Research Article  Open Access
Tao Liu, Changliang Xia, Xin Gu, Tingna Shi, "An ObserverBased Finite Control Set Model Predictive Control for ThreePhase Power Converters", Mathematical Problems in Engineering, vol. 2014, Article ID 351814, 9 pages, 2014. https://doi.org/10.1155/2014/351814
An ObserverBased Finite Control Set Model Predictive Control for ThreePhase Power Converters
Abstract
Finite control set model predictive control (FCSMPC) for threephase power converters uses a discrete mathematical model of the power converter to predict the future current value for all possible switching states. The circuit parameters and measured input currents are necessary components. For this reason, parameter error and time delay of current signals may degrade the performance of the control system. In the previous studies of the FCSMPC, few articles study these aspects in detail and almost no method is proposed to avoid these negative influences. This paper, first, investigates the negative impacts of inductance inaccuracy and ACside current distortion due to the time delay caused by filter on FCSMPC system. Then, it proposes an observerbased FCSMPC approach with which the inductance error can be corrected, the current signal’s time delay caused by filter can be compensated, and therefore the performance of FCSMPC will be improved. At last, as an example, it illustrates the effectiveness of the proposed approach with experimental testing results for a power converter.
1. Introduction
Power converters based on pulsewidth modulation (PWM) technology have been widely applied in modern industry field [1–3]. PWM converters are becoming more and more popular in recent years. Comparing with the traditional diodebased rectifiers, the PWM converters have many advantages such as unity power factor, bidirectional power flow, and low harmonic distortion [4]. Finite control set model predictive control (FCSMPC) is an alternative control strategy for power converters and drives. This method has developed rapidly in the past few years [5–8]. Taking into account the discrete nature of power converters, FCSMPC predicts the future behaviors for all possible switching states over a time frame (finite sampling instants), and the optimal one that minimizes the cost function is selected to obtain the desired output. FCSMPC has attracted experimental applications due to its advantages such as flexibility to control different variables, inherent decoupling behavior, fast dynamic response, and easy inclusion of nonlinearities and constraints [9–14].
In FCSMPC based system, parameter uncertainty and measurement noise may affect the behaviors of the control system. The negative influences of the errors of equivalent series resistance and input inductance have been addressed in several publications [15, 16], but extensive studies on this issue were not performed and method to avoid the negative influence introduced by parameter error has not been addressed so far.
The classical FCSMPC algorithm only chooses one switching state in each sampling period, so the switching frequency is not constant. Although the transient response is better, it demands high sampling frequency and the impact of control system time delay is more obvious on the operating performance. In practical applications, time delays can be caused by signal measurement, filtering, calculation, and actuation. Among them the calculation and filtering have an obvious impact on the performance of the control system. A twostepahead prediction has been proposed to compensate the calculation time delay [15, 17, 18]. Article [18] provides a clear and detailed explanation of the twostepahead prediction.
Filtering is another factor which will cause the time delay of the control system. To remove the unwanted highfrequency noise, the current signals need to be filtered before the A/D conversion. The signal filtering introduces time delay and causes deviation on sampling signals, which consequently cause a high current ripple of the power converter. Finally, the performance of the system is significantly deteriorated.
Although the calculation time delay has been discussed in some articles, the time delay caused by filter almost has not been discussed in the FCSMPC strategy. In the classical linear control methods for power converters, such as PI current controllers with pulsewidth modulation based methods, the filter time delay can be solved by adjusting the parameters in the PI or other linear controllers, but, in FCSMPC system, the situation is different. The FCSMPC is a nonlinear control algorithm without modulation, so it has no parameters similar with those in PI controller. For this reason, it is necessary to find a new method to avoid the negative influence caused by filter time delay.
In this paper, the impacts of the input inductance error and filter time delay are analyzed in detail, respectively. A novel observerbased FCSMPC strategy is proposed in the 2level power converter. This algorithm can compensate ACside inductance error and time delay caused by the filter. Finally, experimental results are presented to validate the theoretical studies.
2. Operating Principle of Classical FCSMPC
Like classical control strategy of power converters, the FCSMPC based power converter, as shown in Figure 1, has two control loops. The voltage control loop includes a PI controller to minimize the difference between the output DC bus voltage and its reference, while the current control loop is based on FCSMPC strategy, which keeps the axis currents tracking its references. In the controller, the reference angle of the source voltage can be estimated by the phaselockedloop (PLL).
Below begins with the development of the discretetime model of a power converter circuit, shown in Figure 2, and then brings up the factors which affect the performance of corresponding FCSMPC strategy.
According to the circuit topology of the 2level power converter shown in Figure 2, the voltage equations of the 2level power converter can be written as where , , and are the input currents; , , and are the source voltages; , , and are the converter voltages; is the equivalent series resistance; and is the ACside inductance.
Performing Clarke transformation on (1), the voltage equations in stationary reference frame can be obtained as where and are the input currents; and are the source voltages; and , are the converter voltages.
Using the finite difference approximation, the derivative term in (2) can be expressed by where and are the current at time instant and , respectively. is the sampling time of the current control loop.
Substituting (3) into (2) and rearranging the equations give the discretetime expression for currents in the power converter
From (4), then, the formula used for current prediction can be obtained as where denotes the switching state of IGBTs, which has eight options; and are the converter’s input voltages at time instant , under IGBT switching state ; and are the input currents predicted for the next time instant, , under the same IGBT switching state .
Table 1 gives the determination of and , corresponding to each IGBT switching state , where denotes the switching signals at three IGBTs at top of power converter circuit, shown in Figure 2, while the other three IGBTs at the bottom are the inverse signals of .

To achieve an optimal prediction, a cost function needs to be defined. The cost functions may take different forms based on different purposes [15]. Here in this paper, the summation of the square of the difference between the predicted currents and the reference currents in axis and axis, respectively, is adopted: where and are the reference currents at time instant .
Finally, the optimal prediction is considered to be achieved by applying the corresponding switching state which gives the smallest over the time range from to .
Figure 1 and the current prediction (5) indicate that the performance of the FCSMPC presented above depends on the accuracy of parameters (such as and ) and input currents and . varies with the saturation status of the inductor core due to the magnetization nonlinearity; the accuracy of input currents is unavoidably affected by the signal filtering process. In the following section, impacts of parameter variation and filteringresulted time delay on the FCSMPC are studied.
3. ObserverBased FCSMPC for Power Converters
3.1. Inductance Observer
Inductance error, discussed here, is the difference between the inductance value used by FCSMPC strategy and the actual inductance value of the physical inductor. In this paper, the inductance value used by FCSMPC strategy is called “model inductance,” and the actual value of the physical inductor is called “actual inductance.” Inductance error, resulted in such a way, may deteriorate the performance of FCSMPC. Experimental results in Figure 3 are an example to illustrate this concern. It can be seen that the input current waveform distortion becomes obvious when model inductance used by the current prediction calculation becomes 40% of its actual value.
(a) Phasea current waveform (3 A/div) when mH
(b) Phasea current waveform (3 A/div) when mH and mH
The function of the proposed inductance observer is to estimate the actual value of the inductance in the power converter.
Considering (4), one can obtain the expression of the axis current at time instant
The estimated value of the inductance can be defined as . With the estimated inductance , the estimated current is
Subtracting (7) from (8), one can obtain the reciprocal of actual inductance as
With the step value , the equation for estimating the value of inductance can be
Substituting (8) into (10) and rearranging the equations, the inductance observer is finally obtained as
Since the can be directly applied into the current prediction in (5), the calculation for is not needed. This will avoid the amplification of noise in the division operation. By changing the parameter , the dynamic response speed of the observer can be adjusted.
3.2. Filter Time Delay Observer
FCSMPC is a control algorithm which uses current signal to perform the current prediction then voltage vector selection for IGBT switching control. In possible future applications, the implementation of FCSMPC requires a filtering operation on the measured current signals to ensure FCSMPC functions as expected. In Figure 1, the filter block is usually placed in front of transformation block to serve such a purpose. A properly designed filter does help to take the unwanted high frequency harmonic noises off the measured current signals. However, the filtering operation results in a time delay between the current signal before filtering and the current signal after filtering. The time delay of the current signal plays a negative role in FCSMPC. The influence of time delay distortion of currents on FCSMPC is investigated and some results are presented below.
Figure 4(a) shows that the current signal time delay increases with the decrease of the filter cutoff frequency. The utilization of filter of lower cutoff frequency cleans the harmonics noise better, but it results in a longer time delay of the current signal, in other words, more time delay distortion of the current signal.
(a) Time delay depends on the cutoff frequency (1order lowpass filter)
(b) The input current waveforms (3.0 A/div) with cutoff frequency 1 kHz (upper figure) and 20 kHz (lower figure), respectively
As a result, Figure 4(b) gives the comparison of the threephase current waveforms obtained by using a 1order lowpass filter of cutoff frequency of 1 kHz and 20 kHz, respectively. It shows that the current waveform obtained with the use of 1 kHz cutoff frequency filtering contains more low frequency harmonics than the one obtained with the use of the 20 kHz cutoff frequency filtering. Such a result tells us that lowpass filtering removes the unwanted high frequency noises but brings in low frequency harmonics due to the time delay distortion of the current signal.
The identification of the problems promotes finding approaches to solve them. A filter time delay observer is proposed to eliminate the corresponding negative impacts of the filteringresulted time delay on FCSMPC. It is presented below starting with the mathematical description of the power converter circuit.
According to (2), the state equation of the power converter can be obtained as
Taking consideration of the 1order lowpass filter operation on the current signals, one has where is the current signal before filtering; is the current signal after filtering; is transfer function of filter; ; and is the cutoff frequency of the filter; this equation is shown in frequency domain.
Equation (13) can be rewritten into the state equation form. Considering the current signals filtering in the FCSMPC based power converters, one has where and are the current signals before filtering and and are the current signals after filtering.
With the consideration of (12) and (14), the proposed filter time delay observer is presented as follows: where and are the estimated current signals before filtering; and are the estimated current signals after filtering; and is the proportional parameter of the observer.
The function of the proposed filter time delay observer is to provide FCSMPC with the currents which do not contain unwanted high frequency noise and without time delay. In other words, the purpose of the filter time delay observer is to eliminate the impacts of the current signals time delay caused by filters. The filters can also cause time delay on the source voltage signals, but the source voltages can be regarded as system disturbance variables instead of state variables of the system; the time delay on these variables ( and ) has lower impact on the performance of the power converter, so, in this paper, the source voltage signals and are thought to not have changed after the process of filtering.
3.3. ObserverBased FCSMPC
The working principle of proposed observers and their integration with the FCSMPC are presented in the following section.
The inductance observer and the filter time delay observer interact with each other to enhance their performance. Inductance observer uses the estimated currents and (without filteringresulted time delay) provided by the filter time delay observer to obtain a closer estimation for ; filter time delay observer uses estimated by the inductance observer to obtain currents and to be more desirable for FCSMPC.
Finally, integration of the combined observer block with the FCSMPC of power converter shown in Figure 1 gives the control diagram of the observerbased FCSMPC of power converter, which is shown in Figure 5.
Comparing the control diagram in Figure 5 with the one in Figure 1, one can restate the advantages of the observedbased FCSMPC over the previous FCSMPC. With the use of the proposed observer, FCSMPC performs the IGBTs switching state determination by using the estimated value of inductance instead of a supposed fixed value and the input currents without unwanted noise and filteringresulted time delay instead of currents with filteringresulted time delay. As a result, the observerbased FCSMPC power converter is expected to provide a better control performance than the previous FCSMPC power converter.
The effectiveness of the proposed inductance observer and the filter time delay observer is verified by the experimental testing. Details are presented below.
4. Experimental Verification
An experimental platform (shown in Figure 6) was developed using TI floating point digital signal processing (DSP) TMS320F28335. The sensor signals are filtered by 1order lowpass filter. The experimental parameters are shown in Table 2. The sampling time of the current control loop is 50 μs ( μs).

4.1. Inductance Error Compensation
Figure 7 shows performance of the inductance observer in the FCSMPC based power converter. To minimize the influence of the time delay caused by filters, the cutoff frequency of 1order filter is 20 kHz.
(a) The estimated inductance (1.0 mH/div)
(b) The change of the phasea current (3.0 A/div) when the inductance observer begins to work
Figure 7(a) shows the accuracy and dynamic performance of the inductance observer. The actual inductance can change rapidly between 5.0 mH and 6.2 mH by cuttingin or cuttingoff an additional inductor (1.2 mH). In the figure, it can be seen that the output result of the inductance observer traces the actual inductance with high speed and well accuracy.
In Figure 7(b), the model inductance used by the current prediction calculation is 40% of its actual value, that is, mH and mH. It can be seen that, because of the inductance error, the current distortion of is obvious. When the estimated value of the inductance observer is added into the control system, the current distortion of becomes much smaller.
4.2. Filter Time Delay Compensation
Figures 8 and 9 show the experimental performance of the filter time delay observer. The cutoff frequency of the lowpass filter is 1 kHz. The model inductance is 5.0 mH and the same as actual value in the physical system.
(a)
(b)
(a) Waveforms of αaxis currents , , and (3.0 A/div) before and after the adding of the estimated results
(b) Threephase currents , , and (3.0 A/div) before and after the adding of the estimated results
In Figure 8, the power converter operates in the intense background noises environment. It can be seen that, because of the time delay caused by filter, the current distortion is more obvious. Moreover, the actual current contains a lot of high frequency noise. With the process of the filter, the current after filtering is smoother than but with the time delay, while the estimated current is not only smooth but also with no time delay compared to the actual current .
In Figures 9(a) and 9(b), it can be seen that, after the addition of the estimated results ( and ) of the observer, the ripple of the currents becomes much lower. The performance of power converter system has been improved with the observer. In the experiment, the background noises have been weakened to make current waveforms easy to be distinguished.
4.3. Startup Sequence of the Observers
Figure 10 presents the current waveforms with different startup sequence of the inductance observer and filter time delay observer. It can be seen that the observers operate stably and well in the following two types of situations. This shows well flexibility and stability of the proposed observerbased FCSMPC for power converters. The background noises have been weakened to make current waveforms easy to be distinguished.
(a)
(b)
4.4. Execution Time of the Observer
The detail execution time of the proposed observerbased FCSMPC is given in Table 3. It can be seen that, compared with the entire execution time of the FCSMPC algorithm, the cost time (3.5 μs) of the observer program is relatively small.

5. Conclusions
A threephase power converter system based on FCSMPC is built in this paper. Considering existing model parameter error and filter time delay in the FCSMPC strategy, this paper analyzes the impacts of the input inductance error and filter time delay, respectively. The input inductance error and filter time delay produced obvious impacts on the performance of power converters.
The designed modelbased observer can estimate the actual value of inductance and the currents without filter time delay. With the addition of the observer, the robustness and performance of FCSMPC strategy have been improved. The experimental results show that, compared with the traditional FCSMPC algorithm, the proposed algorithm can effectively eliminate the influences produced by inductance error and filter time delay, such as the distortion and harmonic of the input current. The proposed FCSMPC based power converter has lower current THD and more concentrated switching frequency distribution with the addition of the observer. The cost time of the observer program is relatively small. These aspects are good for the practical implementation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the financial support from the National Key Basic Research Program of China (973 Program) under Grant 2013CB035600, the National Natural Science Foundation of China under Grant 51307121, and the Key Technologies Research and Development Program of Tianjin under Grant 13ZCZDGX01100.
References
 K.H. Chao and C.T. Hsieh, “Mathematical modeling and fault tolerance control for a threephase softswitching mode rectifier,” Mathematical Problems in Engineering, vol. 2013, Article ID 598130, 13 pages, 2013. View at: Publisher Site  Google Scholar
 J. Tlustý, J. Škramlík, J. Švec, and V. Valouch, “Analytical modeling and simulation of fourswitch hybrid power filter working with sixfold switching symmetry,” Mathematical Problems in Engineering, vol. 2012, Article ID 292178, 17 pages, 2012. View at: Publisher Site  Google Scholar
 L. Ge, X. Yuan, and Z. Yang, “Control system design of shunt active power filter based on active disturbance rejection and repetitive control techniques,” Mathematical Problems in Engineering, vol. 2014, Article ID 864989, 6 pages, 2014. View at: Publisher Site  Google Scholar
 J. R. Rodríguez, J. W. Dixon, J. R. Espinoza, J. Pontt, and P. Lezana, “PWM regenerative rectifiers: state of the art,” IEEE Transactions on Industrial Electronics, vol. 52, no. 1, pp. 5–22, 2005. View at: Publisher Site  Google Scholar
 S. Thielemans, T. J. Vyncke, and J. Melkebeek, “Weight factor selection for modelbased predictive control of a fourlevel flyingcapacitor inverter,” IET Power Electronics, vol. 5, no. 3, pp. 323–333, 2012. View at: Publisher Site  Google Scholar
 K. G. Pavlou, M. Vasiladiotis, and S. N. Manias, “Constrained model predictive control strategy for singlephase switchmode rectifiers,” IET Power Electronics, vol. 5, no. 1, pp. 31–40, 2012. View at: Publisher Site  Google Scholar
 M. Rivera, C. Rojas, J. Rodriguez, and J. Espinoza, “Methods of source current reference generation for predictive control in a direct matrix converter,” IET Power Electronics, vol. 6, no. 5, pp. 894–901, 2013. View at: Google Scholar
 J. Rodriguez, M. P. Kazmierkowski, J. R. Espinoza et al., “State of the art of finite control set model predictive control in power electronics,” IEEE Transactions on Industrial Informatics, vol. 9, no. 2, pp. 1003–1016, 2013. View at: Google Scholar
 J. Scoltock, T. Geyer, and U. K. Madawala, “A comparison of model predictive control schemes for MV induction motor drives,” IEEE Transactions on Industrial Informatics, vol. 9, no. 2, pp. 909–919, 2013. View at: Google Scholar
 R. P. Aguilera, P. Lezana, and D. E. Quevedo, “Finitecontrolset model predictive control with improved steadystate performance,” IEEE Transactions on Industrial Informatics, vol. 9, no. 2, pp. 658–667, 2013. View at: Google Scholar
 V. Yaramasu, M. Rivera, B. Wu, and J. Rodriguez, “Model predictive current control of twolevel fourleg inverters—part I: concept, algorithm, and simulation analysis,” IEEE Transactions on Power Electronics, vol. 28, no. 7, pp. 3459–3468, 2013. View at: Publisher Site  Google Scholar
 P. Cortes, A. Wilson, S. Kouro, J. Rodriguez, and H. AbuRub, “Model predictive current control of twolevel fourleg inverters—part II: experimental implementation and validation,” IEEE Transactions on Power Electronics, vol. 28, no. 7, pp. 3469–3478, 2013. View at: Google Scholar
 P. E. Kakosimos and A. G. Kladas, “Implementation of photovoltaic array MPPT through fixed step predictive control technique,” Renewable Energy, vol. 36, no. 9, pp. 2508–2514, 2011. View at: Publisher Site  Google Scholar
 H. A. Young, M. A. Perez, J. Rodriguez, and H. AbuRub, “Assessing finitecontrolset model predictive control: a comparison with a linear current controller in twolevel voltage source inverters,” IEEE Industrial Electronics Magazine, vol. 8, no. 1, pp. 44–52, 2014. View at: Google Scholar
 S. Kouro, P. Cortés, R. Vargas, U. Ammann, and J. Rodríguez, “Model predictive control—a simple and powerful Method to control power converters,” IEEE Transactions on Industrial Electronics, vol. 56, no. 6, pp. 1826–1838, 2009. View at: Publisher Site  Google Scholar
 J. Rodríguez, J. Pontt, C. Silva et al., “Predictive current control of a voltage source inverter,” IEEE Transactions on Industrial Electronics, vol. 54, no. 1, pp. 495–503, 2007. View at: Google Scholar
 P. Cortés, J. Rodríguez, P. Antoniewicz, and M. Kazmierkowski, “Direct power control of an AFE using predictive control,” IEEE Transactions on Power Electronics, vol. 23, no. 5, pp. 2516–2523, 2008. View at: Publisher Site  Google Scholar
 P. Cortes, J. Rodriguez, C. Silva, and A. Flores, “Delay compensation in model predictive current control of a threephase inverter,” IEEE Transactions on Industrial Electronics, vol. 59, no. 2, pp. 1323–1325, 2012. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2014 Tao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.