A Predictive Current Control Technique for Three-Level NPC Voltage Source Inverters G.S. Perantzakis, F.H. Xepapas, S. A. Papathanassiou, S.N. Manias NATIONAL TECHNICAL UNIVERSITY OF ATHENS Department of Electrical and Computer Engineering Laboratory of Electrical Machines Iroon Polytechniou 9, T.K. 15773 & 15700 Athens Greece Tel.: +301-772-3503, Fax.: +301-772-3593 Email:
[email protected] P (2)
Abstract— In this paper a predictive current control technique for three-level Neutral Point Clamped (NPC) Voltage Source Inverters (VSI) is presented. The method is based on the prediction of the future value of load current by using a discrete-time model of the system and can be applied to any multilevel inverter for high-voltage high-power applications. The inverter is considered as a non-linear system with its 19 switching states. The proposed controller predicts the load current for all voltage vectors generated by the inverter. The current error for each voltage vector is calculated and the vector that ensures the smallest value of current error is selected as the inverter voltage vector for the next sampling time. The method allows optimum reference current tracking for all operating conditions under low current harmonic distortion as a result of many voltage vectors that participate in the control process. Furthermore, the proposed control ensures fast dynamic response and decoupling effect between the load current components. Finally, neutral-point voltage balancing with low ripple is also obtained.
I.
INTRODUCTION
In this paper a predictive current control technique for three-level Neutral Point Clamped (NPC) Voltage Source Inverters (VSI) is presented. The technique is based on the prediction of the future value of load current by using a discrete-time model of the system [1]-[10]. The inverter is considered as a non-linear system with its 19 different switching states. The proposed controller predicts the load current for all voltage vectors generated by the inverter. The current error signal for each voltage vector is calculated and the vector which ensures the smallest value of current error is selected as the inverter voltage vector for the next sampling time. This technique allows optimum reference current tracking for all operating conditions under low current harmonic distortion. The proposed technique exhibits a fast dynamic response and ensures decoupling effect between the load current components. Neutral-point (NP) voltage balancing is also obtained. Three-level inverters exhibit two main advantages against
Sa1 C1 Vdc
Da5
Sa2
Sb1
Db5
Sc1
ia
Dc5 Sb2
Sc2
R
L
ea
R
L
eb
L
ec
a
0 (1)
b Sa3
Sb3
Sc3
c
R
C2
Da6
Db6 Sa4
~
~
n
~
Dc6 Sb4
Sc4
N (0)
Fig. 1: Conventional three-level NPC VSI
the conventional two-level inverters: a) they permit higher dc-link voltage values because only half voltage should be switched by the power semiconductors and b) they exhibit lower harmonic contents for the same switching frequency. Owing to these advantages, multilevel inverters and especially three-level VSI have been extensively used in medium and high power applications the last years [11], [12]. A three-level inverter topology is presented in fig.1. Each phase consists of four by-directional switches Sx1– Sx4 and two clamping diodes Dx5, Dx6, where x=a, b, c the three phases. Table I shows the switching states of a threelevel inverter. TABLE I. SWITCHING STATES OF A THREE-LEVEL INVERTER Switching States P (2) 0 (1) N (0)
Sx1 ON OFF OFF
Sx2 ON ON OFF
Sx3 OFF ON ON
Sx4 OFF OFF ON
VxN Vdc Vdc/2 0
The inverter output voltage can be represented by a twodimensional (α, β) voltage space vectors, defined as: r r r 2 u(t)= [ uaN (t)+a ubN (t)+a 2 ucN (t)] 3
(1)
r u(t)=ua (t)+ j uβ (t)
(2)
r I [(k +1)T ]=
=e
2p
1 3 v j a= e 3 - + j 2 2
(3)
Where: ua (t) , uβ (t ) : real and imaginary components of space
R − T L
( k +1)T − R T [( k +1)T − λ ] r r 1 r I(kT)+ ∫ e L d λ ⎡⎣V(kT) - E(kT)⎤⎦ (6) L kT
r
or: I [(k +1)T ]= e
voltage vector in complex notation, uaN ,ubN , ucN : inverter output phase voltages between terminals a, b, c and N (fig.1), v
α
R - T L
R r - T 1 I(kT)+ (1 - e L R
j β (Imag.)
V9
output phase voltages ua, ub, uc are α, β variables by the axis coordinate
V15
1⎤ - ⎥ 2 ⎥ 3⎥ - ⎥ 2 ⎦
⎡ua ⎤ ⎢u ⎥ ⎢ b⎥ ⎣⎢uc ⎦⎥
(4)
V14
V8 V13
V3 V2
V10
1 2 3 2
(7)
ubN
: complex operator.
The inverter transformed to transformation ⎡ 1 ⎡uα ⎤ 2 ⎢ = ⎢ ⎢u ⎥ ⎣ β ⎦ 3 ⎢0 ⎢⎣
r rˆ ) ⎡V(kT)- E(kT)⎤ ⎢⎣ ⎥⎦
V4
V16
V1
V19
V6
V5 V11 V17
V7
uaN
α ( Real)
V18
V12
ucN 3
Table II shows the twenty seven (3 ) switching states of the three-level NPC inverter. Using eq.(4), the inverter voltage vector for each switching state is calculated. The inverter voltage vectors and their switching states are shown in Fig.2. According to their magnitude, the voltage vectors are divided into four groups: the zero voltage vector (ZVV) V19 , the small voltage vectors (SVV) V1 – V6 , the middle voltage vectors (MVV) V13 – V18 and the large voltage vectors (LVV) V7 – V12. The NP voltage control [point (1) in fig.1] is usually achieved by selecting proper redundant switching states from the SVV, so that the charging effect diminishes the discharging effect of the capacitors during one fundamental period. The ZVV and LVV do not affect the NP voltage, while the MVV increase or decrease the NP voltage according to the load condition. II.
PREDICTIVE CURRENT CONTROL
Referring to the ac side of the inverter (fig.1), the load current dynamics for each phase in vector notation is described by the differential equation r r r r dI Vxn = R I x + L x + Ex dt
(5)
r r The I x , Ex are described from similar equations like r r eq.(1). Taking into account that Ex and Vxn remain constant between sampling instants kT and (k+1)T, then eq.(5) can be written in discrete-form as:
Fig. 2: Space voltage vectors of a three-level inverter.
Using values of voltage vector and measured current at TABLE II. SWITCHING STATES OF EACH VOLTAGE VECTOR Vector V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 Switching state: 2 1 0
Switc.St. Vector 211/100 V11 221/110 V12 121/010 V13 122/011 V14 112/001 V15 212/101 V16 200 V17 220 V18 020 V19 022 corresponds to Vdc corresponds to Vdc/2 corresponds to 0
Switc.St. 002 202 210 120 021 012 102 201 000/111/ 222
(kT) sampling time then, with the aid of eq.(7), the load current at sampling time (k+1)T can be predicted. The rˆ vector E ( kT ) is the estimated back-emf and is calculated r from eq.(7) if the current I [(k +1)T ] is replaced from the r predicted reference value I * [(k +1)T ] : -
R
T
rˆ r ReL r R r* E(kT)= V(kT)+ I(kT)I [(k +1)T ] R R - T - T 1- e L 1- e L
(8)
The predicted value of reference current is approximated by the Lagrange quadratic extrapolation formula [4], as: r r r r I * [(k +1)T ]= 3 I * (kT)- 3 I * [(k - 1)T ]+ I * [(k - 2)T ]
(9)
The discrete-time model (eq.7) is used to predict the load current for each of the nineteen voltage vectors. For each voltage vector, a quality function q is evaluated q= ia* - ia + i*β - iβ
(10)
happens in pulse width modulation. The line load current spectrum exhibits very low harmonic content as a result of many voltage vectors that participate in control process. Fig. 12 shows the inverter space voltage vectors, which take part during controller operation, while the distribution of space voltage vectors in time is presented in fig. 13. As it can be seen from fig. 12, the three-level NPC VSI offers a large number of space voltage vectors and this fact improves the current tracking effect. The simulation results have been taken by considering: R=1 Ohm, L=5 mH , Ts=10-4 sec and dc-link voltage Vdc=1400 V.
Where Iα*,Iβ* and Iα ,Iβ are the component parts of reference and load current respectively. The voltage vector which minimizes the quality function (eq.10) is selected as the inverter voltage vector for the next sampling time. The block diagram of the proposed predictive control technique is shown in fig.3.
Load Current Prediction Model (Eq.7)
Min (Eq.10) I[(k+1)T] Iα(kT)
V (V1~V19)
Switching States Selection
I*[(k+1)T]
Iβ(kT)
α β
3-Level NPC VSI
a b c
3M
Fig. 4. Reference current tracking effect of load current.
~
I (kT)
Fig. 2: Figure in a two columns paper
III.
SIMULATION RESULTS
Figs 4 to 13 present simulated waveforms demonstrating the effectiveness of the proposed predictive control. The simulation results have been taken by considering: R=1 Ohm, L=5 mH, Ts=10-4 sec and dc-link voltage Vdc=1400 V. Fig. 4 shows the ability of load current components iα, iβ to track exactly the corresponding reference current components i*a, i*β, while a detail of these currents waveforms at the time of changing (0.4 sec) their amplitudes is presented in fig. 5. Fig. 6 shows the decoupling effect that exists between the load currents components iα, iβ. The amplitude of load current iα, changes at 0.3 sec and the current iβ remains unchanged. Similarly, the current iα remains unaffected as the amplitude of load current iβ is changed at 0.34 sec. Thus, it can be realized from figs 4 and 6 that the proposed predictive controller exhibits fast dynamic response as well as excellent decoupling effect between load current components iα, iβ. Neutral voltage balance is almost obtained as it is shown in fig. 7. The load phase voltage van waveform and a detail of it are presented in figs 8 and 9 respectively. The harmonic content of load phase voltage van is low (fig. 10). The harmonic spectrum is mainly composed of discrete spectral lines groups concentrated around central values, as it
Fig. 5. Enlargement of currents i*ref,α , iα and iref,β , iβ in fig.4.
Fig. 6. Decoupling effect between load current components iα , iβ . Current changes at 0.3 s and 0.34 s
Fig. 7. Effect on neutral-point voltage balancing.
Fig. 8. Load output phase voltage, van.
Fig. 13. Inverter voltage vectors used during control.
IV. Fig. 9. Enlargement of load phase voltage, van (fig.8).
Fig. 10. Spectrum of inverter phase voltage, van.
EXPERIMENTAL RESULTS
The experimental results have been taken by using a prototype in laboratory. The controller is implemented by using the Matlab/Simulink software, which is running in real time. The connection between Simulink and prototype is accomplished by an I/O card of National Instruments (NI 6025 E). The used load values are: R = 15 Ohms, L = 150 mH, Vdc = 200 V and Ts = 10-4 sec. The experimental results are presented in figs 14 to 21. Figs 14 to 16 confirm the tracking ability of load current ia and load current components iα, iβ, when their amplitudes and frequency change. The waveforms of load phase voltage van for 50 Hz and 20 Hz are presented in figs 17 and 18 respectively, while fig. 22 shows its harmonic spectrum for fundamental frequency 50 Hz. The harmonic spectrum of load current ia is given in fig. 21. The waveforms of load line voltage vab for 50 Hz and 20 Hz are presented in figs 19 and 20 respectively. Referring to figs 14 to 22, it is concluded that there is a good agreement between simulation and experimental results.
va/Vdc
Fig. 11. Spectrum of load current, ia.
vβ/Vdc Fig. 12. Normalized components vα ,vβ of participated voltage vectors during control .
Fig. 14. Amplitude change of line load current ia under constant frequency 50 Hz(50 mV/div, probe scale 100 mv/A, 20 ms/div).
Fig. 19. Load line voltage vab for 50 Hz (5*10 V/div with probe scale 10, 20 ms/div).
Fig. 15. Frequency change from 50 Hz to 20 Hz for the line load current ia under constant amplitude (50 mV/div, probe scale 100 mv/A, 20 ms/div).
Fig. 20.Load line voltage vab for 20 Hz (5*10 V/div with probe scale 10, 20 ms/div). Fig. 16. Amplitude change of load current components ia and iβ under constant frequency 50 Hz(50 mV/div, probe scale 100 mv/A, 20 ms/div).
100 90 80 70 60 50% 40 30 20 10 1
2
3
4
5
6
7
8
9
10 KHz
Fig. 21. Harmonic spectrum of load current ia.
Fig. 17. Load phase voltage van for 50 Hz (5*10 V/div with probe scale 10, 20 ms/div).
100 90 80 70 60 50% 40 30 20 10 1
Fig. 18. Load phase voltage van for 20 Hz (5*10 V/div, probe scale 10).
2
3 4 5 6 7 8 9 10 KHz Fig. 22. Harmonic spectrum of loadphase voltage van.
V.
CONCLUSION
In this paper a predictive current control technique for three-level NPC VSI, suitable for high-voltage high-power applications, was presented. From the simulation and experimental results was concluded that, the proposed nonlinear controller ensures: a) reference current tracking by varying current magnitude and frequency, b) decoupling effect between load current components, c) low harmonic content as a result of the large number of voltage vectors participating in control and d) NP voltage with a low ripple. The experimental results was taken by using a prototype in laboratory. The predictive current controller was implemented with the Matalab/Simulink software, which was running in real time. REFERENCES [1]
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