Electrical Power & Energy Systems, vol. 143, p. 108508, Dec. 2022. [Baidu Scholar]
Z. Ma, X. Li, Z. Tang et al., “Integrated control of primary frequency regulation considering dead band of energy storage,” Transactions of China Electrotechnical Society, vol. 34, no. 10, pp. 2102-2115, May 2019. [Baidu Scholar]
F. Zhang, H. You, and L. Ding, “Influential mechanism modelling of dead band in primary frequency regulation of renewable energy and its coefficient correction strategy,” Automation of Electric Power Systems, vol. 47, no. 6, pp. 158-167, Mar. 2023. [Baidu Scholar]
L. Harnefors, M. Schweizer, J. Kukkola et al., “Generic PLL-based grid-forming control,” IEEE Transactions on Power Electronics, vol. 37, no. 2, pp. 1201-1204, Feb. 2022. [Baidu Scholar]
T. Li, Y. Li, X. Chen et al., “Research on AC microgrid with current-limiting ability using power-state equation and improved Lyapunov-function method,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 6, pp. 7306-7319, Dec. 2021. [Baidu Scholar]
M. A. Zamani, A. Yazdani, and T. S. Sidhu, “A control strategy for enhanced operation of inverter-based microgrids under transient disturbances and network faults,” IEEE Transactions on Power Delivery, vol. 27, no. 4, pp. 1737-1747, Oct. 2012. [Baidu Scholar]
A. Pal, D. Pal, and B. K. Panigrahi, “A current saturation strategy for enhancing the low voltage ride-through capability of grid-forming inverters,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 70, no. 3, pp. 1009-1013, Mar. 2023. [Baidu Scholar]
F. Salha, F. Colas, and X. Guillaud, “Virtual resistance principle for the overcurrent protection of PWM voltage source inverter,” in Proceedings of 2010 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT Europe), Gothenburg, Sweden, Oct. 2010, pp. 1-6. [Baidu Scholar]
Z. Wang, F. Zhuo, H. Yi et al., “Analysis of dynamic frequency performance among voltage-controlled inverters considering virtual inertia interaction in microgrid,” IEEE Transactions on Industry Applications, vol. 55, no. 4, pp. 4135-4144, Jul. 2019. [Baidu Scholar]
Z. Wang, H. Yi, F. Zhuo et al., “Active power control of voltage-controlled photovoltaic inverter in supporting islanded microgrid without other energy sources,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 10, no. 1, pp. 424-435, Feb. 2022. [Baidu Scholar]