Enhanced Voltage Sensorless Control for a PWM Converter with DSOGI-FLL Under Grid Disturbances

Abstract

This paper presents Enhanced Voltage Sensorless Control for PWM converter with DSOGI-FLL under grid disturbances. Even without grid voltage sensors, the proposed method accurately estimates the grid angle and voltage, which are necessary for power transfer between the DC link of the PWM converter and the grid. The estimated grid voltage obtained through observer design is separated into positive and negative sequence components, and the grid frequency is estimated using the Dual Second-Order Generalized Integrator Quadrature Signal Generator (DSOGI-QSG) and Dual Second-Order Generalized Integrator Frequency-Locked Loop (DSOGI-FLL). The estimated positive and negative sequence voltages were effectively controlled using a dual current controller. The method operates effectively under normal, balanced AC source conditions and in various grid fault scenarios, including unbalanced voltage, harmonic distortion, voltage sag, and frequency step changes. The validity of the proposed method was evaluated through experimental results by using a grid simulator to implement the fault condition.

1. Introduction

With the increasing deployment of distributed energy resources (DERs) and growing electricity demand, conventional distribution networks are facing significant challenges in maintaining power quality. In particular, bidirectional power flow, along with voltage and frequency fluctuations, is becoming more widespread [1]. Moreover, the growing integration of direct current (DC) loads imposes additional burdens on traditional alternating current (AC) distribution systems [2]. To address these issues, three-phase PWM converters have been implemented in industrial systems, offering bidirectional power regulation and harmonic reduction [3].

Usually, the control scheme of the three-phase PWM converter needs a grid voltage sensor, an AC-line current sensor, and a DC-link voltage sensor. A detailed description of the PWM converter is introduced in [4,5,6]. Different control methods, such as sensorless control, have been proposed to reduce the cost of a PWM converter. In [7,8,9,10,11], a source voltage sensorless control method that estimates source voltages to reduce the source voltage sensors was investigated. In addition to the obvious benefits of sensor reduction, there are several other advantages, like the elimination of noise, resolution limitations, offset, various disturbances related to sensors, and a decrease in hardware complexity [12,13,14]. Figure 1 illustrates the structure of a three-phase source-voltage sensorless PWM converter.

To implement sensorless control, various strategies have been developed. In [15], a Kalman filter was adopted. The nonlinearity of these methods makes control design and stability analysis more difficult. As an alternative, linear estimators have also been used to predict grid voltage. Among these, methods based on a change in variables [16] and virtual flux [17] have been reported. A voltage sensorless controller using a source voltage observer was proposed in [18] to account for grid impedance and abnormal grid voltages. It is effective in handling grid disturbances during steady-state operation. However, it experiences significant current overshoots during grid voltage disturbances.

Generally, when source voltage unbalance occurs in a three-phase PWM converter, it causes DC-link ripples, higher reactive power, and poorer current regulator performance [19]. These conditions result in greater system losses, output current distortion, and electromagnetic interference [20]. Furthermore, the resulting distortions can also lead to estimation errors, as the source voltage observer designed for sensorless operation is based on the mathematical model of the overall system and relies on measured current and frequency for voltage estimation.

To address this problem, control methods are required to separate positive and negative sequences in an unbalanced grid condition. To separate the positive and negative sequences, DSOGI-QSG can be used [21]. A sequence calculator positioned prior to the PLL effectively suppresses low-order harmonics and enables accurate estimation of positive- and negative-sequence components [22]. Although many PLL techniques have been studied, synchronous-reference frame phase-locked loop (SRF-PLL) is arguably the most widely used synchronization technique in three-phase power systems. However, its phase estimation may become inaccurate in the presence of low-order harmonics such as the 5th and 7th. Moreover, it tends to show poor performance under grid fault conditions [23].

According to [3], sensorless control is proposed under unbalanced grid conditions using a voltage observer along with SRF-PLL. However, a voltage observer based on the mathematical model of the overall system has difficulty in accurately estimating grid voltage when harmonics are present due to current distortion. Furthermore, since voltage estimation relies on parameters such as estimated frequency, grid disturbance conditions can lead to estimation errors.

To address similar issues, research on sensorless control has been actively progressing in the motor control field, which shares similarities with grid models. In recent years, PLL and FLL have gained significant attention for their simple and fast response characteristics in position and speed estimation [24,25,26,27]. In [28], a second-order SOGI-frequency-locked loop (SOGI-FLL)-based flux observer for PMSM sensorless control was proposed.

Specifically, the authors of [29] introduced a sensorless observer based on Sliding Mode Observer (SMO) operating with SOGI-FLL. SOGI-FLL enables phase synchronization by providing positive sequence extraction, phase angle estimation, and inherent filtering capabilities. Unlike conventional methods, the proposed control method enables reliable inverter control under various grid conditions, owing to its stable and fast transient characteristics [30,31].

In this paper, a three-phase voltage sensorless PWM converter with DSOGI-FLL is proposed. This method enhanced sensorless control to improve dynamic performance under grid disturbance conditions. Additionally, a full-state source voltage observer based on the stationary reference frame mathematical model of the overall system, along with a dual current controller for controlling the positive and negative sequences, is designed.

2. Conventional Sensorless Control for a Three-Phase PWM Converter

2.1. Source Voltage Observer

Conventional voltage sensorless control [32] for a three-phase PWM converter uses only an AC line current sensor and a DC voltage sensor. By using a full-state source voltage observer, the AC line voltage can be estimated. The following Equation (1) represents the voltage equation for the three-phase PWM converter in the stationary reference frame, where $V_{abc}$ is the converter output voltage, ${\widehat{E}}_{abc}$ is the grid phase estimated voltage, $I_{abc}$ is the phase current, and $R_s$ and $L_s$ represent the resistance and inductance of the input reactor, respectively.

$$V_{abc}=R_s{i}_{abc}+L_s{\displaystyle \frac{d{i}_{abc}}{dt}}+{\widehat{E}}_{abc}$$

(1)

When the above Equation (1) is transformed into the stationary reference frame through coordinate transformation, it can be expressed as follows:

$$V_{dqs}=R_s{i}_{dqs}+L_s{\displaystyle \frac{d{i}_{dqs}}{dt}}+{\widehat{E}}_{dqs}$$

(2)

When the above Equation (2) is expressed (3) as a state-space equation [32], we obtain

$${\displaystyle \frac{d}{dx}}\widehat{x}=\left[\begin{array}{cccc}-{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}& 0\\ 0& -{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}\\ 0& 0& 0& -\widehat{\omega }\\ 0& 0& \widehat{\omega }& 0\end{array}\right]\widehat{x}+\left[\begin{array}{cc}{\displaystyle \frac{1}{L_s}}& 0\\ 0& {\displaystyle \frac{1}{L_s}}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V}_{ds}\\ V_{qs}\end{array}\right]y=Cx=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{c}{i}_{ds}\\ i_{qs}\\ {\widehat{E}}_{ds}\\ {\widehat{E}}_{qs}\end{array}\right]$$

(3)

where $x=\left[i_{ds,} i_{qs, } {\widehat{E}}_{ds, }{\widehat{E}}_{qs}\right], y=\left[i_{ds,} i_{qs }\right]$. When the state-space equation of the system is observable, a full-state source voltage observer is represented (Equation (4)). Figure 2 shows a block diagram of a source voltage observer.

$${\displaystyle \frac{d}{dt}}\widehat{x}=\left[\begin{array}{cccc}-{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}& 0\\ 0& -{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}\\ 0& 0& 0& -\widehat{\omega }\\ 0& 0& \widehat{\omega }& 0\end{array}\right]\widehat{x}+\left[\begin{array}{cc}{\displaystyle \frac{1}{L_s}}& 0\\ 0& {\displaystyle \frac{1}{L_s}}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V}_{ds}\\ V_{qs}\end{array}\right]+\left[\begin{array}{c}{L}_1\\ L_2\end{array}\right](y-\widehat{y})$$

(4)

Here, $L1, L2$ is the proportional gain of the full-state source voltage observer. If there is no error in the system matrix used in the state observer, the dynamic characteristics of the state variable estimation error are determined by Equation (5).

$$\det \left[sI-(A-LC)\right]==\det \left[\begin{array}{cccc}s-{\displaystyle \frac{R_s}{L_s}}+l_{11}& l_{12}& -{\displaystyle \frac{1}{L_s}}& 0\\ l_{21}& s-{\displaystyle \frac{R_s}{L_s}}+l_{22}& 0& -{\displaystyle \frac{1}{L_s}}\\ l_{31}& l_{32}& s& -\widehat{\omega }\\ l_{41}& l_{42}& \widehat{\omega }& s\end{array}\right]$$

(5)

The characteristic polynomial that determines the dynamic characteristics of the state variable estimation error can be obtained from Equation (5). Once the desired characteristic polynomial is defined, the coefficients of the two characteristic polynomials can be compared to determine the elements of the gain matrix. Since the target system is a fourth-order system, it is necessary to specify four poles. In the given paper, the poles are determined as conjugate complex double poles, as shown in Equation (6), and the required gain is calculated accordingly.

$$\begin{array}{l}{S}_{1,2}=-\varsigma {\omega }_{obs}+j{\omega }_{obs}\sqrt{1-}{\varsigma }^2\\ S_{3,4}=-\varsigma {\omega }_{obs}-j{\omega }_{obs}\sqrt{1-}{\varsigma }^2\end{array}$$

(6)

$$C(s)={(s^2+2\varsigma {\omega }_{obs}s+{\omega }_{obs})}^2$$

(7)

When the poles are determined as in Equation (6), the characteristic polynomial is given by Equation (7), and the corresponding gain can be obtained, as shown in Equation (8).

$$L={\left[\begin{array}{cccc}{L}_{11}& L_{21}& L_{31}& L_{41}\\ L_{12}& L_{22}& L_{32}& L_{42}\end{array}\right]}^T==\left[\begin{array}{cc}-{\displaystyle \frac{R_s}{L_s}}+2\varsigma {\omega }_{obs}& -\widehat{\omega }\\ \widehat{\omega }& -{\displaystyle \frac{R_s}{L_s}}+2\varsigma {\omega }_{obs}\\ ({\widehat{\omega }}^2-{{\omega }_{obs}}^2)L_s& 2\varsigma \widehat{\omega }{\omega }_{obs}{L}_s\\ -2\varsigma \widehat{\omega }{\omega }_{obs}{L}_s& ({\widehat{\omega }}^2-{{\omega }_{obs}}^2)L_s\end{array}\right]$$

(8)

2.2. Grid Synchronization

The SRF PLL is likely the most popular type of PLL for grid synchronization techniques. Figure 3 shows a diagram of the SRF PLL. It consists of a standard PI feedback controller with a feedforward nominal grid frequency. The PI controller is used to nullify the quadrature component of the grid voltage $E_d$, ensuring continuous synchronization between the d–q reference frame and the rotating voltage vector [33].

2.3. Grid Synchronization Sensorless Control for a Three-Phase PWM Converter

Figure 4 shows the three-phase PWM converter sensorless control structure using a source voltage observer. The grid voltage is estimated by the observer, and the phase angle is extracted using the SRF-PLL.

However, the conventional voltage sensorless control method has disadvantages. When grid harmonics are present, though the phase angle is tracked precisely, the harmonics are not completely canceled. Moreover, frequency deviations occur, and a small error is obtained in the output signal. Significantly, under an unbalanced grid condition, SRF PLL produces high errors since it can track only the positive sequence component [34].

Hence, though SRF PLL is most suitable under balanced conditions, it is unsuitable for operation under grid disturbance conditions. To overcome this limitation, frequency-locked loop (FLL) has been proposed [35]. Frequency-locked loop (FLL) can estimate the input signal frequency without being influenced by instantaneous changes in phase angle, offering performance advantages over PLL-based methods under unbalanced conditions.

3. Proposed Sensorless Control of a Three-Phase PWM Converter Under Grid Disturbance Conditions

The proposed control method enables sensorless control even in grid disturbance conditions such as unbalanced, harmonics, voltage sag, or frequency sag. To achieve a good dynamic response under grid disturbance conditions, DSOGI-FLL is applied [35]. DSOGI-FLL enables the precise extraction of the positive and negative voltage components by effectively eliminating the 2ω oscillation that occurs under unbalanced voltage conditions. Additionally, it exhibits fast dynamics when the phase angle changes.

In this paper, the voltage sensorless strategy for the proposed three-phase PWM converter is as follows. First, for sensorless control of a three-phase PWM converter under grid disturbance conditions, the source voltage observer individually estimates the positive and negative sequences of the grid voltage. To implement this, DSOGI-QSG is used to separate the positive and the negative sequences of the current and output voltage of the converter. Second, DSOGI-FLL was adopted to improve stability and transient characteristics when the grid is unbalanced or when grid voltage fluctuations, frequency variations, and voltage drops occur. Third, to address grid harmonic conditions, a harmonic compensator is implemented within the current controller. Fourth, Dual Vector Current Control is adopted to implement sensorless control for the PWM converter under grid disturbance conditions. Figure 5 shows the proposed sensorless control of a three-phase PWM converter.

In relation to this study, grid parameters such as resistance and inductance also influence the observer, as they are incorporated into the estimation algorithm. In [36], the impact of parameter variations was analyzed using a state observer, assuming an L-type grid filter. It was demonstrated that the estimation performance remains stable even with a 20% variation in grid inductance. However, since the control method adopted in this study differs from the approach presented in [36], the robustness of the proposed method against parameter variations cannot be ensured. Therefore, this study assumes that the grid parameters are precisely known.

3.1. Source Voltage Observer for Proposed Sensorless Control of Three-Phase PWM Converter

Under grid disturbance conditions, the input of the phase-locked loop fluctuates, which makes the estimated grid angle fluctuate. This fluctuation in the angle leads to an AC-side harmonic current with the current regulation loop of the PWM converter, where a positive and negative sequence voltage observer has been devised.

To address this issue, the positive and negative sequence voltages are estimated separately. By applying the source observer equation, the positive sequence and negative sequence voltages can be separated and defined to estimate them individually as follows. The proposed source voltage observer is represented by the following Equations (9) and (10).

$${\displaystyle \frac{d}{dt}}\widehat{x}=\left[\begin{array}{cccc}-{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}& 0\\ 0& -{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}\\ 0& 0& 0& -\omega \\ 0& 0& \omega & 0\end{array}\right]\widehat{x}+\left[\begin{array}{cc}{\displaystyle \frac{1}{L_s}}& 0\\ 0& {\displaystyle \frac{1}{L_s}}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V_{ds}}^{+}\\ {V_{qs}}^{+}\end{array}\right]+\left[\begin{array}{c}{L}_1\\ L_2\end{array}\right](y-\widehat{y})$$

(9)

where $x=\left[i_{ds}^{+},i_{qs}^{+},E_{ds}^{+}{E}_{qs}^{+}\right],y=\left[i_{ds}^{+},i_{qs}^{+}\right]$

$${\displaystyle \frac{d}{dt}}\widehat{x}=\left[\begin{array}{cccc}-{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}& 0\\ 0& -{\displaystyle \frac{R_s}{L_s}}& 0& -{\displaystyle \frac{1}{L_s}}\\ 0& 0& 0& -\omega \\ 0& 0& \omega & 0\end{array}\right]\widehat{x}+\left[\begin{array}{cc}{\displaystyle \frac{1}{L_s}}& 0\\ 0& {\displaystyle \frac{1}{L_s}}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V_{ds}}^{-}\\ {V_{qs}}^{-}\end{array}\right]+\left[\begin{array}{c}{L}_1\\ L_2\end{array}\right](y-\widehat{y})$$

(10)

where $x=\left[i_{ds}^{-},i_{qs}^{-},E_{ds}^{-} E_{qs}^{-}\right], y=\left[i_{ds}^{-},i_{qs}^{-}\right]$.

To achieve this, it is necessary to have the currents and converter voltages separated into positive and negative sequences, which is achieved using DSOGI-QSG, as described in Section 3.2.

3.2. Separation of Positive and Negative Sequences Using DSOGI-QSG for the Proposed Sensorless Control of a Three-Phase PWM Converter

In the previously proposed positive and negative sequence voltage observers, it is essential to separate the positive and negative sequence components of the current. To achieve this, this paper adopts DSOGI-QSG [37].

DSOGI-QSG consists of two quadrature signal generators (QSGs) based on a second-order generalized integrator (SOGI). Figure 6 shows the structure of SOGI, in which $k$ is the damping ratio and $\widehat{\omega }$ is the undamped natural frequency. The block diagram of SOGI is expressed in Figure 6, and its transfer functions are given in Equations (11) and (12). In these equations, the bandwidth of SOGI is decoupled from the natural frequency $\widehat{\omega }$, but it is affected by the damping ratio gain $k$. Additionally, $qI’$ and $I’$ have a phase difference of 90 degrees. Figure 7 is a bode plot for the above equation. The top part is the magnitude bode plot, and the bottom part is the phase bode plot. From the figure, it can be seen that a lower gain k leads to improved filtering performance but reduces stability in response to frequency variations. At the natural frequency, the amplitude reaches its maximum, and the phase is zero. As the frequency increases, the amplitude gradually decreases.

$$D(s)={\displaystyle \frac{I’}{I}}(s)={\displaystyle \frac{k\widehat{\omega }s}{s^2+k\widehat{\omega }s+{\omega }^2}}$$

(11)

$$Q(s)={\displaystyle \frac{qI’}{I}}(s)={\displaystyle \frac{k{\widehat{\omega }}^2}{s^2+k\widehat{\omega }s+{\omega }^2}}$$

(12)

DSOGI-QSG consists of two SOGI-QSGs, each formed by a second-order generalized integrator with unity feedback. By applying the previously described characteristics of SOGI-QSG, DSOGI-QSG can effectively separate the positive and negative sequence components. To achieve this, the symmetrical components method is applied to extract the respective components from the balanced three-phase current, as shown in Equations (13)–(15).

$$\left[\begin{array}{c}{I}_a^{+}\\ I_b^{+}\\ I_c^{+}\end{array}\right]={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ {\alpha }^2& 1& \alpha \\ \alpha & {\alpha }^2& 1\end{array}\right]\left[\begin{array}{c}{I}_a\\ I_b\\ I_c\end{array}\right]\left(I_{abc}^{+}=[T_{+}]I_{abc}\right)$$

(13)

$$\left[\begin{array}{c}{I}_a^{-}\\ I_b^{-}\\ I_c^{-}\end{array}\right]={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}1& {\alpha }^2& \alpha \\ \alpha & 1& {\alpha }^2\\ {\alpha }^2& \alpha & 1\end{array}\right]\left[\begin{array}{c}{I}_a\\ I_b\\ I_c\end{array}\right]\left(I_{abc}^{-}=[T_{-}]I_{abc}\right)$$

(14)

$$\left[\begin{array}{c}{I}_a^0\\ I_b^0\\ I_c^0\end{array}\right]={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}{I}_a\\ I_b\\ I_c\end{array}\right]\left(I_{abc}^0=[T_0]I_{abc}\right)$$

(15)

where $a=e^{j{\displaystyle \frac{2\pi }{3}}}=-{\displaystyle \frac{1}{2}}+j{\displaystyle \frac{\sqrt{3}}{2}},a^2=e^{j{\displaystyle \frac{4\pi }{3}}}=-{\displaystyle \frac{1}{2}}-j{\displaystyle \frac{\sqrt{3}}{2}}$.

In this paper, it is assumed that there are no zero sequence components. Therefore, the balanced three-phase current in the stationary reference frame is given by Equation (16).

$$\left[\begin{array}{c}{I}_{\alpha }\\ I_{\beta }\\ I_0\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\end{array}\right]\left[\begin{array}{c}{I}_a\\ I_b\\ I_c\end{array}\right]\left(I_{\alpha \beta 0}=[T_{\alpha \beta 0}]I_{abc}\right)$$

(16)

Based on Equations (13)–(16), the positive and negative sequences in the stationary reference frame are derived as shown in Equations (17) and (18).

$$I_{\alpha \beta }^{+}=[T_{\alpha \beta }]\cdot I_{abc}^{+}=[T_{\alpha \beta }]\cdot [T_{+}]\cdot I_{abc}=[T_{\alpha \beta }]\cdot [T_{+}]\cdot {[T_{\alpha \beta }]}^T\cdot I_{\alpha \beta }={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}1& -e^{-j{\displaystyle \frac{\pi }{2}}}\\ e^{-j{\displaystyle \frac{\pi }{2}}}& 1\end{array}\right]\cdot I_{\alpha \beta }$$

(17)

$$I_{\alpha \beta }^{-}=[T_{\alpha \beta }]\cdot I_{abc}^{-}=[T_{\alpha \beta }]\cdot [T_{-}]\cdot I_{abc}=[T_{\alpha \beta }]\cdot [T_{-}]\cdot {[T_{\alpha \beta }]}^T\cdot I_{\alpha \beta }={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}1& e^{-j{\displaystyle \frac{\pi }{2}}}\\ -e^{-j{\displaystyle \frac{\pi }{2}}}& 1\end{array}\right]\cdot I_{\alpha \beta }$$

(18)

In Equations (17) and (18), $e^{-j{\displaystyle \frac{\pi }{2}}}$ represents a $90°$phase delay. Therefore, by using the characteristics of DSOGI-QSG along with Equations (17) and (18), a block diagram for separating the positive and negative sequence components can be created, as shown in Figure 8.

3.3. DSOGI-QSG-Based Frequency-Locked Loop for the Proposed Sensorless Control of a Three-Phase PWM Converter

To control the three-phase PWM converter, the full-state source voltage observer needs to be used to estimate the phase angle of the grid. In conventional SRF-PLL systems used for grid phase angle synchronization, the grid voltage value is measured to extract the grid phase angle. This method can efficiently extract the grid phase angle and accurately track it even under parameter and frequency variations. However, under grid fault conditions, transient oscillations in the frequency measurement can occur, which may affect DC link control and power factor control. However, DSOGI-FLL is capable of providing stable grid angle estimation with filtering even under grid fault conditions. In addition, it demonstrates reliable frequency tracking performance. Figure 9 shows the DSOGI-QSG-based frequency-locked loop.

In the case of the proposed DSOGI-FLL, it can stably and accurately track grid frequency, even under grid disturbance conditions. The transfer function relating to the input and error signals is presented in Equation (19), and Figure 10 is a bode plot of Equation (19). As shown in Figure 10, the center frequency $\widehat{\omega }$ should be aligned with the input frequency ${\omega }_{ff}$ in order to generate a balanced set of in-quadrature outputs with equal amplitudes. To enable automatic tuning of the SOGI structure, it is necessary to analyze the error signal ${\epsilon }_v$, which is used to regulate the center frequency $\widehat{\omega }$.

$$E(s)={\displaystyle \frac{{\epsilon }_v}{\widehat{v}}}(s)={\displaystyle \frac{s^2+{\widehat{\omega }}^2}{s^2+k\widehat{\omega }s+{\widehat{\omega }}^2}}$$

(19)

It can function as an adaptive filter if an external circuit or algorithm is capable of measuring or detecting this frequency. According to [22], The sign of the frequency error ${\epsilon }_f$ indicates whether the estimated frequency $\widehat{\omega }$ is below, equal to, or above the reference frequency ${\omega }_{ff}$. This sign information is applied by an integral controller with a negative gain $-\gamma$ to continuously adjust $\widehat{\omega }$ to match ${\omega }_{ff}$. According to its operational characteristics, FLL uses $\gamma$ as its only tuning parameter. To define this parameter, ref. [38] demonstrates that, with gain normalization, FLL exhibits a first-order transfer function. The parameter γ can be selected based on the expected settling time of the system response. In this paper, $\gamma$ is set to 40. This control provides faster dynamics compared to other methods in the case of a grid fault condition, such as a phase angle jump or transient grid unbalanced conditions [31].

3.4. Current Control Strategy for a Three-Phase PWM Converter Under Grid Disturbance

An unbalanced three-phase voltage without a zero-sequence component can be described by Equations (20) and (21), which are expressed in the positive and negative synchronous reference frames, respectively [39].

$${\widehat{E}}_{dq}^p=L_s{\displaystyle \frac{d{I}_{dq}^p}{dt}}+R_s{I}_{dq}^p+j\widehat{\omega }{L}_s{I}_{dq}^p+V_{dq}^p$$

(20)

$${\widehat{E}}_{dq}^n=L_s{\displaystyle \frac{d{I}_{dq}^n}{dt}}+R_s{I}_{dq}^n-j\widehat{\omega }{L}_s{I}_{dq}^n+V_{dq}^n$$

(21)

Here, ${{\widehat{E}}^p}_{dq},{{\widehat{E}}^n}_{dq}$ represent the estimated grid voltage using the observer. ${{\widehat{I}}^p}_{dq},{{\widehat{I}}^n}_{dq}, {{\widehat{V}}^p}_{dq},{{\widehat{V}}^n}_{dq}$ represents the positive and the negative sequences of components using DSOSI-QSG. Under unbalanced grid conditions, the apparent power can be expressed as follows:

$$S=\left(e^{j\widehat{\omega }t}{\widehat{E}}_{dq}^p+e^{-j\widehat{\omega }t}{\widehat{E}}_{dq}^n\right){\left(e^{j\widehat{\omega }t}{I}_{dq}^p+e^{-j\widehat{\omega }t}{I}_{dq}^n\right)}^{*}$$

(22)

Here, the real power P and reactive power Q are expressed as follows. Thus, the active and reactive power can be obtained as shown in [40].

$$P(t)=P_0+P_{c2}\cos (2\omega t)+P_{s2}\sin (2\omega t)$$

(23)

$$Q(t)=Q_0+Q_{c2}\cos (2\omega t)+Q_{s2}\sin (2\omega t)$$

(24)

where $P_{c2},P_{s2}, Q_{c2},Q_{s2}$ are caused by unbalanced grid conditions.

The real power P is delivered to the DC link and affects the DC voltage level. However, if $P_{c2},P_{s2}$ (which are caused by unbalanced grid conditions) are not controlled to zero, P(t) will vary with time, resulting in a DC voltage ripple.

Therefore, a control strategy is required to bring $P_{c2},P_{s2}$ to zero. Similarly, to achieve a power factor of 1, control is necessary to reduce Q(t) to zero. This can be expressed in equation form as follows:

$$\left[\begin{array}{c}{\displaystyle \frac{2}{3}}{P}_0\\ {\displaystyle \frac{2}{3}}{Q}_0\\ {\displaystyle \frac{2}{3}}{P}_{s2}\\ {\displaystyle \frac{2}{3}}{P}_{c2}\end{array}\right]=\left[\begin{array}{cccc}{\widehat{E}}_d^p& {\widehat{E}}_q^p& {\widehat{E}}_d^n& {\widehat{E}}_q^n\\ {\widehat{E}}_q^p& -{\widehat{E}}_d^p& {\widehat{E}}_q^n& -{\widehat{E}}_d^n\\ {\widehat{E}}_q^n& -{\widehat{E}}_d^n& -{\widehat{E}}_q^p& {\widehat{E}}_d^p\\ {\widehat{E}}_d^n& {\widehat{E}}_q^n& {\widehat{E}}_d^p& {\widehat{E}}_q^p\end{array}\right]\left[\begin{array}{c}{I}_d^p(t)\\ I_q^p(t)\\ I_d^n(t)\\ I_q^n(t)\end{array}\right]$$

(25)

To eliminate DC link voltage ripple and achieve a power factor of 1, the following current reference values can be obtained:

$$\begin{array}{c}\left[\begin{array}{c}{I}_d^p(t)\\ I_q^p(t)\\ I_d^n(t)\\ I_q^n(t)\end{array}\right]={\left[\begin{array}{cccc}{\widehat{E}}_d^p& {\widehat{E}}_q^p& {\widehat{E}}_d^n& {\widehat{E}}_q^n\\ {\widehat{E}}_q^p& -{\widehat{E}}_d^p& {\widehat{E}}_q^n& -{\widehat{E}}_d^n\\ {\widehat{E}}_q^n& -{\widehat{E}}_d^n& -{\widehat{E}}_q^p& {\widehat{E}}_d^p\\ {\widehat{E}}_d^n& {\widehat{E}}_q^n& {\widehat{E}}_d^p& {\widehat{E}}_q^p\end{array}\right]}^{-1}\left[\begin{array}{c}{\displaystyle \frac{2}{3}}{P}_0\\ 0\\ 0\\ 0\end{array}\right]={\displaystyle \frac{2P_0}{3D}}\left[\begin{array}{c}{\widehat{E}}_d^p\\ {\widehat{E}}_q^p\\ -{\widehat{E}}_d^n\\ -{\widehat{E}}_q^n\end{array}\right]\end{array}$$

(26)

where$D=\left[{({\widehat{E}}_d^p)}^2+{({\widehat{E}}_q^p)}^2\right]-\left[{({\widehat{E}}_d^n)}^2+{({\widehat{E}}_q^n)}^2\right]$, and $D\ne 0$ is assumed. Here, P can be considered as the product of the voltage controller output and the DC link voltage reference value. According to the above equation, the negative-sequence current is included in the case of an unbalanced grid. Therefore, the current controller command can be determined as follows through Equation (27):

$$\left[\begin{array}{c}{I}_d^{p*}(t)\\ I_q^{p*}(t)\\ I_d^{n*}(t)\\ I_q^{n*}(t)\end{array}\right]={\displaystyle \frac{2}{3D}}{P}^{*}\left[\begin{array}{c}{\widehat{E}}_d^p\\ {\widehat{E}}_q^p\\ -{\widehat{E}}_d^n\\ -{\widehat{E}}_q^n\end{array}\right]$$

(27)

Figure 11 shows the control block diagram, which employs two independent current regulators.

In the current controller, the separation of positive and negative sequences is not performed using DSOGI-QSG. If DSOGI-QSG is used for separation, its filtering effect can degrade the transient response characteristics of the current controller. Therefore, the separation of the positive and negative sequences for current control was conducted according to the equations described earlier. For this reason, in grid harmonic conditions, a harmonic compensator is required to compensate the current controller command voltage. Thus, a harmonic compensator consisting of 5th and 7th PR controllers was used [41].

3.5. Grid Synchronization Strategy During Initial Operation

When the converter operation starts, the nominal voltage of the grid is already known; however, the initial grid angle error may result in poor source voltage feedforward. This may invoke a high in-rush current and an undesirable boost of the DC link voltage. Thus, the initial grid angle has to be estimated to prevent overcurrent and DC link overvoltage trip. To initialize grid angle estimation, the output voltage reference of the PWM converter is temporarily set to zero at the start of switching, lasting briefly to avoid triggering overcurrent protection. During this zero-voltage period, the voltage equation can be simplified, as expressed in Equation (28):

$${\widehat{E}}_{dq}^s\approx -L_s{\displaystyle \frac{d}{dt}}{i}_{dq}^s+v_{dq}^s\Rightarrow {\widehat{E}}_{dq}^s\approx -L_s{\displaystyle \frac{\mathrm{\Delta }{i}_{dq}^s}{\mathrm{\Delta }t}}$$

(28)

where$\Delta t$ is the PWM zero-voltage period. From this equation, the initial source voltage can be estimated [3], and the corresponding grid angle can be estimated using Equation (29) based on the current measurements.

$${\widehat{\theta }}_{e.\mathrm{initial}}\approx \arctan \left({\displaystyle \frac{-\mathrm{\Delta }{i}_d^s}{\mathrm{\Delta }{i}_q^s}}\right)$$

(29)

Additionally, if there is an initial charging circuit, the initial grid angle can be obtained using Equation (29) through the DSOGI-QSG current flowing during initial charging. However, before errors in the initial grid angle occur, the PWM converter should be operated and controlled using the grid angle obtained from the observer. Therefore, as soon as the initial charging is completed, a signal is received to operate the PWM converter, allowing for proper control. This paper implements control using this method.

4. Experimental Results

To validate the proposed control method, an experiment was performed. The target three-phase PWM converter was connected to an isolated AC power supply (TC.ACS, Regatron, Rorschach, Switzerland) [42] to simulate grid disturbance conditions. The parameters of the experiment system are shown in

Table 1. Parameters of the experiment system.

Specification	Value
Rated Grid Voltage	220 Vrms/60 Hz
DC Link Voltage	350 V
Interface Inductor	2 mH
DC Link Capacitor	4700 μF
Switching Frequency	5 kHz
Rated Power	10 kW

.

Table 2. Experiment condition.

No.	Test Condition
1	Balanced grid condition
2	Unbalanced grid condition
3	Harmonic injection
4	Frequency step change
5	Voltage sag with unbalance

shows the experimental conditions used for experimenting with the three-phase PWM converter control.

The experiments were conducted under balanced grid conditions, unbalanced grid conditions, harmonic conditions, and frequency step changes. The test condition of No. 1 represents the balanced grid condition, which refers to a three-phase balanced system with a line-to-line voltage of 220 Vrms at 60 Hz. For the unbalanced grid condition in No. 2, a voltage deviation of ±20% from the nominal voltage was assumed. In No. 3, harmonic injection was set based on the limits specified in EN 50160 [43], with 5th harmonics at 6% and 7th harmonics at 5%. The frequency step change in No. 4 was assumed to be ±10% of the nominal frequency. For the voltage sag under unbalanced conditions in No. 5, a sag of up to 50% lasting for more than two cycles was assumed.

Figure 12 presents the Equipment Under Test (EUT) and the load equipment utilized in the experiments. During the experiments, an oscilloscope (Model: HDO6054, Firmware Version: 9.2.0.4) from Teledyne LeCroy was used.

Figure 13 shows the estimated initial angle obtained using Equation (29). It can be observed that the estimated grid angle matches the actual grid angle calculated using the grid voltage sensor. It is important to achieve rapid synchronization and stabilization when starting up a PWM converter under a grid voltage sensorless scheme due to the absence of a voltage sensor. As mentioned in Section 3.5 in this paper, the initial value of the grid angle was obtained through the phase of the current flowing during the initial charging.

Figure 14 shows the experimental results of grid voltage and current under balanced grid conditions with a line-to-line voltage of 220 Vrms at 60 Hz. Figure 15 shows the experimental results of the estimated positive source voltage and estimated grid angle. The negative sequence voltage is not attacheds because it does not exist under balanced grid conditions. In Figure 16, the actual grid angle measured by the voltage sensor is compared with the estimated grid angle. The estimated grid angle demonstrates close tracking of the actual angle, with a time difference of approximately 200$\mathsf{\mu }\mathrm{s}$.

Figure 17 shows the experimental results under unbalanced grid conditions. The unbalanced grid condition is defined with phase B as the reference, where phase A is reduced by 20%, phase C is increased by 10%, and the system operates at 60 Hz.

Figure 18a shows the estimated positive sequence voltage. Due to the extraction of only the positive sequence under unbalanced conditions, a difference in voltage magnitude is observed. Figure 18b shows the estimated positive sequence voltage and estimated positive grid angle. Figure 18c shows the estimated negative sequence voltage, which is detected under unbalanced conditions.

Figure 19a shows the estimated grid voltage and current in the synchronous reference frame using the conventional control method under unbalanced grid conditions.

As previously discussed, when the positive and negative sequence components are not decoupled and conventional control strategies are adopted under unbalanced grid conditions, ripples are observed in the DC link voltage, grid voltage, and current in the synchronous reference frame. These disturbances result in oscillations in the active power, as shown in Figure 19b. However, as shown in Figure 19c,d, the proposed control method adopted to estimate the positive and negative sequence voltages suppresses active power ripples.

Figure 20 presents the experimental results under harmonic distortion conditions. In Figure 20a, the phase voltage is distorted due to the injection of 5th and 7th harmonics, which leads to a distorted current waveform, as shown in Figure 20c. Figure 21a illustrates both the measured grid voltage and the estimated positive sequence voltage in the stationary dq frame. As shown in Figure 21b, the proposed method is capable of accurately estimating the positive sequence voltage even in the presence of harmonic components in the grid voltage.

Figure 22 shows the experimental results under frequency step changes. Figure 22a,b show the grid voltage frequency change and estimated grid phase angle of the positive sequence when the grid frequency changes. It can be observed that the proposed control method exhibits a fast response to a frequency step change. In Figure 22c,d, the displayed dynamic response of frequency estimation is shown using the proposed DSOGI-FLL and SRF-PLL. Additionally, since no ripple is observed in the DSOGI-FLL output, it facilitates more effective control. In Figure 23, when the frequency changes from 60 Hz to 64 Hz and 56 Hz, the voltage in the state observer is estimated using the updated frequency parameter. Due to the fast dynamic response of DSOGI-FLL, no overshoot is observed during this process.

Figure 24 shows the experimental results under voltage drop and unbalanced grid conditions. At 60 Hz, for a duration of four cycles, phase A decreased by 50%, phase B increased by 10%, and phase C decreased by 35%. In Figure 25b,c, from the moment the voltage drops and unbalanced conditions occur, the estimated negative grid voltage in the stationary dq frame and estimated negative grid angle can be observed. In Figure 26, the estimated positive and negative sequence grid voltage in the synchronous reference frame and the active power are shown when the voltage drops and unbalanced conditions occur.

5. Conclusions

In this paper, an Enhanced Voltage Sensorless Control for a PWM converter with DSOGI-FLL is proposed and analyzed. To respond to grid disturbance, a DSOGI-QSG-based frequency-locked loop was adopted. The performance of the proposed method has been verified through experimental results. The operation of the PWM converter was validated not only under a three-phase balanced grid condition but also under a 20% unbalanced condition, severe harmonic conditions with 5th harmonics at 6% and 7th harmonics at 5%, a 6% frequency step change, and voltage sag conditions with a maximum imbalance of 50%. The experiment was tested under various grid conditions, and the results confirmed its capability for current control and voltage estimation.