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Improved tracking of shunt active power filter by sliding mode control
Electrical Power and Energy Systems 78 (2016) 916–925
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improved tracking of shunt active power filter by sliding mode control Anirban Sinha Ray ⇑, Avik Bhattacharya Department of Electrical Engineering, Indian Institute of Technology-Roorkee, India
a r t i c l e
i n f o
Article history: Received 7 December 2014 Received in revised form 19 September 2015 Accepted 10 November 2015 Available online 29 January 2016 Keywords: Shunt Active power filter CC-VSI Non linear load Sliding mode control Feedback linearization Variable structure system
a b s t r a c t An improved tracking technique of the reference current waveform in a shunt type active power filter is proposed in this paper using sliding mode control and feedback linearization. Feedback linearization approach helps to reduce the complexity of the controller. Excellent tracking is achieved in both for steady state and dynamic performance. A three-phase four-wire system is considered. The proposed algorithm is simulated first in MATLAB/SIMULINK. An experimental prototype using a dspace1104 based controller is also produced in the laboratory. Results from simulation match well with the corresponding results from experimental prototype confirming the usefulness of the proposed technique. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction In recent times most of the ac electrical power systems include several kinds of non-linear load. Non linear load involves harmonics that contaminate the sinusoidal supply voltage and current. Harmonics not only increase the losses in the system but also produce unwanted disturbance to the communication network, more voltage and/or current stress, etc. The uses of passive filters are not desirable as they are bulky and de-rate itself with age. Moreover this passive element may cause resonance with source impedance. This has motivated the introduction of the Active Power Filter (APF) for improving power quality. Shunt APF is used to eliminate the current harmonics, whereas, the series compensation does the same job for the voltage harmonics. Fig. 1 shows the system where a Voltage Source Inverter (VSI) operating as an APF and connected in parallel with the load [1]. Discusses the prospects of research in power quality improvement [2]. Provides extensive discussions on the robust performance of HPFC using digital simulation. While three phase reactive power compensation has been discussed in [3]. Singh et al. [4] have developed a fuzzy rule based generalized unified power flow controller. Sasaki [5] worked on systematic nonlinear control approach to a power factor corrector design. The basic principle of operation is to inject a correct nature of current to compensate for the load harmonics. This ⇑ Corresponding author. E-mail addresses: [email protected] (A. Sinha Ray), [email protected] (A. Bhattacharya). http://dx.doi.org/10.1016/j.ijepes.2015.11.015 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
requires detection of harmonic component of the load current following which the reference-current needs to be produced. Different methods of reference generation have been proposed. These include different variants of p–q theory [12,19,21,22], synchronous reference frame [20], FFT, RDFT, wavelet based techniques and more recently ANN, GA and soft computing based approaches [27–33]. Reference [6] provides a survey of such methods. Once the reference is generated, the VSI needs to track the reference current. Many controller strategies are employed to explore better performance of tracking. Most popular techniques are PI controller, standard hysteresis controller, one cycle control, sliding mode control, etc. Adaptive control and negative feedback based repetitive control are also employed to improve the dynamic performance and stability of the overall system [17,18]. Sliding mode controller is more immune to parameter uncertainties. This feature encourages the researcher to apply sliding mode controller (SMC) to mitigate tracking error in APF [6,9,13–17,23]. Singh et al. [8,9] applied sliding mode on a three phase four wire system. The SMC is used to control the dc bus voltage. Furga et al. [10,11] reported an improved dynamic performance of the APF by applying SMC and a passive LC filter with and without active source. Feedback linearization based control is also applied to shunt APF to reduce the computation burden [24–26]. Matas et al. [7] reported feed back linearization by Tellegen’s theorem [34–36] of single phase APF and then SMC is applied to simply the overall design. This paper has applied SMC to a three phase four wire system to study tracking and also to take up unbalance in the three phase load. Contrary to the available approach [11], this paper utilizes two
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Fig. 1. Active power filter to compensate for a non-linear load.
sliding mode controllers, one for dc bus regulation and another for tracking the current reference. A dspace1104 based controller is produced to implement the control-algorithm and generate the switching pattern to the power circuit. Fig. 1 shows the active power topology used for experimental and simulation purpose. The system may be modeled as a set of the current sources that injects reactive and harmonic current to the source. The paper is organized in six sections. Section ‘Modeling of shunt active power filter for sliding mode control’ deals with theoretical aspects of feedback linearization, sliding mode control and modeling of shunt active power filter with the same is presented. Simulation result is presented in Section ‘Simulation results’. Experimental verification of proposed scheme is presented in Section ‘Experimental results’. Section ‘Conclusions’ concludes the work.
From (A13) (in Appendix) tracking error may be defined as follows:
½uab idc :
ð6Þ
Modeling of shunt active power filter for sliding mode control
qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi Where T r ¼ 2p= Lf C f and T rd ¼ 2p= Rf C f
ð7Þ
Tracking of current reference
ð4Þ
Detail derivation is available in Appendix. The reference is generated by capacitor based predictive algorithm and then followed by THD minimization technique [18]. For tracking the quantities in abc frame is transformed in ab frame. Thus (4) may be expressed in the following way in ab frame:
½eab ¼ ½ic ab þ ½ic ab
ð5Þ
From (1)–(4) the state space representation of the APF may be expressed
½€eab ¼
1 T 2r
½eab
T rd T 2r
½eab þ ½f ab
1 T 2r
The tracking function ½f ab may be expressed as follows:
From Fig. 1 the following equation may be written for shunt active power filter. It is also assumed that source inductance is small and thus the drop across the source inductance may be neglected. The voltage at the point of common coupling is same as that of the source voltage.
Lf
dic ¼ ½v s abc ½v cf þ Rf ic dt
ð1Þ
Cf
d½V cf abc ¼ ½uabc idc ½ic abc dt
ð2Þ
ua , ub and uc are the switching function for the leg a, b and c for the voltage source inverter. Following way switching function is defined: u ¼ 1: If upper switch is on and lower switch is closed. u ¼ 0: If upper switch is off and lower switch is on. Applying KCL we may establish the relation between source current, load current and compensating current as:
½is abc ¼ ½iL abc þ ½ic abc ½is abc ¼ ½iL abc þ ½ic abc
½eabc ¼ ½ic abc þ ½ic abc
ð3Þ
T 1 rd ½f ab ¼ ½€ic ab ½€ic ab þ 2 ½i_c ab ½i_c ab þ 2 ½ic ab ½ic ab : Tr Tr
ð8Þ
The resonance frequency (xr ) of the L-C filter is much higher than supply frequency, i.e xr xs . Where
1 LF C F
xr ¼ pffiffiffiffiffiffiffiffiffiffi
ð9Þ
Eq. (9) may be further simplified considering the fact that the resistance present in the shunt path is very low. Thus modified expression of ½f ab is:
1 ½f ab ½€ic ab ½€ic ab þ 2 ½ic ab ½ic ab Tr
ð10Þ
The use of sliding mode control and design of sliding surface has been discussed in detail in [19,21]. For active damping of L-C filter and reduction in tracking error, following relation is established between the sliding surface ‘‘s” and nonlinear control command ‘‘u”.
½ui ab ¼ signð½sab Þ
ð11Þ
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The sliding surface may represented as:
½sab ¼ ½eab þ ½K d ab ½e_ ab
ð12Þ
K d is constant and is responsible for active damping of L-C filter. According to variable structure system, the proposed controller will be asymptotically stable if:
½sa s_ a < 0 and ½sb s_ b < 0
ð13Þ
From (1)–(5) and (6)–(12) ½s_ ab may be expressed as:
½s_ ab ¼
½K d ab
"
T 2r
! # T 2r ½eab þ T rd ½e_ ab þ ½f ab idc signð½Sab ½K d ab ð14Þ
Thus from (13) and (14), the condition for stability is determined as:
! dv dc T 2r C dc T rd ½e_ ab þ ½f ab ¼ idc > ½eab dt Kd
ð15Þ Fig. 2. The capacitor voltage stabilization circuit.
If the gain K d is selected such that:
1
T 2r T rd Kd
ð16Þ
Using (16), (15) may be rewritten as:
dv dc C dc dt
¼ idc > ½eab þ ½f ab
Eq. (10) may be reduced as:
½f ab
T 2r
½ic ab
½ic ab
ð18Þ
Thus the system will be asymptotically stable if (19) is satisfied.
C dc
dv dc ¼ idc > ½f ab dt
ð19Þ
This sliding mode controller is employed for tracking the reference current and active damping of the network. Sliding mode controller for maintaining dc link voltage
dic ¼ Rf ic v c1 þ v s dt
Let x1 ¼ ic ; x2 ¼ v c1 and x3 ¼ v c2
C 1 x_ 2 ¼ 0 and C 2 x_ 3 ¼ x1
ð26Þ
Eq. (27) represents the state space equation in matrix form:
3 Rf x_1 Lf 6 _ 7 4 x2 5 ¼ 0 1 x_3 C 2
2
0 0 0
ð20Þ ð21Þ
Thus (20) may be expressed as:
Rf 1 vs x_ ¼ x1 x2 þ Lf Lf Lf
ð22Þ
C 1 x_ 2 ¼ x1 and C 2 x_ 3 ¼ 0
ð23Þ
3 20 6 7 6 1 0 4 x2 5 þ 6 4 C1 1 0 x3 C
1 2 L f x1
2
1 Lf
0 0
32 3 2 vs 3 x1 Lf 76 7 6 7 07 u þ x 4 5 4 05 2 5 x 0 3 0
1 Lf
ð27Þ
From (27) it may be seen that the relative degree of the system is r = 1. The dc link capacitor voltage and the current through inductor is considered as state of the system. From the power balance equation, the following condition may be derived. The dc link capacitors C1 and C2 are considered same (i.e. C1 = C2 = C) in magnitude.
2 d 12 ½C v 2c 12 þ 12 Lf ic dt
The sliding mode controller derived in this section is to maintain dc link voltage. As is discussed in Introduction and also reported in [18], the control of dc link voltage is highly nonlinear. Non linear sliding surface will increase controller complexity. Such computational complexity will increase the delay in DSP based system. Fig. 2 shows the equivalent circuit diagram of APF per phase. The low frequency filter connected at the pole of the inverter is neglected for simplification. Two capacitors, C1 and C2 form the dc bus as shown in Fig. 2. The state space equation of the circuit diagram shown in Fig. 2 may be derived. For u ¼ 1, following switching function is obtained:
Lf
ð25Þ
ð17Þ
r
ð24Þ
Rf 1 vs x1 x3 þ Lf Lf Lf
x_ 1 ¼
The second order term present in ½f ab may be neglected since the amplitude of harmonic current decreases with increase in frequency. Also note that T12 >> 1.
1
For u ¼ o
¼ v s ic ic Rf 2
ð28Þ
Let us consider, output function as:
Z
yðtÞ ¼
v s ic dt
ð29Þ
Thus
_ yðtÞ ¼ v s ic
ð30Þ
Rx v uv uv V €ðtÞ ¼ v_ s ic þ v s i_c ¼ v s f 1 þ c2 c1 c2 þ s y Lf Lf Lf Lf Lf
ð31Þ
Therefore by critically observing (31), it may be concluded that the output function yðtÞ, described in (29) achieves a relative degree of r = 2, since the control input is second order time derivative of (30). Now following transformation is applied to achieve diagonal form (19)–(21).
_ z ¼ ðz1 z2 ÞT ¼ ðy; yÞ
ð32Þ
Assuming variation of v s with respect to switching frequency is negligible, following equation is obtained.
€¼ z_ 2 ¼ y
Rf x1 v c2 u ðv c1 þ v c2 Þ þ Lf 2 Lf Lf
v 2s
ð33Þ
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Again from (27)
v_c1 ¼
x1 u C
v_ c2 ¼
x1 x1 u 1 u ¼ x1 þ þ C C C C
v c2 ¼ ð34Þ ð35Þ
From (28), (35) and (36) following equations are derived:
v c1 ¼
Z
ic z u dt ¼ u C v sC
Z iC
1 u z 1 u þ þ dt ¼ C C vs C C
ð37Þ
Substituting (36) and (37) in (38) following equation is obtained.
€¼ z_2 ¼ y
v 2s Lf
Rf i c
vs Lf
z ð2u2 2u þ 1Þ CLf
ð38Þ
Eq. (38) may be further be simplified as:
ð36Þ
€¼ z_2 ¼ y
v 2s Lf
Rf i c
vs Lf
zx20 ð2u2 2u þ 1Þ
ð39Þ
Fig. 3. Source current before and after compensation. Waveform (a) source voltage of phase A. Waveform (b) source current of phase A. Waveform (c) source current of phase B. Waveform (d) source current of phase C. Time scale 20 ms/Div.
Fig. 4. Steady state operation of the controller. Waveform (a) source voltage. Waveform (b) source current. Waveform (c) load current. Waveform (d) compensating current. Time scale 20 ms/Div.
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The degree of system being two, following transformation may be applied to simplify the system:
v 2s Lf
Rf ic
vs Lf
zx20 ð2u2 2u þ 1Þ ¼ w
ð40Þ
‘‘w” is the new input signal to the system. This transformation enables us to establish linear relationship between the output ‘‘y” and new control signal (from the discussion made in (A6)). Applying the feedback linearization in (40) and (43) is obtained. From (41) following expression may be obtain.
z_ 1 ¼ z2
ð41Þ
z_ 2 ¼ c1 z2 c0 z1 þ w
ð42Þ
where
w ¼ €z1 þ c1 z_ 1 þ c0 z_ 1
ð43Þ
It is clear from (47) that the closed loop control dynamics is totally linearized. This linearization makes the controller simple. According to the theory of sliding mode control the system will be stable if coefficients of polynomial expressed in (51) are all positive. The error in tracking may be expressed as:
€e ¼ c1 e_ þ c0 e ¼ 0
ð44Þ
Fig. 5. Source, load and compensating current waveforms for step change in load with feedback linearization. Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current time scale 20 ms/Div.
Fig. 6. Source, load and compensating current waveforms for step change in load with feedback linearization. Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current (d) Capacitor voltage time scale 20 ms/Div.
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Since shunt active power filter will ensure that the source current will be in phase with source voltage. Thus source current may be considered as: is
¼a
vs
ð45Þ
where ‘‘a” is the proportionality constant. Thus ic may be express as follows:
ic ¼ iL av s
ð46Þ
Thus tracking error after feedback linearization may be express as:
e ¼ ic ic
ð47Þ
After feedback linearization, the sliding surface may be modified as:
e ¼ z1 z1 ¼
Z
v s ðic ic Þds
ð48Þ
The new sliding surface may be express as:
s ¼ v s ðic ic Þ þ k1 ðic ic Þds þ k0
ZZ
v s ðic ic Þds
ð49Þ
The sliding surface will be stable if the following condition is satisfied:
ss_ ¼ 0
ð50Þ
Fig. 7. Source, Load and compensating current waveforms for step change in load without feedback linearization. Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current. (d) Capacitor voltage time scale 50 ms/Div.
Fig. 8. Source, load, compensating current and dc link capacitor voltage waveforms for PI controller Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current. (d) Capacitor voltage time scale 50 ms/Div.
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Table 1 Comparison of control strategies. Control strategy
Value of capacitance (lF)
DC voltage of ripple (V)
THD of source current
PI Sliding mode without feedback Sliding mode with feedback linearization
1200 800 800
8 6 6
3.2 6.2 3.47
From this condition control logic may be obtained as follows:
uj ¼
1; for s > 0 0;
for s < 0
ð51Þ
This simplified controller is very easy to implement and shows excellent dynamic performance. Both the control signal ui and uj are ‘‘ORed” to obtain resultant control signal. This twin sliding mode controllers are found to be simple, reliable and insensitive to parameter uncertainty. Simulation results For the purpose of simulation the system is configured in SIMULINK. The powersim block is used to construct inverters and diode bridge rectifiers. Inverter dead-time is set at 2 ls. Simulations have been carried out for balanced and unbalance loads. Fig. 3 shows transformation of source current with and without APF. A three phase diode bridge feeding a R-C kind of load is considered. The source voltage and three source currents are shown in top to bottom order. Fig. 4 shows the steady state performance of APF. The source voltage, source current, load current and compensating current are shown in descending order. Dynamic performance is shown in Fig. 5. A step load change is indicated at 84 ms.This result shows excellent dynamic performance. The source current load current and compensating current are shown at the top, middle
and bottom of the waveform. The simulation results shown in Figs. 3–5 are obtained by linearization of the controller. Fig. 6 shows dynamic performance of APF with feedback linearization. Fig. 7 shows dynamic performance without feedback linearization. It is clear from the Fig. 7 that even though the system is stable by fulfilling the condition shown in (47), however chattering is more. The source current, load current, compensating current and capacitor voltage are shown in top to bottom order. Fig. 8 shows the functionality of PI controller based APF. It has been observed that the value of dc link capacitor is 50% more than the sliding mode controller. Comparison of these three controllers are provided in Table 1. The source, load, compensating current and capacitor voltage are shown in same sequence as of Fig. 7. To study the unbalance system, a star connected unbalanced R-L load is added in parallel to nonlinear load. Thus the system is now consists of an unbalanced load connected in parallel to the non-linear load (i.e. a diode rectifier feeding a R-C network and with a parallel R-L network). Two phases for the source and load currents are shown in top and bottom of the figures respectively in Fig. 9. The dynamic performance is also tested in case of unbalanced load. A step change in load current has occurred at 66 ms. This controller shows excellent performance also in unbalanced load. Fig. 10 shows transition of balance to unbalance load. Dc link capacitor voltage, source current, load current and compensating currents of any two phases are shown in top to bottom order.
Fig. 9. Transition of non-linear balanced load current Unbalanced load current. Top waveform: Source current of Phase A and Phase B. Bottom waveform: Unbalanced Load current of phase A and phase B. Time scale 20 ms/Div.
A. Sinha Ray, A. Bhattacharya / Electrical Power and Energy Systems 78 (2016) 916–925
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Fig. 10. Step load change in balance to unbalance condition. (a) Capacitor voltage. (b) Source current of phase A&B waveform. (c) Load current phase A&B. (d) Compensating current time scale 50 ms/Div.
Experimental results The proposed algorithm is also tested in the laboratory. An experimental prototype is made with two level inverter. Voltage source current controlled inverter using IGBTs that is available in the laboratory is used for this purpose. 110 V, 50 Hz supplying a
load of 3 kVA is considered. The whole system is built in SIMULINK where the ANN-routine is called whenever necessary for reference generation purpose. This facilitates the use of all the built-in blocks in SIMULINK. The clock frequency of the system is set at 10 kHz. Fig. 11 shows three source current before after compensation. Fig. 12 shows the steady state performance of APF with feedback linearization. The source voltage, source current, source current, load current and compensating currents are shown top to bottom order. Fig. 13 shows steady state performance of the controller without feedback linearization. For this reason it may be seen that considerable amount of chattering is present. Source, load and compensating currents are shown in descending order. Fig. 14 shows dynamic performance of proposed controller. Fig. 14 shows source voltage, source current, load current and compensating current in same sequence as shown in Fig. 12. Load change is occurred at 56 ms.This linear controller tracks the trajectory very fast and tracking error is minimal. Fig. 15 is the Fourier series(harmonic spectrum) of the compensated and the uncompensated current waveforms (see Fig. 16).
Fig. 11. Phase currents of the three phases.
Fig. 12. Steady state operation of shunt active power filter. Waveform (a) source voltage. Waveform (b) source current. Waveform (c) load current. Waveform (d) compensating current.
Fig. 13. Source, load and compensating current waveforms for step change in load without feedback linearization. Waveform (a) source current. Waveform (b) load current. Waveform (c) compensating current.
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Fig. 14. The dynamic performance of the controller. Waveform (a) source voltage. Waveform (b) source current. Waveform (c) load current. Waveform (d) compensating current.
tracking is actively damped by a derivative controller. From Fig. 14, we can see that the APF current can quickly track the harmonic current, thus to achieve the harmonic compensation. The reference currents tracking behavior has been improved, and the power supply harmonic current has been reduced with proposed sliding mode control. The system being non-linear and computationally extensive, it is difficult to implement in hardware. This is mainly because of delay in computation. To overcome this drawback feedback linearization is applied to the sliding mode controller. Table 1. Shows the comparison of three type of controller. Even though PI controller provides require THD the dc link ripple and size of capacitor is higher than the sliding mode controller. Feedback linearization based sliding mode control ensures desire THD with reduced values of capacitor and it is observe that THD is higher without feedback linearization based sliding mode control for greater chattering. The system is extensively simulated in MATLAB/SIMULINK. An experimental prototype is produced in the laboratory using dspace1104. The results from experiments match well with the simulation confirming usefulness of the proposed controller. Appendix A Feedback linearization is an important approach for nonlinear control design. The essential idea of this approach is to algebraically transform a nonlinear system into a (fully and partly) linear one. A.1. Feedback linearization Let us consider a non linear single input and single output system. The characteristics equation is shown below
Fig. 15. The uncompensated Fourier series of the current waveform is series1 and the corrected waveform is shown in series2.
x_ ¼ AðxÞ þ BðxÞ u þ C
ðA1Þ
y ¼ DðxÞ
ðA2Þ
This is called system input matrix and C is called constant matrix of vector field, u is the control input variable, and y is the system output. Operating grad operation on y the following equation is obtained.
y_ ¼ rDðA þ BuÞ ¼ LA DðxÞ þ LB DðxÞ u
ðA3Þ
The above terms are the derivatives with respect to time and if the relative degree r coincides with the system order (r = n), we must differentiate r times the system output, yielding the following equation.
yk ¼ LKA DðxÞ for all k < r 1
ðA4Þ
yr ¼ LrA DðxÞ þ LB Lr1 A DðxÞ u
ðA5Þ
Which implies that the result is zero for all ‘‘k < r 1” and that when r = n. Then system in (A1) may be linearized by using the following input substitution.
/i ðxÞ ¼ LAi1 DðxÞ
Fig. 16. Experimental set up.
Conclusions This paper has presented sliding mode based tracking error minimizing controller for shunt active power filter. One controller is used for tracking current reference with active damping of the network. Another controller is used to maintain dc link voltage. The proposed dc link controller also includes integral controller to eliminate steady state error. More over the oscillation in
ðA6Þ
This transformation leads to a linear relation between the system output Y and the input control U. However this linearization cannot be achieved if the relative degree is less than the system order. For the case of (r = n) and allows system in (1) can be transformed in a normal form by the following local change of co-ordinates:
For 1 6 i 6 n
ðA7Þ
Considering z1 = y, the (A1) may be rewritten as shown in below:
z_ ¼ aðzÞ þ bðzÞ u þ c
ðA8Þ
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A.2. Sliding mode control After feedback linearization a sliding surface may be designed. The sliding surface should be restricted by following equation:
! ! r j r j r1 r1 dy X dy d y X d y ¼0 kj : j þ kj r þ j j dt dt dt dt j¼0 j¼0
ðA9Þ
This last relation must be satisfied when the sliding rule is achieved.
s_ ¼
! ! r j r j r1 r1 dy X dy d y X d y ¼0 kj j þ kj r þ j j dt dt dt dt j¼0 j¼0
ðA10Þ
Thus from (A10) sliding surface may derived by simple integration. r1
s¼
d
ðy y Þ
dt
r1
þ
! Z j1 r1 X d ðy y Þ þ k kj ðy y Þds 0 j1 dt j¼1
ðA11Þ
The integrating term on sliding surface is useful to remove the steady state error of the system. From (A9) the tracking error dynamics of the closed-loop may be expressed as shown in (A12).
d eðtÞ d eðtÞ þ þ k0 ¼ 0 r þ kr1 r1 dt dt r
r1
ðA12Þ
where error is given as:
eðtÞ ¼ y ðtÞ yðtÞ
ðA13Þ
This error dynamics is exponentially stable. The co-efficient of the characteristic equation in Laplace domain satisfy the Routh– Hurtwitz criteria. The characteristic equation is shown below.
sr þ kr1 s þ þ k0 ¼ 0
ðA14Þ
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improved tracking of shunt active power filter by sliding mode control Anirban Sinha Ray ⇑, Avik Bhattacharya Department of Electrical Engineering, Indian Institute of Technology-Roorkee, India
a r t i c l e
i n f o
Article history: Received 7 December 2014 Received in revised form 19 September 2015 Accepted 10 November 2015 Available online 29 January 2016 Keywords: Shunt Active power filter CC-VSI Non linear load Sliding mode control Feedback linearization Variable structure system
a b s t r a c t An improved tracking technique of the reference current waveform in a shunt type active power filter is proposed in this paper using sliding mode control and feedback linearization. Feedback linearization approach helps to reduce the complexity of the controller. Excellent tracking is achieved in both for steady state and dynamic performance. A three-phase four-wire system is considered. The proposed algorithm is simulated first in MATLAB/SIMULINK. An experimental prototype using a dspace1104 based controller is also produced in the laboratory. Results from simulation match well with the corresponding results from experimental prototype confirming the usefulness of the proposed technique. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction In recent times most of the ac electrical power systems include several kinds of non-linear load. Non linear load involves harmonics that contaminate the sinusoidal supply voltage and current. Harmonics not only increase the losses in the system but also produce unwanted disturbance to the communication network, more voltage and/or current stress, etc. The uses of passive filters are not desirable as they are bulky and de-rate itself with age. Moreover this passive element may cause resonance with source impedance. This has motivated the introduction of the Active Power Filter (APF) for improving power quality. Shunt APF is used to eliminate the current harmonics, whereas, the series compensation does the same job for the voltage harmonics. Fig. 1 shows the system where a Voltage Source Inverter (VSI) operating as an APF and connected in parallel with the load [1]. Discusses the prospects of research in power quality improvement [2]. Provides extensive discussions on the robust performance of HPFC using digital simulation. While three phase reactive power compensation has been discussed in [3]. Singh et al. [4] have developed a fuzzy rule based generalized unified power flow controller. Sasaki [5] worked on systematic nonlinear control approach to a power factor corrector design. The basic principle of operation is to inject a correct nature of current to compensate for the load harmonics. This ⇑ Corresponding author. E-mail addresses: [email protected] (A. Sinha Ray), [email protected] (A. Bhattacharya). http://dx.doi.org/10.1016/j.ijepes.2015.11.015 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
requires detection of harmonic component of the load current following which the reference-current needs to be produced. Different methods of reference generation have been proposed. These include different variants of p–q theory [12,19,21,22], synchronous reference frame [20], FFT, RDFT, wavelet based techniques and more recently ANN, GA and soft computing based approaches [27–33]. Reference [6] provides a survey of such methods. Once the reference is generated, the VSI needs to track the reference current. Many controller strategies are employed to explore better performance of tracking. Most popular techniques are PI controller, standard hysteresis controller, one cycle control, sliding mode control, etc. Adaptive control and negative feedback based repetitive control are also employed to improve the dynamic performance and stability of the overall system [17,18]. Sliding mode controller is more immune to parameter uncertainties. This feature encourages the researcher to apply sliding mode controller (SMC) to mitigate tracking error in APF [6,9,13–17,23]. Singh et al. [8,9] applied sliding mode on a three phase four wire system. The SMC is used to control the dc bus voltage. Furga et al. [10,11] reported an improved dynamic performance of the APF by applying SMC and a passive LC filter with and without active source. Feedback linearization based control is also applied to shunt APF to reduce the computation burden [24–26]. Matas et al. [7] reported feed back linearization by Tellegen’s theorem [34–36] of single phase APF and then SMC is applied to simply the overall design. This paper has applied SMC to a three phase four wire system to study tracking and also to take up unbalance in the three phase load. Contrary to the available approach [11], this paper utilizes two
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Fig. 1. Active power filter to compensate for a non-linear load.
sliding mode controllers, one for dc bus regulation and another for tracking the current reference. A dspace1104 based controller is produced to implement the control-algorithm and generate the switching pattern to the power circuit. Fig. 1 shows the active power topology used for experimental and simulation purpose. The system may be modeled as a set of the current sources that injects reactive and harmonic current to the source. The paper is organized in six sections. Section ‘Modeling of shunt active power filter for sliding mode control’ deals with theoretical aspects of feedback linearization, sliding mode control and modeling of shunt active power filter with the same is presented. Simulation result is presented in Section ‘Simulation results’. Experimental verification of proposed scheme is presented in Section ‘Experimental results’. Section ‘Conclusions’ concludes the work.
From (A13) (in Appendix) tracking error may be defined as follows:
½uab idc :
ð6Þ
Modeling of shunt active power filter for sliding mode control
qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi Where T r ¼ 2p= Lf C f and T rd ¼ 2p= Rf C f
ð7Þ
Tracking of current reference
ð4Þ
Detail derivation is available in Appendix. The reference is generated by capacitor based predictive algorithm and then followed by THD minimization technique [18]. For tracking the quantities in abc frame is transformed in ab frame. Thus (4) may be expressed in the following way in ab frame:
½eab ¼ ½ic ab þ ½ic ab
ð5Þ
From (1)–(4) the state space representation of the APF may be expressed
½€eab ¼
1 T 2r
½eab
T rd T 2r
½eab þ ½f ab
1 T 2r
The tracking function ½f ab may be expressed as follows:
From Fig. 1 the following equation may be written for shunt active power filter. It is also assumed that source inductance is small and thus the drop across the source inductance may be neglected. The voltage at the point of common coupling is same as that of the source voltage.
Lf
dic ¼ ½v s abc ½v cf þ Rf ic dt
ð1Þ
Cf
d½V cf abc ¼ ½uabc idc ½ic abc dt
ð2Þ
ua , ub and uc are the switching function for the leg a, b and c for the voltage source inverter. Following way switching function is defined: u ¼ 1: If upper switch is on and lower switch is closed. u ¼ 0: If upper switch is off and lower switch is on. Applying KCL we may establish the relation between source current, load current and compensating current as:
½is abc ¼ ½iL abc þ ½ic abc ½is abc ¼ ½iL abc þ ½ic abc
½eabc ¼ ½ic abc þ ½ic abc
ð3Þ
T 1 rd ½f ab ¼ ½€ic ab ½€ic ab þ 2 ½i_c ab ½i_c ab þ 2 ½ic ab ½ic ab : Tr Tr
ð8Þ
The resonance frequency (xr ) of the L-C filter is much higher than supply frequency, i.e xr xs . Where
1 LF C F
xr ¼ pffiffiffiffiffiffiffiffiffiffi
ð9Þ
Eq. (9) may be further simplified considering the fact that the resistance present in the shunt path is very low. Thus modified expression of ½f ab is:
1 ½f ab ½€ic ab ½€ic ab þ 2 ½ic ab ½ic ab Tr
ð10Þ
The use of sliding mode control and design of sliding surface has been discussed in detail in [19,21]. For active damping of L-C filter and reduction in tracking error, following relation is established between the sliding surface ‘‘s” and nonlinear control command ‘‘u”.
½ui ab ¼ signð½sab Þ
ð11Þ
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The sliding surface may represented as:
½sab ¼ ½eab þ ½K d ab ½e_ ab
ð12Þ
K d is constant and is responsible for active damping of L-C filter. According to variable structure system, the proposed controller will be asymptotically stable if:
½sa s_ a < 0 and ½sb s_ b < 0
ð13Þ
From (1)–(5) and (6)–(12) ½s_ ab may be expressed as:
½s_ ab ¼
½K d ab
"
T 2r
! # T 2r ½eab þ T rd ½e_ ab þ ½f ab idc signð½Sab ½K d ab ð14Þ
Thus from (13) and (14), the condition for stability is determined as:
! dv dc T 2r C dc T rd ½e_ ab þ ½f ab ¼ idc > ½eab dt Kd
ð15Þ Fig. 2. The capacitor voltage stabilization circuit.
If the gain K d is selected such that:
1
T 2r T rd Kd
ð16Þ
Using (16), (15) may be rewritten as:
dv dc C dc dt
¼ idc > ½eab þ ½f ab
Eq. (10) may be reduced as:
½f ab
T 2r
½ic ab
½ic ab
ð18Þ
Thus the system will be asymptotically stable if (19) is satisfied.
C dc
dv dc ¼ idc > ½f ab dt
ð19Þ
This sliding mode controller is employed for tracking the reference current and active damping of the network. Sliding mode controller for maintaining dc link voltage
dic ¼ Rf ic v c1 þ v s dt
Let x1 ¼ ic ; x2 ¼ v c1 and x3 ¼ v c2
C 1 x_ 2 ¼ 0 and C 2 x_ 3 ¼ x1
ð26Þ
Eq. (27) represents the state space equation in matrix form:
3 Rf x_1 Lf 6 _ 7 4 x2 5 ¼ 0 1 x_3 C 2
2
0 0 0
ð20Þ ð21Þ
Thus (20) may be expressed as:
Rf 1 vs x_ ¼ x1 x2 þ Lf Lf Lf
ð22Þ
C 1 x_ 2 ¼ x1 and C 2 x_ 3 ¼ 0
ð23Þ
3 20 6 7 6 1 0 4 x2 5 þ 6 4 C1 1 0 x3 C
1 2 L f x1
2
1 Lf
0 0
32 3 2 vs 3 x1 Lf 76 7 6 7 07 u þ x 4 5 4 05 2 5 x 0 3 0
1 Lf
ð27Þ
From (27) it may be seen that the relative degree of the system is r = 1. The dc link capacitor voltage and the current through inductor is considered as state of the system. From the power balance equation, the following condition may be derived. The dc link capacitors C1 and C2 are considered same (i.e. C1 = C2 = C) in magnitude.
2 d 12 ½C v 2c 12 þ 12 Lf ic dt
The sliding mode controller derived in this section is to maintain dc link voltage. As is discussed in Introduction and also reported in [18], the control of dc link voltage is highly nonlinear. Non linear sliding surface will increase controller complexity. Such computational complexity will increase the delay in DSP based system. Fig. 2 shows the equivalent circuit diagram of APF per phase. The low frequency filter connected at the pole of the inverter is neglected for simplification. Two capacitors, C1 and C2 form the dc bus as shown in Fig. 2. The state space equation of the circuit diagram shown in Fig. 2 may be derived. For u ¼ 1, following switching function is obtained:
Lf
ð25Þ
ð17Þ
r
ð24Þ
Rf 1 vs x1 x3 þ Lf Lf Lf
x_ 1 ¼
The second order term present in ½f ab may be neglected since the amplitude of harmonic current decreases with increase in frequency. Also note that T12 >> 1.
1
For u ¼ o
¼ v s ic ic Rf 2
ð28Þ
Let us consider, output function as:
Z
yðtÞ ¼
v s ic dt
ð29Þ
Thus
_ yðtÞ ¼ v s ic
ð30Þ
Rx v uv uv V €ðtÞ ¼ v_ s ic þ v s i_c ¼ v s f 1 þ c2 c1 c2 þ s y Lf Lf Lf Lf Lf
ð31Þ
Therefore by critically observing (31), it may be concluded that the output function yðtÞ, described in (29) achieves a relative degree of r = 2, since the control input is second order time derivative of (30). Now following transformation is applied to achieve diagonal form (19)–(21).
_ z ¼ ðz1 z2 ÞT ¼ ðy; yÞ
ð32Þ
Assuming variation of v s with respect to switching frequency is negligible, following equation is obtained.
€¼ z_ 2 ¼ y
Rf x1 v c2 u ðv c1 þ v c2 Þ þ Lf 2 Lf Lf
v 2s
ð33Þ
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Again from (27)
v_c1 ¼
x1 u C
v_ c2 ¼
x1 x1 u 1 u ¼ x1 þ þ C C C C
v c2 ¼ ð34Þ ð35Þ
From (28), (35) and (36) following equations are derived:
v c1 ¼
Z
ic z u dt ¼ u C v sC
Z iC
1 u z 1 u þ þ dt ¼ C C vs C C
ð37Þ
Substituting (36) and (37) in (38) following equation is obtained.
€¼ z_2 ¼ y
v 2s Lf
Rf i c
vs Lf
z ð2u2 2u þ 1Þ CLf
ð38Þ
Eq. (38) may be further be simplified as:
ð36Þ
€¼ z_2 ¼ y
v 2s Lf
Rf i c
vs Lf
zx20 ð2u2 2u þ 1Þ
ð39Þ
Fig. 3. Source current before and after compensation. Waveform (a) source voltage of phase A. Waveform (b) source current of phase A. Waveform (c) source current of phase B. Waveform (d) source current of phase C. Time scale 20 ms/Div.
Fig. 4. Steady state operation of the controller. Waveform (a) source voltage. Waveform (b) source current. Waveform (c) load current. Waveform (d) compensating current. Time scale 20 ms/Div.
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The degree of system being two, following transformation may be applied to simplify the system:
v 2s Lf
Rf ic
vs Lf
zx20 ð2u2 2u þ 1Þ ¼ w
ð40Þ
‘‘w” is the new input signal to the system. This transformation enables us to establish linear relationship between the output ‘‘y” and new control signal (from the discussion made in (A6)). Applying the feedback linearization in (40) and (43) is obtained. From (41) following expression may be obtain.
z_ 1 ¼ z2
ð41Þ
z_ 2 ¼ c1 z2 c0 z1 þ w
ð42Þ
where
w ¼ €z1 þ c1 z_ 1 þ c0 z_ 1
ð43Þ
It is clear from (47) that the closed loop control dynamics is totally linearized. This linearization makes the controller simple. According to the theory of sliding mode control the system will be stable if coefficients of polynomial expressed in (51) are all positive. The error in tracking may be expressed as:
€e ¼ c1 e_ þ c0 e ¼ 0
ð44Þ
Fig. 5. Source, load and compensating current waveforms for step change in load with feedback linearization. Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current time scale 20 ms/Div.
Fig. 6. Source, load and compensating current waveforms for step change in load with feedback linearization. Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current (d) Capacitor voltage time scale 20 ms/Div.
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Since shunt active power filter will ensure that the source current will be in phase with source voltage. Thus source current may be considered as: is
¼a
vs
ð45Þ
where ‘‘a” is the proportionality constant. Thus ic may be express as follows:
ic ¼ iL av s
ð46Þ
Thus tracking error after feedback linearization may be express as:
e ¼ ic ic
ð47Þ
After feedback linearization, the sliding surface may be modified as:
e ¼ z1 z1 ¼
Z
v s ðic ic Þds
ð48Þ
The new sliding surface may be express as:
s ¼ v s ðic ic Þ þ k1 ðic ic Þds þ k0
ZZ
v s ðic ic Þds
ð49Þ
The sliding surface will be stable if the following condition is satisfied:
ss_ ¼ 0
ð50Þ
Fig. 7. Source, Load and compensating current waveforms for step change in load without feedback linearization. Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current. (d) Capacitor voltage time scale 50 ms/Div.
Fig. 8. Source, load, compensating current and dc link capacitor voltage waveforms for PI controller Wave form (a) source current. Wave form (b) load current. Wave form (c) compensating current. (d) Capacitor voltage time scale 50 ms/Div.
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Table 1 Comparison of control strategies. Control strategy
Value of capacitance (lF)
DC voltage of ripple (V)
THD of source current
PI Sliding mode without feedback Sliding mode with feedback linearization
1200 800 800
8 6 6
3.2 6.2 3.47
From this condition control logic may be obtained as follows:
uj ¼
1; for s > 0 0;
for s < 0
ð51Þ
This simplified controller is very easy to implement and shows excellent dynamic performance. Both the control signal ui and uj are ‘‘ORed” to obtain resultant control signal. This twin sliding mode controllers are found to be simple, reliable and insensitive to parameter uncertainty. Simulation results For the purpose of simulation the system is configured in SIMULINK. The powersim block is used to construct inverters and diode bridge rectifiers. Inverter dead-time is set at 2 ls. Simulations have been carried out for balanced and unbalance loads. Fig. 3 shows transformation of source current with and without APF. A three phase diode bridge feeding a R-C kind of load is considered. The source voltage and three source currents are shown in top to bottom order. Fig. 4 shows the steady state performance of APF. The source voltage, source current, load current and compensating current are shown in descending order. Dynamic performance is shown in Fig. 5. A step load change is indicated at 84 ms.This result shows excellent dynamic performance. The source current load current and compensating current are shown at the top, middle
and bottom of the waveform. The simulation results shown in Figs. 3–5 are obtained by linearization of the controller. Fig. 6 shows dynamic performance of APF with feedback linearization. Fig. 7 shows dynamic performance without feedback linearization. It is clear from the Fig. 7 that even though the system is stable by fulfilling the condition shown in (47), however chattering is more. The source current, load current, compensating current and capacitor voltage are shown in top to bottom order. Fig. 8 shows the functionality of PI controller based APF. It has been observed that the value of dc link capacitor is 50% more than the sliding mode controller. Comparison of these three controllers are provided in Table 1. The source, load, compensating current and capacitor voltage are shown in same sequence as of Fig. 7. To study the unbalance system, a star connected unbalanced R-L load is added in parallel to nonlinear load. Thus the system is now consists of an unbalanced load connected in parallel to the non-linear load (i.e. a diode rectifier feeding a R-C network and with a parallel R-L network). Two phases for the source and load currents are shown in top and bottom of the figures respectively in Fig. 9. The dynamic performance is also tested in case of unbalanced load. A step change in load current has occurred at 66 ms. This controller shows excellent performance also in unbalanced load. Fig. 10 shows transition of balance to unbalance load. Dc link capacitor voltage, source current, load current and compensating currents of any two phases are shown in top to bottom order.
Fig. 9. Transition of non-linear balanced load current Unbalanced load current. Top waveform: Source current of Phase A and Phase B. Bottom waveform: Unbalanced Load current of phase A and phase B. Time scale 20 ms/Div.
A. Sinha Ray, A. Bhattacharya / Electrical Power and Energy Systems 78 (2016) 916–925
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Fig. 10. Step load change in balance to unbalance condition. (a) Capacitor voltage. (b) Source current of phase A&B waveform. (c) Load current phase A&B. (d) Compensating current time scale 50 ms/Div.
Experimental results The proposed algorithm is also tested in the laboratory. An experimental prototype is made with two level inverter. Voltage source current controlled inverter using IGBTs that is available in the laboratory is used for this purpose. 110 V, 50 Hz supplying a
load of 3 kVA is considered. The whole system is built in SIMULINK where the ANN-routine is called whenever necessary for reference generation purpose. This facilitates the use of all the built-in blocks in SIMULINK. The clock frequency of the system is set at 10 kHz. Fig. 11 shows three source current before after compensation. Fig. 12 shows the steady state performance of APF with feedback linearization. The source voltage, source current, source current, load current and compensating currents are shown top to bottom order. Fig. 13 shows steady state performance of the controller without feedback linearization. For this reason it may be seen that considerable amount of chattering is present. Source, load and compensating currents are shown in descending order. Fig. 14 shows dynamic performance of proposed controller. Fig. 14 shows source voltage, source current, load current and compensating current in same sequence as shown in Fig. 12. Load change is occurred at 56 ms.This linear controller tracks the trajectory very fast and tracking error is minimal. Fig. 15 is the Fourier series(harmonic spectrum) of the compensated and the uncompensated current waveforms (see Fig. 16).
Fig. 11. Phase currents of the three phases.
Fig. 12. Steady state operation of shunt active power filter. Waveform (a) source voltage. Waveform (b) source current. Waveform (c) load current. Waveform (d) compensating current.
Fig. 13. Source, load and compensating current waveforms for step change in load without feedback linearization. Waveform (a) source current. Waveform (b) load current. Waveform (c) compensating current.
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Fig. 14. The dynamic performance of the controller. Waveform (a) source voltage. Waveform (b) source current. Waveform (c) load current. Waveform (d) compensating current.
tracking is actively damped by a derivative controller. From Fig. 14, we can see that the APF current can quickly track the harmonic current, thus to achieve the harmonic compensation. The reference currents tracking behavior has been improved, and the power supply harmonic current has been reduced with proposed sliding mode control. The system being non-linear and computationally extensive, it is difficult to implement in hardware. This is mainly because of delay in computation. To overcome this drawback feedback linearization is applied to the sliding mode controller. Table 1. Shows the comparison of three type of controller. Even though PI controller provides require THD the dc link ripple and size of capacitor is higher than the sliding mode controller. Feedback linearization based sliding mode control ensures desire THD with reduced values of capacitor and it is observe that THD is higher without feedback linearization based sliding mode control for greater chattering. The system is extensively simulated in MATLAB/SIMULINK. An experimental prototype is produced in the laboratory using dspace1104. The results from experiments match well with the simulation confirming usefulness of the proposed controller. Appendix A Feedback linearization is an important approach for nonlinear control design. The essential idea of this approach is to algebraically transform a nonlinear system into a (fully and partly) linear one. A.1. Feedback linearization Let us consider a non linear single input and single output system. The characteristics equation is shown below
Fig. 15. The uncompensated Fourier series of the current waveform is series1 and the corrected waveform is shown in series2.
x_ ¼ AðxÞ þ BðxÞ u þ C
ðA1Þ
y ¼ DðxÞ
ðA2Þ
This is called system input matrix and C is called constant matrix of vector field, u is the control input variable, and y is the system output. Operating grad operation on y the following equation is obtained.
y_ ¼ rDðA þ BuÞ ¼ LA DðxÞ þ LB DðxÞ u
ðA3Þ
The above terms are the derivatives with respect to time and if the relative degree r coincides with the system order (r = n), we must differentiate r times the system output, yielding the following equation.
yk ¼ LKA DðxÞ for all k < r 1
ðA4Þ
yr ¼ LrA DðxÞ þ LB Lr1 A DðxÞ u
ðA5Þ
Which implies that the result is zero for all ‘‘k < r 1” and that when r = n. Then system in (A1) may be linearized by using the following input substitution.
/i ðxÞ ¼ LAi1 DðxÞ
Fig. 16. Experimental set up.
Conclusions This paper has presented sliding mode based tracking error minimizing controller for shunt active power filter. One controller is used for tracking current reference with active damping of the network. Another controller is used to maintain dc link voltage. The proposed dc link controller also includes integral controller to eliminate steady state error. More over the oscillation in
ðA6Þ
This transformation leads to a linear relation between the system output Y and the input control U. However this linearization cannot be achieved if the relative degree is less than the system order. For the case of (r = n) and allows system in (1) can be transformed in a normal form by the following local change of co-ordinates:
For 1 6 i 6 n
ðA7Þ
Considering z1 = y, the (A1) may be rewritten as shown in below:
z_ ¼ aðzÞ þ bðzÞ u þ c
ðA8Þ
A. Sinha Ray, A. Bhattacharya / Electrical Power and Energy Systems 78 (2016) 916–925
A.2. Sliding mode control After feedback linearization a sliding surface may be designed. The sliding surface should be restricted by following equation:
! ! r j r j r1 r1 dy X dy d y X d y ¼0 kj : j þ kj r þ j j dt dt dt dt j¼0 j¼0
ðA9Þ
This last relation must be satisfied when the sliding rule is achieved.
s_ ¼
! ! r j r j r1 r1 dy X dy d y X d y ¼0 kj j þ kj r þ j j dt dt dt dt j¼0 j¼0
ðA10Þ
Thus from (A10) sliding surface may derived by simple integration. r1
s¼
d
ðy y Þ
dt
r1
þ
! Z j1 r1 X d ðy y Þ þ k kj ðy y Þds 0 j1 dt j¼1
ðA11Þ
The integrating term on sliding surface is useful to remove the steady state error of the system. From (A9) the tracking error dynamics of the closed-loop may be expressed as shown in (A12).
d eðtÞ d eðtÞ þ þ k0 ¼ 0 r þ kr1 r1 dt dt r
r1
ðA12Þ
where error is given as:
eðtÞ ¼ y ðtÞ yðtÞ
ðA13Þ
This error dynamics is exponentially stable. The co-efficient of the characteristic equation in Laplace domain satisfy the Routh– Hurtwitz criteria. The characteristic equation is shown below.
sr þ kr1 s þ þ k0 ¼ 0
ðA14Þ
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