Optimal switching sequence model predictive control for three‐phase Vienna rectifiers
2018; Institution of Engineering and Technology; Volume: 12; Issue: 7 Linguagem: Inglês
10.1049/iet-epa.2018.0033
ISSN1751-8679
AutoresShiming Xie, Yao Sun, Mei Su, Jianheng Lin, Qiming Guang,
Tópico(s)Advanced Battery Technologies Research
ResumoIET Electric Power ApplicationsVolume 12, Issue 7 p. 1006-1013 Research ArticleFree Access Optimal switching sequence model predictive control for three-phase Vienna rectifiers Shiming Xie, Shiming Xie School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorYao Sun, Yao Sun School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorMei Su, Corresponding Author Mei Su sumeicsu@mail.csu.edu.cn School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorJianheng Lin, Jianheng Lin School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorQiming Guang, Qiming Guang School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this author Shiming Xie, Shiming Xie School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorYao Sun, Yao Sun School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorMei Su, Corresponding Author Mei Su sumeicsu@mail.csu.edu.cn School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorJianheng Lin, Jianheng Lin School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this authorQiming Guang, Qiming Guang School of Information Science and Engineering, Central South University, Changsha, People's Republic of ChinaSearch for more papers by this author First published: 18 May 2018 https://doi.org/10.1049/iet-epa.2018.0033Citations: 13AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Vienna rectifier is a typical three-level rectifier with complicated operating constraints. Also, the constraints pose a challenge for designing controllers with good dynamic performance. As predictive control is good at dealing with constraints, an optimal switching sequence model predictive control (OSS-MPC) strategy for the three-phase Vienna rectifier is proposed. A proportional–integral controller is designed to regulate the dc-link voltage. Also, an improved OSS-MPC method is utilised to control the input currents. Compared to the conventional finite control set model predictive control, it has the extra advantages of improved steady-state performance, fixed switching frequency, and elimination of weight factors. Simulation and experimental results verify the correctness and effectiveness of the proposed control scheme. 1 Introduction Vienna rectifier has the advantages of low input current total harmonic distortion (THD), low blocking voltage stress, high power density, and so on [1, 2]. Hence, it has been widely used in various applications, such as electric aircrafts, telecommunication, and wind turbine systems [1, 3, 4]. The related researches of the Vienna rectifier are mainly focused on modulation strategies and control methods. In [5], a space vector modulation (SVM) is presented, which achieves neutral-point voltage balance by adjusting the duty cycles of redundant vectors. In [6], the relationship between three-level voltage vectors and two-level voltage vectors has been discovered, which makes the traditional two-level SVM method applicable to Vienna rectifier. A carrier-based pulse width modulation (CBM) is presented in [7], where the neutral-point voltage is balanced through regulating the zero-sequence signals of reference voltages. Two discontinuous pulse width modulation schemes are proposed [8]. They achieve a wide range of modulation ratio and low switching losses by avoiding switching the switches with maximum phase current. Moreover, zero average neutral-point current within a mains period is guaranteed. Meanwhile, the minimum switching action is achieved [9]. In addition, a lot of control schemes are presented for good dynamic performance. Usually, the main control targets of the three phase Vienna rectifier are: (i) to keep the output voltage at the desired value under time-varying loads or grid voltages; (ii) to achieve sinusoidal input currents and given input reactive power; and (iii) to maintain the neutral-point voltage balance. Hysteresis current control method is proposed to control the ac current and zero-sequence current [1], which is a useful control schemes with the advantages of strong robustness and simple design procedure. In [10], classic proportional–integral (PI) controllers are used based on the average d–q model of the Vienna rectifier. An optimal zero-sequence component is selected for dc-link voltage balance, which is based on the relationship between the controlled duty cycle and the dc-link neutral-point voltage. In [11], the modified one-cycle control based on three-phase Vienna rectifier is presented, which introduces the neutral-point voltage loop. Then, it improves the utilisation ratio of dc-link voltage and balances the neutral-point voltage. To improve dynamic performances during both start-up and step-load and achieve good performances in steady state of the Vienna-type rectifiers, an improved direct power control (DPC) strategy based on sliding mode control with dual-closed-loop is presented [12]. The control strategies above are designed for Vienna rectifier in continuous conduction mode (CCM). Lately, an open-loop current control scheme without current measurement is proposed for Vienna rectifier working in discontinuous conduction mode, which improves efficiency further [13]. Recently, finite control set model predictive control (FCS-MPC) has been well developed in power electronics community [14, 15]. It has changed the conventional control frameworks which often includes modulation and control. The modulation has been eliminated from FCS-MPC. Also, it is particularly suitable for the system such as Vienna rectifier which suffers from strict constraints. As there are only eight switching states in the three-phase Vienna rectifier, the calculation burden of the FCS-MPC is small. Therefore, the idea of FCS-MPC control is introduced in the inner loop of the Vienna rectifier [16]. Optimal switching vector MPC (OSV-MPC) is a commonly used control method in FCS-MPC [15]. However, it would lead to uncertain switching frequency, which increases the difficulty of designing input filters. Moreover, it needs higher sampling frequencies to achieve a better performance. Optimal switching sequence model predictive control (OSS-MPC) [15] is another control method in FCS-MPC. It uses an optimal switching sequence in a control period instead of an optimal switching state in OSV-MPC. Thus, OSS-MPC has a better performance than OSV-MPC. Moreover, it has the advantage of constant switching frequency. It has been applied in the DPC of two-level converters [17, 18]. However, the selections of optimal switching sequence and optimal duty cycles are more complicated, so the computational burden of OSS-MPC is heavy. To overcome the shortcoming of OSS-MPC above, an improved OSS-MPC with low computational burden for Vienna rectifiers is presented. In the proposed control scheme, OSS-MPC only takes charge of input current control. Also, the dc-link voltage regulation and the neutral-point voltage balance are realised by a PI control and a redundant vector pre-selection technique, respectively. Compared to the conventional FCS-MPC for Vienna rectifiers [16], it has the advantages of improved steady-state performance, fixed switching frequency, and elimination of weight factors. The remainder of this paper is organised as follows: Section 2 presents the system model of the three-phase Vienna rectifier. In Section 3, the proposed control scheme is described in detail. In Section 4, the simulation and experimental results are presented and discussed. Finally, the main points of this paper are summarised in Section 5. 2 Model of three-phase Vienna rectifier Fig. 1 shows the topology of the three-phase Vienna rectifier, which involves a three-phase diode rectifier, three bidirectional switches, three boost inductors, and two dc capacitors. Fig. 1Open in figure viewerPowerPoint Circuit configuration of the three-phase Vienna rectifier Assume that the Vienna rectifier is operating in current CCM. The dynamics of input currents are expressed as follows: (1) where L is the boost inductance, are the input phase voltage and current, is the zero-sequence voltage, are the semi-controllable voltage, which is determined by not only the switching state of the bidirectional switch, but also the direction of the input current. For convenience, denote as switching states of the three bidirectional switches of the Vienna rectifier. Also, means the switch is off; means the switch is on. Then the semi-controllable voltage can be given as: (2) where vc1 and vc2 are the voltage across the capacitor C1 and C2, respectively. Also, is the sign function to distinguish the direction of currents. Assume , then voltage error of the two capacitors equation can be written as: (3) where are the currents passing through the capacitors, is the neutral-point current, and the capacitor voltage deviation, i.e. . 3 Proposed OSS-MPC Fig. 2 shows the proposed control block diagram of the Vienna rectifier. The dc-link voltage is controlled by a PI controller. Also, input currents and voltage balance control controlled by the proposed predictive controller. Fig. 2Open in figure viewerPowerPoint Block diagram of the proposed control scheme 3.1 Redundant vector pre-selection The Vienna rectifier is a highly constrained three-level rectifier. According to (2), its semi-controllable voltages depend on the polarity of the input currents. Assume that , the possible controllable voltage vectors lie in sector I and are shown in Fig. 3 (cf. Fig. 2 in [9]). In fact, for a certain input current vector, there are only eight controllable voltage vectors. Fig. 3Open in figure viewerPowerPoint Vienna rectifiers voltage vectors in sector I According to the principle of SVM, to obtain good current quality, it is better to apply a proper switching sequence rather than only one switching state applied in one control period like OSV-MPC which is a traditional FCS-MPC. OSS-MPC is such a predictive method based on this idea. As Fig. 3 depicts, sector I could be divided into six small sectors (A–F). Based on the criterion that the switching sequence made up of adjacent vectors, all the switching sequences are listed in Table 1 where is the vector sequence and are the corresponding switch states used in the modulation period. Vectors or are called the redundant vectors as they have the same effect on input currents but an opposite effect on the neutral-point voltage. Table 2 shows the switch states of the redundant vectors in each sector. Since vectors or are the vertices of any small sector like (A–F), the neutral-point voltage balance could be well achieved. From Table 1, there are 12 feasible switching sequence candidates. Table 1. Vector sequence to be applied Sector A , , ( or ), ( or ), , B , , ( or ), ( or ), , C , , ( or ), ( or ), , D , , ( or ), ( or), , E , , ( or ), ( or ), , F , , ( or ), ( or ), , Table 2. Switch state of redundant vectors in each sector Sector I [0, 1, 1] [1, 0, 0] II [0, 0, 1] [1, 1, 0] III [0, 1, 0] [1, 0, 1] IV [1, 0, 0] [0, 1, 1] V [1, 1, 0] [0, 0, 1] VI [1, 0, 1] [0, 1, 0] Obviously, selecting an optimal switching sequence from the 12 candidates would consume amounts of time. Therefore, a pre-selected algorithm based on the polarity of voltage error is proposed to reduce the computation efforts. The principle of redundant vector pre-selection is that the selected redundant vector must satisfy . For example, select the redundant vector as when in sector I. After the redundant vector pre-selection, there are six feasible switching sequence candidates. Hence, the amount of computation is almost reduced by half. 3.2 Duty cycles Transform (1) in coordinate into the equation in coordinate as follows: (4) where are the input voltages and currents in coordinate. Assume the control period is short enough, then it is reasonable to view increments of the input current as constants. The duty cycles of in a control cycle are defined as , and . Then, the predicted input currents of Vienna rectifier at ()th instant can be expressed as follows: (5) where is the control period, and (6) where are the increments for the input current ; are the predicted input currents at ()th instant, and are the measured value of input currents at kth instant. The optimal duty cycles are solved by minimising the input current tracking errors, which are defined as: (7) where and are current tracking errors, and they are expressed as: (8) The optimisation problem can be formulated as follows: According to the following conditions: (9) This yields the following optimal duty cycles: (10) 3.3 Algorithm implementation To implement the OSS-MPC, the cost function (shown in Fig. 4a) of six feasible switching sequence candidates should be calculated in each control cycle. The switching sequence with minimum cost function is the optimal switching sequence. The flowchart of the proposed OSS-MPC strategy is shown in Fig. 4a. is the value of cost function calculated for the tth switching sequence, and is the minimum value of the calculated cost functions. Besides, denote the optimal switching sequence and the duty cycles of , respectively. Assume that the Vienna rectifier is operating in sector I and the optimal switching sequence is , then the double-side switching pattern is adopted for the modulation process, which is illustrated in Fig. 4b. Fig. 4Open in figure viewerPowerPoint Schematic of the control process and modulation (a) Flowchart of the proposed OSS-MPC, (b) Schematic diagrams of the modulation scheme However, the delay between the measurements and the actuation is an inherent shortcomings of digital control, which has a great impact on the performance. In this paper, the first-step prediction is repeated every sampling time to compensate the time delay. The first-step prediction are as follows: (11) where is the first prediction values replacing the measured value, and are the increments for the input current and the duty cycles calculated by the previous control period. The third terms of (11) are correction term which improves accuracy of the prediction. 3.4 Stability discussion Stability is an important issue in the research of MPC. The terminal constraint method and terminal cost method are two commonly used methods to guarantee the stability of the MPC. According to the results in [19], if there exists a control law that admits (12), then the control system is stable: (12) where . As many existing methods for Vienna rectifier have been proved to be locally stable [10-12], the control law which admits (12) exists clearly. Thus, the proposed OSS-MPC control system is stable with a certain region of attraction. A long prediction horizon is advantageous for stability [20]. However, the proposed OSS-MPC is a one-step prediction. Therefore, it could be inferred that the region of attraction may not be large, which is one shortcoming of the proposed OSS-MPC. 4 Simulations and experimental results The proposed control scheme has been tested on the three-phase Vienna rectifier system. The main system parameters are summarised in Table 3. Table 3. Parameters used in the simulations and experiments Symbol Description Value input line voltage input angular frequency L boost inductance C dc-link capacitance R load resistance dc-link voltage sample period 4.1 Simulation results Numerical simulations have been carried out in MATLAB/Simulink platform to illustrate the steady-state and dynamic performance of the proposed control scheme. Besides, the method using OSV-MPC [21] is utilised for comparison. First, the operating condition of the unity power factor is considered. The reference voltage is set to 250 V, and the control system starts up at t = 0.2 s. Fig. 5 shows the simulation results during start-up with the proposed control. As seen, before start-up, the capacitor voltages are pulsating and the input current is severely distorted. After start-up, each capacitor voltage is regulated to half of the desired dc-link voltage quickly. Also, the input current also is sinusoidal and in phase with the input phase voltage. Fig. 5Open in figure viewerPowerPoint Simulation results of the OSS-MPC during start-up To show the dynamic performance of the proposed control scheme, the voltage reference is changed from 200 to 250 V at t = 0.2 s. Figs. 6a and b show the simulation results of the proposed method and the OSV-MPC method, respectively. The sampling period of the OSV-MPC method is 100 μs. They have almost the same dynamic response with regard to input currents, dc-link voltage, and neutral-point voltage balance. However, it is easy to find that the steady-state performance of the proposed control scheme is better than that of the OSV-MPC. Fig. 6Open in figure viewerPowerPoint Simulation results (a) The proposed for the dc-link voltage reference changes from 200 to 250 V, (b) The OSV-MPC for the dc-link voltage reference changes from 200 to 250 V, (c) The proposed for the load step change from 80 to 60 , (d) The OSV-MPC for the load step change from 80 to 60 , (e) The proposed for the load step change from 80 to 2.4k , (f) The OSV-MPC for the load step change from 80 to 2.4k In addition, the dynamic response of the dc-link voltage to the step load changing from 80 to 60 is tested. As observed in Figs. 6c and d, both the proposed controller and the OSV-MPC controller are able to maintain the desired dc-link voltage in the presence of the load disturbance. It is clear that they have the similar dynamic performance by selecting the optimal result in the finite set. Moreover, the capacitor voltages still keep the balance. It is proved that the pre-selection of vector can work well. To compare the input current quality of OSV-MPC and OSS-MPC, the THD values of the input currents under different load conditions are recorded and shown in Fig. 7. Clearly, the proposed method has better performance in input current quality. Fig. 7Open in figure viewerPowerPoint THD of input current under different loads (dc-link voltage reference of 250 V) Figs. 6e and f show the simulation results for the step load changing from 80 to 2400 at t = 0.1 s. A large overshoot is found in dc-link voltage as seen in Fig. 6e. Also, the overshoot voltage disappears after ∼3 period because Vienna rectifier is a unidirectional one. When the system is in steady state under light loads, approximately sinusoidal input currents are obtained still but some large spikes exist. The same experiment is tested with OSV-MPC; the results are shown in Fig. 6f. Clearly, the dc-link voltage is out of control in this case. From the comparative results, it can be inferred that the proposed method has a larger stable attractive region than the OSV-MPC. Since mathematical model is required in the FCS-MPC, the robustness against parameter uncertainties should be considered. The related tests are carried out with the desired dc-link voltage of 250 V. Figs. 8a and b show the simulation results of the controller in which the nominal parameters of the converter are , while the real parameters are . From (3), the variation of capacitance will affect the ripple of the neutral-point voltage but has little effect on the stability, which is verified in Fig. 8. As the actual capacitance increases, the voltage error of dc-link voltage becomes smaller. However, the input current ripple become larger. The reasons are twofold: (i) the predictive model is inaccurate; (ii) the filtering performance degrades as the inductance decreases. Fig. 8Open in figure viewerPowerPoint Simulation waveforms of the input currents and error between the capacitor voltages in (a) , (b) , (c) THD of the input current under filter inductance value (L) variation Next, the inductance parameter uncertainty is considered. A series of different inductance values are tested. As seen in Fig. 8c, the input current THD is 0.55 times and <2 times of its nominal value. The results show that the proposed control is robust to parameter uncertainty. 4.2 Experimental results Apart from the simulation studies, experiments are also performed to validate the effectiveness of the proposed method. A prototype of the three-phase Vienna rectifier is built in the laboratory as shown in Fig. 9. The specifications of this system are the same with those in simulation, which are listed in Table 3. For simplicity, IGBT-Module FS3L25R12W2H3_B11 (Infineon) is used to construct the power stage of the Vienna rectifier. Besides, a control board (32 bit floating digital signal processor TMS320F28335 and field-programmable gate array EP2C8T144C8N) has been developed to execute the proposed control methods. The control period is . Fig. 9Open in figure viewerPowerPoint Laboratory prototype of the Vienna rectifier The experimental waveforms before and after the start-up are shown in Fig. 10. The measured waveforms include a-phase voltage and current, and two capacitor voltages. As observed, the proposed controller has fast dynamic response and takes <10 ms to go into steady state. Fig. 11a shows the experimental results in the case of the dc-link voltage reference changing from 200 to 250 V. The THD of the input current is 3.89 and 2.55% under and , respectively. The experimental results are in good agreement with the simulation results in Figs. 5 and 6a. Fig. 10Open in figure viewerPowerPoint Experimental results of the proposed control scheme when the control system starts up Fig. 11Open in figure viewerPowerPoint Experimental results under the dc-link voltage changing from 200 to 250 V (a) OSS-MPC, (b) OSV-MPC, (c) Method in [10] Besides OSV-MPC, the conventional control method in [10] is used for comparison in experiments. All of them are tested under the same experimental conditions. Fig. 11 shows the experimental results in the case of the dc-link voltage reference changing from 200 to 250 V. As seen, three methods have basically the same dynamic performance. The corresponding spectrum analysis results of the input current with are illustrated in Fig. 12 and the THDs are listed in Table 4. The spectrum analysis results show that the proposed method has better steady-state performance. Table 4. THD of the input current in the three methods Method OSS-MPC OSV-MPC Method in [10] THD, % 3.89 6.02 6.93 Fig. 12Open in figure viewerPowerPoint Spectrum graphics of input currents In addition, experimental results under the load resistance changing from 80 to 60 are shown in Fig. 13. As observed, the dc-link voltages of the three methods dip a little and recover to normal in a short time. However, voltage ripple of the method in [10] is larger than those of the other methods. The distinct voltage ripple is caused by the approximated zero-sequence component [10]. Fig. 13Open in figure viewerPowerPoint Experimental results under the load resistance changing from 80 to 60 (a) OSS-MPC, (b) OSV-MPC, (c) Method in [10] Furthermore, Figs. 14a–d show the corresponding experimental waveforms under . As seen, the OSS-MPC still achieve a good performance in these cases. The THD of the input current are given in Table 5. It can be found that the THD of input current in the case of is higher than that in the case of . This conforms to the basic operating characteristics of the Vienna rectifier [16]. Table 5. THD of the input current under different input power factor THD, % 2.65 6.79 2.36 2.30 Fig. 14Open in figure viewerPowerPoint Experimental results under different desired input power factor (a) , (b) , (c) , (d) 5 Conclusion In this paper, an OSS-MPC control method is proposed for the three-phase Vienna rectifier. Owing to the proposed redundant vector pre-selection, the computational burden of the proposed OSS-MPC is greatly reduced. Compared with traditional OSV-MPC method, OSS-MPC has better steady-state performance. In addition, the proposed scheme overcomes the drawback of OSV-MPC that the switching frequency is not fixed. Meanwhile, it eliminates the weight factors in the cost function. 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