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energies Article

Model Predictive Direct Power Control for Nonredundant Fault Tolerant Grid-Connected Bidirectional Voltage Source Converter Nan Jin 1,2 , Leilei Guo 1, * and Gang Yao 3 1 2 3

*

Department of Electrical Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China; [email protected] Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37996, USA Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; [email protected] Correspondence: [email protected]; Tel.: +86-0371-63556790

Received: 7 July 2017; Accepted: 28 July 2017; Published: 2 August 2017

Abstract: This paper proposes a model predictive direct power control scheme for nonredundant fault tolerant grid-connected bidirectional voltage source converter (BVSC) with balanced dc-link split capacitor voltage and high reliability. Based on the operation analysis of fault-tolerant BVSC with phase leg faults, a power predictive model of three-phase four-switch fault-tolerant topology in αβ coordinates is established, and the space voltage vectors with unbalanced dc-link split capacitor voltage are analyzed. According to the power predictive model and cost function, the optimal space voltage vector is selected to achieve a flexible, smooth transition between inverter and rectifier mode with direct power control. Pulse width modulation and phase locked loop are not required in the proposed method. The constraint of dc-link voltage constraint is designed for the cost function to achieve a central point of dc-link voltage offset suppression, which can reduce the risk of electrolytic capacitor failure for over-voltage operation. With the proposed control method, the converter can work continuously in both inverter mode and rectifier mode, even if phase leg faults occur. The simulation and experimental results show good steady-state and dynamic performance of the proposed control scheme to enhance the reliability of bidirectional power conversion. Keywords: bidirectional voltage source converter; model predictive control; three-phase four-switch; fault tolerant; open circuits faults

1. Introduction The bidirectional voltage source converter (BVSC) can integrate various ac/dc loads, distributed storages, distributed generation, and ac grid with high efficiency and flexible power regulation. Due to the increasing development of switching devices and renewable energy power generation, there will be more applications of bidirectional alternating current (ac)/direct current (dc) conversion, such as electric vehicles, which can store power in the night and generate power to the grid in the daytime. The growth of BVSC has been promoted by the environment pollution issues caused by the traditional fossil energy resources such as oil and coal. This has received great attention in developing countries [1,2]. For the high performance of the bidirectional power conversion between the ac and dc sides, the reliability of the BVSC in different working conditions, such as under unbalanced power grid, paralleled applications, or islanding mode, has been studied to ensure safe and continuous operation [3–5]. However, much research shows that switching devices in power converters, such as an insulated gate bipolar transistor (IGBT) or a metal-oxide field-effect transistor (MOSFET), are prone

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to have faults caused by the surges and spikes in the conditions of high voltage and high switching frequency transition, which are a major challenge to the reliability of the power converter. On the other hand, most power converters are not designed to be redundant. Once there is a switching device fault, the power conversion will be interrupted. Therefore, the study of nonredundant fault-tolerant bidirectional power conversion is urgent and significant to enhance the reliability of the BVSC [6–9]. The three-phase four-switch (TPFS) topology was first presented for its cost-effective design, and it became a fault-tolerant topology for the switching devices open or short circuits faults of the conventional three-phase six-switch converter (TPSS) for bidirectional power conversion [10–15]. As a promising fault-tolerant topology for the widely used TPSS plan, the control scheme study of TPFS topology has drawn great attention. The TPFS converter is applied in motor drive applications as a low cost fault-tolerant topology. A control strategy–based single current sensor is proposed for the fault-tolerant brushless dc motor to lower cost and improve performance [10]. A compensation scheme with different forward voltage drop values is proposed for the direct torque control of the TPFS converter to correct the stator flux imbalance and reduce the total harmonic distortion [11]. The voltage unbalance of the dc-link split capacitors is another problem for the reliable operation of a fault-tolerant converter. The spatial repetitive controller is proposed to eliminate the dc-link central point voltage fluctuation of the TPFS converter in microgrids application [12]. The double Fourier integral analysis is used to investigate the phase-leg switched voltage spectrum of TPFS converter. The dc-link central point voltage fluctuation can be neutralized by injecting certain terms to the modulating waveforms [13]. A predictive torque control for TPFS inverter-fed induction motor with dc-link voltage offset suppression is proposed to balance the phase current [14]. The TPFS inverter based on the topology of the single-ended primary-inductance converter is proposed to provide the higher output voltage, which enhances the utilization of the dc voltage [15]. The above control method of TPFS converter is based on the space voltage vector modulation method, which needs the coordinating transformation, the complex calculation of the vector sector, and the duty ratio. Furthermore, the bidirectional power conversion of the fault-tolerant operation is not considered. Model predictive control (MPC) has the advantages of simplicity and flexibility, and the cost function can be designed with different control objectives. Compared to classical control methods, MPC has a fast, dynamic response, good adaptability, and robustness without using the phase locked loop control and pulse width modulation (PWM) [16–20]. MPC has been used in a grid-connected inverter system under normal conditions without faults. A model predictive direct power control strategy for a grid-connected inverter in a photovoltaic system is proposed, which achieve flexible power regulation and switching frequency reduction [18]. In addition, coordinate transformation and proportional-integral regulators are not necessary. The switching table and PWM modulation module are not included either. The delay compensation method is proposed to reduce the influence of the calculation delay when there are a large number of voltage vectors [19]. A model predictive power control method is proposed for the PWM rectifier that is able to operate under both balanced and unbalanced grid voltages [20]. Neither complicated sequence extraction of grid voltage/current nor the phase locked loop is needed. However, this does not consider the power switching faults conditions. Although there are many studies on the model predictive control scheme of the power converter, the switching device open or short circuit faults are not considered. Therefore, the control scheme of the fault-tolerant topology needs further study. This paper proposes a model predictive direct power control method for nonredundant fault tolerant grid-connected BVSC with high reliability. With high-penetration renewable energy and microgrids integration, the proposed method offers a higher reliability for power conversion. The main contributions of this paper are as follows: 1. 2.

The fault tolerant grid-connected BVSC model is established. The impacts of the unbalanced dc-link capacitor voltage on the voltage vectors are analyzed in detail. The model predictive direct power control (MPDPC) method is designed considering direct power control and dc-link voltage balance control, which does not need double loop control,

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3. 3.

4. 4.

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PWM modulation, or phase locked loop, and iscompensation easy to implement. The split capacitor voltage balance control is achieved by adding term to thedc-link cost function, which reduces risk of electrolytic capacitor failure for over-voltageterm operation. voltage the balance control is achieved by adding compensation to the cost function, which With thethe proposed MPDPC, the fault-tolerant BVSC has good operation. steady-state performance with reduces risk of electrolytic capacitor failure for over-voltage sinusoidal output current waveforms. When the reference power changes,performance the fault-tolerant With the proposed MPDPC, the fault-tolerant BVSC has good steady-state with working modes of inverter and rectifier can be switched smoothly with good dynamic sinusoidal output current waveforms. When the reference power changes, the fault-tolerant working performance. modes of inverter and rectifier can be switched smoothly with good dynamic performance. When fault-tolerant converter work continuously continuously When there there are are phase phase leg leg faults faults with with BVSC, BVSC, the the fault-tolerant converter can can work without without disconnecting disconnecting the the dc dc side side and and ac ac grid, grid, which which enhances enhances the the reliability reliability of of the the bidirectional bidirectional power power conversion. conversion.

2. Operation Principles of the Nonredundant Fault-Tolerant Fault-Tolerant BVSC BVSC The reliability of the BVSC depends on many elements, such as the hardware hardware design, design, working working conditions, and adopted power devices. As a result of switch open circuits, short circuits, and driver signal faults, it has been estimated that more than 80% of faults are caused caused by switching switching device failures [8]. Therefore, Therefore, fast fast fuse fuse devices devices can can be connected in series with the switching devices to convert the short circuits to open circuits circuits faults. faults. The topology of nonredundant fault-tolerant BVSC is shown in Figure 1a. The switch devices in the phase leg are IGBTs IGBTs with with antiparallel antiparallel freewheeling freewheeling diodes. The The phase phase legs legs are are also also connected connected with with the the central central point of the series capacitor by three bidirectional switches, switches,which whichcan canbebe triode alternating current (TRIAC) or IGBTs with singlebidirectional triode forfor alternating current (TRIAC) or IGBTs with single-phase phase rectifier. diode rectifier. In normal operations, the bidirectional switches thestate. open When state. When diode In normal operations, the bidirectional switches are in are the in open short short circuits orcircuits open circuits occur in oneleg phase legas(such a fast fuse(Fdevice F2) is circuits or open occur in one phase (such phaseasa),phase a fasta), fuse device opened, 1 or F2 )(Fis1 or opened, the corresponding bidirectional Ta is conducted to achieve continuous operation and the and corresponding bidirectional switch switch Ta is conducted to achieve continuous operation [9]. [9]. The reconstructed fault-tolerant BVSC is shown in Figure The reconstructed fault-tolerant BVSC is shown in Figure 1b. 1b.

AC grid ea ia eb ib n ec ic

F3 S3

F1 S1 R

L

F5 S5

a b c

Ta Tb Tc

P

S4 F4

(a)

S6 F6

S5

S3

C1 DC M Side

n

ea ia eb ib ec ic

R

idc P Udc1 a

L b c S4

S2 F2

idc1 C1

C2 N

S6

idc2 C2

Udc2 N

(b)

Figure 1. The fault-tolerant topology of BVSC. (a) Nonredundant fault-tolerant structure; (b) threeFigure 1. The fault-tolerant topology of BVSC. (a) Nonredundant fault-tolerant structure; phase four-switch (TPFS) structure with leg fault of phase a. (b) three-phase four-switch (TPFS) structure with leg fault of phase a.

There are four switching devices in fault-tolerant BVSC (see Figure 1b). The relationship between There are four switching in fault-tolerant (see Figure 1b). The between the output voltage vector anddevices switching states of theBVSC converter is analyzed. Si (irelationship = b, c) is defined as the output voltage vector and switching states of the converter is analyzed. S (i = b, c) is defined as the i the switching state for fault-tolerant BVSC converter as, switching state for fault-tolerant BVSC converter as, upper bridge of i phase is on and lower bridge is off ( 1 Si = (1) 1 upperbridge bridgeofofi phase i phaseisisoff onand andlower lowerbridge bridgeisison off 0 upper Si =  (1) 0 upper bridge of i phase is off and lower bridge is on The relationship between the output voltage and switching states of fault-tolerant BVSC can be expressed as [14], The relationship between the output voltage and switching states of fault-tolerant BVSC can be expressed as [14],

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 U dc 1 U −Sb − Sc ) + dc 2 ( 2 − Sb − Sc ) (  uan = 3 3 Energies 2017, 10, 1133 4 of 16  U dc 1 U dc 2  ( 2Sb − Sc ) + 3 ( 2Sb − Sc − 1) (2) ubn =   3 Udc1 Udc2  u = U  3 (− Sb − Sc ) +U 3 (2 − Sb − Sc )  an ucn =Udc1dc 1 ( 2Sc − Sb ) + Udcdc22 ( 2Sc − Sb − 1)  (2) ubn  = 3 3 (2Sb − Sc ) + 33 (2Sb − Sc − 1)    Udc1 Udc2 u = 3 (2Sc − Sb ) + 3 (2Sc − Sb − 1) output voltages of the converter, and dc voltage of capacitor C1, C2, where uan, ubn, ucn, Udc1, Udc2 are cn respectively. where uan , ubn , ucn , Udc1 , Udc2 are output voltages of the converter, and dc voltage of capacitor C1 , The voltage components uα, uβ of the stationary coordinate system are obtained by Clark C2 , respectively. coordinate transformation. There switching states of (0, 0), (0,system 1), (1, 0), 1) in fault-tolerant The voltage components uα ,are uβfour of the stationary coordinate are(1,obtained by Clark BVSC. The transformation. voltages of TPFSThere structure in αβ stationary frame fault are1)shown in Table 1. coordinate are four switching states of with (0, 0),phase (0, 1),a(1, 0), (1, in fault-tolerant BVSC. The voltages of TPFS structure in αβ stationary frame with phase a fault are shown in Table 1. Table 1. Voltage vectors of fault-tolerant bidirectional voltage source converter (BVSC) with phase a fault. 1. Voltage vectors of fault-tolerant bidirectional voltage source converter (BVSC) with phase Table a fault.

U(Sb, Sc) Switch on S4, Son 6 U(Sb , ScU ) 1(0, 0) Switch , S5 U (0, 0)U2(0, 1) SS,4S 1

4

U2 (0, 1)U3(1, 0) U3 (1, 0) U4(1, 1) U4 (1, 1)

6

uα 2Udc2/3u

α

uβ 0

uβ

− 3(Udc1 + Udc2 ) / 3 (Udc2 − U dc1)/3 /3 0 2U dc2

√

− )/3 Udc1 )/33(U − Udc32()U/ dc1 3 + Udc2 )/3 (Udc2(U−dc2 Udc1 dc1 +√

SS 4 ,3S , 5S6 S3 , S6 S3, S5 S3 , S5

(U

− Udc1 )/3 −2Udc2 /3

−2Udc2 dc2/3

0

3(Udc1 + Udc2 )/3 0

The voltage vectors divide the vector space into four sectors, which are shown in Figure 2. The The voltage divide the vector into fourAccording sectors, which are shown in Figure 2. amplitudes of thevectors four basic voltage vectorsspace are not equal. to different conditions under The amplitudes of the four basic voltage vectors are not equal. According to different conditions under Udc1 = Udc2, Udc1 < Udc2, and Udc1 > Udc2, the basic voltage vectors are different, shown in Figure 3. Udc1 = Udc2 , Udc1 < Udc2 , and Udc1 > Udc2 , the basic voltage vectors are different, shown in Figure 3.

β U3

U4

β U4

α Ⅱ Ⅰ Ⅲ Ⅳ U1

U3 Ⅰ

Ⅱ

(a)

Ⅱ Ⅲ Ⅰ

α

Ⅳ

Ⅲ

α

Ⅳ U2

U4

U1

U2

U2

β U1

U3

(b)

(c)

Figure 2. Voltage space vectors. (a) Phase a leg fault; (b) Phase b leg fault; (c) Phase c leg fault. Figure 2. Voltage space vectors. (a) Phase a leg fault; (b) Phase b leg fault; (c) Phase c leg fault.

β U3

β U3 α

U4

U1

U3

α U4

U1

U2 (a)

β

U2 (b)

α U4

U1 U2 (c)

Figure 3. vectors of the BVSCBVSC with phase leg fault. (a)fault. Udc1 = U(a) dc2; (b) Udc1 < Udc2; Figure 3. Voltage Voltage vectors of fault-tolerant the fault-tolerant with aphase a leg Udc1 = Udc2 ; dc1 > Udc2. (c) U (b) Udc1 < Udc2 ; (c) Udc1 > Udc2 .

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3. Mathematic Model of Fault Tolerant BVSC 3.1. Power Predictive Model of Fault-Tolerant BVSC The reconstruction topology of fault-tolerant BVSC with leg fault of phase a is illustrated in Figure 1. It is connected to the grid through the filter inductor Lf and line resistance R. The dc side includes two capacitors: C1 and C2 with equal value C. The bidirectional power conversion of BVSC contains rectifier mode and inverter mode. The state equation of the converter in the αβ two phase stationary coordinates can be expressed as follows: Lf

diαβ = uαβ − eαβ − Riαβ dt

(3)

where uαβ , iαβ , eαβ are the αβ components of the converter output voltage, current and grid voltage. Equation (3) can be discretized and predictive current of tk+1 instant is given as follows: T iαβ (k + 1) = S uαβ (k) − eαβ (k) + Lf

RTS 1− Lf

! iαβ (k)

(4)

where uαβ (k), iαβ (k), eαβ (k) are αβ components of the converter output voltage, current, and grid voltage at the tk instant. iαβ (k + 1) are αβ components of predictive current value at tk+1 instant. TS is sampling period. Based on the instantaneous power theory, the complex power S of the power grid can be expressed as, 3 3 S = ei∗ = (eα + je β )(iα − ji β ) = P + jQ (5) 2 2 where * represents a conjugate. Active power and reactive power of can be obtained by, "

P(k) Q(k)

#

3 = 2

"

eα eβ

eβ − eα

#"

iα (k ) i β (k)

# (6)

Therefore, the predictive power model of the fault-tolerant BVSC at tk+1 sampling instant is, "

P ( k + 1) Q ( k + 1)

#

3T = S 2L

"

eα eβ

eβ − eα

#"

uα (k ) − eα − Riα (k) u β (k) − e β − Ri β (k)

#

"

+

P(k) Q(k)

# (7)

3.2. Central Point of dc-Link Voltage Offset As shown in Figure 1b, the two capacitors C1 and C2 are in series connected in dc-link, where C1 = C2 . Then the current in the faulty phase is equally divided into the two capacitors. The capacitor currents idc1 and idc2 can be obtained by:  1    idc1 = −idc2 = 2 i a ; switching f ault with leg a idc1 = −idc2 = 12 ib ; switching f ault with leg b    i = −i = 1 i ; switching f ault with leg c dc1 dc2 2 c The capacitor voltage Udc1 and Udc2 change with currents as follows:

(8)

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   # " # 1 " dU   dc1  C idc −ib −ic    dt  =   Sb ; switching f ault with leg a  dUdc2  i + i + i − i − i c c dc b b C dt   Sc        # " # "  Sb dUdc1  C dt −i a idc −ic   =  1 ; switching f ault with leg b dUdc2  − i i + i + i − i a a c c dc C dt   Sc        " # " #  S dUdc1 a   C dt −i a −ib idc     =   Sb ; switching f ault with leg c  dUdc2  − i − i i + i + i a a b dc b C dt  1

(9)

where idc is the dc-link current, shown in Figure 1b. The offset component of dc-link voltage can be shown as,  d (U −U )  C dc1dt dc2 = −(ic + ib ) = i a ; switching f ault with leg a   d(Udc1 −Udc2 ) dt d (U −U ) C dc1dt dc2

C

  

= −(ia + ic ) = ib ; switching f ault with leg b

(10)

= −(i a + ib ) = ic ; switching f ault with leg c

The predictive offset component at tk+1 instant is obtained by discretization of (10). In order to keep the central point of dc-link voltage to Udc /2, the offset component ∆Udc should be minimized.

min[∆Udc (k + 1)] =

  ∆Udc (k) +   ∆Udc (k) +

  

∆Udc (k) +

TS C ia ; TS C ib ; TS C ic ;

switching f ault with leg a switching f ault with leg b

(11)

switching f ault with leg c

Then the offset component constraint of (11) can be added to the cost function to achieve the central point of dc-link voltage offset suppression. 4. Model Predictive Control of Fault Tolerant BVSC In recent years, MPC has been widely used in power electronics application fields such as motor drives, active filter, power regulation, and distributed generation. The main principle of MPC is shown in Figure 4. The MPC system is designed to make the input variable x be equal to the reference value x*. In each sampling period, the optimal switching states of the power devices are solved online. x(k) is the input value of the kth sampling period and is obtained by discretization of input variables. When the sampling frequency is high, the input variables in a sampling period can be considered as a constant. If the number of switching states Sw is n, the value of variable xp i (k) in the next period can be determined by function xp i (k) = fp {x(k), Swi }, i = 1, 2 . . . n. Based on the working principle of the power converter, the predictive function fp is established. The cost function g need to be designed to select the optimal switching states Sw*(k) for the power converter, where g = fg {x*(k + 1), xp i (k + 1)}, i = 1, 2 . . . n. x*(k + 1) is the reference value of (k + 1)th sampling instant. According to different control objectives, the cost function can be changed to achieve optimal performance. The output switching states of MPC can be applied to power devices directly. Therefore, the pulse width modulation (PMW) is not necessary.

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reference value x*(k+1) Cost Function

x(k)

Prediction Function x pi(k+1)= fp{x(k), Swi} i=1,…n

xpi(k+1)

gi= fg{x∗(k+1), xpi(k+1)}

i=1,…n Select the optimal value

Sw*(k)

Figure 4. 4. The basic basic principle principle of of model model predictive predictive control control (MPC). (MPC). Figure

For the selection of the optimal switching vector to realize model predictive control, a cost For the selection of the optimal switching vector to realize model predictive control, a cost function function g is designed to compare all the predictive power values and select the voltage vector to g is designed to compare all the predictive power values and select the voltage vector to minimize the minimize the cost function. The sum of the absolute error value between the predictive power, cost function. The sum of the absolute error value between the predictive power, reference power, and reference power, and dc-link voltage offset component is chosen as cost function, dc-link voltage offset component is chosen as cost function,

g = Pref − P( k + 1) + Qref − Q( k + 1) + λ ΔUdc ( k + 1) (12) g = Pre f − P(k + 1) + Qre f − Q(k + 1) + λ|∆Udc (k + 1)| (12) where Pref, Qref are active power and reactive power reference value. P(k + 1), Q(k + 1) are power where Pref , values Qref areatactive power andλ reactive power reference value. 1),+ Q(k 1) are power predictive the tk+1 instant. is the weighting coefficient, andP(k ΔU+dc(k 1) is+the predictive predictive values at the t instant. λ is the weighting coefficient, and ∆U (k + 1) is the predictive voltage offset at central point k+1 M of the dc-link. dc voltage at implementation, central point M ofthere the dc-link. In aoffset digital is usually a one-step delay between the selected voltage vector In aapplied digital implementation, there has is usually a one-step delay between the selected vector and the voltage vector, which a significant impact on the dynamic and staticvoltage performance and thesystem applied voltage vector, which a significant impact oncontrol, the dynamic and static performance of the [19]. To compensate thishas one-step delay in digital the cost function considering of systemerror [19]. To compensate this should one-step in digital the cost function considering thethe tracking at (k + 2)th instant bedelay evaluated andcontrol, minimized to select the best voltage the tracking error as, at (k + 2)th instant should be evaluated and minimized to select the best voltage vector, expressed vector, expressed as, g = Pref − P( k + 2) + Qref − Q( k + 2) + λ ΔUdc ( k + 2 ) (13) g = P P(the k +cost 2) + Q(k + 2)the λ|∆Udcripple. (13) (k + 2)|The third term is used Qre f −penalize + power re f − in The first and second terms function to minimize the unbalanced dc-link capacitor voltages. The first and second terms in the cost function penalize the power ripple. The third term is used The predictive power and dc-link voltage offset at (k + 2)th sampling period can be obtained to minimize the unbalanced dc-link capacitor voltages. from (k + 1)th sampling instant and shown as, The predictive power and dc-link voltage offset at (k + 2)th sampling period can be obtained from e β   uα ( k +1) − eα − Riα ( k +1)   P( k +1)  2) shown (k + 1)th sampling instant and 3T  eas, α  P( k + = + (14)    − eα  uβ ( k +1) − e β − Riβ ( k +1)  Q #Q( k + 2)" 2 L  e β #" " #( k +1) "  # 3T eα e β uα (k + 1) − eα − Riα (k + 1) P ( k + 2) P ( k + 1) = + (14)  T T 2L e β −eα Q ( k + 2) u β (ks + 1) − e βs − Ri β (k + 1) Q ( k + 1) Δ + Δ + + + U k U k i k i k leg a fault 2 = 1 ( ) ( ) ( ) ( )  dc dc C a C a    (k + 2) = ∆Udc (k ) +TTsCs i a (k ) +TsTCs i a (k + 1) leg a f ault  ∆U  ( k ) + ib ( k ) + ib ( k + 1) leg b fault   ΔdcU dc ( k + 2 ) =ΔU dc (15)  dc (k + 2) = ∆Udc (k ) +CTCs ib (k ) +CTCs ib (k + 1) leg b f ault ∆U (15)  TsTs TsTs    ∆U U dcdc((k )) ++ Cicic((kk))++ Cicic((kk++1)1)leg legc cfault f ault ) =Δ∆U  ΔdcU(dck (+k 2+)2= C C  The MPDPC structure of fault-tolerant BVSC is shown in Figure 5. The grid voltage and current ea , The MPDPC structure of fault-tolerant BVSC is shown in Figure 5. The grid voltage and current eb , ec , ia , ib , ic can be acquired by signal sampling circuits. After Clark transformation, eα , eβ , iα , iβ can ea, eb, ec, ia, ib, ic can be acquired by signal sampling circuits. After Clark transformation, eα, eβ, iα, iβ can be obtained. Predictive functions (14) and (15) calculate the power and dc-link offset voltage predictive be obtained. Predictive functions (14) and (15) calculate the power and dc-link offset voltage values P(k + 2), Q(k + 2), ∆Udc (k + 2). The voltage vectors can be evaluated by the cost function (13). predictive values P(k + 2), Q(k + 2), ΔUdc(k + 2). The voltage vectors can be evaluated by the cost Then, the switching states, which minimizes the cost function, are selected and applied at tk+1 instant function (13). Then, the switching states, which minimizes the cost function, are selected and applied to achieve direct power control. The flow chart of the MPDPC algorithm is shown in Figure 6, where at tk+1 instant to achieve direct power control. The flow chart of the MPDPC algorithm is shown in m, n, j are variable parameters and gj is the calculation result of the cost function. Figure 6, where m, n, j are variable parameters and gj is the calculation result of the cost function.

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LL DC DC bus bus

Udc1 Udc1 Udc2 Udc2

TT

ia R R ia iibb

AC AC grid grid

iicc U Udcdc

SSaa Sb SScc Sb switching states switching states

uuα uuβ P(k+2) α β P(k+2) predictive

cost cost function function (16) (16)

PPref ref

predictive function function

(17)(18) (17)(18)

Q(k+2) Q(k+2) ΔUdc(k+2) Q ref ΔUdc(k+2) Qref

iiα iiβα eeαβ abc/αβ eeaaeebbeecc eeβα abc/αβ β MPDPC MPDPC

Figure Model predictive direct power control BVSC with leg Figure 5. 5. Model predictive (MPDPC) structurefor forfault-tolerant fault-tolerant BVSC with Figure 5. Model predictivedirect directpower powercontrol control (MPDPC) (MPDPC) structure structure for fault-tolerant BVSC with legleg fault of phase a. fault of phase a. fault of phase a. sample and calculate sample and calculate e α、 e β、i α、 i β α、 i β e α、 e β、i m=1044, j=0 m=10 , j=0 apply optimal voltage vector V apply optimal voltage vector V (k+1)th instant predictive (k+1)th instant predictive function (7) (11) function (7) (11)

(k+2)th instant predictive (k+2)th instant predictive function (14) (15) function (14) (15)

j=j+1 j=j+1

cost function (13) cost function (13) if gj