Generalized design of shunt active power filter with output LCL filter.
Nowadays, the power quality in the distribution system deteriorates due to the excessive application of reactive, nonlinear and unbalanced load , . To resolve this problem, shunt active power filter (SAPF) plays an important role and has been an area of intense investigation in recent years. Unlike passive power filter (PPF), which is sensitive to the parameters of components and apt to resonate with other loads in the grid, APF provides a flexible and rounded solution .
SAPF has been used extensively for harmonic suppression, reactive power compensation and grid current equilibrium in the distribution system , . However, switching ripples produced by APF inject to the grid and result in a considerable harm. For example, capacitor loads will increase their losses and reduce service lives; moreover, high frequency noise existing in the common voltage will disturb the sensitive equipment . To address this problem, switching noise filter is indispensable for APF.
Compared with L or LC filter, LCL filter ensures a better smoothing output current from APF. As a result, decreasing the inductance is easy to achieve, which guarantees the dynamic performance of APF . Despite increasing the complexity, LCL filter has been widely used in medium and high power applications. Meanwhile, the design of LCL filter is a complicated procedure and attracts much research attentions -. Most of literature on LCL designing is aimed at applications in grid-connected PWM rectifiers or inverters -. Little literature research design methods in APF applications . Despite some algorithms can be references, to design LCL filters for APF is much more difficult due to the high bandwidth of output current. In ref , a set of methods on design, control, and implementation of LCL-filter-based SAPF are presented. Unfortunately, the part of design methods for LCL filter is not detailed.
Despite having many advantages, resonance peak of LCL filter system enlarges harmonic seriously at certain frequency, which distorts grid current and may make system unstable . Fortunately, this problem can be resolved by applying damping technologies. In recent years, active damping has received lots of attention for reducing losses of system. There are several mainstream control methods for active damping. The first one is to detect the current of filter capacitor and generate a virtual damping resistor -. Another method is to construct an element with negative resonance peak characteristic. On one hand, some state variables can be fed back to construct notch filter element to achieve this goal -; on the other hand, notch filter or double band-pass filter can be applied directly to generate negative resonance peak without additional sensors .
Despite active damping methods have some advantages, unfortunately, the stability of system is decreased due to the function of feed-back or filter elements, which means the damping coefficient can be regulated only in a small scale to guarantee system's stability margin. Meanwhile, active methods have lower bandwidth, poor performance on dynamic response and noise immunity. Moreover, additional sensors are usually needed and the complexity of control is increased. The last but not the least, due to the restriction of system's bandwidth, effect of active damping is much inferior to corresponding passive damping . Consequently, most of damping strategies in actual industrial applications adopt passive damping or combined method. Hence, it is still of positive engineering significance to research design methods for passive damping.
This paper is organized in six sections. A single-phase equivalent circuit model is established and characteristics of LCL filter system are analysed in Section II. Section III presents a set of systemic design methods for APF with LCL filter. Another contribution in Section III is providing the relationship between damping ratio, resonant frequency and attenuation degree of switching ripples, which is used for system's generalized design. In Section IV, an improved control strategy correcting magnitude and phase for output current is proposed. Simulation and experimental results are shown in Section V to demonstrate the validity of proposed methods. Section VI concludes this paper.
II. STRUCTURE AND CHARACTERISTICS OF SYSTEM
Figure 1 shows the topology of SAPF system. SAPF consists of two basic units: one is three-phase voltage source inverter (VSI) with a capacitor in DC side, the other one is LCL output filter.
Figure 2 shows the model of single-phase equivalent circuit. Where [L.sub.1], [L.sub.2] and C comprise LCL output filter. [u.sub.c] is the output voltage of three-phase VSI; us denotes the grid phase voltage. [L.sub.S], R and [Z.sub.L] are respectively grid equivalent inductor, damping resistor and load. The capacitor presents a low resistance for high frequency signal, which will shunt the switching ripple current. Unlike LC filter, which require a large capacitance and is influenced by [L.sub.S] seriously, LCL filter has a larger grid-side equivalent resistance and better filter performance.
No-load condition ([Z.sub.L] is open circuit) is the most significant situation for analysis and design. The transfer function can be expressed by
G(s) = [i.sub.2]/[u.sub.c] = RCs+1/[L.sub.1] [L.sub.3]Cs([s.sup.2] + R [L.sub.1]+[L.sub.3]/[L.sub.1] [L.sub.3] s + [[omega].sup.2.sub.res]) (1)
[[omega].sub.res] = [square root of ([L.sub.1] + [L.sub.3]/[L.sub.1][L.sub.3]C]) (2)
[L.sub.3] = [L.sub.2]+ [L.sub.S], (3)
From (1), there is an oscillating element in the transfer function. A resonant peak exists and can be weakened by damping resistor R.
Different type of [Z.sub.L] can impact upon characteristics of LCL filter system. It is not the focus of this paper; hence, some relevant conclusions are given here directly. When [Z.sub.L] represents a capacitive load, extra resonant peaks will be induced; When [Z.sub.L] is active resistive load, system's damping ratio can be increased and a variety of resonance is suppressed; In addition, the status of an inductive load is much the same as the condition of no load.
Considering that the current inner loop for APF usually concentrates on the current [i.sub.1], analysis mentioned above is not applicable to the design for APF system. When system's sampling frequency and switching frequency are both equal to [T.sub.s], the transfer function of current inner loop can be approximately expressed as
[i.sub.1](s)/[i.sup.*.sub.1] = 1/1.5[T.sub.s]s+1. (4)
The precision of current inner loop tends to be high enough, consequently, we can consider both the VSI and [L.sub.1] as a controlled current source. Furthermore, the improved single-phase equivalent circuit model is as in Fig. 3.
In Fig. 3, considering no-load situation, the model is optimized and current transfer function is expressed as
G(s) = [i.sub.2](s)/[i.sub.1](s) = sRC+1/[s.sup.2] [L.sub.3] + sRC + 1 (5)
From (5), the resonant frequency of system can be obtained as
[[omega].sub.res] = [square root of (1/[L.sub.3]C]). (6)
It is denoted that the resonant frequency of system depends upon total grid-side inductance [L.sub.3] with no relation to [L.sub.1]. As a matter of fact, the resonant frequency obtained from (2) is recessive due to the control strategy, whereas that from (6) is dominant and significative.
III. DESIGN METHODS OF LCL FILTER
A. Design of Total Inductance
Total inductance means the sum of [L.sub.1] and [L.sub.2]. There is a similarity between LCL filter and LC filter to design. When the inductance is low, RMS of current ripple will be biggish; whereas traceability will turn to be unsatisfactory with a high inductance. However, the effect of LCL filter tends to be much better than LC filter. Hence, the principle consideration to design LCL filter is traceability limit. In other words, to determine the upper limit of total inductor is crucial.
For the sake of simplicity, a reduced single-phase equivalent circuit model which ignores the influence of filter capacitor is as follows in Fig. 4.
The voltage on inductor can be obtained as
[u.sub.L] = L [di.sub.c]/dt = [u.sub.c]-[u.sub.s]. (7)
When [U.sub.dc] denotes voltage on DC bus, output voltage of three-phase VSI uc has five electrical levels with 0, [+ or -]1 / 3[U.sub.dc] and [+ or -]2 / 3[U.sub.dc] respectively. Obviously, the worst situation for traceability of APF occurs at the peak of [u.sub.s]. Without loss of generality, assumed that us is equal to positive peak value [square root of (2[U.sub.s]]), where [U.sub.s] means the RMS value of grid phase voltage. An average voltage of [u.sub.c] can be obtained as 0.5[U.sub.dc] considering the coupling of three phases. Provided that A is the maximum slope of APF output current, the formula required is as follows
0.5[U.sub.dc] - [square root of 2][U.sub.S]/[omega]L [greater than or equal to] A. (8)
Besides, regarding fundamental wave, (8) can be converted to the following one
[square root of 2]/2[U.sub.dc] - [U.sub.S]/[[omega].sub.1] L [greater than or equal to] [A.sub.1], (9)
where [A.sub.1] means the maximum RMS value of APF output current for fundamental wave. We can convert harmonic to fundamental wave by multiplying relative coefficient for the sake of simplicity. Take 100 A 5th harmonic for example, it can be considered as 500 A fundamental wave. Then we can choose the maximum corrected fundamental wave to calculate total inductance.
B. Design of Damping Ratio
Grid inductance [L.sub.s] can be estimated by transformer's parameters. Define [xi] as the damping ratio, (5) can be converted into the standard form as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[[omega].sub.res] = [square root of (1/[L.sub.3]C]), (11)
[xi] = R/2 [square root of (C/[L.sub.3]]). (12)
We can obtain the magnitude-frequency characteristic as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
It is indicated that the oscillating element influences the part lg [square root of ([(1 - [[omega].sup.2]/[[omega].sup.2.sub.res]).sup.2] + [(2[xi] [omega]/[[omega].sub.res]).sup.2]). Define [omega] = k[[omega].sub.res], and the magnitude gain at the frequency of [omega] is -20 lg[square root of ([(1-[k.sup.2]).sup.2] + [(2[xi]k).sup.2]]) dB. When [omega] [much less than] [[omega].sub.res] or [omega] [much greater than] [[omega].sub.res], the gain is about 0 dB. Nevertheless, when [omega] = [[omega].sub.res] the gain will trend to be infinity if [xi] = 0. When we introduce damping, the gain is -20 lg 2[xi] dB and the variation tendency is as in Fig. 5.
It is indicated in Fig. 5 that magnitude is amplified when [xi] < 0.5 and the larger gain the less [xi]. When [xi] [greater than or equal to] 0.5, no amplification occurs for the oscillating element. In this sense, we should design [xi] larger than 0.5. However, that will increase thermal losses of system by enlarging the value of R. If a magnification below 50 % at the resonant frequency is required, [xi] must be larger than 0.33.
C Design of Resonant Frequency
It is important to note that there are two typical elements with inertia differential element and oscillating element respectively. Owing to a tiny time constant, inertia differential element mainly performs a role in high frequency. While in low and medium frequency, oscillating element dominantly affects the characteristics of system. From (10), we can obtain the phase-frequency characteristic as
[PHI](j[omega]) = arctan(RC[omega]) - arctan(2[xi][omega][[omega].sub.res]/[[omega].sup.2.sub.res]-[[omega].sup.2]). (14)
Define [omega] = k[[omega].sub.res]. From (13) and (14), magnitude gain and phase excursion are respectively
-20 lg [square root of ([(1-[k.sup.2]).sup.2] + [(2[xi]k).sup.2]) dB and -arctan(2[xi]k/1-[k.sup.2]) radian.
The characteristics when [xi] = 0.5 or [xi] = 0.33 and k [less than or equal to] 1 are as in Fig. 6.
From Fig. 6(a), it is indicated that magnitude is enlarged when k [less than or equal to] 1 and a higher damping can restrain the gain better. From Fig. 6(b), it is noteworthy that phase excursion occurs across LCL filter and a higher damping leads to a more serious delay. Specifically, if [omega] = 2/3[[omega].sub.res], when [xi]=0.33 magnitude reach about 3 dB which means current is enlarged by 41 %, and phase delay is 38[degrees]; when [xi] = 0.5, magnitude is 1.23 dB which means current is enlarged by 15 %, and phase delay is 50[degrees]. Consequently, resonant frequency must exceed APF's compensation bandwidth by 50 % in order to reduce the adverse effect from the oscillating element.
In addition, Fig. 7 shows the magnitude characteristics when k [greater than or equal to] 1. It is indicated that magnitude characteristics tend to turn superposition for different damping. If k = 2, gain is -10.3 dB and -11.14 dB with [xi] = 0.33, [xi] = 0.5 respectively, which means an attenuation by 70 % at least. Hence, resonant frequency should be below half of switching frequency.
D. Attenuation Degree of Switching Ripple
The chief purpose to use LCL filter is to prevent switching ripple from injecting into grid. Therefore attenuation degree of switching ripple is an important indicator to design and evaluate LCL filter.
From (10), magnitude at the switching frequency is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
where [[omega].sub.s] is switching frequency and B is the gain to achieve. Define attenuation degree of switching ripple as [eta] and it can be obtained from (10) as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
h = [[omega].sub.res]/[[omega].sub.s]. (17)
From (11), (12), (16) and (17) simultaneously, [eta] can be determined uniquely by [xi] and h as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
A 3D graphic can be plotted as in Fig. 8.
From Fig. 8, it is indicated that [eta] increases rapidly as we enhance [xi]. When [xi] is small, [eta] is always low however much h is. However, when [xi] is big such as 0.5, [eta] is at least 50 % when h = 0.1. As a result, we usually determine [xi] and h reasonably in order to reduce the value of [eta]. In general, [eta] should be below 0.2 to ensure the switching ripple filter effect. In some fastidious occasions, [eta] may be chosen below 0.15.
E. Global Design Methods
How to design LCL filter in APF integrally?
To begin with, we should make certain of the capacity and application. Total inductance can be obtained from methods indicated above. The maximum value of capacitor usually is chosen below 5 % of APF's capacity. Switching frequency of APF can be determined by the capacity and heat-sinking capability which is generally below 10 kHz for actual applications. Target value of [eta] should be chosen according to actual conditions.
Moreover, we can obtain resonant frequency and damping ratio from methods proposed above.
Last but not least, check the value of [eta] using (18). If [eta] is dissatisfactory, try reducing h or [xi] . ALL the parameters of LCL filter are achieved.
IV. OPTIMIZATION CONTROL METHODS
If grid load is harmonic source, the model of single-phase equivalent circuit of system is as follows as in Fig. 9.
In Fig. 9, [i.sub.Lh] is harmonic current source. Because ideal harmonic voltage source is inexistent, practical nonlinear load can be equivalent to harmonic current source with different peaks. Compensation coefficient g is usually equal to -1.
If [i.sub.2] = [i.sub.1] = -[i.sub.Lh], harmonic will be compensated completely. Unfortunately, [i.sub.2] changes not only magnitude but also phase compared with [i.sub.1] as indicated above. Consequently, traditional feed-forward control methods which ii is equal to -[i.sub.Lh] cannot achieve nice effect especially when resonate frequency is low. Factually, due to a low switching frequency, this situation often occurs.
If used adequate magnitude and phase compensation to reference value of [i.sub.1], to make [i.sub.2] = -[i.sub.Lh] can eliminate the adverse effect of LCL filter.
Harmonic can be extracted by synchronous rotating transformation methods which can easily distinguish positive-order and negative-order components from current. The schematic diagram of harmonic extraction is showed in Fig. 10.
Load current is transformed by Clarke and Park matrix into [i.sub.pk] and [i.sub.qk] which represent active harmonic current and reactive harmonic current respectively. The harmonic order k is positive value when we are to extract positive-order harmonic; k is negative value when we are to extract negative-order harmonic. 0 is the phase angel of A-phase fundamental voltage obtained from phase-locked loop. Through Park transformation, k-order harmonic turns into DC component which can be extracted by a Low-pass filter, and the other order harmonics are transformed into AC component. Traditionally, Park inverse transformation uses the same phase information with direct transformation. Here a novel control strategy is proposed, which makes a lead compensation for Park inverse transformation. [[theta].sub.kc] is the lead angle obtained by (14). Through Clarke inverse transformation, k-order harmonic current reference can be obtained by multiply -1 considered reference direction traditionally. Here magnitude compensation is introduced, where [M.sub.ck] is a compensation factor obtained by (13). Through feeding forward magnitude and phase compensation, the adverse effect of LCL filter can be reduced.
V. SIMULATION AND EXPERIMENT RESULTS
Here an example to design APF with 200A capacity is provided. Filter capacitor chooses 60 uF delta connected which introduce reactive current of 12.4 A, and this meets the rule that reactive current should be below 5 % of capacity mentioned above approximately. The total inductance should be below 0.28 mH when [U.sub.dc] = 800V using the rule on total inductance. Switching frequency of system is decided to be 5 kHz for the purpose of reducing switching loss. The maximum order harmonic to compensate is 13th, and resonant frequency should be chosen between 1 kHz and 2.5 kHz according to the rule proposed above. Based on the parameter of transformer in experiment place, an estimated value of Ls is 0.04 mH. We choose [L.sub.2] =0.07 mH and resonate frequency is 1.13 KHz calculated by (6). From (17), it can be obtained that h = 0.226 . Damping ratio [xi] is chosen to be 0.32 and [eta] =0.16 can be obtained from (18). As a result, we can choose [L.sub.1] = 0.2mH, R = 0.5[ohm] (Y connection).
A model is established with Matlab Simulink according to parameters above. Nonlinear load adopts harmonic current source. The RMS of harmonic is shown in Table I.
Firstly, resonant frequency is 1.13 KHz according to (6), whereas it is 1.41 KHz obtained from (2). A contrast of the gain is listed in Table II. It is shown that the magnification times with the (3) are much larger and validity of the proposed method to obtain resonant frequency is proved. Meanwhile, the gain is about 1.6, which is accordant to design method on damping ratio mentioned above.
Moreover, magnitude compensation factor [M.sub.ck] and the lead angle [[theta].sub.kc] for various orders harmonic can be obtained from (13) and (14) which are listed in Table III.
From Fig. 11, it is indicated that load current has a serious distortion. After compensation by APF, grid current becomes approximately sinusoidal but not ideal. There are several reasons influencing compensation performance. Firstly, the harmonic RMS of load current is very large, which is a rather awful occasion and difficult to compensate by APF due to a steep changing current. Secondly, the resonant characteristics of LCL filter make APF's output current distorted. Last but not least, the control strategy is imperfect and system's inertial and delay elements decrease the accuracy of current tracking. On the other hand, from Table I, compensation effect is obviously improved when using correction. With individual magnitude or phase correction, THD of grid current reduces to 12.2 %, when we correct magnitude and phase, integrated compensation effect is the best during the methods provided. THD of grid current reduces to 10.18 % and total harmonic elimination rate reaches up to 94.7 % which is very satisfactory.
To further determine the effectiveness and validity of the proposed method, an experiment is accomplished in the laboratory. Figure 12 is the photo of the laboratory prototype. As is showed, APF and thyristor switched capacitor (TSC) compose a hybrid system to compensate the reactive power and harmonic current generated by inductors and SCR rectifier loads in the grid respectively. In the experiment, only SCR rectifier loads are connected to the grid and the RMS value of harmonic current in load side is showed in Table IV. Then APF prototype with the compensation capacity of 100 KVA is switched on to verify the proposed method. Parameters of LCL filter for APF are as follows: [L.sub.1] = 0.3mH, [L.sub.2] = 0.075mH, C=90uF, R = 0.5[omega] (Y connection). Inductors and capacitors in LCL filter are available in the laboratory but not with the best value, which induces the damping ratio to be only 0.15. The RMS value of grid line voltage is 380 V, and DC-side voltage for APF is 760 V accordingly. Switching frequency for APF is 5 KHz, and 5th, 7th, 11th, 13th harmonic current is selected to be compensated.
From Fig. 13, grid current after compensation is not ideal. This is mainly because inductors and capacitors in LCL filter are existing but not with the best value, which induces the damping ratio to be only 0.15. Resonant frequency is approximately 1.5 kHz and the circumambient harmonic is enlarged obviously. Another reason is the adoption of selective harmonic compensation strategy which means only 5th, 7th, 11th, 13th harmonic is chosen. Nevertheless, it is indicated that after magnitude and phase correction grid current is more sinusoidal. From Table IV, after magnitude and phase correction harmonic elimination has a better performance for all the selected harmonics. THD of load current reaches up to 73.2 %, and THD of grid current decreases from 22.2 % to 15.1 % using the strategy proposed
This paper has made the following conclusions on the basis of previous studies:
A single-phase equivalent circuit model has been established to analyse the characteristics of LCL filter applied in APF. oscillating element leads to a resonance peak enlarging harmonic. Resonant frequency depends upon total grid-side inductor with no relation to inverter-side inductor owing to the function of inverter current loop.
We proposed a set of methods to design LCL filter. Total inductance is restricted by the capability of APF. Damping ratio, attenuation degree of switching ripples and resonant frequency are the crucial factors to determine the performance of LCL filter. Intrinsic relation of them has been presented and a comprehensive consideration is essential.
LCL filter has an adverse effect on magnitude and phase of compensation current. A correction method using synchronous rotating transformation has been proposed and demonstrated by simulation and experimental results. Further studies may be needed on adapting to varying parameters of system.
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Manuscript received June 3, 2013; accepted December 10, 2013.
Rui Hou (1), Jian Wu (1), Yuchao Liu (1), Dianguo Xu (1)
(1) Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001. P.R. China, Phone: +8613654587937 email@example.com
TABLE I. LIST OF HARMONIC. 5th 7th 11th 13th RMS(A) RMS(A) RMS(A) RMS(A) THD (%) Load side 132 88.5 36 15 149.5 % Grid side without 5.47 11.64 11.16 4.76 15.97 % correction With magnitude 5.29 9.24 6.53 4.67 12.2 % correction With phase correction 5.02 8.73 7.74 2.96 12.16 % With magnitude 4.65 7.93 3.77 4.00 10.18% and phase correction TABLE II. MAGNITUDE GAIN AROUND RESONANCE FREQUENCY. 1125Hz 1150Hz 1400Hz 1425Hz Peak of [i.sub.1] 0.3 0.78 0.51 0.94 Peak of [i.sub.2] 0.46 1.26 0.66 1.08 Gain 1.53 1.61 1.29 1.15 TABLE III. MAGNITUDE AND PHASE COMPENSATION FACTOR. Order 5th 7th 11th 13th [M.sub.ck] 1/1.0503 1/1.1013 1/1.2702 1/1.3946 [[theta].sub.kc] 0.0071 0.0201 0.0853 0.1497 (radian) TABLE IV. LIST OF HARMONIC AFTER COMPENSATION. 5th 7th 11th 13th Load current(A) 28.2 15.9 6.1 5.6 Without correction 5.1 3.5 3.8 3.0 With correction 2.1 1.6 2.8 1.6
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|Author:||Hou, Rui; Wu, Jian; Liu, Yuchao; Xu, Dianguo|
|Publication:||Elektronika ir Elektrotechnika|
|Date:||May 1, 2014|
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