A COMPARATIVE STUDY OF THREE-PHASE AND SINGLE-PHASE PLL ALGORITHMS FOR GRID-CONNECTED SYSTEMS Rubens Marcos dos Santos Filho
Paulo F. Seixas, Porfírio C. Cortizo
Centro Federal de Educação Tecnológica - CEFET-MG Coord. Eletrônica - Av. Amazonas 5253 Belo Horizonte - MG - CEP. 30.480-000 - BRASIL Fax: +55.31 3319.5135 - e-mail:
[email protected]
Universidade Federal de Minas Gerais - UFMG EE.UFMG-DELT - Av. Antônio Carlos 6627 Belo Horizonte - MG - CEP. 31.270-901 - BRASIL Fax: +55.31 3499.5480 - e-mail:
[email protected]
Abstract – This work presents a comparative study of two three-phase and five single-phase PLL structures. It is shown through analytical and simulation results that the three-phase SRF PLL and the instantaneous power based three-phase PLL behave exactly equal, what is corroborated by tests of voltage and frequency disturbances, harmonic injection and line unbalance. However, the power PLL algorithm is more efficient since it requires fewer computations in its phase detector. The five different types of single-phase PLL structures are briefly reviewed and their performances are evaluated under several line disturbances. Index terms – Phase-locked loop (PLL), synchronous reference frame (SRF), adaptive linear combiner (ALC). I. INTRODUCTION The correct phase angle is a very important information in grid-connected systems such as UPS, controlled rectifiers, active filters, dynamic voltage restorers and also in the emerging distributed generation systems such as eolic and photo-voltaic. To estimate the phase angle open loop and closed loop methods are available [7]. The closed loop methods are commonly known as Phase-Locked Loops or PLLs. The figures of merit of a PLL are the steady state phase angle error, speed of response to phase, frequency and voltage amplitude disturbances, harmonic rejection and line unbalance in the case of three-phase systems. Generally, the line frequency varies within a limited range even in isolated systems, and its rate of change is limited by generators’ mechanical inertia. But when (unbalanced) grid faults occur, equipments are subjected to phase angle jumps and voltage sags [13]. The unbalanced situation may last for several cycles before the fault is cleared. Thus, the development of robust synchronizing algorithms is needed for the growing performance requirements of modern grid-connected equipments. In recent years, several PLL algorithms have been developed and presented in the literature [1]-[15]. The main objective of this work is to briefly review and evaluate some PLL algorithms for grid-connected systems under diverse line disturbances. In Section II, analytical results of the three-phase synchronous reference frame PLL structure (dqPLL) [1] and three-phase instantaneous power based PLL (pPLL) [2]-[4], will be presented. Section III will present five different single-phase structures. The first two are based on the SRF method, the third is based on the instantaneous power PLL, and the two later are based on adaptive filter theory.
II. THREE-PHASE PLLS In the following subsections the block diagrams and equations of the phase detector of the two three-phase PLL structures will be shown. A. Three-Phase SRF PLL (dqPLL) The block diagram of the dqPLL is shown in Fig. 1 [1]. The grid voltage samples are va, vb and vc. Comparing this diagram to the conventional PLL used in telecommunications, it can be seen that the PI controller is analogous to the low pass filter, the integrator is analogous to the voltage controlled oscillator, and the SRF transformation blocks scheme is analogous to the phase detector. Its working principle relies on regulating to zero the direct component of the rotating frame. This component is calculated using the estimated phase angle θˆ , closing the loop. kp + V =0 * d
-
ωff +
ki s
+
+ +
Vd
1 s
θˆ
dq
αβ
Vq va vb vc
Vα abc
Vβ
αβ Fig. 1. Three-Phase SRF PLL
Assuming balanced and harmonic free input voltages, the expression of the d-axis component which is feed to the PI controller is:
Vd = Vα sinθˆ + Vβ cosθˆ
(1)
Vd = V cos θ sin θˆ − V sin θ cosθˆ
(2)
Vd = − V sin(θ − θˆ )
(3)
or leading to: where: Vd is the phase detector output signal; V is the amplitude of the input voltages; θ is the angle of phase A; θˆ is the estimated angle.
Using the same procedure for the q-axis component Vq, we find its amplitude: Vq = V cos(θ − θˆ ) (4) When θˆ approximates θ , Vd in (3) will approximate zero and the PLL will be locked. In this situation, according to (4), Vq will be equal to the input voltage amplitude. B. Instantaneous Power Three-Phase PLL (pPLL) The block diagram of the three-phase pPLL is shown in Fig. 2 [2]-[4]. The working principle of its phase detector is based on regulating to zero the fictitious instantaneous threephase power. This power is calculated using the voltages samples and the fictitious currents ia and ic depicted in Fig. 2 and generated through the estimated angle θˆ . kp p*=0 + p
ωff +
ki s
-
+
+
+ ic
+ +
ia va vb vc
1 s
θˆ
The kp and ki gains determine the speed of response and disturbance rejection of the PLL in a direct relation. However, there is a trade off between noise, harmonic and voltage unbalance rejection and speed of response. The higher the gains, the worse the noise, harmonic and line unbalance rejection. The adequate bandwidth will depend on the application purposes. The following simulation results have been obtained for the discretized three-phase dqPLL and pPLL. There have been used 64 samples per line period or 3840Hz. The input voltages amplitude and frequency were normalized at 1p.u./60Hz. The same PI gains were set for both PLLs: kp=200V-1s-1, ki=20,000 V-1s-2. The PI gains were adjusted such that a suitable settling time and damping and an acceptable harmonic rejection were obtained. The 3/2 factor has been included in the dqPLL. 1) Phase Angle Jump Response – The PLL response to a 30 degrees phase angle step at t=1s is shown in Fig. 3. Both PLLs behave exactly equal as pointed out by (3) and (7) with a settling time of about 40ms and zero steady-state error.
− cos (θˆ + 120 ) − cos(θˆ )
+ + Fig. 2. Three-phase pPLL
Assuming balanced and harmonic free input voltages, the expression of the phase detector output signal p which is fed to the PI controller can be found by writing the instantaneous three-phase power expression:
p = v a i a + vb i b + vc ic
(5)
Since i a + i b + ic = 0 , (5) can be rewritten as:
p = ( v a − v b ) i a + ( vc − v b )i c
(6)
Fig. 3. Response of the three-phase pPLL and dqPLL to a 30degrees phase-angle jump in the input voltages.
2) Frequency Step Response – Fig. 4 shows the response of the three-phase PLLs to a +5Hz frequency step at t=1s. Although the line utility frequency commonly changes in a narrow range even in isolated systems and its rate of change
Substituting v a = V sin( θ ) , and v b = V sin( θ − 2π / 3 ) v c = V sin( θ + 2π / 3 ) in (6) and making the simplifications we obtain:
3 p = − V sin(θ − θˆ ) 2
(7)
C. Simulation Results and Performance Comparison The phase detector expressions of both three-phase PLLs depend on input voltage amplitude and phase. Equations (3) and (7) are similar, except for the factor 3/2 in (7). As they were obtained for balanced and harmonic free grid voltages, the following tests will evaluate the PLLs’ behavior under non ideal and transient situations. Fig. 4. Response of the three-phase PLLs to a +5Hz frequency step.
is limited due to the generators’ inertia, the frequency step test intends to give an idea of systems’ speed and adaptivity. The overlap of both responses can be seen again. The PLLs achieve zero error in steady-state after the frequency change. 3) Harmonic Rejection – The presence of harmonics in the line voltages leads to oscillating phase angle errors at PLLs’ output with near to zero average value. The test was performed applying one harmonic at a time. The peak phase angle error obtained for both PLLs as a function of harmonic order and amplitude is shown in Fig. 5. As stated before, both PLLs behave equally. It can be seen that higher harmonic frequencies lead to lower errors.
The inability of the dqPLL to accurately track unbalanced line voltages is well known in the literature. Several schemes have been proposed in order to improve the unbalance rejection of the dqPLL [7],[11],[14]. Those schemes usually compute the instantaneous positive sequence component of the input voltages and feed them to the dqPLL. The simulation and analytical results obtained suggest that such schemes are also applicable to the three-phase pPLL and would produce the same results. 5) Computational Load – Table I displays the number of computations needed in the phase detector stages of the three-phase dqPLL and pPLL. The PI and the integrator were not included because they are equal in both PLLs. As can be seen, the phase detector of the three-phase pPLL is more efficient since it requires fewer computations. Actually, effective computational load of the PLLs algorithms will depend on used processor’s architecture. Table I Computational Load of Three-Phase Algorithms Number of Operations in the Phase Detector Algorithm
Mult.
Addition
Trigon.
Total
dqPLL
7
4
2
13
100%
pPLL
2
3
2
7
54%
III. SINGLE-PHASE PLLS Fig. 5. Peak phase angle error of the three-phase PLL in the presence of harmonic distortion in input voltages.
4) Phase Unbalance Response – This test has been performed using the ANSI/IEEE standard 141-1986 which quantifies the three-phase unbalance as the relative maximum individual line rms voltage deviation in relation to the average rms value of the three phases:
Unbalance% =
(
)
a ,b ,c max |VRMS − VRMS | × 100% VRMS
(8)
where VRMS is the average rms value of the three phases. Fig. 6 shows the peak and average phase angle errors of both PLLs as a consequence of line unbalance, which has been calculated through (8).
Five different single-phase PLL structures have been studied: two single-phase versions of the three-phase dqPLL (PLL-dqFIFO and PLL-dq-Park) [5],[6]; one single-phase version of the three-phase pPLL (pPLL) [15]; the enhanced PLL (EPLL) [7],[8]; and finally one PLL which employs an adaptive filter to estimate line phase angle (PLL-ALC) [12]. A. Single-phase Transport Delay PLL (PLL-dq-FIFO) Fig. 7 shows the block diagram of the PLL-dq-FIFO [5],[6]. Its working principle is the same of its three-phase version. It uses a first-in-first-out register to build the quadrature component to the dq transformation. Due to the fixed length delay, it is not able to adjust to input voltage frequency deviations, leading to phase angle errors as will be shown in the simulation results. Other alternative to generate the quadrature component is the Hilbert transformer, but it is very difficult to implement in low frequency (50/60Hz). kp + Vd* = 0
ki s
-
ωff + +
+
1 s
+
Vd Vq
dq
αβ
Vβ PLL input
Fig. 6. Peak and average phase angle errors of the three-phase PLLs as a function of input voltage unbalance.
Delay 1/4 Cycle (FIFO)
Vα
Fig. 7. Single-phase transport delay PLL
θˆ
B. Single-phase Inverse Park PLL (PLL-dq-Park) Fig. 8 displays the block diagram of the single-phase dq
inverse Park based PLL [5],[6]. It is also based on frame orientation. The quadrature component Vα is build through the inverse Park transformation.
kp
u +
e
+
^
Vd* = 0
+
-
+
+ +
Vd
1 ps + 1
Vα
Vβ PLL input
αβ dq
C. Single-phase Power PLL (pPLL) Fig. 9 displays the block diagram of the single-phase pPLL [15]. As its three-phase version, it is also based on annulling the fictitious instantaneous power. Assuming purely sinusoidal input voltage in the form V sinθ , the expression of the phase detector output signal is:
p = −V sinθ cosθˆ
system’s output (the desired signal), as depicted in Fig. 11. In the PLL context, the desired signal y(t) is the grid voltage. The estimated signal yˆ (t) is built with the estimated phaseangle, as well as the inputs x(t) to the filter. yˆ (t) (estimated signal)
Adaptive Filter
(9)
e(t)
or
e
+ -
2nd order filter
State Fbk. Controller
x(t)
(10)
y(t)
System
(desired signal)
As pointed out by (10), there is a strong drawback to this structure: the product of input voltage and “virtual” current has a second harmonic component which has to be filtered out. Thus, a low pass filter with low cutoff frequency is needed, slowing down system’s speed. The adopted approach in the performed simulations was to use the state feedback technique to allocate closed loop poles and hence system’s dynamics, as presented in [15].
p*=0
+
-
V V p = − sin(θ − θˆ ) − sin(θ + θˆ ) 2 2
ωff
x2
cos
x2
reconstructed (or estimated) modifying the gains of a linear combiner as a function of an error e(t), which in turn is the difference between the estimated signal yˆ (t) and the
Fig. 8. Single-phase inverse Park PLL
x1
sin
This PLL is based on adaptive filter theory. Basically, it reconstructs in real time the fundamental component of the input signal by estimating its amplitude, phase and frequency through the steepest descent algorithm. In other words, this PLL has a non-linear phase detector. The gain K controls the ˆ . convergence speed of the estimated amplitude A Roughly speaking, the adaptive filter theory is based on the idea that an output signal y(t) of a system can be
αβ
1 ps + 1
x1
+ ωˆ +
Fig. 11. Adaptive filter structure
E. Single-phase Adaptive Linear Combiner PLL (PLL-ALC) Fig. 12 displays the block diagram of the PLL-ALC, which is also based on adaptive filter theory [12]. kp
W 2* = 0
1 s
+
ki s
-
θˆ
ωff + +
+ ˆ& θ +
u(t)
is(t)
1 s
W2 W12 + W22
x3
x1
W1
PLL input
θˆ
Fig. 10. Single-phase Enhanced PLL
dq
Vq
1 s
+
θˆ
1 s
+ ˆ& θ
+
K s
ωff
ki s
+
ki s
A
kp
ωff
PLL input
sin
+
− cos(θˆ )
+
x2
W2
cos
Fig. 9. Single-phase power PLL PLL input
D. Single-phase Enhanced PLL (EPLL) Fig. 10 displays the block diagram of the EPLL [7],[8].
u
-
e
Adaptation Algorithm
+
Fig. 12. Single-phase Adaptive Linear Combiner PLL
θˆ
The filter inputs x1 and x2 are built with the estimated phase angle θˆ , what is analogous to the EPLL scheme in Fig. 10. The linear combiner gains W1 and W2 are updated on-line using the delta rule. The PI controller regulates the gain W2 to zero. Thus the W1 gain becomes equal to the input voltage amplitude when θˆ equals θ. The normalizing block which computes W12 + W22 improves transient response to voltage
2) Phase-Angle Jump Response – Fig. 14 shows the results for this test, and Table III shows the ISE of all five responses. The pPLL has the slowest response due to its filter with small cutoff frequency. PLL-dq-FIFO has the smallest undershoot, and PLL-dq-Park has the smallest settling time.
disturbances but it is not essential. F. Simulation Results and Performance Comparison All single-phase PLL were discretized with a sample rate of 64 samples per 60Hz cycle or 3840Hz. The same PI gains were set to the PLL-dq-FIFO, PLL-dq_Park, and EPLL: kp=200V-1s-1, ki=20,000V-1s-2. Half this value had to be set to the PLL-ALC PI controller due to stability issues. The EPLL gain K was set to 200. The pPLL filter was implemented by two cascaded first order filters with cutoff frequency of 24Hz, resulting in 28dB of attenuation at 120Hz. The state feedback gains were set so that the closed loop poles were at 15Hz and 20Hz with optimum damping factor. In the dqPLL-Park the cutoff frequencies of the two first order filters were set to 120Hz. In the PLL-ALC the convergence gain α of the adaptation algorithm was set to 0.25. 1) Voltage Sag Response – Fig. 13 shows the simulation results for this test. It can be seen that the PLL-dq-FIFO has the best response to voltage disturbances. The closed loop poles chosen to the pPLL led to a noticeable oscillation in the phase angle error in steady state, though smaller than 0.5 degree peak-to-peak. The PLL-dq-Park presented the highest overshoot with the shortest settling time. Table II shows a more objective comparison criterion of all five responses through the ISE – integral of square error index.
Fig. 14. Single-phase PLLs responses to a phase-angle jump of 30 degrees
Table III Phase-Angle Jump Test ISE – Single-Phase Algorithms ISE – Integral of Square Phase-Angle Error Algorithm PLL-dq-FIFO PLL-dq-Park pPLL EPLL ISE
3.96
3.18
23.3%
18.7%
16.96
5.19
100% 30.6%
PLL-ALC
7.10 41.8%
3) Frequency Step Response – Fig. 15 shows the simulation results for this test. As can be seen, PLL-dq-FIFO can not deal with frequency deviations due to its fixed delay, which have been adjusted to the line nominal frequency. Again, the pPLL presented the slowest response but zero steady state error. The best response was achieved by the PLL-dq-Park, as indicated in Table IV.
Fig. 13. Single-phase PLLs responses to a voltage sag of 30%
Table II Voltage Sag Test ISE – Single-Phase Algorithms ISE – Integral of Square Phase-Angle Error Algorithm PLL-dq-FIFO PLL-dq-Park pPLL ISE
EPLL
PLL-ALC
0.12
1.12
0.04
0.28
0.63
10.7%
100%
3.5%
25%
56.2%
Fig. 15. Single-phase PLLs responses to a frequency step of +5Hz
Table IV
Table V
Frequency Step Test ISE – Single-Phase Algorithms
Computational Load of the Single-Phase Algorithms Total Number of Operations
ISE – Integral of Square Phase-Angle Error Algorithm PLL-dq-FIFO PLL-dq-Park pPLL EPLL ISE
PLL-ALC
Algorithm
Mult
Add.
Trig.
Div
Shift
Total
%
5.53
0.62
53.5
1.85
2.25
PLL-dq-FIFO
2
2
2
0
2
8
36%
10.3%
1.2%
100%
3.5%
4.2%
PLL-dq-Park
13
7
2
0
0
22
100%
4) Harmonics Response – Fig. 16 shows the time response to 5% 3rd harmonic injection in the line input voltage. In some responses there is a dc error superimposed to the oscillating phase-angle error.
pPLL
9
7
1
0
0
17
77%
EPLL
4
3
2
0
0
9
41%
PLL-ALC
14
7
2
2
0
38*
172%
* Division has been weighted by a factor of 10.
IV. CONCLUSIONS
Fig. 16. Single-phase PLLs responses to 5% 3rd harmonic injection in the input voltage
Fig. 17 shows how each PLL behave in presence of the 3rd, 5th and 7th harmonics. The PLL-ALC has the lowest sensitivity (degrees of deviation per harmonic %).
In this work, it was shown that the phase detector stage of the two main three-phase PLL structures for grid-connected systems found in the literature have practically the same equation, leading to identical behavior under line disturbances, harmonics and line unbalance. Furthermore, the three-phase power PLL algorithm is more efficient from the computational point of view, demanding about half the number of calculations on its phase detector stage. Although not tested in this work, the simulation and analytical results suggest that the positive sequence detection schemes used in the traditional three-phase SRF PLL are also applicable to the three-phase power PLL and would produce the same results. Five different types of single-phase PLLs found in the literature have been briefly reviewed and its simulation results have been presented. The single-phase power PLL algorithm presented the slowest responses to frequency and phase disturbances but it is the stiffest to voltage sags and harmonics due to the low cutoff frequency of its filters. The transport delay PLL requires the lowest computational effort, followed by the EPLL. The PLL-ALC is the heaviest algorithm from processing time viewpoint, but is less sensitive to harmonics and so fast as the EPLL and PLL-dqPark. V. REFERENCES [1]
[2] Fig. 17. Single-phase PLLs sensibilities to harmonics
5) Computational Load – The total number of calculations for each single-phase PLL algorithm is shown in Table V. The normalizing block was not considered in the PLL-ALC computational load, and the division operation has been weighted by a factor of 10 due to its inherent implementation complexity.
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