An Integrated Design Approach for LCL-Type Inverter to Improve Its Adaptation in Weak Grid

Abstract

To improve the robustness of grid-connected inverter against grid impedance in a weak grid an integrated design method of LCL-filter parameters and controller parameters is proposed. In the method the inherent relation of LCL-filter parameters and controller parameters is taken into consideration to realize their optimized match. A parameter normalization scheme is also developed to facilitate the system stability and robustness analysis. Based on the method all normalization parameters can be designed succinctly according to the required stability and robustness. Additionally, the LCL parameter and controller parameter can be achieved immediately by restoring normalization parameters. The proposed design method can guarantee the inverter stability and robustness simultaneously without needing any compensation network, additional hardware, or the complicated iterative computations which cannot be avoided for the conventional inverter design method. Simulation and experiment results have validated the superiority of the proposed inverter design method.

1. Introduction

LCL-type inverters have been widely utilized in renewable energy applications to convert DC power into AC power [1,2] as renewable energy expands. One of the challenges encountered by grid-connected inverters is the instability due to the uncertainty and variation grid impedance in a weak grid [3,4,5,6]. It has been revealed that the interaction between inverter and grid impedance is the main reason for that problem [7,8,9,10]. Additionally, some solutions have been proposed to cope with this issue from different perspectives. The existing solutions for this problem can be classified as two categories. One is to add additional compensation network in the inverter or hardware equipment outside the inverter after the instability phenomenon occurs in the power plant [11,12,13,14,15,16,17,18,19,20,21,22]. Additionally, the other is the optimization design by taking grid impedance into account before the inverter is produced [23,24,25,26,27,28,29,30].

The first class of methods belongs to repair strategies. For example, a zero-compensation element is inserted in the conventional current controller to improve the phase response [11]. An online phase compensation method was proposed to ensure that the inverter has enough stability margins and bandwidth against varied grid impedance [12]. The notch filter was designed to generate an anti-resonance peak to offset the resonance of the LCL filter [13,14]. Since LCL resonant frequency varies along with grid impedance, the center frequency of the notch filter has to be regulated from the detected resonance frequency in real time. Given that the feedforward of the voltage at the point of common coupling (PCC) is equivalent to adding a virtual impedance in parallel with the inverter output impedance [15,16,17,18], possible solutions are to enhance the phase margin by reshaping the magnitude or phase of the ratio of the inverter output impedance and the grid impedance. Based on the feedforward PCC voltage, the grid current feedback has been used through the estimated grid impedance to alleviate the influence of grid impedance on system stability [19]. Reference [20] proposed an optimized capacitor-current-feedback active damping to improve inverter robustness against grid impedance. Additional equivalent impedance is added to the LCL filter in [21] to enhance the inverter robustness in a weak grid. A rectifier connected to the PCC was controlled as an active damper to damp the resonance [22]. Since this kind of method needs additional compensation networks or hardware devices it is complicated in practice.

The second class of methods is to take the grid impedance into consideration while the inverter is designed originally. A controller parameter design guideline based on the current loop transfer function was proposed to obtain better system performance against grid impedance variation [23]. An optimal capacitor-current-feedback for a digitally-controlled inverter was investigated so that the gain margin for stability was always satisfied as grid impedance varies [24]. A robust active damping factor and current controller parameters were designed to ensure that the inverter stability margins satisfy the requirement as grid impedance varies from the minimum to the maximum value [25]. Two stability criteria were proposed to limit the controller parameters [26]. Then collection of controller parameters fulfilling the criteria is derived so that the inverter can function well in a weak grid. Others attempt to improve the inverter robustness by the advanced controller [27,28]. The robust partial state feedback [27] and a robust H∞ controller [28] have been employed in the single-loop inverter to improve the inverter adaptability to grid impedance. An optimized range of the ratio between the switching frequency and resonant frequency was derived to ensure that the active damping is always effective under the grid impedance variation [29]. Moreover, the ratio between the grid-side inductance and the converter-side inductance was optimized to reduce the influence of LCL parameter variation on resonant frequency. An LCL-filter parameter design method was developed to ensure the resonant frequency always in the tolerance range as LCL-filter parameter variation due to the internal and external factors [30]. Although this kind of method can prevent instabilities for grid-connected inverters to some degree, it always designs LCL parameters and controller parameters separately rather than taking their inherent relation into consideration. Therefore, this kind of method cannot realize the optimal match between LCL parameters and controller parameters to prevent instabilities.

In fact, the stability and robustness of the grid-connected inverter are closely related to the LCL parameters as well as the controller parameters. Only when LCL parameters and controller parameters complement each other perfectly will the inverter be characterized with strong stability and robustness. Therefore, the paper is aimed at developing a robust inverter design method by taking the inherent relation of LCL parameters and controller parameters into consideration. Additionally, the salient contributions are summarized as follows: (1) An integrated inverter design method by taking the inherent relation of LCL-filter parameters and controller parameters into consideration is proposed; (2) A parameter normalization scheme is developed to facilitate the system stability and robustness analysis; (3) Parameter constraints and design guidelines for robust inverter design are derived.

The rest of this paper is organized as follows. In Section 2, the inverter is modeled and stability constraints are investigated. Section 3 derives the limitations of control parameters such that the current loop transfer function and the inverter admittance contain no RHP poles. The proposed integrated design method is described in Section 4. Section 5 presents a case study to illustrate the design procedure. Simulation and experimental results are presented in Section 6. Section 7 states the conclusion.

2. System Modeling and Stability Constraint

Single loop control structure of the LCL-inverter with grid-side current feedback is shown in Figure 1. L₁ is the inverter-side filter inductance, L₂ the grid-side inductance, and C is the filter capacitor. U_dc is the dc-link voltage. i₁ is the inverter-side current and i_s is the inverter current to the grid. i_c is the filter capacitor current and u_p is the PCC voltage. Z_g is the grid impedance and u_g is the ideal power grid voltage. In the control structure, i_r is the reference current generated by the magnitude of I* and the PCC phase locked by PLL. G_c is the quasi-proportional plus resonant (quasi-PR) current controller [23]

$$G_{\mathrm{c}}(s)=k_{\mathrm{p}}+\frac{2k_{\mathrm{r}}{\omega }_{\mathrm{i}}s}{s^2+2{\omega }_{\mathrm{i}}s+{({\omega }_0)}^2}$$

(1)

where k_p is the proportional coefficient, k_r is the resonant coefficient, ω_i is the bandwidth coefficient, and ω₀ is the fundamental angular frequency.

The s-domain model of the inverter is depicted in Figure 2 where k_pwm $\approx$ U_dc/2 is the transfer function of the pulse width modulation (PWM) modulator and G_d(s) is the transfer function of the total delay due to digital control. The delay includes the computation delay, zero-order holder and the sampling switch and can be expressed as [31]

$$G_{\mathrm{d}}(s)=\frac{1}{T_{\mathrm{s}}}\frac{1-e^{-T_{\mathrm{s}}s}}{s}{e}^{-T_{\mathrm{s}}s}$$

(2)

According to Figure 2, the open-loop transfer function from the current (i_r $-$ i_s) to i_s is to be derived as

$$G_{\mathrm{os}}(s)=\frac{i_{\mathrm{s}}}{i_{\mathrm{r}}-i_{\mathrm{s}}}=\frac{k_{\mathrm{pwm}}{G}_{\mathrm{c}}(s)G_{\mathrm{d}}(s)}{s^3{L}_1{L}_{2}C+s(L_1+L_2)}$$

(3)

and the inverter admittance can be expressed as

$$Y_{\mathrm{es}}(s)=\frac{i_{\mathrm{s}}}{u_{\mathrm{p}}}=\frac{s^2{L}_{1}C+1}{s^3{L}_1{L}_{2}C+s(L_1+L_2)+k_{\mathrm{pwm}}{G}_{\mathrm{c}}{G}_{\mathrm{d}}}$$

(4)

The Norton model of the inverter is depicted in Figure 3 [15]. Y_g is the equivalent grid admittance and U_g the grid voltage. G_e is the closed-loop transfer function from the reference current to the inverter output current and can be expressed

$$G_{\mathrm{e}}(s)=\frac{G_{\mathrm{os}}(s)}{G_{\mathrm{os}}(s)+1}$$

(5)

It has been shown [32] that stability of the system in Figure 3 can be assessed by the admittance ratio K_o = Y_es/Y_g as long as G_e contains no RHP (right-half plane) poles. If G_os satisfies the Nyquist criterion, then G_e contains no RHP poles. In accordance to Nyquist criterion, more conditions have to be satisfied to guarantee the closed-loop system stable if the open-loop transfer function contains RHP poles. If G_os contains RHP poles, sufficient stability margin is difficult to obtain even though G_os satisfies the Nyquist criterion [24]. The inverter design method developed in the following will be based on the condition that G_os contains no RHP poles.

Y_g does not contain any RHP poles or RHP zeros because it is passive. So the admittance ratio K_o has no RHP poles as long as Y_es does not contain any RHP poles. Additionally, the system will be stable as long as K_o has sufficient phase margin (PM). For the grid impedance, since the resistive part tends to make the system stable, in this study, only the inductive component is considered. Therefore, Y_g = 1/(sL_g), where L_g is equivalent grid inductance. The phase of Y_g can be viewed as −90° for all frequencies. Then the PM of K_o can be expressed as

$$\mathrm{PM}={180}^{°}-\left|\arg \left[Y_{\mathrm{es}}(j{\omega }_{\mathrm{in}})\right]+{90}^{°}\right|$$

(6)

where ω_in denotes the frequency at which the magnitude responses of Y_es and Y_g intersect. Generally, PM should be no less than 30° to thwart the adverse effects due to harmonics [16]. Therefore, arg[Y_es(jω_in)] should be in the range of (−240°, 60°). However ω_in varies with the variation of L_g. To keep PM > 30° as L_g varies, the phase response of Y_es is required to be in the range of (−240°, 60°). Therefore, one of the design objectives is to keep the phase arg[Y_es(jω)] in (−240°, 60°) for a wide frequency range.

3. Frequency Response Analysis

This section develops alternative expressions of the inverter admittance so that the frequency response can be easily analyzed for stability improvement. Instead of using the admittance, we investigate the frequency response of the impedance. It follows from Equation (4) that the inverter impedance can be expressed as

$$Z_{\mathrm{es}}=s{L}_2+\frac{1}{\underset{Z_{\mathrm{c}1}}{\underbrace{s^2{L}_{1}C+1}}}\cdot \underset{Z_{\mathrm{a}}}{\underbrace{(s{L}_1+k_{\mathrm{pwm}}{G}_{\mathrm{c}}{G}_{\mathrm{d}})}}=Z_2+\underset{Z_{\mathrm{cs}}}{\underbrace{Z_{\mathrm{c}1}\cdot Z_{\mathrm{a}}}}$$

(7)

It can be seen from Equation (7) that Z_es is expressed as two impedances: Z₂ the impedance due to the grid-side filter inductance and Z_cs = Z_c1$·$Z_a the equivalent impedance seen from the filter capacitor, i.e., $Z_{\mathrm{cs}}=u_{\mathrm{c}}/i_{\mathrm{s}}$.

Since Z_es is the reciprocal of Y_es, it follows from (6) that the phase of Z_es should be in the range of (−60°, 240°) in order to keep PM of K_o no less than 30°. It is desired that Y_es contains no RHP poles. Then Z_es should contain no RHP zero. Since Z₂ does not contain any RHP zero or RHP pole, if Z_cs has no RHP zero or RHP pole, then Z_es does not have any RHP zero.

The Padé approximation is widely adopted to linearize G_os and Z_a because they are nonlinear functions [31]. Figure 4 shows the frequency responses of the first-order to the fourth-order Padé approximations and the original function. It can be seen from Figure 4 that the fourth-order approximation is sufficiently accurate for a broad range of frequencies. Therefore, in the following the fourth-order approximation is brought in to conduct the qualitative analysis, such as obtaining the variation tendency of root locus zeros or Bode diagrams. Meanwhile, Euler’s equation is brought in to conduct the quantitative analysis such as obtaining the boundary value of k_p.

The following is to find the range of k_p such that Z_cs contains no RHP zeros. Since the zeros of Z_cs are the same as that of Z_a, and it is easier to find k_p from Z_a, Z_a is used to conduct the analysis.

Figure 5 illustrates the zero locus of Z_a with a fourth-order Padé approximation. Additionally, there are 15 zeros in Z_a. It is observed from Figure 5 that two crossover points occur at the imaginary axis. One appears when moving from RHP to the left-half plane (LHP), another from LHP to the RHP.

Figure 6 shows the Bode diagrams of Z_a for various k_p. It is observed from Figure 6 that two resonant peaks occur at ω_t and ω_e no matter what the value of k_p is. This implies that |Z_a(jω)| is nearly zero at ω_t or ω_e. To find ω_e, we assume that ω_e is much larger than ω₀ such that k_r is ignored. Then the frequency response Z_a(jω_e) can be expressed as

$$Z_{\mathrm{a}}(j{\omega }_{\mathrm{e}})=\frac{k_{\mathrm{pwm}}{k}_{\mathrm{p}}B-j(k_{\mathrm{pwm}}{k}_{\mathrm{p}}A-{\omega }_{\mathrm{e}}^2{L}_1)}{{\omega }_{\mathrm{e}}}$$

(8)

where $A=\frac{\cos T_{\mathrm{s}}\omega -\cos 2T_{\mathrm{s}}\omega }{T_{\mathrm{s}}}$, $B=\frac{\sin 2T_{\mathrm{s}}\omega -\sin T_{\mathrm{s}}\omega }{T_{\mathrm{s}}}$, T_s is the sampling period.

Set the real part of Z_a(jω_e) to zero. We have k_p = 0 or B = 0. When k_p = 0, the imaginary part of Z_a(jω_e) is not zero. Therefore k_p = 0 does not make |Z_a(jω_e)| = 0. Since B = 0 occurs at ω = ω_s/6, then the imaginary part of Z_a(jω_e) becomes zero at ω_e = ω_s/6 and k_p = k_pcr where

$$k_{\mathrm{pcr}}=\frac{{\omega }_{\mathrm{s}}^2{L}_1{T}_{\mathrm{s}}}{36k_{\mathrm{pwm}}}.$$

(9)

To find ω_t, k_r cannot be ignored because ω_t is close to ω₀. Under this circumstance, the real part and the imaginary part of Z_a(jω_t) are expressed as

$$\{\begin{array}{c}\mathrm{Re}\left[Z_{\mathrm{a}}(j{\omega }_{\mathrm{t}})\right]=\frac{\left[AJD{k}_{\mathrm{r}}+B{J}^2{k}_{\mathrm{p}}+B{D}^2(k_{\mathrm{p}}+k_{\mathrm{r}})\right]k_{\mathrm{pwm}}}{(J^2+D^2){\omega }_{\mathrm{t}}}\\ \mathrm{Im}\left[Z_{\mathrm{a}}(j{\omega }_{\mathrm{t}})\right]={\omega }_{\mathrm{t}}{L}_1-\frac{\left[A{J}^2{k}_{\mathrm{p}}+A{D}^2(k_{\mathrm{p}}+k_{\mathrm{r}})-JDB{k}_{\mathrm{r}}\right]k_{\mathrm{pwm}}}{(J^2+D^2){\omega }_{\mathrm{t}}}\end{array}$$

(10)

where $J={\omega }_0^2-{\omega }_{\mathrm{t}}^2$ and $D=2{\omega }_{\mathrm{i}}{\omega }_{\mathrm{t}}$.

By setting Re{Z_a(jω_t)} = 0 and Im{Z_a(jω_t)} = 0, the solution yields the relationship between k_r and ω_t

$$(J^2+D^2){\omega }_{\mathrm{t}}^2{L}_{1}B+(B^2+A^2)DJ{k}_{\mathrm{pwm}}{k}_{\mathrm{r}}=0.$$

(11)

Combining Re{Z_a(jω_t)} = 0 with equation (11), we obtain the lower bound of k_p as

$$k_{\mathrm{pt}}=-\frac{k_{\mathrm{r}}D(3{\omega }_{\mathrm{t}}{T}_{\mathrm{s}}J+2D)}{2(J^2+D^2)}.$$

(12)

It is concluded from Equations equation (9) and (12) that the range of k_p that ensures Z_a does not have RHP zero is k_pt $\le$ k_p $\le$ k_pcr.

Moreover, it can be derived from Equation (10) that the phase of Z_a(jω_t) can be expressed as

$$\phi ({\omega }_{\mathrm{t}})=-\frac{3{\omega }_{\mathrm{t}}{T}_{\mathrm{s}}}{2}$$

(13)

It is observed from Figure 6 that when 0 < k_p < k_pt, all phase curves cross φ(ω_t) − 180° at ω_t and tend to be −270° at the high-frequency range. When k_pt < k_p < k_pcr, all phase curves cross φ(ω_t) at ω_t and cross 90° at ω_e. For k_p > k_pcr, all phase curves cross φ(ω_t) at ω_t and cross −90° at ω_e.

In summary, since Z_c1 does not contain RHP poles. To guarantee that Y_es does not contain any RHP poles, k_p should be in the range of (k_pt, k_pcr).

4. Integrated Design Method

To improve grid-connected inverter robustness to grid impedance, this section presents an integrated design method to design the LCL-filter parameters and inverter controller parameters. The following parameter normalization is used for the development: k_p = λ_pk_pcr where k_pt/k_pcr < λ_p < 1.

4.1. Design ω_res and ω_c According to Stability Margin Constraints of G_os

Generally, the cutoff frequency ω_c is far away from ω₀ to guarantee enough bandwidth. Therefore, the resonant coefficient of the current controller k_r can be ignored. On the other hand, the phase curve of G_os crosses 180^° at the frequency below ω_r. Therefore, to obtain enough stability margins, ω_c is usually set much lower than ω_r. Therefore, the influence of the filter capacitance C on |G_os(jω_c)| can be neglected. That is to say both k_r and C can be neglected when calculating |G_os(jω_c)|. Then k_p can be derived as Equation (14) by setting |G_os(jω)| = 1

$$k_{\mathrm{p}}=\frac{{\omega }_{\mathrm{c}}(L_1+L_2)}{k_{\mathrm{pwm}}}$$

(14)

The open-loop transfer function can be expressed as Equation (15) according to Equation (3).

$$G_{\mathrm{os}}(s)=\frac{1}{L_1{L}_{2}C}\frac{k_{\mathrm{pwm}}{G}_{\mathrm{c}}(s)G_{\mathrm{d}}(s)}{s^3+s{\omega }_{\mathrm{res}}^2}$$

(15)

where ${\omega }_{\mathrm{res}}=\sqrt{\frac{L_1+L_2}{L_1{L}_{2}C}}$.

To normalize ω_res and ω_c we mark ω_res = δω_e and ω_c = ξω₀. To ensure the stability, δ should be larger than 1 and ω_c should be lower than ω_e, i.e., ξ < ω_e/ω₀. ω_res is generally lower than half of the switching frequency, i.e., ω_res < ω_sw/2, to ensure the proper filtering performance on the switching harmonics. Since ω_sw is equal to ω_s/2 for the regular-sampling method, the range of δ is (1, 1.5). ω_c should be larger than 10ω₀ to obtain sufficient bandwidth. Therefore the feasible range of ξ is (10, ω_e/ω₀).

Substituting the above normalization parameters into Equation (15) a normalized open-loop transfer function of the single-loop inverter can be derived as

$$G_{\mathrm{os}}(s)=\frac{\xi {\delta }^2{\omega }_{\mathrm{e}}^2{\omega }_0{G}_{\mathrm{d}}(s)}{s^3+s{\delta }^2{\omega }_{\mathrm{e}}^2}$$

(16)

Since ω₀, ω_e, and T_s are known, G_os can be determined by δ and ξ. Therefore, the inverter stability margins is also dependent on δ and ξ. These normalization parameters and their feasible region are known. Therefore, PM and GM can be analyzed based on the normalized model of Equation (16).

Suppose that the required stability margins are GM > 6dB and PM > 30°. The regions of ξ and δ that meet the stability margin requirements are shown in Figure 7. It is observed from Figure 7 that both GM and PM increase as δ rises. Nevertheless, both GM and PM decrease as ξ rises. It is found that δ = 1.5 is the optimal value such that both GM and PM are the maximum. As can be seen from Figure 7, under this circumstance, ξ should be in (10, 19.4) to guarantee GM > 6 dB and PM > 30°. For sufficient stability margins, ξ should not be set too large. Therefore, care should be taken to select ξ and δ to obtain large stability margins as well as sufficient bandwidth.

4.2. Design ω_r1 and k_p According to Phase Constraints of Z_es

It can be seen from Equation (7) that Z_c1 only contains a pair of conjugate poles on imaginary axis. Additionally, a resonant peak appears at ${\omega }_{\mathrm{r}1}=\sqrt{1/(L_{1}C)}$. In the range (0, ω_r1), the phase of Z_c1 is zero which does not affect the phase of Z_cs. However, Z_c1 has a phase jump of −180° at ω_r1. When ω > ω_r1, the phase of Z_c1 becomes −180°. The frequency response of Z_cs(s) can be expressed as

$$Z_{\mathrm{cs}}(j\omega )=\frac{{\omega }_{\mathrm{r}1}^2{Z}_{\mathrm{a}}(j\omega )}{{\omega }_{\mathrm{r}1}^2-{\omega }^2}$$

(17)

It can be seen from Equation (17) that arg[Z_cs(jω)] is the same as arg[Z_a(jω)] for ω < ω_r1. On the other hand, arg[Z_cs(jω)] is equal to arg[Z_a(jω)]−180° for ω > ω_r1. Bode diagrams of Z_cs(s) for various ω_r1 are shown as Figure 8. It is observed from Figure 8 that arg[Z_cs(jω)] approaches to −90° even lower than −90° in the vicinity of ω_r1. Figure 8 also shows the Bode diagrams of Z₂ for different L₂. It shows that |Z_cs(jω)| is much larger than |Z₂(jω)| in the vicinity of ω_r1. When ω approaches to ω_r1 from the left hand side, arg[Z_es(jω_r1)] ≈ arg[Z_a(jω_r1)] and when ω approaches ω_r1 from the right, we have arg[Z_es(jω_r1)] ≈ arg[Z_a(jω_r1)]−180°. To have arg[Z_es(jω_r1)] in (−60°, 240°), both arg[Z_a(jω_r1)] and arg[Z_a(jω_r1)]−180° should be in (−60°, 240°), that is, arg[Z_a(jω_r1)] should be in (120°, 240°).

It has been shown in the above analysis that the phase of Z_c1 is below 90° in (ω₀, ω_e), but above 90° in (ω_e, ω_s/2). The phase curve shifts up with the increase of k_p. So ω_r1 should be set in (ω_e, ω_s/2) to make the phase of Z_a(jω_r1) higher than 120°. Since ω_e is much larger than ω₀, k_r can be neglected when analyzing the frequency characteristic of Z_a in (ω_e, ω_s/2). Without considering k_r the phase frequency characteristic of Z_a can be derived as Equation (18). To normalize the phase function, we define ω_r1 = βω_e where 1 > β > 0. Then arg[Z_a(jω_r1)] can be obtained as Equation (19) by substituting ω = ω_r1 into (18).

$$\arg \left[Z_{\mathrm{a}}(j\omega )\right]={180}^{°}+\arctan \left(\frac{9{\omega }^2{T}_{\mathrm{s}}^2}{2{\lambda }_{\mathrm{p}}{\pi }^2\cos \frac{3T_{\mathrm{s}}\omega }{2}\sin \frac{T_{\mathrm{s}}\omega }{2}}-\tan \frac{3T_{\mathrm{s}}\omega }{2}\right)$$

(18)

$$\arg \left[Z_{\mathrm{a}}(j{\omega }_{\mathrm{r}1})\right]={180}^{°}+\arctan \left(\frac{{\beta }^2}{2{\lambda }_{\mathrm{p}}\cos \frac{\pi \beta }{2}\sin \frac{\beta \pi }{6}}-\tan \frac{\beta \pi }{2}\right),$$

(19)

By combining Equation (9) with Equation (14), λ_p can be rewritten as Equation (20). Substituting Equation (20) into Equation (19) yields arg[Z_a(jω_r1)] as Equation (21).

$${\lambda }_{\mathrm{p}}=\frac{36{\delta }^2\xi {\omega }_0}{{\omega }_{\mathrm{s}}^2{T}_{\mathrm{s}}({\delta }^2-{\beta }^2)}.$$

(20)

$$\arg \left[Z_{\mathrm{a}}(j{\omega }_{\mathrm{r}1})\right]={180}^{°}+\arctan \left(\frac{{\beta }^2{\omega }_{\mathrm{s}}^2{T}_{\mathrm{s}}({\delta }^2-{\beta }^2)}{72{\delta }^2\xi {\omega }_0\cos \frac{\pi \beta }{2}\sin \frac{\pi \beta }{6}}-\tan \frac{\pi \beta }{2}\right).$$

(21)

This phase arg[Z_a(jω_r1)] is a function of β only because ξ and δ have been known. Figure 9 illustrates the relationship of arg[Z_a(jω_r1)] and β for β in (1, δ). It shows that the phase is a monotonic increasing function of β. The phase arg[Z_a(jω_r1)] is equal to 120° at β = β_s1, which can be obtained by solving the equation arg[Z_a(jω_r1)] = −120°. Therefore, we have arg[Z_es(jω_r1)] $\ge -60°$ if β is in (β_s1, δ).

In addition, it can be seen from Equation (20) that λ_p is also a monotonic increasing function of β. There is an upper bound β_s2 to make λ_p = 1. It is derived from Equation (20) that β_s2 can be expressed as

$${\beta }_{\mathrm{s}2}=\delta \sqrt{1-\frac{\xi {\omega }_0}{{\omega }_{\mathrm{e}}^2{T}_{\mathrm{s}}}}.$$

(22)

Therefore, the value of β should be in (β_s1, β_s2) to have arg[Z_es(jω_r1)]$\ge -60°$. In addition, since L₂ is getting large when β approaches δ, β should be selected as β_s1 or the value slightly higher than β_s1 so that L₂ will not be too large. After β is determined, λ_p can be obtained according to Equation (20).

4.3. LCL Filter Computation According to Normalization Parameters

In this part, an LCL filter design scheme is developed by combing ω_r1 and ω_res with constraints of the switching harmonics on LCL filter. The LCL filter design can be started with the design of converter-side inductance. In the three-phase four-wire system, the maximum current ripple at the inverter output is given by equation [33]

$$\Delta I_{\max }=\frac{U_{\mathrm{dc}}}{6L_1{f}_{\mathrm{sw}}}.$$

(23)

To restrict the switching current and the core loss of L₁, the current ripple on the inverter side is recommended to be around 20% of the rated inverter current. Then the inverter-side inductance can be determined by

$$L_1\ge \frac{U_{\mathrm{dc}}}{6\times 20\%I_{\mathrm{s}}{f}_{\mathrm{sw}}}$$

(24)

where $I_{\mathrm{s}}=\frac{\sqrt{2}{P}_{\mathrm{n}}}{3U_{\mathrm{g}}}$. It is recommended that L₁ be selected at or slightly larger than the lower bound in order to keep the cost low and reduce the core loss. It follows from the relation ω_r1 = βω_e that the capacitance C can be selected by

$$C=\frac{1}{L_1{\beta }^2{\omega }_{\mathrm{e}}^2}.$$

(25)

Moreover, since the reactive power absorbed by the filter capacitance should not be more than 5%, C is limited by

$$C\le \frac{5\%P}{3{\omega }_0{U}_{\mathrm{g}}^2}.$$

(26)

Finally, the grid-side inductance L₂ can be derived as Equation (27) from the relation ω_res = δω_e.

$$L_2=\frac{1}{C{\omega }_{\mathrm{e}}^2({\delta }^2-{\beta }^2)}$$

(27)

4.4. Design of k_r

It has been shown [23] that |G_os(jω₀)| higher than 50 dB and |Z_es(jω₀)| larger than 40 dB would enable the system to have a good tracking performance. If the influence of L₂, C, and G_d on |Z_es(jω₀)| is negligible, then |Z_es(jω₀)| can be expressed as

$$\left|Z_{\mathrm{es}}(j{\omega }_0)\right|=\sqrt{{({\omega }_0{L}_1)}^2+{\left[k_{\mathrm{pwm}}(k_{\mathrm{p}}+k_{\mathrm{r}})\right]}^2}.$$

(28)

Similarly, since C and G_d only have a slight effect on |G_o(jω₀)|, ignoring C and G_d yields

$$\left|G_{\mathrm{os}}(j{\omega }_0)\right|=\frac{k_{\mathrm{pwm}}(k_{\mathrm{p}}+k_{\mathrm{r}})}{{\omega }_0(L_1+L_2)}.$$

(29)

If it is required that |Z_es(jω₀)| > 40 dB and |G_os(jω₀)| > 50 dB, then k_r should satisfy the following condition

$$k_{\mathrm{r}}\ge \max \left\{\frac{\sqrt{{10}^4-{({\omega }_0{L}_1)}^2}}{k_{\mathrm{pwm}}}-k_{\mathrm{p}},\begin{array}{c}\end{array}\frac{{10}^{2.5}{\omega }_0(L_1+L_2)}{k_{\mathrm{pwm}}}-k_{\mathrm{p}}\right\}.$$

(30)

Although ω_c is far away from ω₀, k_r still affects the phase of G_os in the controller bandwidth. With the increase of k_r, the phase of G_os in the controller bandwidth decreases. This leads to the reduction of stability margins. Therefore, an upper limit k_rm1 for k_r needs to be set to guarantee GM > 6 dB and PM > 30°.

In accordance with Equations (11) and (12), k_pt and k_r has a positive linear relationship, as shown in Figure 10. K_pt rises as k_r increases. Therefore, there is an upper bound k_rm2 that satisfies k_pt < k_p. k_rm2 can be obtained by a numerical method. Therefore, the overall constraint of k_r can be obtained by

$$\max \left\{\frac{\sqrt{{10}^4-{({\omega }_0{L}_1)}^2}}{k_{\mathrm{pwm}}}-k_{\mathrm{p}},\begin{array}{c}\end{array}\frac{{10}^{2.5}{\omega }_0(L_1+L_2)}{k_{\mathrm{pwm}}}-k_{\mathrm{p}}\right\}<k_{\mathrm{r}}<\min \left\{k_{\mathrm{rm}1},k_{\mathrm{rm}2}\right\}$$

(31)

4.5. Detailed Design Procedure

Table 1. Parameter Constraint and Design Guideline.

Parameter	Constraint	Design Guideline
δ, ξ	Region shown in Figure 7	Consider the desired bandwidth and make PM and GM as large as possible
β	(βs1, βs2)	β should be close to βs1
λp	${\lambda }_{\mathrm{p}}=\frac{36{\delta }^2\xi {\omega }_0}{{\omega }_{\mathrm{s}}^2{T}_{\mathrm{s}}({\delta }^2-{\beta }^2)}$	--
L1	$L_1\ge \frac{U_{\mathrm{dc}}}{6\times 20\%I_{\mathrm{s}}{f}_{\mathrm{sw}}}$	L1 should be as close to the lower bound as possible
C	$C=\frac{1}{L_1{\beta }^2{\omega }_{\mathrm{e}}^2}$	$C\le \frac{5\%P}{3{\omega }_0{U}_{\mathrm{g}}^2}$
L2	$L_2=\frac{1}{C{\omega }_{\mathrm{e}}^2({\delta }^2-{\beta }^2)}$	--
kp	$k_{\mathrm{p}}=\frac{{\lambda }_{\mathrm{p}}{\omega }_{\mathrm{s}}^2{L}_1{T}_{\mathrm{s}}}{36k_{\mathrm{pwm}}}$	--
kr	$\begin{array}{l}\max \left\{\frac{\sqrt{{10}^4-{({\omega }_0{L}_1)}^2}}{k_{\mathrm{pwm}}}-k_{\mathrm{p}},\frac{{10}^{2.5}{\omega }_0(L_1+L_2)}{k_{\mathrm{pwm}}}-k_{\mathrm{p}}\right\}\\ <k_{\mathrm{r}}<\min \left\{k_{\mathrm{rm}1},k_{\mathrm{rm}2}\right\}\end{array}$	kr should be as close to the upper bound as possible

. summarizes the parameter constraints and design guideline of the integrated design method. Procedure of the design is presented below.

(1)

Initialize the power converter parameters: the rated power P_n, rated ac voltage U_g, fundamental frequency f₀, dc-link voltage U_dc, sampling frequency f_s, and the switching frequency f_sw.

(2)

Refer to Figure 7 to obtain stability margins as large as possible. Set δ as 1.5.

(3)

According to Figure 7, ξ should be selected from (10, 19.4) for δ = 1.5 to satisfy the stability margin, PM > 30° and GM > 6 dB. The final ξ should be selected according to the desired stability margins and bandwidth.

(4)

Select β from (β_s1, β_s2) and β should be close to β_s1.

(5)

Compute λ_p from Equation (20). Select L₁ from Equation (24) near the lower bound.

(6)

Calculate C using Equation (25), verify that Equation (26) is satisfied, and obtain L₂ from Equation (27).

(7)

Select k_r according to Equation (31).

As seen, compared with traditional inverter design method the proposed method is simple and does not need complicated iterative computation and the trial and error method to design LCL filter and controller parameters. Additionally, a set of normalization parameters can be used to design many inverters no matter what the power level is. More importantly, the inverter will be characterized with strong stability and robustness to grid impedance by using the proposed method.

5. Case Study

A 500 kW inverter was considered to evaluate the performance of the proposed integrated design method. In the test, the sampling frequency was 16 kHz. The DC-link voltage and the grid voltage were U_dc = 700 V and U_g = 220 V, respectively. The asymmetric regular-sampling method was employed and the switching frequency was 8 kHz. The LCL filter and controller parameters designed by the proposed integrated design method and the conventional method are shown in

Table 2. New and conventional inverter parameters.

Item	L1 (μH)	C (μF)	L2 (μH)	kp	kr
new inverter	70	33.6	143.7	0.0029	1
conventional inverter	70	40	75	0.0014	0.73

. The conventional methods design LCL filter and controller parameters separately and do not consider the inverter impedance [23,34,35,36].

5.1. Parameter Design

(1)

Start with P_n = 500 kW, f_s = 16 kHz, U_dc = 700 V, U_g = 220 V, and f_sw = 8 kHz.

(2)

δ is set as 1.5 in order to have the stability margins as large as possible.

(3)

ξ is set as 15 to obtain enough bandwidth and stability margins.

(4)

According to Equations (21) and (22) the range of β is calculated as (1.23, 1.28). A tradeoff between L₂ and λ_p yields β = 1.23.

(5)

Then λ_p is calculated as 0.82 and the lower bound of L₁ is 68 μH according to Equation (24). L₁ is set as 70 μH.

(6)

C is calculated as 33.6μF, which is less than the upper bound of 548 μF from Equation (26) and L₂ as 143.7 μH.

(7)

The range of k_r is (0.2828, 1.47) obtained by Equation (31). k_r is set as 1. Additionally, k_p is calculated as 0.0029.

5.2. Performance Evaluation

Bode diagrams of G_os for new and conventional inverters are shown in Figure 11 using the parameters in Table 2. It is observed from Figure 11 that both inverters provide sufficient stability margins. Additionally, the bandwidth is large enough to produce satisfactory dynamic performance.

Bode diagrams of inverter admittance and grid admittance for new and conventional inverters are shown in Figure 12 to evaluate the robustness to grid impedance interaction. It can be seen from Figure 12 that the phase curve of the new inverter admittance is lower than 90° for the whole frequency range. Therefore, the admittance ratio K_o = Y_e/Y_g satisfies the Nyquist criterion as the grid impedance varies. Close examination reveals that the intersection frequency of the grid admittance and the inverter admittance becomes small as L_g increases. However, the phase difference between the grid and inverter admittance at the intersection frequency is always lower than 150° even for very large L_g. Therefore, the cascade system can obtain sufficient stability margin even though the inverter is connected to a weak grid.

For the conventional inverter, there is a frequency range where the admittance phase exceeds 90°. When the frequency at which the magnitude response of the inverter admittance and that of the grid admittance intersect falls in that frequency band, the PM of K_o = Y_e/Y_g becomes negative. This indicates that the grid-inverter system loses stability. Since the inverter impedance is not considered in the conventional controller design, even though the current open-loop transfer function is designed with sufficient stability margins, the inverter impedance may exist in the unstable region, where the phase of inverter admittance is larger than 90°.

6. Simulation and Experimental Results

Extensive simulations and experiments have been conducted to verify the effectiveness and robustness of the proposed integrated inverter design method in a weak grid.

6.1. Simulation Results

Parameters of the inverter used in the test are listed in Table 2. In MATLAB simulation, both current and voltage signals are sampled by ZOH and the computation delay is emulated by a delay module with one sampling period. To demonstrate the transient performance of the inverter, the reference currents of the current loop are alternated between the full load and half load.

Simulation results of the inverter output currents under different SCRs (Short Circuit Capacity Ratio) for new and conventional inverters are shown as Figure 13 and Figure 14 respectively. Reference currents step between full and half load to illustrate the transient performance. It is evidenced from Figure 13 that the steady-state current is smooth with satisfactory total harmonic distortion (THD). All THDs of the steady-state current are not larger than 2%, which satisfies with the IEEE 519 Std. However, for the conventional inverter, all THDs of the steady-state current are much larger than that of the new inverter. In particular, when L_g is larger than 61μH the THD for half load exceeds 5% which does not meet the IEEE 519 Std. Both inverters exhibit oscillation and overshoot current after the reference current steps. Additionally, the overshoot current of the new inverter is smaller than that of the conventional inverter for large grid impedance. Even worse, when L_g increases to 400 μH, currents of the conventional inverter are completely out of control, indicating that the inverter loses stability.

6.2. Experimental Results

Experimental test was conducted to evaluate the effectiveness of the proposed inverter design method. The test is based on a 10 kW experiment setup shown as Figure 15. The three-phase inverter bridge is realized by a CCS050M12CM2 module. The program of the control algorithm for the inverter is generated through a z-domain MATLAB/Simulink model and is implemented in the TMS320F28335 system. In the experiment, an inductor is inserted at the PCC in series with the programmable AC source to emulate the inductive grid impedance. Inverters designed by the proposed methods and the conventional method are shown as

Table 3. New and conventional experiment inverter parameters.

Item	L1 (mH)	C (μF)	L2 (mH)	kp	kr
new inverter	1.5	1.0	4.3	0.0776	18
conventional inverter	4.0	2.0	0.7	0.06	35

.

Output currents of the new inverter for various SCR are shown in Figure 15. In the figures, the middle panel is the zoom-in view of the left panel showing the step-down detail and the right panel is that of the step-up. It can be seen from Figure 16 that all currents are stable with small THDs in response to load changes. All THDs of the steady-state current are less than 3% satisfied with the IEEE 519 Std. The step-down transients are excellent with nearly no oscillation and no overshoot. Small overshoots appear in the step-up transients. The transient time of the step up is slightly longer than that of the step down. Additionally, longer response time is observed as the grid impedance increases.

Experimental results of such output currents are shown in Figure 17. It can be seen from that the output currents are satisfactory at L_g = 0. A disastrous high-frequency resonance happens when L_g is only increased to 3.1 mH. For protection, the inverter is disconnected to the grid. Simulation and experiment results demonstrate that the proposed integrated design method improves the inverter robustness to grid impedance.

7. Conclusions

This paper has presented an integrated design method that carries out the design of LCL filter and controller by taking their inherent relation into account. This design improves the inverter performance on stability and robustness with respect to grid impedance. A parameter normalization scheme based on parameter constraints has also been developed to facilitate the system stability and robustness analysis. Investigations have shown that the inherent LCL resonant frequency and the cutoff frequency of the current open-loop transfer function are crucial to the inverter stability margins. The proportional factor of the current controller and the resonant frequency of the inverter-side LC filter are critical to the inverter robustness. Procedures that integrate the design of LCL filter and controller parameters have been described. Additionally, parameter constraints and design guidelines for robust inverter design are derived. Simulation and experimental results have demonstrated that the proposed design method improves the inverter robustness to grid impedance, whereas the conventional design methods may fail.