Sliding Mode Observer-Based Phase-Locking Strategy for Current Source Inverter in Weak Grids

Abstract

The current source inverter (CSI) has become the main grid-connected interface of distributed generation systems due to its advantages, such as boost capability, current controllability, and short-circuit protection capability. However, in weak grids, the grid-connected CSI that uses a phase-locked loop to achieve grid voltage synchronization has problems, such as instability in the fundamental positive-sequence voltage phase detection at the point of common coupling and instability in the current loop control, which seriously hamper the promotion and application of the CSI and its interconnected systems. For this reason, this paper proposes a sliding mode observer-based phase-locking strategy for the CSI. The strategy proposes a sliding mode observer for grid voltage phase detection, so that the grid current can directly follow the grid voltage, solving the problem of inconsistency or distortion between the voltage phase of the point of common coupling and the grid voltage phase in weak grids. On this basis, the grid impedance is regarded as part of the CL filter, and a robust parameter design method is proposed for the grid current closed-loop control in weak grids, which achieves robust operation of a CSI in weak grids. Finally, an experimental platform for a single-phase grid-connected CSI is built to verify the effectiveness and feasibility of the proposed scheme.

1. Introduction

The current source inverter (CSI), due to its advantages, such as boost capability, current controllability, and short-circuit protection capability, has been gradually applied to scenarios such as uninterruptible power supply systems [1], electric aircraft power systems [2], photovoltaic power systems [3], and wind power generation systems [4]. However, the public power grid with power electronics exhibits weak grid characteristics [5], which seriously affects the phase detection of the grid fundamental voltage at the point of common coupling (PCC) by the phase-locked loop (PLL), resulting in instability problems in the PLL-based grid voltage following a control strategy in weak grids [6,7,8,9].

Due to the duality of the topological structures of the CSI and voltage source inverter (VSI) [10], the research on the grid-connected VSI with an L or LCL filter in weak grids can be used as a reference. According to the small signal impedance stability criterion [11], a PLL will introduce negative incremental resistance, resulting in instability of the grid-connected system, and the increase in the bandwidth of the PLL will expand the frequency range of instability. In addition, the interaction between the PLL, current controller, and grid impedance will damage the stability of the system [12].

The grid following a control structure based on a PLL plus a current loop enables the solution to improve the stability of the grid-connected system to be developed through a PLL or grid-connected current closed loop. In the solution based on a PLL, reducing the PLL bandwidth is a way to improve the PLL equivalent impedance [13], but the fast dynamics of a PLL are contradictory to reducing the bandwidth, which reduces the dynamic response of the system. On the other hand, optimizing the PLL control structure to reshape the PLL equivalent impedance can reduce the instability caused by the negative resistance characteristics of the PLL. Khaled Mohammad Alawasa et al. [14,15] introduced the notch filter or the bandpass filter in a PLL to reshape the inverter output impedance. However, the design of the filter depends on an accurate system impedance model. Dongsheng Yang et al. [16] used a symmetrical PLL to eliminate the frequency coupling effect and combined impedance shaping to offset the negative resistance characteristics caused by the PLL. Xianfu Lin et al. [17] solved the problem of grid-connected system stability by decoupling a PLL and grid impedance through an improved PLL. Xianfu Lin et al. [18] effectively reduced the coupling between the PLL and the control system through a new type of PLL with a constant coupling effect, making the system more robust to grid impedance changes. However, the decoupling effect depends on the construction of an accurate system model, which increases the complexity of system design. Matias Berg et al. [19] compensated for the phase difference between the PCC voltage and the grid voltage caused by grid impedance through a PLL phase compensation method. However, its compensation phase strongly depends on the grid impedance measurement accuracy, which can easily lead to system instability. Leilei Guo et al. [20,21] used a PLL-independent online grid voltage observation method based on the sliding mode observer (SMO) to achieve grid voltage sensorless control for the single L-filtered grid-connected inverter, but they did not consider the impact of grid impedance on the grid current closed loop.

Improving the current control loop is also the key to improving the stable operation of the CSI. Adjusting the current controller parameters is the simplest and most feasible method. Hong Gong et al. [12] designed the controller parameters to improve the system stability based on the influence of the proportional gain of the current controller on the instability of the PLL in weak grids. However, this method will reduce the current loop control bandwidth and affect the dynamic performance of the system. Introducing additional control links is also a main method. Guoning Wang et al. [22] introduced a grid voltage feedforward control link to reshape the inverter output impedance and offset the negative resistance characteristics of the PLL. Xuehua Wang et al. [23] extracted the grid voltage from the PCC voltage through a double sampling mode and fed it forward to the current controller output. Xin Chen et al. [24] introduced the PCC voltage feedforward to the current loop to achieve dynamic compensation of the impedance phase. Donghai Zhu et al. [25] introduced a small signal compensation function based on a PLL to reduce the negative resistance effect of the PLL on the current control loop. Bin Guo et al. [26] introduced a virtual series impedance and moved the virtual impedance action point to the current controller output through an equivalent transformation of the series impedance reshaping scheme. However, in the above schemes, the multi-loop control system will increase the burden on the design of control parameters. In addition, in the existing PLL-based current control scheme, it is difficult to completely eliminate the interaction between control loops.

To address the above problems, the contributions of this paper are as follows. On the one hand, an SMO for grid voltage phase detection is proposed for a single-phase CSI, so that the grid current can directly follow the grid voltage, solving the problem of inconsistency or distortion between the phases of PCC voltage and grid voltage in weak grids. On the other hand, based on the SMO phase-locking strategy, the grid impedance is regarded as part of the CL filter, and a robust parameter design method is proposed for the grid current closed-loop control, solving the problem of robust operation of CSI in weak grids.

The comparison between the proposed SMO-based phase-locking strategy and the traditional PLL-based phase-locking strategy is shown in

Table 1. Comparison between the proposed SMO-based phase-locking strategy and the traditional PLL-based phase-locking strategy.

	Voltage Phase Tracking Speed	Grid Frequency Adaptability	Whether Having Coupling Problems or Not	Stability in Weak Grids
SMO-based phase-locking strategy	Fast	Strong	Yes	Weak
PLL-based phase-locking strategy	Fast	Strong	No	Strong

. Both phase-locking strategies can achieve fast tracking of the observed voltage phase and are not affected by grid frequency fluctuations. However, the PLL-based phase-locking strategy has a coupling problem between the PLL, current controller and grid impedance, which leads to the issue of weak stability of the PLL-based grid-connected system in weak grids. In comparison, the proposed SMO-based phase-locking strategy can solve the coupling problem caused by the inconsistency between the PCC voltage and grid voltage phases, thereby stabilizing the SMO-based grid-connected system in weak grids.

The rest of the paper is organized as follows. In Section 2, the unstable effect of a PLL-based phase-locking strategy on a grid-connected CSI in weak grids is discussed. In Section 3, an SMO for grid voltage phase detection is proposed to solve the problem of inconsistency between the PCC voltage phase and grid voltage phase in weak grids. In Section 4, the grid impedance is regarded as part of the CL filter, and a robust design method for grid current closed-loop parameters in weak grids is proposed to achieve the robustness of a grid-connected CSI to grid impedance changes under an SMO-based phase-locking strategy. In Section 5, an experimental platform is built to verify the effectiveness of the proposed strategy and parameter design method. Finally, a conclusion is drawn in Section 6.

2. Traditional Control Strategy Based on PLL

The topology of a single-phase grid-connected CSI using a traditional PLL-based grid following a control strategy is shown in Figure 1. Since the current pulsation in the large inductor on the DC side of the CSI is minimal, the DC side can be regarded as a constant current source, and its impact on the control characteristics of the AC-side grid-connected current during the PWM modulation process can be ignored. To have reverse voltage resistance, the power switch device is composed of an IGBT and a diode in series. The inverter output is connected to the infinite AC grid through a CL filter and line impedance. The PLL is used to detect the phase θ_v of the PCC voltage v_pcc, and θ_v is further combined with the current amplitude given I_ref and the power factor angle φ to generate a current reference i_ref. To suppress the CL resonance spike, active damping is achieved through capacitor voltage proportional feedback, and K is the damping coefficient. G_PR(s) represents a proportional and resonant (PR) controller. PWM represents pulse width modulation. I_dc is the DC current, L_f and C_f are the filter inductance and capacitance, respectively, Z_g is the equivalent impedance of the grid, i_c and i_L are the inverter output current and grid current, respectively, and v_C and v_g are the filter capacitor voltage and grid voltage, respectively.

As a typical PLL, the PLL based on the second order generalized integrator (SOGI) has good harmonic suppression capability [27], and its control block diagram is shown in Figure 2. k is the proportional factor for adjusting the SOGI bandwidth, ω₀ is the fundamental angular frequency, V_gm is the grid voltage amplitude, and K_pp and K_ip are the proportional coefficient and integral coefficient of the regulator, respectively.

According to the system structure shown in Figure 1 and the SOGI-PLL control block diagram shown in Figure 2, affected by the PI regulator in the PLL, the PCC phase information θ_v obtained by the PLL has phase angle fluctuations or deviations under small signal disturbances, which makes the inverter synchronous rotating coordinate system and the grid rotating coordinate system corresponding to the PCC voltage not coincident, and further leads to the deterioration of the tracking effect of the grid current on the voltage phase. In addition, the grid current will affect the PCC voltage through the grid impedance, resulting in a vicious cycle and system oscillation or divergence. As the grid impedance increases, the coupling degree of the PLL, current controller, and grid impedance intensifies, making the grid-connected CSI less stable in weak grids.

3. Stable Control Strategy Based on SMO

In weak grids, the relationship between the PCC voltage v_pcc and the grid voltage v_g can be derived as

$$L_g{\displaystyle \frac{d{i}_L}{dt}}=v_{pcc}-R_g{i}_L-v_g$$

(1)

where L_g and R_g are the equivalent inductance and resistance of the power grid.

According to the mathematical model shown in Equation (1), the SMO is established as

$$L_g{\displaystyle \frac{d\widehat{i_L}}{dt}}=v_{pcc}-R_g\widehat{i_L}-M\mathrm{sgn}(\widehat{i_L}-i_L)$$

(2)

where $\widehat{i_L}$ is the estimated value of the grid current, M is the sliding mode gain, and sgn( ) is the sign function.

According to Equations (1) and (2), the relation satisfied by the grid current observation error $\overline{i_L}=\widehat{i_L}$−i_L can be obtained as

$$L_g{\displaystyle \frac{d\overline{i_L}}{dt}}=v_g-R_g\overline{i_L}-M\mathrm{sgn}(\overline{i_L})$$

(3)

For the stability analysis of the SMO, the Lyapunov function is selected as

$$V={\overline{i_L}}^2/2$$

(4)

The derivative of Equation (4) can be obtained as

$${\displaystyle \frac{dV}{dt}}=-{\displaystyle \frac{R_g{\overline{i_L}}^2}{L_g}}+{\displaystyle \frac{v_g\overline{i_L}}{L_g}}-{\displaystyle \frac{M\left|\overline{i_L}\right|}{L_g}}$$

(5)

To satisfy the stability condition of the Lyapunov function, dV/dt < 0 must be satisfied. According to Equation (5), the sufficient condition for the sliding mode gain M to satisfy dV/dt < 0 can be derived as

$$M>\max (\left|v_g\right|)=V_{gm}$$

(6)

Under this condition, the convergence of the SMO can be ensured, that is, $\overline{i_L}$ = 0. The larger the value of M, the faster the SMO converges, but the larger the sliding mode noise; the smaller the value of M, the slower the SMO converges, but the smaller the sliding mode noise. Therefore, the selection of M needs to take into account both rapidity and anti-interference. However, under the action of the low-pass filter (LPF) mentioned later, the sliding mode noise can be effectively filtered out. Therefore, the value of M can be much larger than V_gm to improve the dynamic performance of the SMO.

Substituting $\overline{i_L}$ = 0 into Equation (3), the estimated value $\widehat{v_g}$ of the grid voltage can be obtained as

$$\widehat{v_g}=M\mathrm{sgn}(\overline{i_L})$$

(7)

The sliding mode term on the right side of Equation (7) contains the grid voltage fundamental signal, background harmonics, and sliding mode noise. To extract the fundamental signal, an LPF is applied to the observation result, shown as

$$\widehat{v_{gL}}={\displaystyle \frac{{\omega }_{cL}}{s+{\omega }_{cL}}}M\mathrm{sgn}(\overline{i_L})$$

(8)

where $\widehat{v_{\mathit{gL}}}$ is the estimated value of the grid voltage after filtering, and ω_cL is the cutoff frequency of the LPF.

However, the LPF will cause the extracted fundamental signal to have amplitude attenuation and phase lag. To obtain the grid voltage phase, phase compensation is required. The phase lag caused by the LPF can be expressed as

$$\mathsf{\Delta }\theta =\arctan \left(\omega /{\omega }_{cL}\right)$$

(9)

According to Equation (9), the phase compensation of the estimated value depends on the grid frequency. In addition, the grid voltage estimate includes two parts, amplitude and phase, and the phase part needs to be extracted separately. When the SMO is applied to a three-phase system, the accurate grid voltage can be obtained through multiple low-pass-filtered α and β components of the three-phase voltage [20]. However, when the SMO is applied to a single-phase system, the construction process of the α and β components of the single-phase voltage is relatively complicated. Therefore, this paper applies a frequency-adaptive SOGI to avoid the construction of multiple α and β components.

For the frequency-adaptive SOGI shown in Figure 2, taking $\widehat{v_{\mathit{gL}}}$ as the input signal, on the one hand, the frequency locking purpose can be achieved to obtain the grid frequency ω_g required for phase compensation, and on the other hand, two completely orthogonal signals, $\widehat{v_{\mathit{gL}\alpha }}$ and $\widehat{v_{\mathit{gL}\beta }}$, can be generated, so that the phase estimate $\widehat{{\theta }_{\mathit{vL}}}$ of the filtered grid voltage can be expressed as

$$\widehat{{\theta }_{vL}}=\arctan \left(\widehat{v_{gL\beta }}/\widehat{v_{gL\alpha }}\right)$$

(10)

Furthermore, the phase estimate $\widehat{{\theta }_v}$ of the grid voltage at the grid frequency can be expressed as

$$\widehat{{\theta }_v}=\widehat{{\theta }_{vL}}+\mathsf{\Delta }{\theta }_g=\arctan \left(\widehat{v_{gL\beta }}/\widehat{v_{gL\alpha }}\right)+\arctan \left({\omega }_g/{\omega }_{cL}\right)$$

(11)

The actual control system is a discrete system, so the SMO shown in Equation (2) needs to be converted into a differential equation form, shown as

$$\widehat{i_L}(k)=\left(1-{\displaystyle \frac{R_g{T}_s}{L_g}}\right)\widehat{i_L}(k-1)-{\displaystyle \frac{T_s}{L_g}}\left[v_{pcc}-M\mathrm{sgn}\left(\widehat{i_L}(k-1)-i_L\right)\right]$$

(12)

where $\widehat{i_L}(k)$ and $\widehat{i_L}(k-1)$ are estimated values of the grid current in the current sampling period and the previous sampling period, respectively, and T_s is the sampling period.

The measured values $\widehat{L_g}$ and $\widehat{R_g}$ of the grid equivalent impedance in the SMO are obtained by online grid impedance measurement [28]. The topological structures of the designed SMO for observing the grid voltage phase and the single-phase grid-connected CSI using the SMO-based stable control strategy are shown in Figure 3 and Figure 4, respectively.

4. Parameter Design under Proposed SMO-Based Phase-Locking Strategy

Focusing on the worst operating conditions of the grid-connected CSI, the grid equivalent resistance R_g is ignored as a passive component. In addition, the role of the SMO is to obtain the grid voltage phase, which is combined with the current amplitude given I_ref and the power factor angle φ to generate the current reference i_ref. Under the premise that the dynamic response of the SMO is fast enough, its influence on the current control loop can be ignored. Therefore, the mathematical model of the SMO is not considered in the following parameter design and is represented by i_ref. Based on the above analysis, the current control block diagram in the mixed s- and z-domain, when considering the influence of the grid impedance on the grid current closed loop, is shown in Figure 5a. This structure of the current outer loop and damping inner loop can be easily found to be equivalent to the structure of the current outer loop and voltage inner loop shown in Figure 5b [29].

It should be highlighted that the PWM inverter has a transfer function, K_PWM = I_dc/I_tri, whereas Itri is the peak-to-peak amplitude of the triangular carrier. In the following analysis for the CSI system, the control output is divided by K_PWM before sending as the PWM modulation reference. Furthermore, the modulated signal can be amplified by Idc and then basically restored to the control output at the inverter output. By doing so, the control parameter design will not be interfered with by the effect of K_PWM, and the control block diagram will also be simplified without containing the parameter K_PWM. Moreover, the one-sampling-period delay z⁻¹ and zero-order hold (ZOH) are utilized to model the PWM generation process. G_h(s) is the transfer function of ZOH, shown as

$$G_h(s)={\displaystyle \frac{1-e^{-T_{s}s}}{s}}$$

(13)

Discretized by the Tustin discretization method with frequency pre-warping, the discrete form G_PR(z) of the PR controller can be obtained as

$$G_{PR}(z)=K_{pc}+{\displaystyle \frac{K_{rc}{\omega }_i\sin (T_s{\omega }_0)(z^2-1)}{\left[{\omega }_0+{\omega }_i\sin (T_s{\omega }_0)\right]z^2-2{\omega }_0\cos (T_s{\omega }_0)z-{\omega }_i\sin (T_s{\omega }_0)+{\omega }_0}}$$

(14)

where K_pc is the proportional gain, K_rc is the resonant gain, and ω_i is the bandwidth term.

Furthermore, the current control block diagram in the z-domain is shown in Figure 6. In the discrete z-domain, the system loop gain G_op(z) can be derived as

$$G_{op}(z)=G_{PR}(z)·{\displaystyle \frac{z^{-1}{G}_{iL}(z)}{1+z^{-1}K{G}_{vC}(z)}}={\displaystyle \frac{G_{PR}(z)·(z+1)\left[1-\cos ({\omega }_r{T}_s)\right]}{z\left[z^2-2z\cos ({\omega }_r{T}_s)+1\right]+\frac{K·\sin ({\omega }_r{T}_s)}{{\omega }_r{C}_f}(z-1)}}$$

(15)

where G_iL(z) and G_vC(z) are the transfer functions from the inverter output current i_c to the grid current i_L and the capacitor voltage v_C after ZOH transformation, shown as

$$G_{iL}(z)={\displaystyle \frac{(z+1)[1-\cos ({\omega }_r{T}_s)]}{z^2-2z\cos ({\omega }_r{T}_s)+1}}$$

(16)

$$G_{vC}(z)={\displaystyle \frac{(z-1)\sin ({\omega }_r{T}_s)}{{\omega }_r{C}_f[z^2-2z\cos ({\omega }_r{T}_s)+1]}}$$

(17)

ω_r is the resonant frequency when the grid equivalent inductance L_g is regarded as part of the CL filter, expressed as

$${\omega }_r=\sqrt{{\displaystyle \frac{1}{(L_f+L_g)C_f}}}=2\pi f_r$$

(18)

When the effect of L_g is not considered, the resonant frequency ω_r0 of the CL filter itself is expressed as

$${\omega }_{r0}=\sqrt{{\displaystyle \frac{1}{L_f{C}_f}}}=2\pi f_{r0}$$

(19)

According to Equation (15), the number of poles outside the unit circle of the system open-loop transfer function depends on the following equation:

$$z\left[z^2-2z\cos ({\omega }_r{T}_s)+1\right]+{\displaystyle \frac{K·\sin ({\omega }_r{T}_s)}{{\omega }_r{C}_f}}(z-1)=0$$

(20)

The resonant frequency f_r is usually expected to be less than half of the switching frequency f_sw to ensure good harmonic attenuation performance. In addition, the sampling frequency f_s is set to twice f_sw in this article. Therefore, the discussion on control parameter design in this article is only within the range of f_r < f_s/4. Use the transformation z = (1 + w)/(1 − w) to map the unit circle in the z-domain in Equation (20) to the imaginary axis in the w-domain, and then apply the Routh–Hurwitz stability criterion to obtain

$$\begin{array}{ll}0<K<K_P={\displaystyle \frac{2\cos ({\omega }_r{T}_s)-1}{\sin ({\omega }_r{T}_s)}}{\omega }_r{C}_f,& 0<f_r<f_s/6\\ {\displaystyle \frac{2\cos ({\omega }_r{T}_s)-1}{\sin ({\omega }_r{T}_s)}}{\omega }_r{C}_f=K_N<K<0,& f_s/6<f_r<f_s/4\end{array}$$

(21)

When the damping coefficient K satisfies Equation (21), the system open-loop transfer function has no poles outside the unit circle. Otherwise, the system open-loop transfer function will have two poles outside the unit circle.

When performing stability analysis, the PR controller can only consider the proportional gain K_pc in the high-frequency band. Substituting K_pc for G_PR(z) in Equation (15), the simplified system loop gain G_so(z) can be obtained as

$$G_{so}(z)={\displaystyle \frac{K_{pc}·(z+1)\left[1-\cos ({\omega }_r{T}_s)\right]}{z\left[z^2-2z\cos ({\omega }_r{T}_s)+1\right]+\frac{K·\sin ({\omega }_r{T}_s)}{{\omega }_r{C}_f}(z-1)}}$$

(22)

According to the logarithmic frequency response stability criterion, the number Z of poles outside the unit circle of the closed-loop system is expressed as Z = P − 2N. P is the number of poles outside the unit circle of G_so(z), and N is the number of times the G_so(z) phase–frequency characteristic curve crosses −180°. If Z = 0, the closed-loop system is stable, otherwise the system is unstable.

Figure 7 shows the Bode diagram for different values of the resonant frequency f_r and the damping coefficient K. The following can be obtained:

① When f_r < f_s/6, 0 < K < K_P, and P = 0, the −180° crossover frequency f_r′ is within the range of (f_r, f_s/6). Adjusting K_pc can make the amplitude of G_so(z) at f_r′ less than 0 dB. At this time, N = 0, Z = 0, and the closed-loop system is stable.

② When f_r < f_s/6, K < 0, and P = 2, the phase of G_so(z) will not cross −180°. At this time, N = 0, Z = 2, and the closed-loop system is unstable.

③ When f_r < f_s/6, K > K_P, and P = 2, the phase of G_so(z) will not cross −180°. At this time, N = 0, Z = 2, and the closed-loop system is unstable.

④ When f_s/6 < f_r < f_s/4, K_N < K < 0, and P = 0, the −180° crossover frequency f_r′ is within the range of (f_s/6, f_r). Adjusting K_pc can make the amplitude of G_so(z) at f_r′ less than 0 dB. At this time, N = 0, Z = 0, and the closed-loop system is stable.

⑤ When f_s/6 < f_r < f_s/4, K < K_N, and P = 2, the phase of G_so(z) will not cross −180°. At this time, N = 0, Z = 2, and the closed-loop system is unstable.

⑥ When f_s/6 < f_r < f_s/4, K > 0, and P = 2, the phase of G_so(z) will not cross −180°. At this time, N = 0, Z = 2, and the closed-loop system is unstable.

Based on the above analysis, only cases ① and ② can make the grid-connected CSI system stable. At this time, the phase of G_so(z) crosses −180° at f_r′, and G_so(z) is a pure real number at f_r′. Therefore, G_so(z) at f_r′ can be derived as

$$G_{so}(z=e^{j{{\omega }_r}^{\prime }{T}_s})={\displaystyle \frac{K_{pc}\left[\cos ({{\omega }_r}^{\prime }{T}_s)+1+j\sin ({{\omega }_r}^{\prime }{T}_s)\right]\left[1-\cos ({\omega }_r{T}_s)\right]}{A+jB}}$$

(23)

The specific expressions of A and B are obtained as

$$\left\{\begin{array}{l}A={\displaystyle \frac{K\sin ({\omega }_r{T}_s)}{{\omega }_r{C}_f}}\left[\cos ({{\omega }_r}^{\prime }{T}_s)-1\right]+2\left[\cos ({{\omega }_r}^{\prime }{T}_s)-\cos ({\omega }_r{T}_s)\right]\left[2{\cos }^2({{\omega }_r}^{\prime }{T}_s)-1\right]\\ B={\displaystyle \frac{K\sin ({\omega }_r{T}_s)}{{\omega }_r{C}_f}}\sin ({{\omega }_r}^{\prime }{T}_s)+4\sin ({{\omega }_r}^{\prime }{T}_s)\cos ({{\omega }_r}^{\prime }{T}_s)\left[\cos ({{\omega }_r}^{\prime }{T}_s)-\cos ({\omega }_r{T}_s)\right]\end{array}\right.$$

(24)

Let the imaginary part of G_so(z) be 0; G_so(z) at f_r′ can be derived as

$$G_{so}(z=e^{j{{\omega }_r}^{\prime }{T}_s})={\displaystyle \frac{2K_{pc}(1-a)}{2a-2b-\sqrt{c}+1}}$$

(25)

where a = cos(ω_rT_s), b = Ksin(ω_rT_s)/ω_rC_f, c = (2a + 1)2 − 8b.

To make the closed-loop system stable, the amplitude of G_so(z) at f_r′ needs to be less than 0 dB. At the same time, to enhance the anti-interference ability of the system, a 3 dB amplitude margin is usually left. Let 20 lg$\left|G_{\mathit{so}}\left(z=e^{j{{\omega }_r}^{\prime }{T}_s}\right)\right|$ < −3 dB; K_pc can be derived as

$$K_{pc}<K_{\max }={\displaystyle \frac{\sqrt{2}(2a-2b-\sqrt{c}+1)}{4(a-1)}}$$

(26)

Figure 8 shows a three-dimensional graph of K_max changing with the damping coefficient K and the equivalent inductance of the grid L_g when f_r0 < f_s/6, 0 < K < K_P. It can be seen that when L_g = 0, that is, under a strong grid, there is an optimal damping coefficient K_op, corresponding to the maximum proportional gain K_pb, which maximizes the system loop gain and improves the system dynamic performance. As L_g increases, K_op and K_pb change, but under the same K, K_max will increase with the increase in L_g. Therefore, the optimal control parameters designed in strong grids can ensure the stability of the grid current closed loop even if the influence of L_g on the CL filter is considered in weak grids.

However, when f_s/6 < f_r0 < f_s/4 and K_N < K < 0, the increase in L_g will reduce f_r, which may cause f_r to cross the boundary of f_s/6, and the stable range of K will change. When the control parameters remain unchanged, the system is prone to instability in weak grids. Therefore, when observing the grid voltage phase through the SMO, to ensure the robustness of the control parameters in weak grids, the system should use a CL filter with f_r0 < f_s/6.

In addition, the value of the resonant gain K_rc needs to be a compromise between the steady-state error and stability margin of the system. Figure 9 shows the Bode diagram corresponding to a different K_rc when the proportional gain K_pc of the PR controller is the same. It can be seen that the larger the K_rc, the higher the fundamental frequency amplitude and the smaller the steady-state error of the system. However, when the −45° corner frequency of the PR controller is much lower than the resonant frequency f_r0 of the CL filter itself, it can not only reduce the impact of the phase offset of the PR controller on the system amplitude margin, but can also have a certain stability margin when the grid impedance causes f_r to shift.

5. Experimental Results

To verify the effectiveness of the proposed control strategy and parameter design method, an experimental platform for a grid-connected CSI was built, as shown in Figure 10, including the following: TMS320F28335 controller, power supply module, Buck circuit, CL filter, programmable AC power supply for simulating the power grid, and grid impedance. The Buck circuit is powered by a 40 V DC voltage source to control the DC bus current of the CSI to be constant. The main circuit parameters are shown in

Table 2. Main parameters of single-phase grid-connected CSI.

Parameters	Values
DC bus current Idc/A	8
Grid RMS voltage Vg/V	110
Grid voltage frequency f0/Hz	50
Sampling frequency fs/kHz	10
Switching frequency fsw/kHz	5
CL filter inductance Lf/mH	2
CL filter capacitance Cf/μF	20

.

In the stability verification experiment based on the two phase-locking strategies of the SMO and PLL, the experimental waveforms under different L_g are shown in Figure 11, Figure 12, Figure 13 and Figure 14. It is worth noting that the parameters of the traditional PLL are designed based on the large-signal stability criterion and boundaries for grid voltage sags [30], and the design results are K_pp = 1.4 and K_ip = 300. As the inductance value of L_g increases, the deviation between the PCC voltage phase and the grid voltage phase increases. When L_g = 0 mH, L_g = 10 mH, and L_g = 20 mH, the PLL-based current control system can still remain stable, and the grid current and PCC voltage maintain the same phase. However, when L_g = 37 mH, the current oscillates, and the system cannot be stable. Under different L_g, the grid current in the SMO-based current control system can accurately track the grid voltage phase and show a higher sinusoidal degree than that in the PLL-based current control system, and the SMO-based system remains stable.

From the dynamic verification experiment based on the two phase-locking strategies of the SMO and PLL, the experimental waveforms under L_g = 10 mH are shown in Figure 15 and Figure 16. When the amplitude reference of the grid current undergoes a step change, the grid current of the SMO-based current control system can transition to a new steady state within one fundamental cycle, just like the grid current of the PLL-based current control system.

From the robustness verification experiment based on the SMO phase-locking strategy, the experimental waveform under L_g = 10 mH is shown in Figure 17. When the grid voltage frequency fluctuates by ±0.5 Hz, the SMO can still keep tracking the grid voltage phase, so that the grid current and the grid voltage can maintain the same phase.

Based on the results of the above three experiments, the following conclusions can be drawn. The PLL-based phase-locking strategy has a significant problem of phase inconsistency between the PCC voltage and the grid voltage when the L_g is large, which makes the coupling degree between the PCC voltage, the grid impedance, and the current controller severe, and the system is prone to instability. However, on the premise that the SMO-based phase-locking strategy can maintain the same dynamic performance as the PLL-based phase-locking strategy, the former avoids the instability caused by coupling phenomena in weak grids by obtaining the grid voltage phase, and has the robustness to frequency changes of the grid voltage. In addition, under the SMO-based phase-locking strategy, the control parameters can be employed under different L_g without changing, which verifies the robustness of the parameters.

From the parameter comparison experiment, the experimental waveforms corresponding to M = 180 and M = 140 under L_g = 0 are shown in Figure 18. When M satisfies Equation (6), that is, M is greater than 155.54, the SMO converges, and the grid-connected CSI operates stably, but when M is less than 155.54, the SMO cannot converge and cannot accurately track the grid voltage phase. The experimental waveforms corresponding to K_pc = 0.41 and K_pc = 0.8 under L_g = 0 are shown in Figure 19. Under the optimal damping coefficient K_op = 0.09, the maximum proportional gain K_pb = 0.41, and the resonant gain K_rc = 190, the grid current can accurately track the current command. However, when K_pc does not satisfy Equation (26), that is, K_pc is greater than 0.41, the system stability margin is low. When K_pc is slightly greater than 0.41, the current tends to diverge, and the system becomes unstable. The above phenomenon verifies the effectiveness of the parameter design results.

In the test experiment of grid impedance measurement deviation, the grid current can remain stable and accurately track the grid voltage phase when there is a ±20% error in the grid impedance measurement value, as shown in Figure 20. Therefore, the SMO-based current control system does not rely on the accuracy of the grid impedance measurement results.

6. Conclusions

To solve the problem of unstable operation of the grid-connected CSI in weak grids, this paper proposes an SMO for grid voltage phase detection, regards grid impedance as part of the CL filter, and further proposes a robust design method for grid current closed-loop parameters in weak grids. The main conclusions are as follows:

1. According to the circuit relationship between PCC voltage, grid current, and grid voltage, the proposed SMO can directly observe the grid voltage phase, so that the grid current can follow the grid voltage, and avoid the coupling problem between the PCC voltage phase detection effect of the PLL and the grid impedance. In addition, the observation result of the SMO does not depend on the accuracy of impedance measurement results, and the grid-connected CSI can operate stably even under ±20% measurement error.

2. Under the premise that the SMO can accurately obtain the grid voltage phase, the grid impedance can be regarded as a part of the CL filter, and a robust design method for the current controller parameters of the grid-connected CSI is further proposed to ensure that the CSI can operate stably under any grid impedance.

The proposed SMO-based phase-locking strategy and robust parameter design method are not only applicable to CSIs, but also to VSIs. The difference in the application is only the modeling of the control plant, but the analysis method is the same.