IET Electric Power Applications

Research Article

Optimal switching sequence model predictivecontrol for three-phase Vienna rectifiers

ISSN 1751-8660Received on 18th January 2018Revised 28th March 2018Accepted on 13th April 2018doi: 10.1049/iet-epa.2018.0033www.ietdl.org

Shiming Xie1, Yao Sun1, Mei Su1 , Jianheng Lin1, Qiming Guang1

1School of Information Science and Engineering, Central South University, Changsha, People's Republic of China E-mail: sumeicsu@mail.csu.edu.cn

Abstract: Vienna rectifier is a typical three-level rectifier with complicated operating constraints. Also, the constraints pose achallenge for designing controllers with good dynamic performance. As predictive control is good at dealing with constraints, anoptimal switching sequence model predictive control (OSS-MPC) strategy for the three-phase Vienna rectifier is proposed. Aproportional–integral controller is designed to regulate the dc-link voltage. Also, an improved OSS-MPC method is utilised tocontrol the input currents. Compared to the conventional finite control set model predictive control, it has the extra advantagesof improved steady-state performance, fixed switching frequency, and elimination of weight factors. Simulation and experimentalresults verify the correctness and effectiveness of the proposed control scheme.

1 IntroductionVienna rectifier has the advantages of low input current totalharmonic distortion (THD), low blocking voltage stress, highpower density, and so on [1, 2]. Hence, it has been widely used invarious applications, such as electric aircrafts, telecommunication,and wind turbine systems [1, 3, 4].

The related researches of the Vienna rectifier are mainlyfocused on modulation strategies and control methods. In [5], aspace vector modulation (SVM) is presented, which achievesneutral-point voltage balance by adjusting the duty cycles ofredundant vectors. In [6], the relationship between three-levelvoltage vectors and two-level voltage vectors has been discovered,which makes the traditional two-level SVM method applicable toVienna rectifier. A carrier-based pulse width modulation (CBM) ispresented in [7], where the neutral-point voltage is balancedthrough 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 lowswitching losses by avoiding switching the switches withmaximum phase current. Moreover, zero average neutral-pointcurrent within a mains period is guaranteed. Meanwhile, theminimum switching action is achieved [9].

In addition, a lot of control schemes are presented for gooddynamic performance. Usually, the main control targets of the threephase Vienna rectifier are: (i) to keep the output voltage at thedesired value under time-varying loads or grid voltages; (ii) toachieve sinusoidal input currents and given input reactive power;and (iii) to maintain the neutral-point voltage balance. Hysteresiscurrent control method is proposed to control the ac current andzero-sequence current [1], which is a useful control schemes withthe advantages of strong robustness and simple design procedure.In [10], classic proportional–integral (PI) controllers are used basedon the average d–q model of the Vienna rectifier. An optimal zero-sequence component is selected for dc-link voltage balance, whichis based on the relationship between the controlled duty cycle andthe dc-link neutral-point voltage. In [11], the modified one-cyclecontrol based on three-phase Vienna rectifier is presented, whichintroduces the neutral-point voltage loop. Then, it improves theutilisation ratio of dc-link voltage and balances the neutral-pointvoltage. To improve dynamic performances during both start-upand step-load and achieve good performances in steady state of theVienna-type rectifiers, an improved direct power control (DPC)strategy based on sliding mode control with dual-closed-loop ispresented [12]. The control strategies above are designed forVienna rectifier in continuous conduction mode (CCM). Lately, an

open-loop current control scheme without current measurement isproposed for Vienna rectifier working in discontinuous conductionmode, 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 frameworkswhich often includes modulation and control. The modulation hasbeen eliminated from FCS-MPC. Also, it is particularly suitable forthe system such as Vienna rectifier which suffers from strictconstraints. As there are only eight switching states in the three-phase Vienna rectifier, the calculation burden of the FCS-MPC issmall. Therefore, the idea of FCS-MPC control is introduced in theinner loop of the Vienna rectifier [16].

Optimal switching vector MPC (OSV-MPC) is a commonlyused control method in FCS-MPC [15]. However, it would lead touncertain switching frequency, which increases the difficulty ofdesigning input filters. Moreover, it needs higher samplingfrequencies to achieve a better performance. Optimal switchingsequence model predictive control (OSS-MPC) [15] is anothercontrol method in FCS-MPC. It uses an optimal switchingsequence in a control period instead of an optimal switching statein OSV-MPC. Thus, OSS-MPC has a better performance thanOSV-MPC. Moreover, it has the advantage of constant switchingfrequency. It has been applied in the DPC of two-level converters[17, 18]. However, the selections of optimal switching sequenceand optimal duty cycles are more complicated, so thecomputational burden of OSS-MPC is heavy.

To overcome the shortcoming of OSS-MPC above, an improvedOSS-MPC with low computational burden for Vienna rectifiers ispresented. In the proposed control scheme, OSS-MPC only takescharge of input current control. Also, the dc-link voltage regulationand the neutral-point voltage balance are realised by a PI controland 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, fixedswitching frequency, and elimination of weight factors.

The remainder of this paper is organised as follows: Section 2presents the system model of the three-phase Vienna rectifier. InSection 3, the proposed control scheme is described in detail. InSection 4, the simulation and experimental results are presentedand discussed. Finally, the main points of this paper aresummarised in Section 5.

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2 Model of three-phase Vienna rectifierFig. 1 shows the topology of the three-phase Vienna rectifier,which involves a three-phase diode rectifier, three bidirectionalswitches, three boost inductors, and two dc capacitors.

Assume that the Vienna rectifier is operating in current CCM.The dynamics of input currents are expressed as follows:

Ldiadt = vsa − van − iaRs − uno

Ldibdt = vsb − vbn − ibRs − uno

Ldicdt = vsc − vcn − icRs − uno

(1)

where L is the boost inductance, vsi, ii i = a, b, c are the inputphase voltage and current, uno is the zero-sequence voltage,vin i = a, b, c are the semi-controllable voltage, which isdetermined by not only the switching state of the bidirectionalswitch, but also the direction of the input current.

For convenience, denote Si i = a, b, c as switching states of thethree bidirectional switches of the Vienna rectifier. Also, Si = 0means the switch Ti is off; Si = 1 means the switch Ti is on. Thenthe semi-controllable voltage vin can be given as:

vin = 1 − Sisign ii + 1

2 vc1 + sign ii − 12 vc2 (2)

where vc1 and vc2 are the voltage across the capacitor C1 and C2,respectively. Also, sign( ) is the sign function to distinguish thedirection of currents.

Assume C1 = C2 = C , then voltage error of the two capacitorsequation can be written as:

C dv~dt = iC1 − iC2 = − i0 = − ∑

i = a, b, cSiii (3)

where iC1, iC2 are the currents passing through the capacitors, io isthe neutral-point current, and v~ the capacitor voltage deviation, i.e.v~ = vC1 − vC2.

3 Proposed OSS-MPCFig. 2 shows the proposed control block diagram of the Viennarectifier. The dc-link voltage is controlled by a PI controller. Also,input currents and voltage balance control controlled by theproposed predictive controller.

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 thepolarity of the input currents. Assume that ia > 0, ib < 0, ic < 0,the possible controllable voltage vectors lie in sector I and areshown in Fig. 3 (cf. Fig. 2 in [9]). In fact, for a certain input currentvector, there are only eight controllable voltage vectors.

According to the principle of SVM, to obtain good currentquality, it is better to apply a proper switching sequence rather thanonly one switching state applied in one control period like OSV-MPC which is a traditional FCS-MPC. OSS-MPC is such apredictive method based on this idea. As Fig. 3 depicts, sector Icould be divided into six small sectors (A–F). Based on thecriterion that the switching sequence made up of adjacent vectors,all the switching sequences are listed in Table 1 where vt is thevector sequence and vr1, vr2, vr3 are the corresponding switch statesused in the modulation period. Vectors V0 or V7 are called theredundant vectors as they have the same effect on input currentsbut an opposite effect on the neutral-point voltage. Table 2 showsthe switch states of the redundant vectors in each sector. Sincevectors V0 or V7 are the vertices of any small sector like (A–F), theneutral-point voltage balance could be well achieved. From Table1, there are 12 feasible switching sequence candidates.

Obviously, selecting an optimal switching sequence from the 12candidates would consume amounts of time. Therefore, a pre-selected algorithm based on the polarity of voltage error isproposed to reduce the computation efforts. The principle ofredundant vector pre-selection is that the selected redundant vectormust satisfy v~io > 0. For example, select the redundant vector asV0 = [0, 1, 1] when v~ < 0 in sector I. After the redundant vectorpre-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 abc coordinate into the equation in αβ coordinateas follows:

Fig. 1  Circuit configuration of the three-phase Vienna rectifier

Fig. 2  Block diagram of the proposed control scheme

Fig. 3  Vienna rectifiers voltage vectors in sector I

Table 1 Vector sequence to be appliedSector vt = vr1, vr2, vr3, vr3, vr2, vr1

A V1, V2, (V0 or V7), (V0 or V7), V2, V1

B V2, V3, (V0 or V7), (V0 or V7), V3, V2

C V3, V4, (V0 or V7), (V0 or V7), V4, V3

D V4, V5, (V0 or V7), (V0 orV7), V5, V4

E V5, V6, (V0 or V7), (V0 or V7), V6, V5

F V6, V1, (V0 or V7), (V0 or V7), V1, V6

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diαdt = 1

L vsα − vαn − iαRs

diβdt = 1

L vsβ − vβn − iβRs

(4)

where vsαβ, isαβ are the input voltages and currents in αβ coordinate.Assume the control period is short enough, then it is reasonable

to view increments of the input current iα, iβ as constants. The dutycycles of vr j( j = 1, 2, 3) in a control cycle are defined as d1, d2, andd3. Then, the predicted input currents of Vienna rectifier at(k + 1)th instant can be expressed as follows:

iα k + 1 = iα k + ∑j = 1

3f α jd jTs

iβ k + 1 = iβ k + ∑j = 1

3f β jd jTs

(5)

where Ts is the control period, and

f α j = diαdt

S = Sjvsβ = vsβ k

iβ = iβ k

= 1L vsα k − vrα j − iα k Rs

f β j = diβdt

S = Sjvsβ = vsβ k

iβ = iβ k

= 1L vsβ k − vrβ j − iβ k Rs

(6)

where f α j, f β j are the increments for the input current iα, iβ;iα k + 1 , iβ k + 1 are the predicted input currents at (k + 1)thinstant, and iα k , iβ k are the measured value of input currents atkth instant.

The optimal duty cycles are solved by minimising the inputcurrent tracking errors, which are defined as:

Fk = eiαk + 1 2 + eiβ

k + 1 2 (7)

where eiαk + 1 and eiβ

k + 1 are current tracking errors, and they areexpressed as:

eiαk + 1 = iα∗ k + 1 − iα k

eiαk

− ∑j = 1

3f α jd jTs

eiβk + 1 = iβ∗ k + 1 − iβ k

eiβk

− ∑j = 1

3f β jd jTs

(8)

The optimisation problem can be formulated as follows:

minimise Fk

According to the following conditions:

∂Fk

∂d1= 0

∂Fk

∂d2= 0

(9)

This yields the following optimal duty cycles:

d1 = f β2 − f β3 eiαk + f α3 − f α2 eiβ

k + f α2 f β3 − f α3 f β2 TsTs f β2 − f β3 f α1 + f β3 − f β1 f α2 + f β1 − f β2 f α3

d2 = f β3 − f β1 eiαk + f α1 − f α3 eiβ

k + f α3 f β1 − f α1 f β3 TsTs f β2 − f β3 f α1 + f β3 − f β1 f α2 + f β1 − f β2 f α3

d3 = 1 − d1 − d2

(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 calculatedin each control cycle. The switching sequence with minimum costfunction is the optimal switching sequence. The flowchart of theproposed OSS-MPC strategy is shown in Fig. 4a. Ft

k is the value ofcost function calculated for the tth switching sequence, and Fmin

k isthe minimum value of the calculated cost functions. Besides,vr j

op, d1, 2, 3op denote the optimal switching sequence and the duty cycles

of vr jop, respectively. Assume that the Vienna rectifier is operating in

sector I and the optimal switching sequence vr jop is

V2, V3, V7, V7, V3, V2 , then the double-side switching pattern isadopted for the modulation process, which is illustrated in Fig. 4b.

However, the delay between the measurements and theactuation is an inherent shortcomings of digital control, which hasa great impact on the performance. In this paper, the first-stepprediction is repeated every sampling time to compensate the timedelay. The first-step prediction are as follows:

i^α(k + 1) = iα k + ∑j = 1

3f̄ α jd̄ jTs + kα iα k − i^α(k)

i^β(k + 1) = iβ k + ∑j = 1

3f̄ β jd̄ jTs + kβ iβ k − i^β(k)

(11)

where i^ is the first prediction values replacing the measured value,f̄ and d̄ are the increments for the input current and the duty cyclescalculated 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 terminalconstraint method and terminal cost method are two commonlyused methods to guarantee the stability of the MPC. According tothe results in [19], if there exists a control law that admits (12),then the control system is stable:

Fk + 1 − Fk = − α ∥ iα∗ − iα, iβ∗ − iβT ∥ (12)

where α ∈ κ∞.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 isstable with a certain region of attraction. A long prediction horizonis advantageous for stability [20]. However, the proposed OSS-MPC is a one-step prediction. Therefore, it could be inferred thatthe region of attraction may not be large, which is one shortcomingof the proposed OSS-MPC.

4 Simulations and experimental resultsThe proposed control scheme has been tested on the three-phaseVienna rectifier system. The main system parameters aresummarised in Table 3.

4.1 Simulation results

Numerical simulations have been carried out in MATLAB/Simulink platform to illustrate the steady-state and dynamic

Table 2 Switch state of redundant vectors in each sectorSector V0 V7

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]

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performance of the proposed control scheme. Besides, the methodusing OSV-MPC [21] is utilised for comparison.

First, the operating condition of the unity power factor isconsidered. The reference voltage is set to 250 V, and the controlsystem starts up at t = 0.2 s. Fig. 5 shows the simulation resultsduring start-up with the proposed control. As seen, before start-up,the capacitor voltages are pulsating and the input current isseverely distorted. After start-up, each capacitor voltage isregulated to half of the desired dc-link voltage quickly. Also, theinput current also is sinusoidal and in phase with the input phasevoltage.

To show the dynamic performance of the proposed controlscheme, 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 proposedmethod and the OSV-MPC method, respectively. The samplingperiod of the OSV-MPC method is 100 μs. They have almost thesame dynamic response with regard to input currents, dc-link

voltage, and neutral-point voltage balance. However, it is easy tofind that the steady-state performance of the proposed controlscheme is better than that of the OSV-MPC.

In addition, the dynamic response of the dc-link voltage to thestep load changing from 80 to 60 Ω is tested. As observed in Figs.6c and d, both the proposed controller and the OSV-MPCcontroller are able to maintain the desired dc-link voltage in thepresence of the load disturbance. It is clear that they have thesimilar dynamic performance by selecting the optimal result in thefinite set. Moreover, the capacitor voltages still keep the balance. Itis proved that the pre-selection of vector can work well. Tocompare the input current quality of OSV-MPC and OSS-MPC, theTHD values of the input currents under different load conditionsare recorded and shown in Fig. 7. Clearly, the proposed method hasbetter performance in input current quality.

Figs. 6e and f show the simulation results for the step loadchanging from 80 to 2400 Ω at t = 0.1 s. A large overshoot is foundin dc-link voltage as seen in Fig. 6e. Also, the overshoot voltagedisappears after ∼3 period because Vienna rectifier is aunidirectional one. When the system is in steady state under lightloads, approximately sinusoidal input currents are obtained still butsome large spikes exist. The same experiment is tested with OSV-MPC; the results are shown in Fig. 6f. Clearly, the dc-link voltageis out of control in this case. From the comparative results, it canbe inferred that the proposed method has a larger stable attractiveregion than the OSV-MPC.

Since mathematical model is required in the FCS-MPC, therobustness against parameter uncertainties should be considered.The related tests are carried out with the desired dc-link voltage of250 V.

Figs. 8a and b show the simulation results of the controller inwhich the nominal parameters of the converter areL = 6 mH and C = 560 μF, while the real parameters areL = 3 mH and C = 1120 μF. From (3), the variation of capacitancewill affect the ripple of the neutral-point voltage but has little effecton the stability, which is verified in Fig. 8. As the actualcapacitance increases, the voltage error of dc-link voltage becomessmaller. However, the input current ripple become larger. Thereasons are twofold: (i) the predictive model is inaccurate; (ii) thefiltering performance degrades as the inductance decreases.

Next, the inductance parameter uncertainty is considered. Aseries of different inductance values are tested. As seen in Fig. 8c,the input current THD is <4% when the practical value of the inputinductors is >0.55 times and <2 times of its nominal value. Theresults show that the proposed control is robust to parameteruncertainty.

4.2 Experimental results

Apart from the simulation studies, experiments are also performedto validate the effectiveness of the proposed method. A prototypeof the three-phase Vienna rectifier is built in the laboratory asshown in Fig. 9. The specifications of this system are the samewith those in simulation, which are listed in Table 3. For simplicity,IGBT-Module FS3L25R12W2H3_B11 (Infineon) is used toconstruct the power stage of the Vienna rectifier. Besides, a control

Fig. 4  Schematic of the control process and modulation(a) Flowchart of the proposed OSS-MPC, (b) Schematic diagrams of the modulationscheme

Table 3 Parameters used in the simulations andexperimentsSymbol Description ValueVs input line voltage 110 V(rms)ω input angular frequency 314 rad/sL boost inductance 6 mHC dc-link capacitance 560 μFR load resistance 80 ΩVdc dc-link voltage 250 VTs sample period 100 μs

Fig. 5  Simulation results of the OSS-MPC during start-up

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board (32 bit floating digital signal processor TMS320F28335 andfield-programmable gate array EP2C8T144C8N) has beendeveloped to execute the proposed control methods. The controlperiod is 100 μs.

The experimental waveforms before and after the start-up areshown in Fig. 10. The measured waveforms include a-phasevoltage and current, and two capacitor voltages. As observed, theproposed controller has fast dynamic response and takes <10 ms togo into steady state. Fig. 11a shows the experimental results in thecase 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 Vdc

∗ = 200 V

and Vdc∗ = 250 V, respectively. The experimental results are in good

agreement with the simulation results in Figs. 5 and 6a. Besides OSV-MPC, the conventional control method in [10] is

used for comparison in experiments. All of them are tested underthe same experimental conditions. Fig. 11 shows the experimentalresults in the case of the dc-link voltage reference changing from200 to 250 V. As seen, three methods have basically the samedynamic performance. The corresponding spectrum analysis resultsof the input current with Vdc

∗ = 200 V are illustrated in Fig. 12 and

Fig. 6  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 forthe load step change from 80 to 60 Ω, (d) The OSV-MPC for the load step change from 80 to 60 Ω, (e) The proposed for the load step change from 80 to 2.4k Ω, (f) The OSV-MPCfor the load step change from 80 to 2.4k Ω

Fig. 7  THD of input current under different loads (dc-link voltage reference of 250 V)

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the THDs are listed in Table 4. The spectrum analysis results showthat the proposed method has better steady-state performance.

In addition, experimental results under the load resistancechanging from 80 to 60 Ω are shown in Fig. 13. As observed, thedc-link voltages of the three methods dip a little and recover tonormal in a short time. However, voltage ripple of the method in[10] is larger than those of the other methods. The distinct voltageripple is caused by the approximated zero-sequence component[10].

Furthermore, Figs. 14a–d show the corresponding experimentalwaveforms under φs = ± π /12 and ± π /6. As seen, the OSS-MPCstill achieve a good performance in these cases. The THD of theinput current are given in Table 5. It can be found that the THD ofinput current in the case of φs = π /6 is higher than that in the case

of φs = − π /6. This conforms to the basic operating characteristicsof the Vienna rectifier [16].

5 ConclusionIn this paper, an OSS-MPC control method is proposed for thethree-phase Vienna rectifier. Owing to the proposed redundantvector pre-selection, the computational burden of the proposedOSS-MPC is greatly reduced. Compared with traditional OSV-MPC method, OSS-MPC has better steady-state performance. Inaddition, the proposed scheme overcomes the drawback of OSV-MPC that the switching frequency is not fixed. Meanwhile, it

Fig. 8  Simulation waveforms of the input currents and error between thecapacitor voltages in(a) L = 6 mH and C = 560 μF, (b) L = 3 mH and C = 1120 μF, (c) THD of the inputcurrent under filter inductance value (L) variation

Fig. 9  Laboratory prototype of the Vienna rectifier

Fig. 10  Experimental results of the proposed control scheme when thecontrol system starts up

Fig. 11  Experimental results under the dc-link voltage changing from 200to 250 V(a) OSS-MPC, (b) OSV-MPC, (c) Method in [10]

Fig. 12  Spectrum graphics of input currents

Table 4 THD of the input current in the three methodsMethod OSS-MPC OSV-MPC Method in [10]THD, % 3.89 6.02 6.93

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eliminates the weight factors in the cost function. Thus, the OSS-MPC is a very attractive alternative for other power converters.

6 AcknowledgmentsThis work was supported in part by the National Natural ScienceFoundation of China under Grants 61622311, in part by the JointResearch Fund of Chinese Ministry of Education under Grant6141A02033514.

Fig. 13  Experimental results under the load resistance changing from 80to 60 Ω(a) OSS-MPC, (b) OSV-MPC, (c) Method in [10]

Fig. 14  Experimental results under different desired input power factor(a) φs = π /12, (b) φs = π /6, (c) φs = − π /12, (d) φs = − π /6

Table 5 THD of the input current under different inputpower factorφs π /12 π /6 −π /12 −π /6THD, % 2.65 6.79 2.36 2.30

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