A Simple and Effective Solution for Superior Performance in Two-Level Four-Leg Voltage Source Inverters: Predictive Voltage Control V. Yaramasu**, J. Rodriguez*, B. Wu**, M. Rivera*, A. Wilson* and C. Rojas* * Departamento de Electrónica, Universidad Técnica Federico Santa María. Avenida España 1680, Casilla 110-V Valparaíso, Chile. Tel.: +56-32-2654214, Fax: +56-32-2797469, E-mail:
[email protected],
[email protected] ** Department of Electrical and Computer Engineering, Ryerson University, Toronto, ON M5B 2K3 Canada. Abstract – In this paper predictive voltage control strategy is proposed to control the output voltage of a three phase four-leg inverter. The four-leg inverter is developed to deliver power to three-phase loads under light, heavy, balanced and unbalanced load and source conditions and it can produce three output voltages independently with one additional leg. To predict the voltage behavior for each valid switching state, discrete model of the converter and load is developed. The control method chooses a state with minimum error between the output voltages and their references. The performance of the proposed predictive voltage control scheme is compared with carrier-based PWM method which is an equivalent to the symmetrically aligned-class I 3-D SVPWM. The feasibility of the proposed method is verified by MATLAB/Simulink. These results show that the proposed method has good performance and the capacity to compensate disturbances compared to classical methods.
I. INTRODUCTION There are many topologies to handle zero sequence voltage and current caused by an unbalanced source and/or load in three-phase four-wire systems [1]. Basically, these topologies control the neutral current of the load by adding a zero sequence voltage in the output [2]. One of these topologies is three-phase four-leg inverter which is a very flexible converter and widely demanded for power electronics applications, such as active power filters, uninterruptible power supplies, military and medical equipment, distributed power systems and rural electrification schemes. One of the most interesting
applications of these topologies is the hybrid power systems, where generation unit is formed by two or more sources. Several modulation schemes [1]-[10] are available for this type of converter. These methods are based on pulse width modulation (PWM) [2]-[5] and on space vector modulation (SVM) [1], [6]-[11]. With SVM techniques a good utilization of the converter capacity to deliver output voltage can be achieved, allowing a more efficient utilization of the PWM controlled converter [12]. In spite of all the above mentioned advantages, these techniques involve a great amount of complex calculations; therefore most of these modulation schemes are complicated and hard to apply. Authors in [13] presented predictive voltage control strategy for three-phase two-level three-leg inverter. In this paper, same concept is extended for three-phase two-level four-leg inverter. As be shown, this concept is a simple and effective solution for superior performance under light, heavy, balanced and unbalanced load and source conditions. Moreover, this concept does not require modulators. This control scheme uses the load model to make a prediction of the load voltage using each valid commutation state of the converter. These predictions are evaluated with a cost function that minimizes the error between the predicted voltage and their references at the end of each sampling time [14], [15]. This paper is organized as follow: in section II a general description of the mathematical
vCw vCv vCu Fig. 1. Three-phase four-leg inverter topology.
978-1-4244-6392-3/10/$26.00 ©2010 IEEE
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1 2 3 4 5 6 7 8 9 10 11 2 13 14 15 16
TABLE I POSSIBLE SWITCHING STATES OF THE FOUR-LEG INVERTER S1 S2 S3 S4 S5 S6 S7 S8 vun vvn vwn 1 1 0 0 0 1 0 1 vdc 0 0 1 1 1 0 0 0 0 1 vdc vdc 0 0 1 1 1 0 0 0 1 0 vdc 0 0 0 1 1 1 0 0 1 0 vdc vdc 0 0 0 1 1 1 0 1 0 0 vdc 1 0 0 0 1 1 0 1 vdc 0 vdc 1 0 1 0 1 0 0 1 vdc vdc vdc 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 vdc 0 0 1 1 1 0 0 0 1 0 vdc vdc 0 0 1 1 1 0 0 1 0 0 vdc 0 0 0 1 1 1 0 1 0 0 vdc vdc 0 0 0 1 1 1 1 0 0 0 vdc 1 0 0 0 1 1 1 0 vdc 0 vdc 1 0 1 0 1 0 1 0 vdc vdc vdc 0 1 0 1 0 1 1 0 0 0 0
voltage relations are indicated in Table I. The output voltages are established by the following equations, according to the switching signals and the dc-link voltage: (1) vun S1vdc ,
vfn 0 0 0 0 0 0 0 0 vdc vdc vdc vdc vdc vdc vdc vdc
vvn S3vdc , vwn S5vdc , v fn S7 vdc .
(2) (3) (4)
As a result, the voltage applied to the load can be defined as follow: (5) vuf vun v fn ,
model of the converter and load is presented, In section II, classical carrier based-PWM method [1], which is an equivalent method to the symmetrically aligned-class I 3-D SVPWM is discussed. The proposed control strategy, predictive voltage control is discussed in section IV. In section V, the performance of four-leg inverter with classical and proposed control strategy is compared and finally in section VI appropriate conclusions are drawn. II. THREE-PHASE FOUR-LEG INVERTER MODEL Fig. 1 shows the power circuit scheme of three-phase fourleg inverter. This converter presents a connection format similar to the conventional three-phase converter, with an additional leg connected to the neutral point of the load [1]. This fourth leg, with its own switching signals, increases the complexity of the control. However, the main advantage of this configuration is the independence of currents on each of the three phases of the inverter, controlled by using a proper control scheme. As mentioned earlier, its best application is the management of the neutral current caused by unbalanced loads and/or sources [6]. A LC filter included in the output of the inverter is considered in this paper. Before defining the converter model, it is necessary to establish a relation between switching signals and the load voltage. The power semiconductors connect each phase of the load to line P or N of the dc-link. Since there are four control signals in this case, the total number of feasible combinations are 24 = 16, i.e. this converter has 16 possible states of commutation. The valid switching states and the phase and line
vvf vvn v fn ,
(6)
vwf vwn v fn .
(7)
The filter can be described by the following model: di
Lf
dt
Cf
v vf R f i ,
(8)
dv f
(9)
dt
i if ,
where vf is the load voltage vector, defined as v f [vCu vCv vCw ]T ,
(10)
and i represents a vector containing the currents of the three phases as below: (11) i [iu iv iw ]T . In the same way, other variables such as if and v can be defined as: (12) i f [iuf ivf iwf ]T ,
v [vuf
vvf
vwf ]T .
(13)
The system in (8) and (9) has the following representation in state space:
v f 0 i 1 / L f
1/ Cf R f / L f
vf 0 i 1 / L f
1 / C f 0
v i , (14) f
which can be represented in discrete-time form as:
Fig. 3. Proposed predictive voltage control method for four-leg inverter.
Fig. 2. Carrier based PWM method for four-leg inverter.
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v fk 1 v fk vk Φ Γ k 1 k k , i i if
(15)
triangular carrier and the offset voltage calculation as shown in the block diagram of Fig. 2. Tu
where,
Φe
ATs
1
, Γ A (Φ I 2 x 2 )B ,
(16) Tw
with, A
0 1 / L f III.
0 , B Rf / Lf 1 / L f 1/ C f
1/ C f 0
.
(17)
CARRIER-BASED PWM METHOD
In four-leg converter, the additional leg provides neutral connection and also the controllability of a zero sequence voltage. The symmetrically aligned-class I 3-D SVPWM sequencing scheme can be achieved by the selecting a zero sequence vector location at desired timing [1]. This optimum switching sequence can be achieved by selecting the offset voltage as shown below: vmax vmin 0 2 , vmin , vmax 0 . v fn 2 vmax + vmin , otherwise 2
(18)
Therefore, ON-times of the upper switch of respective legs can be obtained as (19), and can be simply implemented with a
v fk , i fk , i k , vdck
for j 1...16
2 Ts 2
vun vdc vwn vdc
Ts
Tv
Ts
2
Tn
T
Ts 2
vvn vdc v fn vdc
Ts
.
(19)
T
By this proposed simple triangular carrier based PWM method, the nonzero voltage vector can be centered during the sampling period identically to the symmetrically aligned-class I 3-D SVPWM, which reveals the lowest harmonics at a given switching frequency. IV. PROPOSED CONTROL STRATEGY The proposed control scheme is basically an optimization algorithm and as such it has to be implemented in microprocessor based hardware. Consequently, the analysis has to be developed using discrete mathematics in order to consider additional restrictions as delays, sampling time, approximations, etc. A. Control Strategy The optimization algorithm and the complete control method are shown in Fig. 3 and Fig. 4, respectively. The control strategy consists of computing switching state in the sampling instant k, which minimizes the output voltage error in the k + 1 sampling instant. Then, the system applies this commutation state during the whole k + 1 sampling period. The algorithm calculates the 16 possible conditions that the state variables can achieve in the instant k + 1. B. Cost Function Minimization The control of the output voltages is done by evaluating the cost function g on each sampling time. It is defined as:
Sij _ op , i 1...8
g op
Ts
* k 1 g k 1 vCu vCu
v 2
* Cv
k 1 vCv
v 2
* Cw
2
k 1 . vCw
(20)
The algorithm chooses the commutation state that minimizes the cost function, delivering the corresponding voltage vfk+1. To achieve this, the algorithm evaluates the 16 possible commutation states presented in Table I. C. Output Voltage Discrete Model The proposed predictive control strategy is based on the prediction of the output voltages with a prediction horizon of one sample time. From the system presented in eq. (15), the capacitor voltage, equivalent to the output voltage, is given by: (21) v fk 1 c1v v k c2 v v fk c3v i k c4 v i fk .
v fk 1
g kj 1 j 16
The inverter can modify the output voltage vk to affect the capacitor voltage vfk in the next sample state and to accomplish this, currents if and i must be known. V. SIMULATION RESULTS
g op ming j j 1...16 jop j g op
Fig. 4. Predictive voltage control flow diagram.
To validate the proposed control method, a simulation model of a three-phase four-leg inverter with the parameters as indicated in Appendix - Table II has been developed. The model was developed in Matlab/Simulink where the
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application of the next switching state is instantaneous, so here it is not necessary any delay compensation. The simulation results with the classical and predictive control strategy with a sample time of Ts = 20s are shown in Fig. 5 and Fig. 6. The simulation model uses classical carrierbased PWM method up to 0.05 seconds and then switches to predictive voltage control. The dc-link voltage is considered to be varying as: vdc random 15 15sin 120 t (22) The reference and fundamental line-to-neutral phase-u voltages are shown in Fig. 5(a) and 6(a)-6(b). With predictive voltage control, the output voltage tracks the reference voltage with less error compared to PWM control, under unbalanced load conditions. The phase voltages between the converter and the filter and fundamental output voltages are shown in Fig. 5(b) and 6(c)-6(d).
(a)
Load Voltage (V)
300
The predictive control uses less commutation frequency compared to PWM control. The dc-link voltage which is considered to be continuously varying (Equation. 22) is shown in Fig. 5(c). The unbalanced three-phase currents can be observed in Fig. 5(d) because of the unbalanced load conditions. The neutral current which is the sum of all line currents is not zero as shown in Fig. 5(e). It is important to mention that the changes on the load parameters are not included in the predictive load model, so, the controller can mitigate these effects only by choosing the switching state which generates the minimum error between the predictions and its references. The simulation results were done with the linear load showed on Table II. A future research is the study of the predictive control operating with a nonlinear load.
PWM Control
Predictive Control
200 100
vCu*
vCv*
vCw*
vCv
vCw
0 -100 -200 -300 0.01
vCu
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
(b)
Load Voltage (V)
400
200
0
-200
-400 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
(c)
dc-link Voltage (V)
400
300
200
100
0 0.01
(d)
Load Currents (A)
30 20 10 0 -10 -20 -30 0.01
(e)
Neutral Current (A)
30 20 10 0 -10 -20 -30 0.01
Fig. 5. Simulation results for classical and predictive control in the four-leg inverter with un-balanced load: (a) load voltage [V]; (b) output voltage [V]; (c) dc-link voltage [V]; (d) load currents [A]; (e) neutral line current [A].
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(a)
Load Voltage (V)
200
100
0
-100
-200 0.03
0.031
0.032
0.033
0.034
0.035
0.036
0.037
0.038
0.039
0.04
(b)
Load Voltage (V)
200
100
0
-100
-200 0.03
0.031
0.032
0.033
0.034
0.035
0.036
0.037
0.038
0.039
0.04
0.0355
0.036
0.0365
0.037
0.0375
0.038
0.0385
0.039
0.0395
0.04
0.0355
0.036
0.0365
0.037
0.0375
0.038
0.0385
0.039
0.0395
0.04
(c)
Load Voltage (V)
300
200
100
0 0.035
(d)
Load Voltage (V)
300
200
100
0 0.035
Fig. 6. Simulation results for classical control in the four-leg inverter with un-balanced load: (a) load voltage with PWM method [V]; (b) load voltage with predictive implementation [V]; (c) output voltage with PWM method [V]; (d) output voltage with predictive implementation [V].
VI. CONCLUSION The Predictive Voltage Control has been proposed in this paper to control three-phase four-leg inverter. The control algorithm tests each of the 16 possible commutation states and then chooses a switching state that minimizes the cost function. The ideal minimum of the cost function is zero and represents the perfect regulation of the output voltages. With predictive control, the output line voltage tracks reference voltage with less error and less commutation frequency and thus with less switching losses compared to PWM control under the light, heavy, balanced and unbalanced load and source conditions. The predictive control scheme can compensate the effect of the uncertainties in the load and dc-link voltage and in consequence the load voltage and current waveforms remain balanced. This compensation can be achieved without any penalty in the transient and steady state operation.
The authors also wish to thank the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through Wind Energy Strategic Network (WESNet) Theme III, Project I.
ACKNOWLEDGMENTS The authors wish to thank the financial support from the Chilean Fund for Scientific and Technological Development (FONDECYT) through project 1100404.
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APPENDIX TABLE II VALUES USED IN THE FOUR-LEG INVERTER AND THE LOAD Variables DC Link vdc Vdc_ripple Input Filter Rf Cf Lf Load RL fref Control Ts
Description
Values
DC link Voltage DC link Voltage ripple
300 (V) 30 (V)
Filter Resistance Filter Capacitor Filter Inductance
0.05 () 80 (F) 2 (mH)
Load Resistance Frequency Reference
20 () 60 (Hz)
Sample Time
20 (s)
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