As the integration of renewable energy sources increases, distributed generation systems with inverters as the interface are becoming increasingly important in the conventional grid (Pan et al., 2013; Xie et al., 2021). The massive use of power electronics and the withdrawal of synchronous machines have led to a lack of inertia in the power system, thus making the frequency stability of the system increasingly problematic (Blaabjerg et al.,2006; Li et al.,2018). To address this problem, a concept of GFM inverter generating distributed virtual inertia was proposed (Lasseter et al.,2019; Quan et al.,2019; Liu et al.,2016; Zhong and Weiss, 2010). The GFM inverter using virtual inertia control can effectively increase the inertia of the power system, reduce the frequency deviation and decrease the rate of change of the grid frequency under large disturbances (Fang et al., 2017). Hence, this paper focuses on the virtual inertia of the GFM inverter.

Virtual inertia control simulates the behavior of a real synchronous generator, but because control parameters such as inertia are virtual, this makes the design of its control parameters both flexible and complex. In order to meet the operational requirements of the system, the parameter design of the virtual inertia control was investigated (Dhingra and Singh., 2018). For example, a step-by-step parameters design method based on the line-frequency-averaged small-signal model of GFM inverter is proposed in (Wu et al.,2016). On the basis of the parallel small-signal model, the effect rules of the eigenvalues by droop coefficient, line parameters, GFM inverter’s parameters, and low pass filter parameters are examined and derived in (Zhang et al., 2017). The parameters of the proposed GFM inverter were optimized in (Shintai et al., 2014). However, these criteria strongly influence each other and it is difficult to balance multiple performance indicators. Therefore, attempts were made to improve the traditional virtual inertia control method in order to better control system. A modified virtual inertial strategy is used for the configuration of GFM inverter in (Shi et al., 2018; Xu et al., 2019), which helps the system to be closer to the theoretical analysis conditions of the traditional virtual inertial algorithm.

However, the GFM inverter also faces stability problems under grid disturbances while benefiting from the operation of simulated synchronous generators. In the event of large signal disturbances, such as transmission line faults, critical grid voltage droop and large load swings, the GFM inverter is exposed to stability risks. The transient behaviors of GFM inverter with power-synchronization control were examined in (Wu and Wang., 2018) under various grid fault scenarios. In (Xin et al., 2016; Huang et al., 2017), a fundamental droop-controlled GFM inverter transient instability phenomenon was discovered in the situation of a current saturation caused by the grid voltage sag. The power-angle curve was used to provide a qualitative examination of reactive power management in (Shuai et al.,2018), which shows how it degrades the GFM inverter’s transient stability. Although obvious, it does not precisely identify how and to what extent the transitory behavior is impacted by the reactive power control and other control elements. Due to the enormous complexity inherent in large-signal non-linear dynamic reactions, this is in fact a basic difficulty in the study of transient stability.

In addition to large disturbances, small disturbances are also an important research element in virtual inertia control (Li et al., 2004; Zhang et al., 2009; Zhang et al., 2010). However, existing studies show that fixed control parameters always lead to a contradiction between the overshoot and the setting time of the system (Chong et al., 2015). To solve this problem, it is necessary to adapt the values of the control parameters to the different scenarios and disturbances in real-time. An adaptive control method based on reinforcement learning and adaptive dynamic programming (ADP) is proposed in (Wang et al., 2021), but its stability was not verified. The GFM inverter-based adaptive damping control strategy in (Zheng et al., 2016) limits the frequency oscillations to a reliable range and attenuates the output power oscillations. However, the effect of virtual inertia on the system is not considered. In (Markovic et al., 2018), a linear quadratic regulator-based (LQR) optimization technique is used to adaptively adjust the emulated inertia and damping constants. However, only a pure simulation platform was used to verify the feasibility of the strategy and no hardware platform verification was performed. An adaptive control scheme combining virtual inertia control with an additional damping controller was proposed in the literature (Wang et al., 2022). This literature establishes a mathematical model of a wind power grid-forming system with a damping controller. However, only the response effect of the system output power was focused on and the frequency stability aspect was not investigated. All the literature above utilize damping adaption. In contrast, this paper focuses on adaptive adjustment of inertia.

In the control method of adaptive virtual inertia, scholars extensively adopted frequency differentiation as a standard of measurement. An adaptive virtual inertia control strategy based on an improved bang-bang control strategy for a micro-grid is presented in (Li M et al., 2019). On the one hand, it can make full use of the variability of virtual inertia to reduce dynamic frequency deviation. On the other hand, the steady-state interval of frequency and the steady-state inertia are set to improve the system frequency stability. Alternating inertia is also proposed in (Alipoor et al., 2014; Cheng et al., 2015; Li et al., 2016; Wang et al., 2018; Zhang et al., 2020), where the main idea is to give the moment of inertia by evaluating the state of the frequency deviation Δf and its rate of change df/dt. However, df/dt is very sensitive to high-frequency noise, which is inevitable in real physical systems. Therefore, the practical performance of this type of control is influenced by high-frequency noise. A communication-free adaptive virtual inertia control to realize the power oscillation suppression of cascaded-type virtual synchronous generators (VSGs) is proposed in (Li et al., 2023), where the adaptive inertia link contains dωDi/dt. But the measurement noises caused by direct calculation is avoided by deriving the formula with rounding roots so that dωDi/dt is obtained without derivative action. In (Li J et al., 2019), A dual-adaptivity inertia control strategy was proposed, but the system stability was not verified when the adaptive regulator was introduced. Moreover, the theoretical analysis of frequency response under power disturbance is not investigated. A practical control method without derivative action is proposed by (Hou et al.,2020). The control method proposed in this paper also does not have a derivative action. The proposed algorithm conquers this chattering deficit without frequency derivative action. It gives tremendous promise for engineering application backgrounds with high-frequency noise.

In this paper, a sigmoid-based adaptive inertia control strategy for GFM inverter is proposed to enhance frequency stability. The frequency response characteristics under different disturbances are theoretically analyzed at first. Then, a non-linear inertia regulator, which is based on an improved sigmoid function, is investigated to adjust the inertia by frequency deviation. Hence, the proposed strategy can achieve a better overall dynamic performance than the fixed inertia regulator. Besides, the full-order small-signal model of the system containing the non-linear inertia regulator is established in this paper to evaluate the stability of the proposed strategy. Compared with (Alipoor et al., 2014; Li et al., 2016; Wang et al., 2018; Zhang et al., 2020), the proposed adaptive strategy is implemented without derivative action (df/dt) and thus the performance is insensitive to high-frequency noise. Finally, the effectiveness of the proposed strategy is verified by the hardware-in-the-loop (HIL) results.

The control scheme of GFM inverter is shown in Figure 1. The main control structure includes an inner voltage and current control loop and an outer power control loop. In the voltage and current control loop, PI control is used to track the reference value. Hence, the control formula under the dq reference system can be obtained as Eq. 1. i c _ r e f d = K v p + K v i s u o _ r e f d − u o d − ω C f u o q + i o d i c _ r e f q = K v p + K v i s u o _ r e f q − u o q + ω C f u o d + i o q u d = K i p + K i i s i c _ r e f d − i c d − ω L f i c q + u o d u q = K i p + K i i s i c _ r e f q − i c q + ω L f i c d + u o q (1) where u ^d and u ^q represent the command voltage for modulating the converter, respectively; i _{c_ref} ^d and i _{c_ref} ^q are the control reference current, respectively; u _o ^d , u _o ^q , i _o ^d and i _o ^q are the inverter output voltage and current, respectively; i _c ^d and i _c ^q denote the inverter bridge arm current; ω is the angular frequency provided by the outer power control loop; C _f and L _f indicate filter capacitor and inductor, respectively; K _vp , K _vi , K _ip and K _ii are the PI control parameters, respectively.

In the P-f control loop, the function of GFM inverter to support the frequency stability is realized through virtual inertia, and the control equation is as follows: J ω ˙ = P s e t − P / ω 0 − D p ω − ω 0 (2) where P and P _set represent the active power and its reference, respectively. ω and ω ₀ denote GFM inverter’s angular frequency and the nominal angular frequency, respectively. J and D _p are the virtual inertia and the damping ratio. Since the objective of this paper is to regulate inertia in P-f control loop, the Q-V control loop is neglected in this paper.

Digital control has an inherent control delay problem because sampling and calculation take a certain amount of time to complete, making it difficult to control the system in real-time. According to (Vukosavic et al.,2017; Wang et al.,2015), the various delays of the system are included in a delay link, which is placed before the PWM loop.

Assuming that there is a delay of one sampling period in the modulated wave update with respect to the sampling moment, the transfer function of the delay and the zero-order retainer in the continuous domain is: G d s = e − τ s , G h s = 1 − e − τ s s G d s ≈ τ e − 0.5 τ s (3) where τ > 0 is the system sampling time. e ^−τs is an irrational function, which is not conducive to subsequent analysis. To evaluate the effect of delay on system stability, the second-order Pade approximation is utilized to simplify the analysis process. G s = G d s G h s ≈ 4.5 τ 3 s 2 − 18 τ 2 s + 24 τ 4.5 τ 2 s 2 + 18 τ s + 24 (4)

Therefore the equation of state for the delayed link is: x ˙ d 1 = − 4 τ x d 1 − 16 3 τ 2 x d 2 + u c d x ˙ d 2 = x d 1 x ˙ d 3 = − 4 τ x d 3 − 16 3 τ 2 x d 4 + u c q x ˙ d 4 = x d 3 u c d , d l = − 8 x d 1 + τ u c d u c q , d l = − 8 x d 3 + τ u c q (5) where x d 1 、 x d 2 、 x d 3 and x d 4 are a set of state variables; u c d , d l and u c q , d l are the voltage control signals obtained from u c d and u c q after a zero-order retainer and a delay time τ , respectively.

J s ⋅ Δ ω = Δ P s e t ω 0 − Δ P ω 0 − D p Δ ω Δ P ≈ 3 V n 2 X Δ δ = A Δ δ Δ δ = Δ ω − Δ ω g s (6) where Δ P and Δ P s e t are the minor change of P and P s e t , respectively; Δ ω and Δ ω g are the minor change of ω and the grid frequency ω g , respectively; Δ δ is the minor change of the phase difference between inverter voltage and grid voltage.

When the system is subject to a small disturbance, GFM inverter can be regarded as a linearized model near the operating point. At this point, the system can be considered a model with two inputs and a single output, as shown in Figure 2. Based on this, a simplified small-signal model of GFM inverter can be obtained, as shown in Eq. 6.

From the analysis in Section 2, it is unrealistic to set a fixed virtual inertia J so that the inverter can meet the requirements of the small overshoot and the short setting time under various disturbances of different amplitude. Therefore, this paper proposes a sigmoid-based adaptive inertia regulator to adjust the virtual inertia. The basic strategy is as follows:

In order to improve the dynamic performance of inverter output frequency, the value of virtual inertia J should be adaptive: when the amplitude of frequency deviation is small, the inverter can ignore the impact of the overshoot and minimize the virtual inertia of the system to achieve rapid response and ensure the system recovery as soon as possible. Once the frequency deviation of the system becomes larger, it should mainly suppress the frequency oscillation and try to increase the virtual inertia of the system to reduce overshoot.

The virtual inertia generated by the sigmoid-based adaptive inertia regulator is shown in Eq. 16, and its function curve is shown in Figure 5. J Δ f = J min + J max − J min 1 1 + e − k Δ f − a (16) where J _min and J _max are the lower limit and upper limit of virtual inertia, respectively. a and k denote the adjustment coefficients. Δf is the input of the sigmoid-based adaptive inertia regulator. By using different values of a and k, the shape of the regulation can be changed. In this paper, a is used to control the curve to move right or left. k is used to control the slope of the curve and can be expressed as the sensitivity of the tuning function. According to Eq. 16, J = (J _min + J _max)/2 when Δf = a. Hence, a is designed to |Δf| _max /2 in this paper, where |Δf| _max is the maximum frequency deviation of the power grid.

In order to balance the overshoot and response speed, the range of system damping ratio ζ is generally required (Wu et al.,2016). According to the expression of ζ in Eq. 9, the upper limit J _min and lower limit J _max of Eq. 16 can be calculated.

Figure 5 shows the curves of sigmoid-based adaptive inertia regulation with different k. There are five curves in Figure 5, where k increases from Υ₀ to Υ₄ and Υ₀ means k = 0. The inertia regulator can be divided into three Zones, which are Zone I with small frequency deviation and small inertia, Zone II with upheaval inertia and Zone III with large frequency deviation and large inertia. When the frequency deviation is small with ordinary state, the inertia regulator works in Zone I. Hence, the corresponding inertia is very small, which can respond to interference as soon as possible. When a disturbance occurs. The increase in frequency deviation causes the inertial regulator to enter Zone II further leading to a rapid growth in virtual inertia. Then, the inertia regulator is transferred to Zone III to maintain a large inertia with the further increase of frequency deviation. Finally, the inertia regulator is returned to work in Zone I with the decrease of the frequency deviation.

The inertia regulator designed in Section 3 is a non-linear part, and its introduction will inevitably affect the system. In order to analyze the influence of the control strategy in this paper on the stability of the system, a full-order small-signal model of GFM inverter containing the inertia regulator is established in this section and thus the eigenvalue analysis is provided to guide the selection of control parameters.

To fully verify the effectiveness of the proposed strategy, a hardware-in-the-loop (HIL) platform is established. As shown in Figure 8, the HIL platform consists of four parts: an OPAL RT-LAB OP5700 real-time digital simulator, three Ti’s TMS320FDSP28377D digital signal processor (DSP)-based control boards, a personal computer (PC) and a Tektronix MOS54 oscilloscope. Firstly, the PC downloads the C-based program to the DSP through the emulator. The RT-LAB is connected to PC via RJ45 Cable. Then, the control commands are sent down from the RT-LAB to the DSP. The DSP implements the control algorithm and generates the modulation signal to the RTLAB. The data transfer between DSP and RTLAB is realized through I/O interface card. Finally, the test results are displayed in oscilloscope which is connected to RT-LAB by the I/O interface card.

In this paper, the frequency response under the disturbances of active power and frequency are theoretically analyzed. Then, the problem that the fixed inertia leads to a contradiction between the maximum frequency deviation and the setting time is presented. Hence, this paper proposed a sigmoid-based inertia control strategy to solve the problem. The stability of proposed strategy is verified by establishing the full-order small-signal model of GFM inverter containing the inertia regulator. By using the proposed control strategy, the system inertia can be varied by frequency deviation to achieve a good dynamic performance on both maximum frequency deviation and setting time. Moreover, the proposed strategy shows strong robustness with high-frequency noises in system frequency due to the absence of derivative action. The effectiveness of the proposed control strategy is verified through HIL results.

Due to the sensitivity of differencing to high-frequency noise, researchers who want to study further in this field should avoid directly employing frequency differentiation for adaptive regulation in the future. One method is to eliminate the operations of frequency differentiation by mathematical method, such as (Hou et al.,2020). Another method is to directly design a virtual inertia regulator without differentiation.