In many of the power electronics devices connected to single-phase utility grids, the synchronization information of the grid is usually estimated by phase-locked loop (PLL) technology to implement the control of the grid current -.
Nowadays, many literatures on single-phase PLLs have been presented -. For example, the power PLL (pPLL), the inverse park transform-based PLL (park-PLL), the EPLL, the quadrature PLL (QPLL), the transport delay-based PLL (TDB-PLL), the second-order generalized integrator PLL (SOGI-PLL) , the synthesis circuit PLL (SC-PLL) , the variable sampling period based PLL-, and so on. Among the aforementioned PLLs, the EPLL has found wide acceptance due to its simple structure and robust performance -.
In practice, grid voltage is usually a non-ideal sine wave while with some harmonics. Meanwhile, the DC offset generated by the measurement devices, conversion processes or a fault is another common disturbance in the input signal. Both harmonics and DC offset result in undesired periodic ripples in the phase and amplitude information estimated by the EPLL.
In recent years, several improved EPLLs have been presented. In , a window-based in-loop filter is introduced in the frequency loop to eliminate the periodic disturbances caused by input harmonics. However, the effect of the input DC offset cannot be eliminated. In , , an integrator is introduced and paralleled with the amplitude loop in the EPLL to estimate the input DC offset. The attached integrator can estimate the input DC offset. However, the effect of the input harmonics cannot be eliminated.
In this paper, the effects of the input harmonics, DC offset and step changes of the input fundamental component are theoretically analyzed. According to the obtained results, a hybrid filter based EPLL (HF-EPLL) is proposed. In the HF-EPLL, a delay signal cancellation filter (DSC filter) is set at the input of the EPLL to reject the input DC offset and even-order harmonics. In addition, a sliding Goertzel transform-based filter (SGT-filter) is introduced into the amplitude and frequency estimation loop to eliminate the periodic ripples generated by the residual input harmonics and step changes.
II. EFFECTS OF HARMONICS, DC OFFSET AND STEP CHANGES ON THE EPLL
A. Review of the Principle of the Conventional EPLL
The system configuration of the conventional EPLL is shown in Fig.1. The differential equations of the system are expressed as :
Where A is the estimated amplitude of the fundamental component. ώ, θ and (y)t denote the derivative with respect to the time of the estimated angular frequency, phase angle and fundamental component of the input signal, KP and KI denote the proportional and integral coefficient of the PI controller in the frequency loop, and Δω denotes the output of the integrator. KA is the gain of the integrator in the amplitude loop.
Fig. 1.Structure of conventional EPLL.
B. Effects of the Input Harmonics and DC Offset
The degraded performance of the conventional EPLL caused by the input harmonics and DC offset is analyzed first. Assume that the input signal, comprising of fundamental components, n-order harmonics and DC offset, is described as:
where Uf, ωf, ϕf, U2n+1, ω2n+1, ϕ2n+1, U2n, ω2n, ϕ2n denote the amplitude, angular frequency and phase angle of the fundamental component, odd-order harmonic, and even-order harmonic component, respectively. In addition, Udc denotes the voltage of the DC offset.
If it is assumed that the phase angle and frequency are estimated correctly, the output of the conventional EPLL is exactly same as the fundamental component. The output signal and error signal can be derived as:
According to (1) and (3), the input and output of the proportional and integral (PI) controller in the frequency loop can be obtained as in (4) and (5) (bottom of the page). Similarly, according to (1) and (3), the input and output of the integrator in the amplitude loop can be given by (6) and (7) (bottom of the next page).
It can be seen from (5) and (7) that the 2nth harmonics cause ripples at 2n+1 times the fundamental frequency and that the (2n+1)th
harmonics cause ripples at 2n and 2n+2 times the fundamental frequency. Meanwhile, the DC offset causes ripple at the fundamental frequency in the outputs of both the frequency and amplitude loops. Therefore, either the harmonics or the DC offset causes a periodic disturbance in both the output frequency and amplitude. In summary, the conventional EPLL cannot obtain both a fast dynamic response and harmonic immunity.
C. Effect of Step Changes of the Fundamental Component
The dynamic performance of the conventional EPLL when amplitude step, phase angle and frequency jumps occur is derived in detail. To simplify the analysis, the following assumptions are made: (1) the input signal is an ideal sinusoidal wave, (2) only one variable changes at the start of each dynamic process, and (3) before the variable changes, the system has reached its steady state.
(1) Amplitude Step
First, considering the situation where the amplitude steps from Uf to Uf1, the input signal is described as:
The error signal is then derived as:
The amplitude and frequency estimated by the conventional EPLL are derived as:
Where A0 and ω0 denote the initial value of the amplitude loop and frequency loop, and ϕAω1 = arctan[KI / (ωfKP)]. When an amplitude step change occurs in amplitude loop, except for DC term, a periodic ripple at the fundamental frequency is generated and in the frequency loop, a double-frequency periodic ripple is generated. Because the amplitudes of the dynamic periodic ripples in both the amplitude and the frequency loops are related to Uf1−Uf, when Uf1=Uf, the periodic ripples disappear. It means that periodic ripples exist only during the dynamic process, and they do not influence the steady state.
(2) Phase Angle Jump
Considering a phase angle jump to ϕf1, the input signal is:
The error signal is derived as:
The estimated amplitude and frequency of the conventional EPLL are derived as in (14) and (15) (bottom of the page).
Where ϕϕω = arctan(2ωfKP/KI). It can be seen from (14) and (15) that when the phase angle of the input signal jumps, a double-frequency ripple is generated in the amplitude loop. A DC term and a double-frequency ripple are generated in the frequency loop. The system will reach a new steady state without a double-frequency ripple in either the amplitude loop or the frequency loop when the estimated phase angle is equal to the input one.
(3) Frequency Jump
Considering a frequency jump to ωf1, the input signal is:
The error signal is derived as:
According to (1), the estimated amplitude and frequency of the conventional EPLL can be obtained as (18) and (19) (bottom of the page). When the frequency steps, a double-frequency ripple and a DC offset are generated in the output of the amplitude loop and a double-frequency ripple is generated in the output of the frequency loop. The system reaches a new steady state without a double-frequency ripple in the amplitude or frequency loop when the estimated frequency is equal to the input one.
III. IMPROVED EPLL BASED ON A HYBRID FILTER
Accordingly, a hybrid filters based EPLL (HF-EPLL) is proposed to eliminate the periodic ripple caused by these factors. Fig.2 shows the system configuration of the proposed HF-EPLL, where DS (delayed sampling) denotes the delayed sampling module. A DSC operator is employed in the input to eliminate the DC offset. In addition, a SGT-filter is introduced into both the frequency and amplitude loops to reject the periodic ripples caused by harmonics and step changes. The principle of the HF-EPLL is described in detail below.
Fig. 2.Diagram of proposed HF-EPLL.
After sampled and saved in DSs, the input signal is delayed by half of the fundamental period(Tp = 2π / ωp). Thus, the new signals consisting only of fundamental component and odd-order harmonics can be obtained by the following mathematical operation between the input signal and its delayed one:
Thus, the DC offset has been eliminated before inputting the EPLL. Only the odd-order harmonics exist in the new input signal. It can be known from the principle of the EPLL that the odd-order harmonics cause periodic ripples at odd-order frequencies in the inputs of the frequency and amplitude loops. This effect is similar to the ones caused by step changes. Correspondingly, a SGT-filter is introduced in the frequency loop and amplitude loop to eliminate the effects. The SGT-filter is derived from the sliding Discrete Fourier Transform (DFT). According to the sliding DFT, the spectral component of the kth frequency at time q for every N input time sample is derived as follows:
Where x(q) and x(q-N) denote the sampling results at times q and q-N, respectively. Therefore, when k=0，the obtained signal only contains a DC component. The z-domain transfer function of the SGT-filter when k=0 is:
Where fs and f0 denote the sampling and fundamental frequencies, respectively. With this filter, frequencies listed in (23) can be rejected while the DC component remains.
After applying the DSC operator to the system, the lowest frequency of the ripples included in the input signals of the frequency and amplitude loops are twice the fundamental frequency. Since the fundamental frequency in the Chinese grid is 50Hz, f0 = 100Hz was chosen for this study.
In addition, after choosing a sampling frequency of 10kHz as fs in this system, N=10k/100=100 can be obtained. Furthermore, the estimated frequency is introduced into the DSC operator and SGT-filter. Thus, the sampling frequency can be adjusted online to enhance the frequency adaptability of the HF-EPLL.
IV. EXPERIMENTAL RESULTS AND COMPARISON
Experiments with a conventional EPLL and the HF-EPLL on a DSP chip (TMS30F28335) - based platform are performed in order to validate the presented analysis. The input signal is programmed in a DSP and transported to the EPLL algorithm. The input signal and estimated information are then transferred to an analog quantity with a DAC chip (DAC7515) so that they can be observed through an oscilloscope. The initial amplitude of the input signal is set with a nominal voltage (1 p.u.), and the frequency is 50Hz. In , the design rule of the various parameters of the EPLL has been presented in detail. According to , the parameters of the controller in the EPLL are selected as KA = 130, KP = 130, and KI = 3000.
First, the performances of the conventional EPLL and the proposed EPLL suffering from the input harmonics and DC offset are tested. First, a 0.2 p.u. DC offset is added, and the 0.1 p.u. 2nd, 0.1 p.u. 3rd, 0.05 p.u. 5th, and 0.03 p.u. 7th harmonics are added after 100 ms. The corresponding voltage total harmonics distortion is 15.8%. Furthermore, after another 100ms, both the DC offset and the harmonics are removed. Fig. 3 shows the waveforms of the two EPLLs. Both of the EPLLs can obtain the correct information with an ideal sinusoidal wave. When the DC offset is added, a periodic disturbance is introduced in the estimated amplitude, phase angle and frequency of the conventional EPLL. Because of the introduction of input harmonics, periodic ripples are also generated in the estimated amplitude, phase angle and frequency. The proposed EPLL can estimate the required information without stead-state errors or ripples under both harmonics and DC offset. The settling time is less than 30 ms.
Fig. 3.Experimental waveforms with DC offset and harmonics. Scale: Input signal: 1.0 p.u./div, y(t): 1.0p.u./div, Amplitude: 0.4p.u./div, Amplitude error: 0.2p.u./div, Frequency: 25Hz/div, Frequency error: 5Hz/div, Phase angle: 144°/div, Phase error: 5°/div, time: 40ms/div. (a) EPLL. (b) HF-EPLL.
Moreover, the dynamic performances of the two EPLLs are tested. The case of an input signal amplitude step of ±30% is performed first and the experimental results are shown in Fig. 4. During the dynamic processes, there is twice frequency periodic ripple in the estimated results of the EPLL. The settling time of the EPLL is about 40ms, the maximum frequency error is about 4Hz and the maximum phase error is about 5°. The proposed EPLL shows a better dynamic response, with a settling time of 30ms, a maximum frequency error of 0.8Hz and a maximum phase error of 1.8°. In addition, there is almost no periodic ripple in the estimated results.
Fig. 4.Experimental waveforms with amplitude step. Scale: Input signal: 1.0 p.u./div, y(t): 1.0p.u./div, Amplitude: 0.4p.u./div, Amplitude error: 0.4p.u./div, Frequency: 25Hz/div, Frequency error: 2Hz/div, Phase angle: 144°/div, Phase error: 5°/div, time: 20ms/div. (a) EPLL. (b) HF-EPLL.
Secondly, the case of a phase angle jump of 40° is tested and the experimental results are shown in Fig. 5. For the dynamic response of the conventional EPLL, the maximum amplitude error is about 0.1p.u., and the maximum frequency error 10Hz. There is twice frequency periodic ripple in the estimated results. As for the proposed HF-EPLL, the dynamic time is about 45ms, the maximum amplitude error is 0.15p.u., and the maximum frequency error is 8Hz. Like the cases of the amplitude steps, the periodic ripples in the estimated results are so tiny that they can be neglected.
Fig. 5.Experimental waveforms with phase angle jump. Scale: Input signal: 1.0 p.u./div, y(t): 1.0p.u./div, Amplitude: 0.4p.u./div, Amplitude error: 0.2p.u./div, Frequency: 25Hz/div, Frequency error: 5Hz/div, Phase angle: 144°/div, Phase error: 20°/div, time: 20ms/div. (a) EPLL. (b) HF-EPLL.
Finally, the case of a frequency jump from 50Hz to 55Hz is tested and the experimental results are shown in Fig. 6. The settling time of the conventional EPLL is 40ms, the maximum amplitude error is 0.05p.u., and the maximum phase angle error is 1°. There is a periodic ripple in the estimated results, and the frequency is about 100Hz. For the HF-EPLL, the dynamic time is about 40ms, the maximum amplitude error is 0.08p.u., the maximum frequency error is 5Hz, and the maximum phase error is about 1°. Just like the cases of the amplitude step and the phase angle jump, the periodic ripples in the estimated results are still tiny.
Fig. 6.Experimental waveforms with Frequency step. Scale: Input signal: 1.0 p.u./div, y(t): 1.0p.u./div, Amplitude: 0.4p.u./div, Amplitude error: 0.1p.u./div, Frequency: 25Hz/div, Frequency error: 5Hz/div, Phase angle: 144°/div, Phase error: 1°/div, time: 20ms/div. (a) EPLL. (b) HF-EPLL.
The conventional EPLL does not have the ability of rejecting the effects of input harmonics and DC offset, which cause periodic ripples in the estimated results. On the other hand, a step change of the input fundamental component also causes a transient periodic ripple in the estimated results. The proposed HF-EPLL achieves correct estimation of the synchronization information of the input signal with an almost equal settling time and a smaller transient oscillation relative to conventional EPLL, even under the presence of harmonics and DC offset. The proposed EPLL can be implemented easily by a digital controller while the burden of the digital controller is not noticeably increased.