New Synchronization Method for Three-Phase Three-Wire PWM Converters under Unbalance and Harmonics in the Grid Voltages Robinson F. de Camargo, Alexandre T. Pereira and Humberto Pinheiro Power Electronics and Control Research Group – GEPOC Federal University of Santa Maria - UFSM ZIP CODE: 97105-900 – Santa Maria, RS – Brazil e-mail:
[email protected],
[email protected] Abstract – This paper proposes a new open-loop synchronization method for three-phase three-wire PWM converters connected to the utility grid. It presents a better performance in terms of distortions in the synchronism signals if compared with other open-loop methods. Moreover, it has a good transient performance due to both angle and frequency disturbances, as well as input currents with low total harmonic distortion (THD) when used to synchronize PWM rectifiers, even under unbalanced and highly distorted grid voltages. In addition, a frequency adaptation algorithm is proposed for applications where large frequency variations are expected, such as in weak grids. Experimental results using a DSP TMS320F2812 are given to demonstrate the good performance of the proposed synchronism method. KEYWORDS – Normalized Voltage, Positive Sequence, Synchronous Frame, Three-Phase Three-Wire PWM Converters.
I. INTRODUCTION Several techniques to synchronize PWM converters to the utility grid have been reported. They can be classified as closed-loop [1-5, 24] and open-loop [6-13] methods. In closed-loop methods the angle of synchronism is obtained through a closed-loop structure, which aimed at locking the estimated value of the phase angle to its actual value. On the other hand, open-loop synchronization methods, the synchronism angle or normalized synchronism vector is obtained directly from the grid voltages [6, 7, 9], virtual flux [8, 10, 11] or estimate grid voltages [12, 13]. Although closed-loop methods have low sensitivity to the grid frequency, a trade off between good transient response and good filtering characteristics must always be considered. Moreover, cycle slips phenomenon is usually present in PLL closed-loop methods. When these methods are used to synchronize PWM converters to grid, this phenomenon can result in large transient currents during the resynchronization [23]. Among the open-loop methods, the modified synchronous reference frame (MSRF) [6, 7] and the low-pass filter based (LPF-B) methods [9] standout for their simplicity. The main attribute of the former is to be independent of the grid frequency. The later and the extend Kalman filter (EKF) [5] are less sensitive to the grid harmonics. Moreover, the weighted least-square estimation (WLSE) [19] rejects the impact of negative-sequence and accommodates frequency variations. However, none of open-loop methods reported, so far, have a good performance in terms of the distortion in synchronism signals and consequently in the current THD in three-phase PWM rectifiers under unbalance and harmonics in the grid voltages.
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In this sense, this paper proposes a new open-loop synchronization method for three-phase three-wire PWM converters connected to the utility grid that provides a good performance in terms of synchronism signals and, consequently, in the input current THD for PWM rectifiers, even in the presence of harmonics and severe unbalance between the grid voltages. Simulation and experimental results demonstrate the good performance of the proposed method using a DSP TMS320F2812 controller. II PROPOSED SYNCHRONIZATION METHOD The proposed synchronization method is called normalized positive sequence synchronous frame, NPSF, and it is shown in the Fig. 1 for three-phase three-wire systems. From the measure of two line-to-line grid voltages, the normalized positive sequence synchronism vector in the fundamental frequency is obtained.
v l −l
LPF1
LPF2
v l −l − f 2 v l −l − f 1
M2
+ +
v αβf +
sin(θ f+) cos(θ f+)
1 f v αβ+
2
M1
Fig. 1 – Proposed normalized positive sequence reference frame for threephase three-wire systems.
As usually the PWM converters are analyzed and controlled considering phase quantities, the line-to-line voltages vector, vl-l, is transformed into a phase voltage vector, vph. This transformation is not unique, so, it will be assumed that the sum of the voltages is zero, then: v ph = Tl − ph v l − l , (1) where:
v ph
⎡ va ⎤ ⎡2 1⎤ 1⎢ ⎡v ⎤ ⎢ ⎥ = ⎢ vb ⎥ Tl − ph = ⎢ −1 1 ⎥⎥ v l − l = ⎢ ab ⎥ . 3 ⎣ vbc ⎦ ⎢⎣ vc ⎥⎦ ⎢⎣ −1 −2 ⎥⎦
(2)
Now, by considering that the grid voltages usually have some degree of unbalance, depending on the synchronization method used, this disturbance generates distortion in synchronism signals [2, 19]. As consequence, the currents that are drained by the rectifier from the grid will also be distorted [14]. To prevent this distortion, the synchronism vector will be generated from the positive sequence components of the grid voltages at the fundamental frequency. The phase voltage positive sequence at the fundamental frequency can be obtained as follows:
f f v ph + = T+ seq v ph ,
(3)
where: f v ph +
⎡ vaf+ ⎤ ⎡1 ⎢ f ⎥ ⎢ = ⎢ vb + ⎥ , T+ seq = ⎢ a 2 ⎢ vcf+ ⎥ ⎢a ⎣ ⎦ ⎣
⎡vaf ⎤ a2 ⎤ ⎢ ⎥ ⎥ f = ⎢vbf ⎥ , (4) a ⎥ , v ph ⎢vcf ⎥ 1 ⎥⎦ ⎣ ⎦
a 1 a2
a = e j 2π 3 = − (1 2 ) +
(
)
3 2 e jπ 2 ,
(5)
In (4), the superscript f and subscript +seq represent the fundamental frequency and the positive sequence, respectively. The operator a in (5) implements in the time domain the 120º phase-shift originally presented by Fortescue in his theory of symmetrical components applied to phasor quantities. This operator implementation will be addressed in Section 2.1. A simple way to obtain the synchronism vector is by transforming grid voltages into αβ stationary coordinates [6,7]. In a similar way, here the positive sequence phase voltages at the fundamental frequency will be transformed to αβ coordinates, that is, f v αβf + = Tαβ v ph + ,
(6)
where:
v
f αβ +
⎡ vαf+ ⎤ = ⎢ f ⎥ , Tαβ = ⎢⎣ vβ + ⎥⎦
2 ⎡1 ⎢ 3 ⎣0
−1 2
−1 2 ⎤ ⎥, 3 2 − 3 2⎦
(7)
Aiming to simplify the transformations presented in (1), (3) and (6), they can be combined as follows:
v αβf + = Tαβ T+ seq Tl - ph v l − l ,
(8)
which can also be expressed as:
v αβf + = M1 v lf− l e − j π 2 + M 2 ( − v lf− l ) ,
(9)
where:
1⎡ M1 = ⎢ 2 ⎣⎢
− 2 2⎤ ⎥, 63 6 6 ⎦⎥ 0
Note that vαβf+ is the grid positive sequence phase voltage vector at the fundamental frequency in αβ coordinates. Thus, its amplitude depends on the grid voltage. A normalized synchronism vector can be obtained dividing vαβf+ by its norm, that is,
v
=
f v αβ+ f v αβ+
,
(11)
2
where this Euclidian norm of the vector is given by: f v αβ+
2
=
(v ) + (v ) f 2 α+
f 2 β+
2.1) Implementation of e-jπ/2 in time domain In [2], the operator ejπ/2 has been implemented in time domain with all-pass filters, which are designed to provide unit gain and 90º phase-shift at the fundamental frequency. However, this approach does not consider the harmonics in the grid voltages, which corrupts the synchronization signals [2,5] and consequently the currents that are drained by PWM rectifiers connected to the grid. As an alternative, this paper considers the use of low-pass filters to reduce the harmonics present in vl-l as well as to implement the 90º phase-shift at the fundamental frequency as required to compute the positive sequence. Therefore, the vl-l vector is firstly filtered using a low-pass filter, LPF1, which generates a filtered vector, vl-l_f1. This vector has its fundamental voltage -90º shifted from the fundamental of vl-l The vl-l-f1 vector is filtered again using another low-pass filter, LPF2. This provides an additional -90º phase-shift, resulting in the filtered vector vl-l-f2, which presents the same amplitude at the fundamental of the original vector vl-l, but with -180º phase-shift. Thereby, the output of the LPF1 filter will be vlf−l e− j π 2 and the LPF2 output will be −vlf−l , which are required to compute
v αβf + . The low-pass filters LPF1 and LPF2 are projected in a similar way. The second order transfer function of the implemented filters in the s domain is: ω2n G (s) = 2 , (14) s + 2ζωn s + ω2n Note that the parameters of the filter must be selected so that: G ( s ) s =jω = 1∠ − 90° , (15) f
(10)
1 ⎡− 6 3 − 6 6⎤ M2 = ⎢ ⎥. 2 ⎢⎣ 0 − 2 2⎥⎦
f αβ+n
In order to obtain vαβf+n, it is required to find vlf−l e− j π 2 e − vlf−l . The next section describes how these variables are derived in the time domain.
.
(12)
The entries of the vector vαβf+n (11) can be understood as being the sine and cosine often used to synchronize PWM converters with the grid, that is,
sin ( θ f + ) = vβ+f n and cos ( θ f + ) = vα+f n ,
(13)
where θf+=ωf t and ωf is the grid fundamental frequency.
507
where, f=60 Hz. To satisfy this condition, ωn =2πf and ζ=0.5. In the discrete domain, the low-pass filters may be implemented by the following discrete state space equation: x ( k + 1) = Gx ( k ) + Hu ( k ) , (16) y ( k ) = Cx ( k ) The equation (16) has been obtained by the discretization of (14), using a zero order hold (ZOH) with a sampling period Ts. The frequency response of the LPF filter implemented with (16) is shown in Fig. 2. It is possible to see that at 60 Hz, the filter presents an unit gain and a -90º phase-shift. In addition, voltage harmonics are significantly reduced for instance at the output of LPF1 the 3 rd is attenuated -19 dB and the 5 th, -28 dB. At the output of LPF2 the 3 rd is attenuated -38 dB and the 5 th, -56 dB. Note that the -90º phase-shift occurs at 60Hz. In cases where the grid frequency varies significantly, the filter parameters must be updated. Next Section proposes an adaptation algorithm for the filter parameters.
ωf ˆ + + ω
Θ
1 kI s
v sc − f
2
sin(θ f+) cos(θ f+)
LPF3
2
Fig. 4 – Proposed frequency adaptation algorithm.
3.1) Frequency Adaptation Algorithm Gain Design
Aiming to design the gain of the frequency adaptation algorithm, a nonlinear model will be developed. The block diagram of the nonlinear model of this adaptation algorithm is shown in Fig. 5, where the dynamics of LPF3 have not been considered. In sinusoidal steady-state, the relationship between ||vsc-f||22, the grid frequency and the estimated frequency can be expressed as follows:
Fig. 2 – Bode diagram of the LPF1 or LPF2.
III. FREQUENCY ADAPTION ALGORITHM In stiff grids, the frequency variations in the considered algorithm is not a concern, since the utility companies usually provide a grid voltage with frequency regulated between ± 1 Hz, as recommended by IEC 61000-2-2. However, in isolated or emergency situation [25] energy systems, the frequency variations can exceed the limits mentioned above. So, to broaden the range of applications of the synchronization method to weak grid, a frequency adaptation algorithm is proposed. Its block diagram is shown in Fig. 3.
v l −l
LPF1
LPF2
v l −l − f 2
M2
v l −l − f 1
Θ
+
v αβ-f +
+
2
=
1 2
⎡ ⎛ ω ⎞2 ⎤ ⎛ ω ⎞2 ⎢1 − ⎜ ⎟ ⎥ + ⎜ ⎟ ⎣⎢ ⎝ ωˆ ⎠ ⎦⎥ ⎝ ωˆ ⎠
By linearizing the system shown in Fig. 5 around the nominal operation point, a linear model is obtained as shown in Fig. 6. This model will be used in the design of the integrator gain. ω ˆ ω
kI s
1
f (ω ˆ ,ω f )
2
ω K2
Details of the frequency adaptation algorithm are given in Fig. 4. It consists of a low-pass filter, LPF3, identical to LPF1 and LPF2. The LPF3 filters the normalized synchronization vector obtained, (11), and from the result is calculated its square norm, that is, 2
2
Fig. 5 – Block diagram of the nonlinear model.
Frequency Adaptation Algorithm
,
v sc − f
2
M1
2
(18)
ωf
Fig. 3 – Proposed normalized positive sequence reference frame for threephase three-wire systems with frequency adaptation algorithm.
v sc - f
2
sin(θf+) cos(θ f+)
1 v αβ-f +
ˆ ω ) = v sc − f f ( ω,
(17)
If the natural frequency of the filter ωn is to equal the grid frequency, then (17) results in 1. However, if ||vsc-f||22>1 it indicates that the natural frequency of the filter ωn is bigger than the grid frequency. On the other hand, if ||vsc-f||22
