# Analysis and Verification of Compound SRFPI-LADRC Strategy for an Off-Grid Single-Phase Inverter

Authors: Liaoyuan Lin; Haoda Li; Kai Zhu; Lingling Shi; Ping Li
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

Summary. The linear active disturbance rejection control (LADRC) is mainly applied for tracking a step signal, and it is inadequate to realize satisfactory performance in the control of a single-phase inverter whose reference signal is sinusoidal. In this paper, an improved voltage control strategy based on compound synchronous reference frame proportional-integral (SRFPI) control and LADRC is proposed. The SRFPI and LADRC remain relatively independent without complicating the control parameter design. Detailed theoretical analyses on the compound strategy and its performance are presented, indicating that it inherits the advantages of the two controllers. Moreover, contrast experiments are carried out to verify the feasibility and superiority of the proposed control strategy. The results show that the system achieves a small steady-state error, voltage tracking error, fast transient response, and low total harmonic distortion (THD) when feeding a highly nonlinear load.


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## Body

## 1. Introduction

Single-phase inverters, as core components, are extensively employed in various applications including industrial facilities such as uninterrupted power supply (UPS) [1,2] and renewable energy fields in terms of distributed generations (DGs) and microgrids (MGs) [3, 4]. To summarize, it is mainly adopted to supply a stable and high-quality sinusoidal voltage for ac loads. [5, 6] Therefore, a proper control technique for regulating the output voltage of single-phase inverters is quite important.Recently, a variety of control methods have been investigated for better performance of inverters. Thanks to implementation simplicity and high reliability, conventional single or dual closed-loop control systems are going with the proportional-integral-derivative (PID) regulators [7, 8]. However, it is difficult to maintain the optimality of system performance in terms of steady errors and total harmonic distortion (THD), especially under nonlinear load conditions. Surrounding the above problems, the deadbeat control (DBC), the proportional-resonant (PR) control, and the repetitive control (RC) are the main solutions. Owing to direct regulation of the instantaneous output voltage and optimal tuning, the DBC provides superior dynamic performance [9, 10]. Despite that, it is more prone to produce a steady-state error and suffers from the sensitivity of parameter variation. The PR control possesses excellent steady-state performance by introducing an infinite gain at the fundamental frequency. [11, 12] Note that multiple resonant units are required for harmonic suppression, which inevitably leads to complexity or even instability, and PR control is also constrained by sensitivity to deviations of sensed signals [13]. On the basis of the internal model principle, the RC can get rid of distortions and steady-state errors in the periodic output voltage [14, 15]. However, it exhibits inadequate performance such as slow transient response and poor tracking accuracy in the presence of aperiodic disturbances, which intensely affects its applications. In addition, some optimal controllers such as the model predictive control (MPC), the linear quadratic Gaussian (LQG) controller, and the optimal PID controllers are presented in the literature. The MPC has been extensively implemented in the control of inverters due to its simplicity and robustness [16]. Nevertheless, the computation and time cost need to be further improved for the real-time and online control platform. The LQG combines the advantages of the Kalman state estimation and the optimal linear quadratic regulator. Thus, it exhibits improved robustness and provides a fast transient response [17]. Moreover, intelligence algorithms have been introduced to realize optimal parameters for the self-turning of PID controllers for inverters [18]. However, these methods show higher complexity and cannot remove the steady-state tracking error.Currently, the synchronous reference frame proportional-integral (SRFPI) control has been broadly adapted and maturely developed in three-phase inverters. This method also increasingly appears in single-phase inverter applications [13, 19, 20]. The steady-state error can be removed by employing PI regulators in the synchronous reference frame (SRF) [21]. However, generating a virtual orthogonal signal, particularly the orthogonal current signal, may result in slow dynamic performance. Meanwhile, it has limited rejection ability with disturbances. Table 1 shows the comparison of the typical control schemes for single-phase inverters.Table 1
Feature comparison of control schemes for single-phase inverters.
Comparative itemDeadbeat controlPR controlRepetitive controlMPCSRFPI controlHardware costHighLowLowLowLowSensitivity of parameter variationStrongMediumLowMediumMediumSteady-state performanceMediumMediumGoodMediumGoodTransient performanceGoodMediumWorseGoodGoodVirtual orthogonal signalsNot requiredNot requiredNot requiredNot requiredRequiredHan comprehensively and systematically introduced the active disturbance rejection control (ADRC) method in 1999 [22]. It does not require the exact system mathematical model. After that, Gao proposed linear ADRC (LADRC) based on the linearization of the extended state observer (ESO) and the controller, which greatly simplified the parameter setting [23, 24]. It treats internal and external disturbances as the total disturbance and utilizes the input and output targets to estimate the disturbances for disturbance rejection [25]. In addition, as a general-purposed control structure [26], the LADRC has been widely used in control systems with step reference signals such as servo systems [27, 28] and three-phase inverters in the SRF [29, 30] but relatively few in single-phase inverters [31–33]. The main reason is that it is unable to eliminate the steady-state error when tracking a sinusoidal signal. In Ref. 31, a step-by-step analysis of the classic LADRC for a single-phase inverter is presented. However, there exists a non-negligible steady-state error in the output voltage. An improved control law using reference differential feedforward for a grid-tied inverter is proposed in Ref. 32 to reduce the steady-state error. Its performance used in off-grid inverters needs further verification. In Ref. 33, a repetitive learning controller is embedded in LADRC to achieve a high-quality output voltage and robustness. However, this approach adds complexity to the LADRC design.To improve the performance of LADRC used in an off-grid single-phase inverter, in this paper, an improved voltage control strategy based on compound SRFPI and LADRC is proposed. The performance verification of this method is focused on the inverter under resistive load, nonlinear load, and step load (no load to resistive load). The salient features of the proposed method are as follows:(1)
The output of SRFPI serves as the voltage reference of LADRC. In terms of control structure, SRFPI and LADRC are relatively independent, which does not complicate the controller design of LADRC.(2)
The proposed method does not contain current loops. Therefore, it does not need to generate the current quadrature signals that may affect the dynamic response. Moreover, high precision and high-cost current sensors are not required for waveform control under normal conditions. It is still necessary to measure the inductor current for overcurrent protection, but more economical ways can be adopted. This part is out of the scope of this paper.(3)
The compound method fuses the merits of both two controllers. Theoretical analysis, simulation, and experimental results are provided to confirm that the proposed scheme brings an overall performance improvement in terms of tracking error, steady-state error, dynamic response as well as voltage THD, especially when feeding highly nonlinear loads.This paper is organized as follows: the implementation of LADRC in a single-phase inverter is introduced in Section2. The proposed compound control strategy and corresponding theoretical analysis are given in Section 3. Experimental results are discussed in Section 4. Finally, conclusions are summarized in Section 5.


## 2. Implementation of LADRC in Single-Phase Inverters

### 2.1. Modelling of Single-Phase Inverters

A single-phase inverter that consists of the insulated-gate bipolar transistor (IGBT) full-bridge frame with anLC output filter is shown in Figure 1. Vdc,vin, vo, iL, and iodenote the dc link voltage, input voltage, output voltage, inductor current, and output current; re, L, and C denote the equivalent series resistance, filter inductor, and filter capacitor, respectively. According to Kirchhoff’s laws, the differential equations describing the dynamics of a single-phase inverter can be written as(1)LdiLdt=vin−vo−reiLCdvodt=iL−io.Figure 1
Single-phase inverter with theLC filter.The relation between the sinusoidal modulation signalvm and vin can be derived as a PWM proportional gain: KPWM = vin/vm = Vdc/Vt, where Vt is the amplitude of the triangular carrier, and its value is set to 1. To obtain a better voltage regulation, a proportional gain 1/Vdc is connected in series with the control signal u. Considering io as the input disturbance, the equivalent model of the inverter is illustrated in Figure 2. The transfer function of the plant can be obtained as(2)Gps=vosvins=vosus=1LCs2+reCs+1.Figure 2
Equivalent block diagram of the single-phase inverter.According to (2), the differential form of the transfer function of the inverter can be described as(3)v¨o=−a1v˙o−a0vo+b0u,where a0 = b0 = 1/LC and a1 = re/L.


### 2.2. Design of LESO

The core idea of LADRC is to implement a linear extended state observer (LESO) for estimating state variables and total disturbance of the system, and then a forward feedback rejection is built for disturbance compensation.As seen in (2) and (3), the control object is a second-order system, and it is a nonstandard integral series system (v¨o =b0u). In general, LESO takes full advantage of the system model to calculate control variables and treats the parts different from the standard integral series system as disturbances. Thus, the known disturbances can be expressed as f0(v˙o,vo) = −a1v˙o−a0vo, where a0 and a1 are the known parameters. There are many unknown internal and external disturbances such as parameter perturbation, sampling noise, and unmodeled dynamics during the actual operation of the inverter. The unclear disturbances are denoted by d. The value of b0 can be directly obtained from the actual inverter system while the unclear part of the system model is described as b1. To sum up, the actual total unknown disturbance can be defined as f1 = d + b1u. Regarding f = f1+ f0(v˙o,vo) as the total disturbance of the system, the actual inverter system can be rewritten as(4)v¨o=b0u+f.Letx = [x1x2x3]T = [vov˙of ]T and y = vo. The disturbance can be added to the following state space representation by introducing an extended state as(5)x˙=Apx+Bpu+Epf˙1,y=Cpx,where(6)Ap=0100010−a0−a1,Bp=0b0−a1b0,Ep=001,Cp=100.From (5), system matrix Ap contains the known model coefficients (a0, a1). Let x^=x^1x^2x^3T=v^ov˙^of^T and Lp=lp1lp2lp3T. The model-assisted LESO (MA-LESO) is designed as(7)x^˙=Apx^+Bpu+Lpy−y^=Ap−LpCpx^+Bpu+Lpy,y^=Cpx^,where x^ and Lp denote the observed state variable and gain matrix of the LESO. The eigenvalues of Ap-LpCp determine the decay rate of the observed error. To simplify the analysis, its poles are arranged at the same position as(8)λs=sI−Ap+LpCp=s+ωo3,where ωo is the bandwidth of the observer. It can be derived that(9)l1=3ωo−a1,l2=3ωo2−3a1ωo−a0+a12,l3=ωo3−3a1ωo2+3a12−a0ωo+2a0a1−a13.By selecting an appropriateωo, the LESO can realize rapid real-time estimations of system state variables and the total disturbance.


### 2.3. Design of the Control Signal

Based on the observed state variables and disturbance, the control signalu and error feedback control law uc are designed as shown in Equation (10). Substituting u into Equation (4), the system turns into standard integral series type (v¨o=b0u+f≈uc).(10)u=−x^3+ucb0=−x^3+k1r−x^1−k2x^2b0,where r is the reference signal, and k1 and k2 denote the error feedback gain coefficient. The bandwidth tuning method in Ref. 24 is adopted to further simplify the controller parameter design, where k1 = 2ωc2, k2 = 2ωc, and ωc denotes the controller bandwidth.In summary, the control block diagram of LADRC for a single-phase inverter mainly contains the error feedback control law, the disturbance compensation block, and MA-LESO, which can be illustrated in Figure3. For investigating the performance of the LESO, the input disturbance δu and the observation noise δy are considered, as advised in Ref. 32.Figure 3
Control block diagram of LADRC for the single-phase inverter.


### 2.4. Performance Analysis of LESO

The deviations between observed values and actual state variables directly affect the control performance of LADRC. The observation error of the output voltage is defined asxe1=x^1−x1. To analyze the effect of the bandwidth ωo on LESO performance, the observation accuracy of output voltage can be studied with the transfer function of xe1/f1. Moreover, the disturbance rejection performance can be investigated through the transfer functions of x^1/δu and x^1/δy. According to Equations (5)–(10), the following three transfer functions can be obtained as(11)xe1f1=−ss+ωo3,x^1δu=b0ss+ωo3,x^1δy=3ωo+a1s2+a0+3ωo2s+ωo3s+ωo3.In this study, the inverter parameters are chosen asL = 0.7mH, C = 40 μF, and re = 0.1 Ω. The Bode diagrams of xe1/f1, x^1/δu, and/δy with ωo ranging from 8000 to 14000 rad/s are depicted in Figures 4 and 5, respectively. From Figure 4, it can be seen a larger ωo makes the observed value x^1 close to x1. In other words, f1 is well estimated and compensated due to the tiny gain in the low-frequency region. However, the estimated ability gets ordinary when the disturbances contain high-frequency signals. As shown in Figure 5(a), when a larger ωo is chosen, the LESO achieves enhanced input disturbance rejection in the low-frequency band, but it has little impact on the high-frequency domain. Clearly, as seen from Figure 5(b), the high-frequency observation noise will be amplified as ωo increases, while the LESO represents insensitivity to observation noise in the low-frequency region. Therefore, a proper bandwidth of the LESO should compromise the disturbance estimation performance and the measurement noise sensitivity. In this study, ωo = 10000 rad/s is selected.Figure 4
Phase-frequency characteristics ofxe1/f1.Figure 5
Phase-frequency characteristics of (a)x^1/δu and (b)x^1/δy.


### 2.5. Analysis of the Steady-State Error

The tracking error of the inverter system is defined ase = r−vo. According to (4) and (10), the transfer function of e can be obtained as(12)es=s2+k2ss2+k2s+k1rs.Clearly, the LADRC-based voltage control for a single-phase inverter cannot eliminate the steady-state error since the reference signalr is sinusoidal.


## 2.1. Modelling of Single-Phase Inverters

A single-phase inverter that consists of the insulated-gate bipolar transistor (IGBT) full-bridge frame with anLC output filter is shown in Figure 1. Vdc,vin, vo, iL, and iodenote the dc link voltage, input voltage, output voltage, inductor current, and output current; re, L, and C denote the equivalent series resistance, filter inductor, and filter capacitor, respectively. According to Kirchhoff’s laws, the differential equations describing the dynamics of a single-phase inverter can be written as(1)LdiLdt=vin−vo−reiLCdvodt=iL−io.Figure 1
Single-phase inverter with theLC filter.The relation between the sinusoidal modulation signalvm and vin can be derived as a PWM proportional gain: KPWM = vin/vm = Vdc/Vt, where Vt is the amplitude of the triangular carrier, and its value is set to 1. To obtain a better voltage regulation, a proportional gain 1/Vdc is connected in series with the control signal u. Considering io as the input disturbance, the equivalent model of the inverter is illustrated in Figure 2. The transfer function of the plant can be obtained as(2)Gps=vosvins=vosus=1LCs2+reCs+1.Figure 2
Equivalent block diagram of the single-phase inverter.According to (2), the differential form of the transfer function of the inverter can be described as(3)v¨o=−a1v˙o−a0vo+b0u,where a0 = b0 = 1/LC and a1 = re/L.


## 2.2. Design of LESO

The core idea of LADRC is to implement a linear extended state observer (LESO) for estimating state variables and total disturbance of the system, and then a forward feedback rejection is built for disturbance compensation.As seen in (2) and (3), the control object is a second-order system, and it is a nonstandard integral series system (v¨o =b0u). In general, LESO takes full advantage of the system model to calculate control variables and treats the parts different from the standard integral series system as disturbances. Thus, the known disturbances can be expressed as f0(v˙o,vo) = −a1v˙o−a0vo, where a0 and a1 are the known parameters. There are many unknown internal and external disturbances such as parameter perturbation, sampling noise, and unmodeled dynamics during the actual operation of the inverter. The unclear disturbances are denoted by d. The value of b0 can be directly obtained from the actual inverter system while the unclear part of the system model is described as b1. To sum up, the actual total unknown disturbance can be defined as f1 = d + b1u. Regarding f = f1+ f0(v˙o,vo) as the total disturbance of the system, the actual inverter system can be rewritten as(4)v¨o=b0u+f.Letx = [x1x2x3]T = [vov˙of ]T and y = vo. The disturbance can be added to the following state space representation by introducing an extended state as(5)x˙=Apx+Bpu+Epf˙1,y=Cpx,where(6)Ap=0100010−a0−a1,Bp=0b0−a1b0,Ep=001,Cp=100.From (5), system matrix Ap contains the known model coefficients (a0, a1). Let x^=x^1x^2x^3T=v^ov˙^of^T and Lp=lp1lp2lp3T. The model-assisted LESO (MA-LESO) is designed as(7)x^˙=Apx^+Bpu+Lpy−y^=Ap−LpCpx^+Bpu+Lpy,y^=Cpx^,where x^ and Lp denote the observed state variable and gain matrix of the LESO. The eigenvalues of Ap-LpCp determine the decay rate of the observed error. To simplify the analysis, its poles are arranged at the same position as(8)λs=sI−Ap+LpCp=s+ωo3,where ωo is the bandwidth of the observer. It can be derived that(9)l1=3ωo−a1,l2=3ωo2−3a1ωo−a0+a12,l3=ωo3−3a1ωo2+3a12−a0ωo+2a0a1−a13.By selecting an appropriateωo, the LESO can realize rapid real-time estimations of system state variables and the total disturbance.


## 2.3. Design of the Control Signal

Based on the observed state variables and disturbance, the control signalu and error feedback control law uc are designed as shown in Equation (10). Substituting u into Equation (4), the system turns into standard integral series type (v¨o=b0u+f≈uc).(10)u=−x^3+ucb0=−x^3+k1r−x^1−k2x^2b0,where r is the reference signal, and k1 and k2 denote the error feedback gain coefficient. The bandwidth tuning method in Ref. 24 is adopted to further simplify the controller parameter design, where k1 = 2ωc2, k2 = 2ωc, and ωc denotes the controller bandwidth.In summary, the control block diagram of LADRC for a single-phase inverter mainly contains the error feedback control law, the disturbance compensation block, and MA-LESO, which can be illustrated in Figure3. For investigating the performance of the LESO, the input disturbance δu and the observation noise δy are considered, as advised in Ref. 32.Figure 3
Control block diagram of LADRC for the single-phase inverter.


## 2.4. Performance Analysis of LESO

The deviations between observed values and actual state variables directly affect the control performance of LADRC. The observation error of the output voltage is defined asxe1=x^1−x1. To analyze the effect of the bandwidth ωo on LESO performance, the observation accuracy of output voltage can be studied with the transfer function of xe1/f1. Moreover, the disturbance rejection performance can be investigated through the transfer functions of x^1/δu and x^1/δy. According to Equations (5)–(10), the following three transfer functions can be obtained as(11)xe1f1=−ss+ωo3,x^1δu=b0ss+ωo3,x^1δy=3ωo+a1s2+a0+3ωo2s+ωo3s+ωo3.In this study, the inverter parameters are chosen asL = 0.7mH, C = 40 μF, and re = 0.1 Ω. The Bode diagrams of xe1/f1, x^1/δu, and/δy with ωo ranging from 8000 to 14000 rad/s are depicted in Figures 4 and 5, respectively. From Figure 4, it can be seen a larger ωo makes the observed value x^1 close to x1. In other words, f1 is well estimated and compensated due to the tiny gain in the low-frequency region. However, the estimated ability gets ordinary when the disturbances contain high-frequency signals. As shown in Figure 5(a), when a larger ωo is chosen, the LESO achieves enhanced input disturbance rejection in the low-frequency band, but it has little impact on the high-frequency domain. Clearly, as seen from Figure 5(b), the high-frequency observation noise will be amplified as ωo increases, while the LESO represents insensitivity to observation noise in the low-frequency region. Therefore, a proper bandwidth of the LESO should compromise the disturbance estimation performance and the measurement noise sensitivity. In this study, ωo = 10000 rad/s is selected.Figure 4
Phase-frequency characteristics ofxe1/f1.Figure 5
Phase-frequency characteristics of (a)x^1/δu and (b)x^1/δy.


## 2.5. Analysis of the Steady-State Error

The tracking error of the inverter system is defined ase = r−vo. According to (4) and (10), the transfer function of e can be obtained as(12)es=s2+k2ss2+k2s+k1rs.Clearly, the LADRC-based voltage control for a single-phase inverter cannot eliminate the steady-state error since the reference signalr is sinusoidal.


## 3. Design and Analysis of SRFPI-LADRC

As described above, a single-phase inverter with LADRC is contaminated by the steady-state error. In this paper, a compound control strategy is proposed as shown in Figure6, where the output voltage vcα of SRFPI is employed as the voltage reference of LADRC. In terms of control structure, the SRFPI and the LADRC are relatively independent. The control structure combines the superiority of both controllers, namely disturbance resisting ability and cancellation of the steady-state error.Figure 6
Proposed voltage control structure based on combined SRFPI and LADRC.


### 3.1. Analysis of the SRPFI Controller

Control of single-phase inverters in direct-quadrature SRF is not as well developed as three-phase applications. More complicated structures and the demand for virtual orthogonal signals are the main reasons. The conventional orthogonal-signal-generation (OSG) approach utilizes a delay block to produce an imaginary orthogonal signal. In this study, the all-pass filter (APF)-based OSG is adopted. It is straightforward and attenuates no higher-order components. [34] The structure of the first-order AFP is illustrated in Figure 7, and its transfer function is as follows:(13)HAPFs=vβsvαs=ωf−sωf+s,where ωf is the fundamental angular frequency.Figure 7
Structure of a first-order APF.To conquer the difficulties of stability analysis of the system when the voltage loop serves in the SRF, the advised method in Ref. 19 is used. The equivalent control block of SRFPI is depicted in Figure8, where vr, kp, and ki are the set reference, proportional, and integral coefficient of the PI controller, respectively. The SRFPI can be derived in the time domain as(14)vcαtvcβt=cosωft−sinωftsinωftcosωft⋅GPIt00GPIt∗cosωftsinωft−sinωftcosωfteαteβt,where ∗ is the convolution operator.Figure 8
The equivalent control block of SRFPI.Taking Laplace transform and substitutingGPI(s) = kp + ki/s, (14) can be described in the s-domain as(15)vcαsvcβs=kp+kiss2+ωf2−kiωfs2+ωf2kiωfs2+ωf2kp+kiss2+ωf2eαseβs.By substituting Equation (13) into (15), the transfer function H(s) can be written as(16)Hs=vcαseαs=c3s3+c2s2+c1s+c0s2+ωf2s+ωf,where c3 = kp, c2 = kpωf + ki, c1 = kpωf2+2ωfki, c0 = kpω3f−kiω2f. According to (16) and Figure 8, the transfer function e(s)/vr(s) can be derived. Substituting s = jωf into e(s)/vr(s) yields(17)esvrs|s=jωf=11+HsGpcs|s=jωf=0.(15) clearly shows that the system realizes zero fundamental voltage tracking error based on the SRFPI control strategy in a steady state.


### 3.2. Design of SRPFI-LADRC

To obtain the transfer function of the whole system, the following equations can be obtained by substituting Equations (11) into (7).(18)x^˙1x^˙2x^˙3=−l110−l2−k1−k20−l3+a1k1−a0+a1k20x^1x^2x^3+0l1k1l2−a1k1l3ry.Taking the Laplace transform from both sides of Equations (18), G1(s) and G2(s) can be derived as(19)G1s=−vosrs=h3s3+h2s2+h1s+h0g2s2+g1s+g0,G2s=us−vos,=g2s2+g1s+g0b0s3+b0k2+b0l1s2+b0k1+b0l2+b0k2l1s,where h3 = k1, h2 = a1k1+k1l1, h1 = a0k1+k1l2+a1k1l1, and h0 = k1l3+a0k1l1+a1k1l2. g2 = l3+k1l1+k2l2, g1 = k1l2+k2l3−a0l2+a1k1l1+a1k2l2, and g0 = k1l3+a0k1l1+a1k1l2, respectively.According to (16) and (19), the control block diagram of the proposed method can be equivalently depicted in Figure 9. According to the superposition theorem, the closed-loop system can be represented as(20)vos=Gcsvrs+Zsios=HsG1sG2sGps1+G2sGps+HsG1sG2sGpsvrs+Gds1+G2sGps+HsG1sG2sGpsios,where Gc(s) and Z(s) denote the no-load voltage gain and the equivalent output impedance, respectively.Figure 9
Block diagram of the proposed control system.By replacingio(s) with vo(s)/Zl (s), where Zl (s) denotes the load, the transfer function e(s)/vr(s) can be obtained. Substituting s = jωf into it gives(21)esvrs|s=jωf=0.Therefore, the proposed control strategy can effectively eliminate the fundamental voltage tracking error, which means the advantage of SRFPI can be retained.


### 3.3. Stability and Dynamic Performance

Based on the stability criteria, the system in Figure9 is stable if the closed-loop transfer function has no right half plane (RHP) poles. From (20), the characteristic polynomial of the closed-loop system can be derived as(22)s5+m4s4+m3s3+m2s2+m1s+m0=0,wherem4 = k2+ωf, m3 = ωf2 + k2ωf + k1+k1kp, m2 = k1ki + k1ωf + k2ω2f+ω3f + k1kpωf, m1 = 2k1kiωf + k1ω2f + k1kpω2f + k2ω3f, and m0 = k1ω3f−k1kiω2f + k1kpω3f, respectively.The stability and dynamic performance of the system with varied control parameters are given. The parameters of the inverter system are listed in Table2. Figure 10 shows the root locus of the system under varied integral gain ki, which changes from 10 to 500. The system remains stable since no poles exist in RHP. The dominant poles λ1 and λ2 move from right to left while λ1 moves towards the imaginary axis indicating that the stability and dynamic performance of the system is improved with increased ki within a certain range.Table 2
Parameters of the inverter system.
ParametersValuesParametersValuesSwitching frequency,fs20 kHzDC-link voltage,Vdc190 VFundamental angular frequency,ωf100π rad/sThe bandwidth of the controller,ωc5500 rad/sFilter inductance,L700μHThe bandwidth of the observer,ωo10000 rad/sFilter capacitance,C40μFThe proportional factor of the PI controller,kp1.5ESR of the inductance,re0.1 ΩIntegral factor of the PI controller,ki100Voltage reference,vd∗/vq∗156 V/0 VDead time,td1.3μsFigure 10
Generalized root locus diagram with a variation ofki (10≤ki ≤ 500, kp = 1.5, and ωc = 5500 rad/s).Figure11 shows the root locus of the system under varied kp, which changes from 0.1 to 1.5. The roots of characteristic polynomials all lie in the left half plane (LHP), which manifests the system and keeps stable. The system becomes more oscillatory since poles λ2 and λ3 are getting close to the imaginary axis, leading to the reduction of the damping ratio.Figure 11
Generalized root locus diagram with a variation ofkp (0.1≤kp ≤ 1.5, ki = 100, and ωc = 5500 rad/s).The root locus withωc increasing from 2000 to 8000 rad/s is shown in Figure 12. The system maintains stability. The damping ratio of the system is almost constant since the positions of the dominant poles λ2 and λ3 change little. The locus of pole λ1 indicates that the settling time slightly gets shorter.Figure 12
Generalized root locus diagram with a variation ofωc (2000≤ωc ≤ 8000 rad/s, ki = 100, and kp = 1.5).


### 3.4. Disturbance Immunity Analysis

The disturbance rejection characteristics of the closed-loop system are investigated. The output currentio is regarded as the external disturbance d. The transfer function vo(s)/d(s) can be obtained as(23)vosds=Gds1+G2sGps+HsG1sG2sGps.tFigure13 shows the amplitude-frequency diagram of vo(s)/d(s) with different controller parameters. As shown in Figure 13(a), with increase in the observer bandwidth ωo and the controller bandwidth ωc, the disturbance suppression ability of the system is enhanced. Furthermore, the parameters of kp and ki have little effect on the disturbance rejection as can be seen from Figure 13(b). Based on the above analysis, in this work, ωc = 5500 rad/s, ωo = 10000 rad/s, ki = 100, and kp = 1.5 are chosen, as listed in Table 2.Figure 13
Bode magnitude plots ofvo(s)/d(s). (a) Variation of ωo and ωc and (b) variation of kp and ki.The comparison of disturbance-resistant performance under three control strategies is shown in Figure14. The SRFPI control includes an inner capacitor current proportional loop [20], and the corresponding proportional gain is set as 3. Notice that the proposed SRFPI-LADRC has a better inhibition effect on disturbance at the low-frequency region, especially at the fundamental frequency.Figure 14
Bode magnitude plots ofvo(s)/d(s) under different control strategies.


### 3.5. Robustness Analysis

In practice, the parameters of theLC filter may vary because of aging and different operating conditions, which may deteriorate the system’s performance. Considering the mismatch between the actual values (LA, CA) and nominal values (L, C), the robustness of the system is investigated. The parameter mismatch factor is defined as n = LA/L = CA/C. Considering that the parametric perturbation is usually within ±20%, [35] the generalized root locus under varied LA and CA, where n varies from -20% to +20%, is shown in Figure 15. The movement trajectory of the poles indicates that the system remains stable, which means the proposed controller is robust to parametric variations.Figure 15
Generalized root locus diagram ofGc(s) with a variation of n.To verify the robustness of the proposed control strategy, a simulation model of a single-phase inverter system is built in MATLAB/Simulink environment, using the parameters listed in Table2. The mismatch of LC parameters is considered under nominal resistive load (20 Ω). The results are shown in Figure 16, where the tracking error e = vo−vr, and e(RMS) denotes the root mean square (RMS) value of e. The system is robust against filter parameter perturbation.Figure 16
Perturbation of parameters under nominal resistive load conditions.


## 3.1. Analysis of the SRPFI Controller

Control of single-phase inverters in direct-quadrature SRF is not as well developed as three-phase applications. More complicated structures and the demand for virtual orthogonal signals are the main reasons. The conventional orthogonal-signal-generation (OSG) approach utilizes a delay block to produce an imaginary orthogonal signal. In this study, the all-pass filter (APF)-based OSG is adopted. It is straightforward and attenuates no higher-order components. [34] The structure of the first-order AFP is illustrated in Figure 7, and its transfer function is as follows:(13)HAPFs=vβsvαs=ωf−sωf+s,where ωf is the fundamental angular frequency.Figure 7
Structure of a first-order APF.To conquer the difficulties of stability analysis of the system when the voltage loop serves in the SRF, the advised method in Ref. 19 is used. The equivalent control block of SRFPI is depicted in Figure8, where vr, kp, and ki are the set reference, proportional, and integral coefficient of the PI controller, respectively. The SRFPI can be derived in the time domain as(14)vcαtvcβt=cosωft−sinωftsinωftcosωft⋅GPIt00GPIt∗cosωftsinωft−sinωftcosωfteαteβt,where ∗ is the convolution operator.Figure 8
The equivalent control block of SRFPI.Taking Laplace transform and substitutingGPI(s) = kp + ki/s, (14) can be described in the s-domain as(15)vcαsvcβs=kp+kiss2+ωf2−kiωfs2+ωf2kiωfs2+ωf2kp+kiss2+ωf2eαseβs.By substituting Equation (13) into (15), the transfer function H(s) can be written as(16)Hs=vcαseαs=c3s3+c2s2+c1s+c0s2+ωf2s+ωf,where c3 = kp, c2 = kpωf + ki, c1 = kpωf2+2ωfki, c0 = kpω3f−kiω2f. According to (16) and Figure 8, the transfer function e(s)/vr(s) can be derived. Substituting s = jωf into e(s)/vr(s) yields(17)esvrs|s=jωf=11+HsGpcs|s=jωf=0.(15) clearly shows that the system realizes zero fundamental voltage tracking error based on the SRFPI control strategy in a steady state.


## 3.2. Design of SRPFI-LADRC

To obtain the transfer function of the whole system, the following equations can be obtained by substituting Equations (11) into (7).(18)x^˙1x^˙2x^˙3=−l110−l2−k1−k20−l3+a1k1−a0+a1k20x^1x^2x^3+0l1k1l2−a1k1l3ry.Taking the Laplace transform from both sides of Equations (18), G1(s) and G2(s) can be derived as(19)G1s=−vosrs=h3s3+h2s2+h1s+h0g2s2+g1s+g0,G2s=us−vos,=g2s2+g1s+g0b0s3+b0k2+b0l1s2+b0k1+b0l2+b0k2l1s,where h3 = k1, h2 = a1k1+k1l1, h1 = a0k1+k1l2+a1k1l1, and h0 = k1l3+a0k1l1+a1k1l2. g2 = l3+k1l1+k2l2, g1 = k1l2+k2l3−a0l2+a1k1l1+a1k2l2, and g0 = k1l3+a0k1l1+a1k1l2, respectively.According to (16) and (19), the control block diagram of the proposed method can be equivalently depicted in Figure 9. According to the superposition theorem, the closed-loop system can be represented as(20)vos=Gcsvrs+Zsios=HsG1sG2sGps1+G2sGps+HsG1sG2sGpsvrs+Gds1+G2sGps+HsG1sG2sGpsios,where Gc(s) and Z(s) denote the no-load voltage gain and the equivalent output impedance, respectively.Figure 9
Block diagram of the proposed control system.By replacingio(s) with vo(s)/Zl (s), where Zl (s) denotes the load, the transfer function e(s)/vr(s) can be obtained. Substituting s = jωf into it gives(21)esvrs|s=jωf=0.Therefore, the proposed control strategy can effectively eliminate the fundamental voltage tracking error, which means the advantage of SRFPI can be retained.


## 3.3. Stability and Dynamic Performance

Based on the stability criteria, the system in Figure9 is stable if the closed-loop transfer function has no right half plane (RHP) poles. From (20), the characteristic polynomial of the closed-loop system can be derived as(22)s5+m4s4+m3s3+m2s2+m1s+m0=0,wherem4 = k2+ωf, m3 = ωf2 + k2ωf + k1+k1kp, m2 = k1ki + k1ωf + k2ω2f+ω3f + k1kpωf, m1 = 2k1kiωf + k1ω2f + k1kpω2f + k2ω3f, and m0 = k1ω3f−k1kiω2f + k1kpω3f, respectively.The stability and dynamic performance of the system with varied control parameters are given. The parameters of the inverter system are listed in Table2. Figure 10 shows the root locus of the system under varied integral gain ki, which changes from 10 to 500. The system remains stable since no poles exist in RHP. The dominant poles λ1 and λ2 move from right to left while λ1 moves towards the imaginary axis indicating that the stability and dynamic performance of the system is improved with increased ki within a certain range.Table 2
Parameters of the inverter system.
ParametersValuesParametersValuesSwitching frequency,fs20 kHzDC-link voltage,Vdc190 VFundamental angular frequency,ωf100π rad/sThe bandwidth of the controller,ωc5500 rad/sFilter inductance,L700μHThe bandwidth of the observer,ωo10000 rad/sFilter capacitance,C40μFThe proportional factor of the PI controller,kp1.5ESR of the inductance,re0.1 ΩIntegral factor of the PI controller,ki100Voltage reference,vd∗/vq∗156 V/0 VDead time,td1.3μsFigure 10
Generalized root locus diagram with a variation ofki (10≤ki ≤ 500, kp = 1.5, and ωc = 5500 rad/s).Figure11 shows the root locus of the system under varied kp, which changes from 0.1 to 1.5. The roots of characteristic polynomials all lie in the left half plane (LHP), which manifests the system and keeps stable. The system becomes more oscillatory since poles λ2 and λ3 are getting close to the imaginary axis, leading to the reduction of the damping ratio.Figure 11
Generalized root locus diagram with a variation ofkp (0.1≤kp ≤ 1.5, ki = 100, and ωc = 5500 rad/s).The root locus withωc increasing from 2000 to 8000 rad/s is shown in Figure 12. The system maintains stability. The damping ratio of the system is almost constant since the positions of the dominant poles λ2 and λ3 change little. The locus of pole λ1 indicates that the settling time slightly gets shorter.Figure 12
Generalized root locus diagram with a variation ofωc (2000≤ωc ≤ 8000 rad/s, ki = 100, and kp = 1.5).


## 3.4. Disturbance Immunity Analysis

The disturbance rejection characteristics of the closed-loop system are investigated. The output currentio is regarded as the external disturbance d. The transfer function vo(s)/d(s) can be obtained as(23)vosds=Gds1+G2sGps+HsG1sG2sGps.tFigure13 shows the amplitude-frequency diagram of vo(s)/d(s) with different controller parameters. As shown in Figure 13(a), with increase in the observer bandwidth ωo and the controller bandwidth ωc, the disturbance suppression ability of the system is enhanced. Furthermore, the parameters of kp and ki have little effect on the disturbance rejection as can be seen from Figure 13(b). Based on the above analysis, in this work, ωc = 5500 rad/s, ωo = 10000 rad/s, ki = 100, and kp = 1.5 are chosen, as listed in Table 2.Figure 13
Bode magnitude plots ofvo(s)/d(s). (a) Variation of ωo and ωc and (b) variation of kp and ki.The comparison of disturbance-resistant performance under three control strategies is shown in Figure14. The SRFPI control includes an inner capacitor current proportional loop [20], and the corresponding proportional gain is set as 3. Notice that the proposed SRFPI-LADRC has a better inhibition effect on disturbance at the low-frequency region, especially at the fundamental frequency.Figure 14
Bode magnitude plots ofvo(s)/d(s) under different control strategies.


## 3.5. Robustness Analysis

In practice, the parameters of theLC filter may vary because of aging and different operating conditions, which may deteriorate the system’s performance. Considering the mismatch between the actual values (LA, CA) and nominal values (L, C), the robustness of the system is investigated. The parameter mismatch factor is defined as n = LA/L = CA/C. Considering that the parametric perturbation is usually within ±20%, [35] the generalized root locus under varied LA and CA, where n varies from -20% to +20%, is shown in Figure 15. The movement trajectory of the poles indicates that the system remains stable, which means the proposed controller is robust to parametric variations.Figure 15
Generalized root locus diagram ofGc(s) with a variation of n.To verify the robustness of the proposed control strategy, a simulation model of a single-phase inverter system is built in MATLAB/Simulink environment, using the parameters listed in Table2. The mismatch of LC parameters is considered under nominal resistive load (20 Ω). The results are shown in Figure 16, where the tracking error e = vo−vr, and e(RMS) denotes the root mean square (RMS) value of e. The system is robust against filter parameter perturbation.Figure 16
Perturbation of parameters under nominal resistive load conditions.


## 4. Experimental Results

According to the previous analysis, the output current caused by the loads or load variations can be seen as the main external disturbance, while the internal disturbances mainly refer to the dead time, parasitic parameters, parameter perturbation, and so on. These disturbances will deteriorate the system’s performance.To verify the effectiveness of the proposed control strategy, a 605-W single-phase inverter prototype was established as shown in Figure17. The experimental setup is composed of a programmable dc power supply, a full-bridge IGBT module, an LC filter, an auxiliary power module, and a set of linear and nonlinear loads. The control algorithm is implemented in a DSP chip (TMS320F28335), and the digital oscilloscope is used to record the output voltage and current waveforms. The related parameters can be found in Table 2. The SRFPI control, the conventional LADRC, and the compound SRFPI-LADRC are tested and compared, and the SRFPI control includes an inner capacitor current proportional loop with a proportional gain of 3. To make the comparison persuasive, parameters of the overlapping parts of the three schemes stay consistent intentionally.Figure 17
Experimental setup.Figure18 shows the experimental steady-state waveforms under a rated resistive load (20 Ω). Compared with the SRFPI and LADRC, the proposed method has the lowest THD (2.44% vs. 2.78% vs. 1.87%).Figure 18
Steady-state waveforms of different control strategies under rated linear load (20 Ω). (a) SRFPI, (b) LADRC, and (c) proposed SRFPI-LADRC.Figure19 depicts the dynamic responses under a step load from no load to a rated load. The proposed method possesses the shortest transient response time (1.6 ms vs. 1.2 vs. 0.8 ms). In addition, it is noted that the voltage amplitude of the proposed method recovers in one period, which further illustrates its great performance against load disturbances.Figure 19
Transient waveforms in response to no load to rated load (20 Ω) step change. (a) SRFPI, (b) LADRC, and (c) proposed SRFPI-LADRC.Figure20 shows the waveforms of output voltage and output current when feeding a highly nonlinear load, which consists of a 1-Ω resistor in series and a diode rectifier bridge feeding a 2700-μF capacitor in parallel with a 30-Ω resistor. All the output currents are significantly distorted with a crest factor approaching 3. Despite that, compared with SRFPI and LADRC, the output voltage of the proposed strategy represented a very small distortion (THD = 2.14%).Figure 20
Steady-state waveforms of different control strategies under highly nonlinear load conditions. (a) SRFPI, (b) LADRC, and (c) proposed SRFPI-LADRC.To verify the robustness of the proposed method, the waveforms under nonlinear load withLA parameter variations (±20%) are shown in Figure 21. The steady-state waveforms indicate that the system remains stable and has small voltage THD (2.18% and 2.23%). In short, the SRFPI-LADRC has good robustness against system parameter perturbation.Figure 21
Steady-state waveforms with differentLA. (a) LA/L=1.2 and (b) LA/L = 0.8.The performances of the three methods, measured in terms of voltage THD, RMS values of output voltage, tracking error, and transient response time under a step load (from no load to a full load), are summarized in Table3. It shows that the proposed SRFPI-LADRC scheme obtains overall performance improvement.Table 3
Performances of the three methods.
Load typeSRFPILADRCSRFPI-LADRCTHD(%)erms(V)Vorms(V)THD(%)erms(V)Vorms(V)THD(%)erms(V)Vorms(V)No load1.601.65110.522.2013.40111.740.871.38110.52Nominal resistive load2.442.89110.452.7816.19110.861.872.81110.47Highly nonlinear load2.919.52110.544.9515.40111.472.143.48110.56No load to rated loadVoltage restore time(ms)1.61.20.8


## 5. Conclusions

In this paper, a novel SRFPI-LADRC-based voltage control strategy is designed for an off-grid single-phase inverter. The main work and conclusions are summarized as follows:(1)
Inadequate ability to track a sinusoidal signal of traditional LADRC was analyzed. Then, an SRFPI controller is combined with LADRC to overcome its shortcomings. The proposed method includes only capacitor voltage loops. Thus, it does not require the generation of current quadrature signals and high-precision current sensors.(2)
The equivalent model of the inverter with the proposed compound control strategy is provided. The system stability, dynamic performance, and robustness are analyzed by means of the generalized root locus method, which show the superiority of the proposed scheme.(3)
Experimental results verified that the performances of the single-phase inverter with the proposed SRFPI-LADRC were markedly improved in comparison to conventional SRFPI and LADRC. The system achieved a very small steady-state error and voltage tracking error, as well as a fast dynamic response under load variations, and low THD (close to 2%) with highly nonlinear load.(4)
As a low-cost but high-performance solution, the proposed method is suitable for off-grid single-phase inverter applications, such as UPS inverters, outdoor portable power supplies, and vehicle inverters.The performance verification of the proposed scheme for single-phase inverters mainly considers resistive loads and nonlinear loads. In future work, its performance under motor loads and during line-to-ground fault and the postfault recovery capability can be investigated. Furthermore, considering a single SRFPI controller only achieves satisfactory voltage control at the fundamental frequency; LADRC and multi-SRFPI-based compound control can be studied to achieve better harmonic compensation performance.



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*Source: 8131611-2022-10-13.xml*