Frequency Estimation for Grid-Tied Inverters Using Resonant Frequency Estimator

Abstract

This paper introduces a new strategy of frequency estimation for grid-tied and stand-alone mode AC/DC power converters. Frequency stabilization is required, especially when operating in stand-alone mode with a Droop Control strategy, or in hybrid mode with a Diesel generator. It is also required to reduce or increase power output along with grid frequency changes. The novel strategy utilizes a resonant filter to estimate the frequency of the voltage grid and is referenced as a resonant-frequency-estimator (RFE). A mathematical background is presented for the proposed estimator and its performance is evaluated. It is compared with three common frequency estimation algorithms: SRF-FLL, DDSRF-FLL, and DSOGI-FLL. Results are presented for three cases: frequency swing, harmonics injection, and type B short-circuit. Results are analyzed and the conclusion shows that the proposed novel strategy has comparable parameters to commonly used frequency estimation algorithms while having a loopless structure.

1. Introduction

The popularization of renewable energy sources, especially photovoltaic panels, while benefiting households, also created a problem with managing the utility grid. As a response, the grid operators introduced restrictions to keep household energy production within allowable, safe limits. All grid-tied inverters have to track frequency, phase, and amplitude of the grid voltage and behave accordingly; the EU Commission Regulation 2016/631 (14 April 2016) introduced, among other codes, LFSM-O mode, in which the generated active power decreases in response to an increase in frequency above a predefined threshold value, or when stand-alone mode is detected. In the case of a microgrid [1,2], the problem of maintaining voltage parameters escalates [3].

Such a system has a natural tendency to fluctuate the frequency as the Diesel generator cannot maintain stable output while under small loads, so it is usually supported by an AC/DC power converter and an energy storage (Figure 1) which will load the generator accordingly in order to maintain stable frequency output [4,5]. A microgrid such as this is a system with relatively high impedance and consequently low short-circuit power [6]. This, in addition, causes voltage to be prone to distortions caused by current harmonics, rendering the frequency estimation even more difficult.

Microgrid operating inverters can be divided into three categories: grid feeding, grid forming, and grid supporting [7]. Out of the three, only the latter can fulfill the roles of the other two: can both create a voltage reference value and become the reference point for other inverters in a microgrid, or slave off another energy source (e.g., a Diesel generator) and work as a current source [6]. Figure 2 presents the basic concept of a grid supporting algorithm known as Droop Control [8]. The commonly used voltage-oriented control algorithm (VOC) [9,10] falls into grid feeding category. The Droop Control algorithm extends the capabilities of the VOC algorithm [11]. Based on measured currents and voltages, apparent power is calculated and compared against reference values. In order to estimate power demand correctly, an accurate grid frequency estimation algorithm is required.

Many of the algorithms used to estimate the frequency of the electric power systems are based on the assumption that the fundamental frequency is practically constant, or that it experiences only minor variations—especially those methods that are based on the Fast Fourier Transform (FFT) [12]. Another group of frequency estimation algorithms is operating in an open-loop scheme [13]: the estimation is based on delaying the measured signal by a set amount of time and comparing it with the actual value of a measured signal. This group is not resistant to distortions such as harmonics, and requires additional filtering with steep-slope characteristics [14] or predictive filtering [15]. A group most commonly used in frequency estimation is the closed-loop group—the phase-locked loop systems. These methods estimate frequency indirectly. The main task of phase-locked loop systems [16] is precise calculation of the grid angle, while the frequency value is recovered from the signal driving the voltage-controlled oscillator (VOC) generator. These methods are highly effective as, along with basic PLL system, other improvements are used, such as the symmetrical components decoupling.

The most common algorithms are FLL-SOGI (frequency-locked loop second order generalized integrator) [17,18] and SRF-PLL (synchronous reference frame phase-locked loop) [19]. FLL-SOGI algorithm is more robust and immune to grid disturbances when compared to SRF-PLL, however, this comes at the cost of increased response delay to rapid frequency changes. Due to this, a novel algorithm of resonant frequency estimation was proposed, based on a resonant filter and a signal scaling model. The algorithm has a loopless form and has a performance comparable with the most advanced frequency locked loop algorithms.

2. Frequency Estimation Algorithms

2.1. Synchronous Reference Frame (SRF–PLL)

The Synchronous Reference Frame PLL algorithm utilizes mathematical transformations in order to extract the grid angle information. First, a three-phase stationary reference system (1) is transformed into a two-phase stationary reference system using Clarke transform (2), which is then released from constraints and rotates with the grid angle, using Park transform (3) [20].

$$\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]=V_m\left[\begin{array}{c}\cos \left(\omega t\right)\\ \cos \left(\omega t-2\pi /3\right)\\ \cos \left(\omega t-4\pi /3\right)\end{array}\right]$$

(1)

$$V_{\alpha \beta 0}=\frac{2}{3}\left[\begin{array}{ccc}1& \cos \left(2\pi /3\right)& \cos \left(4\pi /3\right)\\ 0& \sin \left(2\pi /3\right)& \sin \left(4\pi /3\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]$$

(2)

Compared against reference voltage v_q, it produces the grid angular frequency and, further, the grid angle (Figure 3).

$$\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\cos \left({\theta }_{PLL}\right)& \cos \left({\theta }_{PLL}-\frac{2\pi }{3}\right)& \cos \left({\theta }_{PLL}+\frac{2\pi }{3}\right)\\ -\sin \left({\theta }_{PLL}\right)& -\sin \left({\theta }_{PLL}-\frac{2\pi }{3}\right)& \sin \left({\theta }_{PLL}+\frac{2\pi }{3}\right)\end{array}\right]\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]$$

(3)

$$v_d=v_m\cos \left({\theta }_g-{\theta }_{PLL}\right)$$

(4)

$$v_q=v_m\sin \left({\theta }_g-{\theta }_{PLL}\right)$$

(5)

The transfer function of SRF-FLL (Figure 3) can be expressed as a second order equation in normalized form with help of a damping factor $\xi$ and a natural frequency ${\omega }_n$.

$$G_{cSRF}\left(s\right)=\frac{s^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}$$

(6)

The natural frequency ${\omega }_n$ and damping factor $\xi$ can be expressed as a function of PLL gain $k_p$ and integration time $T_i$ and vice versa:

$${\omega }_n=\sqrt{\frac{k_p}{T_i}}\to k_p=2\xi {\omega }_n$$

(7)

$$\xi =\frac{\sqrt{k_p{T}_i}}{2}\to T_i=\frac{2\xi }{{\omega }_n}$$

(8)

where $\xi$ is a constant damping factor 0.707 [18].

2.2. Double Decoupled Synchronous Reference Frame (DDSRF–FLL)

Similar to the previous SRF-FLL approach, the DDSRF–FLL uses coordinate transformation from the three-phase stationary system into the two-phase orthogonal rotating system. This method, however, performs an additional decoupling and is able to split the grid voltage signals into positive and negative components (9) [21] (Figure 4).

This is achieved by splitting the Clarke and Park transformation into two separate transformations, for positive and negative sequences (10)–(12).

$$v=V^{+1}\left[\begin{array}{c}\cos \left(\omega t+{\phi }_1\right)\\ \cos \left(\omega t-\frac{2\pi }{3}+{\phi }_1\right)\\ \cos \left(\omega t-\frac{4\pi }{3}+{\phi }_1\right)\end{array}\right]+V^{-1}\left[\begin{array}{c}\cos \left(\omega t+{\phi }_{-1}\right)\\ \cos \left(\omega t-\frac{4\pi }{3}+{\phi }_{-1}\right)\\ \cos \left(\omega t-\frac{2\pi }{3}+{\phi }_{-1}\right)\end{array}\right]+V^0\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$$

(9)

Taking the Clarke transform and ignoring the zero component and the zero sequence

$$v_{\alpha \beta }=V^{+1}\left[\begin{array}{c}\cos \left(\omega t+{\phi }_{+1}\right)\\ \sin \left(\omega t+{\phi }_{+1}\right)\end{array}\right]+V^{-1}\left[\begin{array}{c}\cos \left(-\omega t+{\phi }_{-1}\right)\\ \sin \left(-\omega t+{\phi }_{-1}\right)\end{array}\right]$$

(10)

then taking the Park transform using the angle locked by the PLL for the positive sequence:

$$v_{dq}=\left(V^{+1}\left[\begin{array}{c}\cos \left(\omega t+{\phi }_{+1}\right)\\ \sin \left(\omega t+{\phi }_{+1}\right)\end{array}\right]+V^{-1}\left[\begin{array}{c}\cos \left(-\omega t+{\phi }_{-1}\right)\\ \sin \left(-\omega t+{\phi }_{-1}\right)\end{array}\right]\right)\left[\begin{array}{cc}\cos \left(\omega t\right)& \sin \left(\omega t\right)\\ -\sin \left(\omega t\right)& \cos \left(\omega t\right)\end{array}\right]$$

(11)

$$v_{dq+}=V^{+1}\left[\begin{array}{c}\cos \left({\phi }_{+1}\right)\\ \sin \left({\phi }_{+1}\right)\end{array}\right]+{\overline{v}}_{d-}\left[\begin{array}{c}\cos \left(2\omega t\right)\\ -\sin \left(2\omega t\right)\end{array}\right]+{\overline{v}}_{q-}\left[\begin{array}{c}\sin \left(2\omega t\right)\\ \cos \left(2\omega t\right)\end{array}\right]$$

(12)

The decoupled value for DDSRF-FLL synchronisation system is:

$$v_{d+decoupled}=v^{+1}\cos \left({\phi }_{+1}\right)=v_{d+}-{\overline{v}}_{d-}\cos \left(2\omega t\right)-{\overline{v}}_{q-}\sin \left(2\omega t\right)$$

(13)

$$v_{q+decoupled}=v^{+1}\cos \left({\phi }_{+1}\right)=v_{q+}-{\overline{v}}_{d-}\cos \left(2\omega t\right)-{\overline{v}}_{q-}\sin \left(2\omega t\right)$$

(14)

Further, the frequency and angle estimation is performed as in the SRF–FLL algorithm (Figure 3). At the cost of an increased number of computations and longer processing time, robustness is increased, and the system is able to estimate frequency and synchronization angle even in the presence of asymmetrical voltages.

2.3. Double Second Order Generalized Integrator FLL (DSOGI-FLL)

An alternative solution to the one mentioned previously is Double Second Order Generalized Integrator PLL (Figure 5). It is also able to remove the negative voltage component from the voltage, but it does so by filtering the signals in the αβ stationary coordinates. The filtrations are carried out using two resonant filters shifted to each other by 90° angle, which removes the negative voltage component. The SOGI-FLL is composed of two blocks, the SOGI-QSG, which is a second-order adaptive filter, and can generate properly filtered in-phase and quadrature phase components (v_α and v_β) of a given input voltage. The transfer function SOGI-FLL can be expressed as two second order equations:

$$G_{\alpha }\left(s\right)=\frac{v_{\alpha }\left(s\right)}{v_{\beta }\left(s\right)}=k{\omega }^{\prime }\frac{s}{s^2+k{\omega }^{\prime }s+{\omega \prime }^2}$$

(15)

$$G_{\beta }\left(s\right)=\frac{v_{\beta }\left(s\right)}{v_{\alpha }\left(s\right)}=k{\omega }^{\prime }\frac{s}{s^2+k{\omega }^{\prime }s+{\omega \prime }^2}$$

(16)

$${\omega }^{\prime }=\frac{T}{\left(s+T\right)}\omega$$

(17)

where $\omega$’ is the estimated frequency, s is the Laplace operator, k = 0.8, and T = 50 is the FLL positive gain.

In the next step, the phase and quadrature-phase components are multiplied by each other. The sum of components is normalized by two constant factors and integrated. The frequency estimated in that way is extremely stable when compared with other presented methods of frequency estimations, even in the presence of voltage disturbances. This high stability is coming at the cost of a slow response to the quick-changing input signal.

2.4. Resonant Frequency Estimator (RFE)

In this paper, we propose a new method of simple and quick estimation of input voltage frequency. The proposed system can be divided into three major components (Figure 6). The first is a three-phase into two-phase orthogonal system conversion. Next, using high-pass filters, higher order harmonics are filtered out and then subtracted from the original signals, leaving only the base component. The following part scales the signals into per-unit values, decoupling the frequency signal from any voltage disturbances. Next, notch filters are used to condition the output signals. Based on the frequency deviation, these signals will be damped. Thanks to the decoupling of frequency and amplitude of the voltage, the difference between input and output signals of notch filters is caused only by the filter damping, and not by voltage amplitude changes. This ensures that the signal output value is only affected by the frequency deviation.

$$V=\sqrt{v_{\alpha }+v_{\beta }}$$

(18)

$$G_{RFE}\left(s\right)=\frac{s^2+2{\xi }_2{\omega }_{n}s+{\omega }_n^2}{s^2+2{\xi }_1{\omega }_{n}s+{\omega }_n^2}$$

(19)

where ${\xi }_2\ll {\xi }_1$ are gain factors.

The resonant frequency of a notch filter cannot match the nominal frequency of the grid; it has to be either slightly higher or slightly lower. If the former was the case, the algorithm would require additional components to detect if the frequency had dropped or risen.

Shifting the frequency to either side causes the filter to always operate higher than its resonant frequency, except for larger drops, in which case the inverter should shut down anyway. It is also imperative to tune the notch filter so the allowable changes in frequency cause the output signal to move along the linear part of the magnitude curve (Figure 7). Setting the resonant frequency of the filter f₁ to 42 Hz ensures that, by the time the inverter reaches this point, detected frequency deviation has caused it to shut down.

3. Frequency Estimation Testing

To analyze the behavior of RFE, a series of tests were concluded comparing this frequency estimation algorithm with FLL, SRF-FLL, and DDSRF-FLL. For research, three test cases were selected:

(a)

10% frequency swing;

(b)

Higher harmonics injection;

(c)

Single-phase short circuit.

The test bench consisted of a controllable three-phase voltage source Chroma 61511 (Lublin University of Technology, Lublin, Poland), a load, a digital oscilloscope Tektronix MSO5034B (Lublin University of Technology, Lublin, Poland) and a real-time system dSPACE DS1102 (Lublin University of Technology, Lublin, Poland). The dSPACE platform was used to implement the algorithms developed in Matlab/Simulink software (Lublin University of Technology, Lublin, Poland). All four algorithms were developed and tested for three cases typical for the utility grid. The analog-digital converters of the dSPACE platform were connected to the voltage source through Hall-effect based voltage sensors (LEM LV25-P) (Figure 8). The voltage source was used to generate the test cases, while the output of the estimation algorithms was observed in the real-time system. The oscilloscope was also used as an auxiliary voltage signal logging system. The test case parameters, as well as the critical algorithm coefficients, are presented in Appendix A. The RMS error value presented in figures in this section was calculated as RMS value of the frequency error (Δ_f) over fundamental frequency of the grid.

3.1. Frequency Swing

Frequency swing test was performed with ±3 Hz limits (Figure 9 and Figure 10). Higher frequency oscillations are a rare occurrence in the grid [22] and qualify for emergency action. SRF-FLL-based algorithms are more accurate than RFE and FLL in stable conditions. Initial oscillations are present in all four algorithms. The resonant estimation algorithm is an improvement over DSOGI-FLL, as it is much faster and less prone to oscillations.

The results for the frequency swing in range of ±3 Hz and slew rate of 60 Hz/s presented in Figure 9 and Figure 10 show that the fastest of the algorithms is the SRF-FLL (Figure 9d). Despite large (0.4 pu) initial error, this algorithm reaches close to zero error in about 0.07 s. The novel RFE algorithm (Figure 9a–c) shows similar properties to DSOGI-FLL (Figure 9d–f); however, after an initial error, it stabilizes and tracks frequency more accurately than the DSOGI-FLL. Additionally, it does not generate any oscillations, which are present in case of the DSOGI-FLL.

3.2. Harmonics Injection

A harmonics test was performed by connecting a three-phase rectifier with PFC to the controllable source. This type of load emulates the most common load type connected to the grid, and forces typical harmonics present in the utility grid to flow (Figure 11 and Figure 12). It is clear that both SRF-based algorithms are prone to harmonic distortion, while DSOGI-FLL is almost immune to them. However, DSOGI-FLL is significantly slower. The resonant estimator RFE shows much less oscillation in estimated frequency in presence of harmonics compared to the first two algorithms; however, it is still worse than DSOGI-FLL-based estimation.

The results for frequency estimation in the presence of harmonics presented in Figure 10 and Figure 11 show that the DSOGI-FLL is the most robust and immune to distortions (Figure 10d–f). The RFE algorithm shows similar properties (Figure 11a–c). The RMS value of the frequency error reached 0.11 (Figure 11c) which, in comparison to SRF-FLL (0.33), is an excellent result. In the case of two of the algorithms (SRF-FLL and DDSRF-FLL) there is a significant initial error which is absent in case of the RFE. The lack of this initial behavior in the case of RFE is caused by the absence of a loop in the structure of this algorithm. In case of all algorithms their reaction can be divided into three parts: the startup, stabilization, and the stable operation, and in case of the RFE, the SRF-FLL, and the DDSRF-FLL algorithms, those periods have similar timing. The DSOGI-FLL proves absolutely immune to the harmonic distortion.

3.3. Asymmetrical Short-Circuit

A single-phase short circuit is an event in a four-wire network when any phase wire is directly connected to the neutral line. It is an asymmetric event of type B [23,24]. For this type of transient short circuit, the operating conditions must be maintained for the time specified in the FRT codes of the grid operator. A voltage observed by an inverter loses signal in one phase (Figure 13), making frequency estimation, grid synchronization, and stable operation of the inverter significantly harder. For this test, DSOGI-FLL shows the best performance, as there is only a minor deviation in the estimated frequency right after short circuit occurs (Figure 14 and Figure 15). SRF-FLL also stabilizes rapidly, but has greater initial deviation. Both RFE and DDSRF-FLL oscillate after short circuit event, with RFE’s oscillation magnitude being several times lower. The DSOGI-FLL and the DDSRF algorithm provide the best frequency tracking in case of an asymmetrical short circuit. This can be explained by their structure: those algorithms utilize the symmetric component decomposition, and are synchronizing only to the positive sequence. In case of short circuit event, a large negative sequence component appears in the grid, which is ignored by those two algorithms. Another noticeable behavior is that in case of SRF-FLL and RFE, the error caused by the short circuit event is similar in both shape and magnitude to the initial startup error.

4. Conclusions

This paper presents a novel grid frequency estimation algorithm based on resonant filtering called resonant frequency estimator (RFE). To provide comparison with commonly used frequency estimators in the utility grid, the three well established algorithms are presented: SRF-FLL, DDSRF-FLL, and DSOGI-FLL. The algorithms were considered in a situation where a grid would operate in an island state; where, as a source of frequency deviation, a Diesel generator connected in parallel to renewable energy sources could be used. The behavior of such a grid was emulated by the programmable three-phase voltage source. A real time system was used to implement and test all four described algorithms in three cases: a 10% frequency swing, presence of higher grid harmonics and asymmetric short circuit. The selection was based on the probability of occurrence of these events in the grid. When comparing the results, the most robust and precise algorithm was DSOGI-FLL (

Table 1. Comparison of four frequency estimation algorithms.

SRF-FLL	DDSRF-FLL	DSOGI-FLL	RFE
Frequency swing
Very high accuracy High oscillations during startup High startup error	Very high accuracy No oscillations The smallest error during stable run	Good accuracy Minor, constant oscillations during stable run	Good accuracy No oscillations Smallest error during startup
Harmonics injection
No harmonics immunity Very high oscillations	No harmonics immunity Very high oscillations	No oscillations Very good accuracy	Good accuracy Minor oscillations, few magnitudes smaller than SRF-FLL & DDSRF-FLL
Asymmetric short-circuit
Oscillations during short-circuit High frequency error right after short-circuit event	High accuracy Reduced error compared to SRF	High accuracy Almost no error No oscillations	Minor oscillations after short circuit event

). It shows the best performance, which is also confirmed when taking into account average error values.

The RFE algorithm introduced by the authors shows promising results, being only slightly worse than the DSOGI-FLL in two out of three test cases (frequency swing and harmonics injection) and relatively accurate in case of a short circuit. The novel algorithm shows potential, as it retains good accuracy of frequency estimation in both static and dynamic states (Figure 16), while being simpler than other three tested algorithms.