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Space Vector Modulation for Single-Phase Transformerless Three-Leg Unified Power Quality
Conditioner
Yong Lu, Guochun Xiao, Xuanlv Wu, Dapeng Lu School of Electrical Engineering and State Key Lab of Electric Insulation and Power Equipment
Xi’an Jiaotong University Xi’an, China
Abstract—Unified power quality conditioner (UPQC) is an ideal compensation device to improve power quality for sensitive end-users. In this paper, a space vector modulation method is proposed for single-phase three-leg UPQC (TL-UPQC) and one possible special switching sequence with reduced switching losses is discussed based on it. The proposed modulation method is similar to the well-known space vector modulation widely used in three-phase converters, thus brings extra flexibility to the TL-UPQC system. Additionally, with the special switching sequence, at least one leg of the TL-UPQC operates at line frequency and improved system efficiency will be obtained. Analysis, along with simulation and experimental results are presented to verify the feasibility and effectiveness of the proposed space vector modulation and switching sequence.
Keywords—Power quality; Unified power quality conditioner; Space vector modulation; Operation efficiency
I. INTRODUCTION Power quality (PQ) problems have obtained increasing
attentions as they can affect lots of sensitive end-users including industrial and commercial electrical consumers [1-3]. Studies indicate that PQ problems may lead to increased power losses, abnormal behavior of power equipment or even interruption of a manufacture process [4-6]. Dynamic Voltage Regulator (DVR) and Active Power Filter (APF) are two typical types of PQ compensation devices [7-10]. In General, voltage-related PQ problems can be efficiently tackled by DVR [11-13] and current-related PQ problems can be effectively compensated with the implementation of APF [14-16]. However, since the modern distribution system usually demand a better PQ of both voltage and current, the installation of DVR and APF becomes less cost effective. Unified Power Quality Conditioner (UPQC) is an ideal device to improve PQ where series and shunt compensation converters are integrated via a common DC-link bus. Due to the unique configuration of UPQC, excellent control over supplied voltage and drawn current can be achieved simultaneously [17, 18].
In recent years, UPQC has drowned the attention of researchers and has been applied to improve PQ in many applications. A comprehensive review on the topic of UPQC about different topologies and control method is provided in
[19]. In single-phase system, there are mainly three different UPQC topologies, namely full-bridge UPQC (FB-UPQC, total eight switches), half-bridge UPQC (HB-UPQC, total four switches) and three-leg UPQC (TL-UPQC, total six switches). TL-UPQC is generally considered to be a preferable choice with high compensation performance in low-cost low-power applications as its switches are reduced and transformer can also be removed [20]. However, coupling between the series and shunt converters introduced by the common leg has always been a great challenge to the control system of TL-UPQC and improper decoupling will result in performance deterioration.
In this paper, a space vector modulation (SVM) method is proposed for TL-UPQC based on the method adopted in three-leg AC-AC converter [21, 22]. The proposed SVM method treats three legs of TL-UPQC as an entire unit and naturally decouples the series and shunt converters. With this method, existing controllers designed for DVR and APF can directly be applied to control TL-UPQC without considering the coupling introduced by the common leg. Additionally, this modulation method is similar to conventional space vector modulation in three phase systems and thus brings the same flexibility to single-phase UPQC. A special switching sequence is also proposed based on the derived SVM method. Under this modulation sequence, one leg of the TL-UPQC will switch at line frequency, and the operation efficiency will be greatly improved.
This paper starts with introducing the topology of TL-UPQC and the coupling problem about it. Then the SVM method is provided and analyzed followed by the derivation of special modulation sequence. At last, the simulation and experimental results are given to verify the feasibility and effectiveness of the proposed SVM method as well as the special switching sequence.
II. TOPOLOGY AND COUPLING PROBLEMS Fig.1 illustrates a typical TL-UPQC configuration, where is,
ia, iL and ic are respectively the source current, shunt compensation current, load current and inductor current of the series filter; US, UL, Uo and Udc represent the grid voltage, load voltage, series compensation voltage and the DC-link voltage
978-1-4799-6735-3/15/$31.00 ©2015 IEEE 2859
separately. As shown in Fig. 1, the TL-UPQC mainly consists of five parts, including a series converter (leg b: V3, V4; leg c: V5, V6), a shunt converter (leg a: V1, V2; leg b: V3, V4), a DC-link storage (Cdc), a shunt filter (La) and a series filter (Lf, Cf). In this configuration, leg b is shared by both the series and shunt converters and Cdc is virtually separated into Cdc1 and Cdc2 (o is the virtual central point of Cdc) to simplify the following analysis.
During steady state operation, the shunt converter is controlled to generate compensation current according to harmonic and reactive currents drawn by the nonlinear load and the series converter will inject a desired missing voltage in series with the grid to ensure the power supply of sensitive loads. Two state equations can be obtained based on Fig.1 and written as:
( )
( )
aS a a b
cO c b f
diU L U U
dtdi
U U U Ldt
⎧ = + −⎪⎪⎨⎪ = − −⎪⎩
(1)
Where Ua, Ub, Uc respectively represent the voltage between the central point of each leg and o. Obviously, Ub exists in both state equations acting as a coupling term and it will cause interaction between the shunt and series converter control. So, traditional control method designed for DVR and APF cannot be adopted directly in TL-UPQC system and coupling introduced by the common leg should be considered.
There are basically two types of decoupling method proposed in publications about TL-UPQC. The control strategy provided in [20] considers one of the converters in TL-UPQC as the main converter and Ub is then controlled according to it. For example, if voltage quality is of the first priority, series converter will be treated as the main converter. In this situation, the operation of the series converter is the same as single-phase H-bridge DVR. So, Ub and Uc can be calculated through the compensation voltage controller, but Ua should be decided by both the control signal of compensation current controller and the obtained Ub. Thus, the performance of shunt converter will be affected by the series converter and certain restrictions will exist.
Another optional control method is proposed in [23], where Ub is directly determined to eliminate the coupling exists in the state equations of TL-UPQC. If Ub=-US, (1) can be decoupled as:
aa a
cf c L
diL U
dtdi
L U Udt
⎧ = −⎪⎪⎨⎪ = +⎪⎩
(2)
With this decoupling action, the shunt and series converters of TL-UPQC can be controlled independently and Ua, Uc will be determined by its respective control signal. However, the control loop with the mentioned decoupling method is relatively complex and it is hard to optimize the design of Ub.
Furthermore, improper controlled Ub will have bad influence on the performance of TL-UPQC.
III. SPACE VECTOR MODULATION FOR TL-UPQC As mentioned before, the series and shunt converters of the
TL-UPQC share a common leg and this will introduce coupling into the system. Traditional decoupling methods discussed previously treat three legs of the TL-UPQC separately and focus on controlling the output voltage of the common leg. In this section, a space vector modulation (SVM) method aiming at controlling output voltages of both series and shunt converters is proposed and a special switching sequence is also discussed based on the derived SVM method.
A. Space vector modulation If three legs of the TL-UPQC are treated as an entire unit,
the voltages supplied by the TL-UPQC can be displayed in the space plane defined by the output voltages of sires and shunt converters. As a result, voltage vector generated by the TL-UPQC in any situation can be described as:
n ab cbU U jU= + (3)
Where nU represents the voltage vector generated by the TL-UPQC; Uab is the output voltage of the shunt converter and Ucb is the output voltage of the series converter. In order to describe the proposed SVM, qn is applied to indicate the conduction states of all three legs. When upper switch is turned on, qn=1 and qn=0 during the opposite situation. Therefore, there are eight possible switching combinations for TL-UPQC and these switching combinations along with their output voltages are given in Table Ι. As can be seen from Table Ι, eight space vector generated contains six valid vectors and two zero vectors. Moreover, the output voltages of both converters are restricted to three different values: Udc, -Udc and 0.
Then, space vectors generated by all the switching combinations can be displayed in the Uab×Ucb space plane as shown in Fig.2. In Fig.2, Six valid vectors define six sectors from n=1 to 6 and Unref represents the reference voltage to be generated in sector n within one switching period TS. According to the space vector synthesis principle described in [24], Unref can be synthesized by two adjacent valid vectors of sector n and two zero vectors. So, Unref can be written as:
Fig.1 TL-UPQC configuration
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TABLE I. SWITCHING COMBINATIONS AND OUTPUT VOLTAGES
qa qb qc Space vector Uab Ucb
0 0 0 0 0U = 0 0
1 0 0 1 dcU U= Udc 0
1 0 1 42 2
j
dcU U eπ
= Udc Udc
0 0 1 23
j
dcU U eπ
= 0 Udc
0 1 1 4j
dcU U e π= -Udc 0
0 1 0 54
5 2j
dcU U eπ
= -Udc -Udc
1 1 0 32
6
j
dcU U eπ
= 0 -Udc
1 1 1 7 0U = 0 0
1 0 71 0 7
n nnref n n
S S S S
t t t tU U U U U
T T T T+
+= + + + (4)
Where tn，tn+1，t0 and t7 are time weights for each vector and TS= tn+tn+1+t0+t7 (if n=6, n+1=1).
tn and tn+1 under different sector n can be obtained according to (4) and Fig.2, and the results are listed in Table ΙΙ. In Table ΙΙ, Uabref and Ucbref are respectively the desired output voltage of shunt and series converters during TS. Before the time weights for two zero vectors can be chosen, an apportioning factor λ (0≤λ≤1) is needed. Thus, t0 and t7 can be chosen as:
7 zt tλ= (5)
0 (1 ) zt tλ= − (6)
1z S n nt T t t += − − (7)
Where, tz is the overall time weights for two zero vectors. Evidently, different value of λ will determine the duration time of U0 and U7 in one switching cycle, which may result in different circuit behaviors. This will be discussed in the third part of this section.
TABLE II. TIME WEIGHTS FOR VALID VECTORS
n tn tn+1
1 ( ) sabref cbref
dc
TU UU
− scbref
dc
TUU
2 sabref
dc
TUU
( ) scbref abref
dc
TU UU
−
3 scbref
dc
TUU
sabref
dc
TUU
−
4 ( ) scbref abref
dc
TU UU
− scbref
dc
TUU
−
5 sabref
dc
TUU
− ( ) sabref cbref
dc
TU UU
−
6 scbref
dc
TUU
− sabref
dc
TUU
B. Realization of SVM Basic concepts about the SVM of TL-UPQC is proposed
and discussed previously, but further analysis about actual duty cycles of each leg is needed before practical implementation. Fig.3 illustrates the wave forms of one possible switching sequence {U7, U2, U1, U0} for n=1. In Fig.3, αa, αb and αc are respectively the pulse widths for three legs. *
aU , *bU and *
cU represents the desired values of Ua, Ub and Uc separately. αa, αb and αc then can be derived based on Fig.3 as:
1 2 7a t t tα = + + (8)
7b tα = (9)
2 7c t tα = + (10)
In sector from 2 to 6, switching sequences are described as {U7, U2, U3, U0}，{U7, U4, U3, U0}，{U7, U4, U5, U0}，{U7, U6, U5, U0}，{U7, U6, U1, U0}. The pulse widths for three legs in these five sectors can be obtained through the same process. In Table ΙΙΙ, pulse widths derived in all 6 sectors based on the described switching sequence are listed. In fact, the total pulse widths for different switching sequences are always the same.
Fig.2 Space vectors in space plane
Fig.3 Pulse widths for each leg when n=1
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TABLE III. PULSE WIDTHS FOR 6 SECTORS
n αa αb αc 1 t1+t2+t7 t7 t2+t7 2 t2+t7 t7 t3+t2+t7
3 t7 t4+t7 t3+t4+t7 4 t7 t4+t5+t7 t4+t7 5 t6+t7 t5+t6+t7 t7 6 t1+t6+t7 t6+t7 t7
Then, execution steps for practical implementation of the proposed SVM can be concluded as:
1) Determine sector n from Fig.2 based on Uabref and Ucbref obtained through the shunt and series converter controller.
2) Compute tn and tn+1 based on the conclusion provided in Table ΙΙ with the decided n.
3) Compute t0 and t7 from (5)-(7) according to the designed apportioning factor λ.
4) Compute pulse widths αa, αb and αc for each leg based on the method given in Table ΙΙΙ.
C. Special switching sequence with improved efficiency Basic principle of the proposed SVM for TL-UPQC is
actually similar to traditional space vector modulation method widely used in three-phase systems. So, analysis and conclusions about the well-known SVM method are also applicable for the proposed SVM strategy, which introduces extra flexibilities to TL-UPQC systems. However, unique features still exist considering the distinctive topology and operation of the TL-UPQC. In this section, a special switching sequence with improved efficiency is derived based on relations between space vector and practical output voltage of each leg.
When two-level symmetric regular-sampled PWM is adopted to realize the proposed SVM method, αi (i=a, b, c) can be derived as [25]:
2
(1 )2S i
idc
T UU
α = + (11)
Consider the situation shown in Fig.3. Ub can be calculated by combining (9) and (11).
7 1( )2b dc
S
tU U
T= − (12)
Based on (5), (7), (12) and Table ΙΙ, Ub then can be rearranged as:
( 0.5)b dc abrefU U Uλ λ= − − (13)
Ub in n=2-6 can be obtained through the same computation process from (11) to (13) and written as:
( 0.5)b dc cbrefU U Uλ λ= − − (14)
( 0.5) (1 )b dc cbref abrefU U U Uλ λ λ= − − − − (15)
( 0.5) (1 )b dc abrefU U Uλ λ= − − − (16)
( 0.5) (1 )b dc cbrefU U Uλ λ= − − − (17)
( 0.5) (1 )b dc abref cbrefU U U Uλ λ λ= − − − − (18)
If relations between Uabref and Ucbref in Fig.2 are considered, equation that describes Ub within the whole space plane can be concluded based on (13)-(18).
max min( 0.5) (1 )b dcU U U Uλ λ λ= − − − − (19)
Where, Umax and Umin are respectively the maximum and minimum value of {Uabref, Ucbref, 0}.
Equations of Ua and Uc under the same circumstances from n=1 to 6 then can be derived based on (19) and written as:
max min( 0.5) (1 )a ab dcU U U U Uλ λ λ= + − − − − (20)
max min( 0.5) (1 )c cb dcU U U U Uλ λ λ= + − − − − (21)
In steady state operation, Uab is controlled to generate desired compensation current according to load nonlinear current and Ucb is controlled based on the missing voltage. So, there are some restrictions for Uabref and Ucbref considering the actual operation of TL-UPQC. As can be inferred from Fig.1, when the grid voltage is lower than the rated voltage, Ucbref should be controlled opposite phase to the grid and when the grid voltage is higher than the rated voltage, Ucbref should be in phase with the source. In low-power applications, the voltage across La is considered to be small enough compared to the source voltage, so Uabref is at the vicinity of the grid voltage and has the same phase as the source.
As a result, tracks of the desired voltage vector generated by the TL-UPQC can be determined according to grid conditions. When the grid voltage is lower than the rated voltage, Uabref is opposite phase to Ucbref and the TL-UPQC will operate in sector 3 and 6. If λ is set to be 0 at sector 3 and be 1 at sector 6, Ua will be constantly controlled to be -0.5Udc and 0.5Udc respectively according to (20). In this situation, leg a switches at line frequency and the other two legs operate at switching frequency. When the grid voltage is higher than the rated voltage, Uabref is in phase with Ucbref and TL-UPQC will operate in sector 1, 2 and sector 4, 5. If λ in this condition is designed to be 0 at sector 1, 2 and to be 1 at sect 4, 5, leg b will operate at line frequency according to (19).
Switching sequence in the pre-described situation can be concluded as {U2, U1, U0}, {U2, U3, U0}, {U4, U3, U0}, {U7, U4, U5}, {U7, U6, U5}, {U7, U6, U1}. With this switching sequence, at least one leg of the TL-UPQC will operate at line frequency and thus the switching losses can be significantly reduced.
IV. SIMULATION AND EXPERIMENTAL VERIFICATION In order to show the validity of the proposed SVM method,
as well as the special switching sequence, simulation and experimental results are presented in this section. In the simulation and experiment, controllers designed for DVR and APF are directly adopted in the control of TL-UPQC. A digital control strategy based on Equivalent Fundamental and Odd Harmonic Resonators proposed in [26] is applied to control the series converter of the TL-UPQC and this control method ensures good performance with improved dynamic response.
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TABLE IV. SYSTEM PARAMETERS
Descriptions Parameters Descriptions Parameters Rate voltage 110V(rms) Lf 1.5mH
Switching frequency 15kHz Cf 20μF Cdc 4700μF La 4.6mH
Line frequency 50Hz Udc 230V
Compound controller based on PI and repetitive control introduced in [27] is adopted to achieve better steady state current compensation for the shunt converter. Structures of two mentioned controllers are given in Fig.4 and Fig.5. In Fig.6, the implementation of the proposed SVM is illustrated, where g1 to g6 respectively represents the switching signals for V1 to V6. System parameters of the simulation and experiment are provided in Table ΙV.
A. Simulation Results Fig.7 and Fig.8 show the simulation results of the TL-
UPQC with nonlinear load and abnormal grid voltage when the proposed SVM is adopted. In Fig.7, the supply voltage drops to 70% of its rated value and the grid voltage increases to 130% of its normal voltage in Fig.8. Apportioning factor λ in these two simulations is set to 0.5. As can be seen from Fig.7 and Fig.8, the proposed SVM is valid in the control of TL-UPQC and the power quality of the simulated system is well compensated.
The simulation results of the TL-UPQC with the discussed special switching sequence are provided in Fig.9 and Fig.10. Grid voltages in Fig.9 and Fig.10 are separately the same as the source voltages in Fig.7 and Fig.8. Source currents in these two simulations are ten times larger than their real value to be properly displayed with the load voltages. As illustrated in these two results, leg a operates well at 50Hz when the grid is lower than its normal condition and leg b switches effectively
Fig.4 Controller for series converter
Fig.5 Controller for shunt converter
Fig.6 Implementation of the proposed SVM
Fig.7 Simulation results with 70% grid voltage
Fig.8 Simulation results with 130% grid voltage
Fig.9 Special switching sequence with 70% grid voltage
Fig.10 Special switching sequence with 130% grid voltage
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at line frequency when the source is higher than its rated value. However, high frequency switching still exists for leg a and leg b at the vicinity of the zero crossing point as the approximation about the relation between Uabref and US is no longer satisfied with small grid voltage value.
B. Experimental Results Experimental results based on a 1kW prototype are
presented in this part. Fig.11 and Fig.12 respectively demonstrate the compensation waveforms with the proposed SVM method (λ=0.5) under 70V (rms value) and 140V (rms value) grid. Fig.13 and Fig.14 separately show the waveforms under the same conditions as in Fig.11 and Fig.12 when the proposed special switching sequence is implemented. Additionally, efficiency curves of both the normal SVM with λ=0.5 and the discussed special switching sequence are presented in Fig.15. Each efficiency curve in Fig.15 is consists of 8 point obtained by HIOKI 3197 (PQ analyzer from Japan) under steady-state compensation.
As illustrated in the experimental results, the proposed SVM is valid in the control of TL-UPQC and the special switching sequence can significantly improve the operation efficiency of the TL-UPQC with reduced switching actions.
V. CONCLUSION This paper has presented a SVM method for single-phase
TL-UPQC and a novel switching sequence based on the SVM is also proposed with improved system efficiency. The proposed SVM method is derived from similar method introduced in three-leg AC-AC converter. The working principle and practical implementation of the SVM are discussed in detail with theoretical analysis. Additionally, analysis and conclusions about the conventional space vector modulation widely used in three phase converters are also applicable for the proposed modulation strategy, which brings extra flexibility for the TL-UPQC system. The special switching sequence discussed is one type of possible application of the SVM. With this special switching sequence, at least one leg operates at line frequency during steady state compensation, thus the system efficiency is greatly improved. Simulation and experimental results are presented to verify the feasibility and effectiveness of the proposed SVM method and special switching sequence.
ACKNOWLEDGMENT This paper and its related research work are supported by
national Natural Science Foundation of China (NSFC) (project no. 51277146).
(CH1, CH2: 200V/div; CH3, CH4: 30A/div; time: 10ms/div）
Fig.11 Experimental results with 70V grid voltage
(CH1, CH2: 200V/div; CH3, CH4: 30A/div; time: 10ms/div）
Fig.12 Experimental results with 140V grid voltage
(CH1, CH2, CH3, CH4: 250V/div; time: 10ms/div)
Fig.13 Special switching sequence with 70V grid voltage
(CH1, CH2, CH3, CH4: 250V/div; time: 10ms/div)
Fig.14 Special switching sequence with 140V grid voltage
Fig.15 Operation efficiency comparison
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REFERENCES [1] R. C. Dugan, M. F. McGranaghan, and H. W. Beaty, “Electrical power
systems quality,” New York: McGraw-Hill, 1996. [2] M. Bollen, “Overview of power quality and power quality standards,”
Wiley: IEEE Press, 2000. [3] A. Ghosh and G. Ledwich, “Power quality enhancement using custom
power devices,” Boston, MA: Kluwer, 2002. [4] A. Bendre, D. Divan, W. Kranz, and W. Brumsickle, “Equipment
failures caused by power quality disturbances,” in Proc. IEEE IAS Annu. Meeting, 2004, pp. 482–489.
[5] P. Curtis, “The fundamentals of power quality and their associated problems,” Wiley: IEEE Press, 2007.
[6] I. Hunter, “Power quality issues-a distribution company perspective,” Power Engineering Journal, vol. 15, no. 2, pp. 75-80, Apr. 2001.
[7] J. G. Nielsen and F. Blaabjerg, “A detailed comparison of system topologies for dynamic voltage restorers,” IEEE Trans. Ind. Appl., vol. 41, no. 5, pp. 1272–1280, Sep./Oct. 2005.
[8] B. Singh, K. Al-Haddad, and A. Chandra, “A review of active filters for power quality improvement,” IEEE Trans. Ind. Electron., vol. 46, no. 5, pp. 960–971, Oct. 1999.
[9] M. El-Habrouk, M. K. Darwish, and P. Mehta, “Active power filters: A review,” IEE Electr. Power Appl., vol. 147, no. 5, pp. 403–413, Sep. 2000.
[10] J. Praveen, P. Bishnu, M. S. Venkateshwarlu and H. V. Makthal, “Review of dynamic voltage restorer for power quality improvement,” in Proc. 30th Annual Conference of the IEEE Industrial Electronics society, 2004, pp. 749-752.
[11] Y. Lu, G. C. Xiao, B. Lei, X. L. Wu, and S. H. Zhu, “A transformerless active voltage quality regulator with parasitic boost circuit,” IEEE Trans. Power Electron., Vol. 29, No. 4, pp. 1746-1756, Apr. 2014.
[12] T. Jimichi, H. Fujita, and H. Akagi, “Dynamic voltage restorer equipped with a high-frequency isolated DC-DC converter,” IEEE Trans. Ind. Appl., vol. 47, no. 1, pp. 169-175, Jan./Feb. 2011.
[13] A. K. Sadigh, E. Babaei, S. H. Hosseini, and M. Farasat, “Dynamic voltage restorer based on stacked multicell converter,” in Proc. IEEE Symp. Ind. Electron., Appl., 2009, pp. 419–424.
[14] Q. Liu, L. Peng, Y. Kang, S. Y. Tang, D. L. Wu and Y. Qi, “A novel design and optimization method of an LCL filter for shunt active power filter,” IEEE Trans. Ind. Electron., vol. 61, no. 8, pp. 4000-4010, Aug. 2014.
[15] P. Acuna, L. Moran, M. Rivera, J. Dixon and J. Rodriguez, “Improved active power filter performance for renewable power generation
systems,” IEEE Trans. Power Electron., vol. 29, no. 2, pp. 687-694, Feb. 2014.
[16] H. L. J, J. C. Wu, Y. J. Chang, and Y. T. Feng, “A novel active power filter for harmonic suppression,” IEEE Trans. Power Delivery, vol. 20, no. 2, pp. 1507– 1513, Apr. 2005.
[17] A. Teke, L. Saribulut and M. Tumay, “A novel reference signal generation method for power-quality improvement of unified power-quality conditioner,” IEEE Trans. Power Delivery, vol. 26, no. 4, pp. 2205-2214, Oct. 2011.
[18] V. Khadkikar, A. Chandra, A. O. Barry, and T. D. Nguyen, “Power quality enhancement utilizing single-phase unified power quality conditioner: digital signal processor-based experimental validation,” IET Power Electron., vol. 4, no. 3, pp. 323-331, Mar. 2011.
[19] V. Khadkikar, “Enhancing electric power quality using UPQC: a comprehensive overview,” IEEE Trans. Power Electron., vol.27, no.5, pp.2284-2297, May 2012
[20] A. Nasiri and A. Emadi, “Different topologies for single-phase unified power quality conditioners,” in proc. 38th IAS Annual Meeting, vol.2, pp.976-981, Oct. 2003.
[21] C. B. Jacobina, T. M. Oliveira, and E. R. C. da Silva, “Control of the single-phase three-leg AC/AC converter, Industrial Electronics, ” IEEE Trans. Ind. Electron., vol.53, no.2, pp.467-476, Apr. 2006.
[22] H. Pinheiro, R. Blume, and P. Jain, “Comparison of SV modulation methods for single phase on-line three-leg UPS,” in Proc. 26th IECON Annual Conference, vol.2, pp.1328-1333, Oct. 2000.
[23] S. J. Chiang, C. Y. Yen, and K. T. Chang, “A multimodule parallelable series-connected PWM voltage regulator,” IEEE Trans. Ind. Electron., vol.48, no.3, pp.506-516, Jun. 2001.
[24] J. Holtz, “Pulsewidth modulation for electronic power conversion,” in Proc. IEEE, vol. 82, no. 8, pp. 1194–1214, Aug. 1994.
[25] K. L. Zhou, and D. W. Wang, “Relationship between space-vector modulation and three-phase carrier-based PWM: A comprehensive analysis,” IEEE Trans. Ind. Electron., vol. 49, no. 1, pp. 186 -196, Feb. 2002.
[26] L. L. Hu, G. X. Xiao, G. F. Teng and Y. Lu, “An equivalent fundamental and odd harmonic resonators controller for a single-phase dynamic voltage restorer,” Proceedings of the Chinese Society for Electrical Engineering, vol. 32, no. 22, pp. 104-112, Aug. 2012.
[27] Z. Zeng, J. Q. Yang, and N. C. Yu, “Research on PI and repetitive control strategy for Shunt Active Power Filter with LCL-filter,” in Proc. 7th IPEMC, vol.4, pp. 2833-2837, Jun. 2012.
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