Variable Switching Point Predictive Torque Control for ... - IEEE Xplore

Variable Switching Point Predictive Torque Control for the Four-Switch Three-Phase Inverter Georgios Patsakis∗, Petros Karamanakos∗, Peter Stolze† , Stefanos Manias∗ , Ralph Kennel†, and Toit Mouton‡ ∗ School

of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece Email: ∗ [email protected], [email protected], [email protected] † Institute for Electrical Drive Systems and Power Electronics, Technische Universität M¨ unchen, Munich, Germany Email: † [email protected], [email protected] ‡ Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch, South Africa Email: ‡ [email protected]

Abstract—This paper presents a model predictive control (MPC) strategy for the four-switch three-phase inverter (FSTPI) driving an induction machine (IM). A state-space model of the plant, that takes into account the capacitor voltage imbalance, is introduced. The objectives of the proposed MPC-based approach are to control the machine torque and stator flux and to minimize capacitor voltage fluctuations. Furthermore, in order to reduce the high torque and flux ripples, a variable switching time point approach is implemented. The algorithm and mathematical calculations for the determination of the switching time point are presented. Simulation data verify the performance of the presented control strategy.

I. I NTRODUCTION Nowadays, the three-phase voltage source inverter utilizing three legs and six power switches is dominant in industrial applications. However, a significant research effort has been made to reduce the cost of the inverter by reducing the number of switches. One of the most promising topologies in terms of performance, compared to the six-switch three-phase inverter (SSTPI), is the four-switch three-phase inverter (FSTPI) [1]. The FSTPI utilizes two legs and four power switches, while the third phase is connected to the dc-link midpoint. Fig. 1 shows the FSTPI driving an induction machine (IM). The FSTPI offers some advantages over the conventional SSTPI. These include, among others, the reduced hardware and thus cost, due to the lower number of switches, and the reduced switching losses. An additional benefit is the increased reliability since there are fewer switches that may suffer damage. Moreover, the decreased number of switches leads to fewer inverter states and therefore to algorithms with less calculation effort [1]–[4]. An interesting application of the FSTPI topology was also introduced in [5], where a standard SSTPI driving an IM suffers a fault in one inverter leg and the system is reconfigured to an FSTPI topology maintaining its performance. On the other hand, there are also disadvantages. Firstly, FSTPI has decreased voltage gain, therefore a higher dc-link voltage is required. One solution in the case of a low-voltage (LV) ac drive would be to reconfigure the machine windings from wye to delta connection. Secondly, there is an inevitable fluctuation of the capacitor voltages.

Regarding the FSTPI-IM drive, a thorough investigation was conducted in [1], but the existing assumption of equal capacitor voltages was considered unrealistic in later works. Schemes to compensate the voltage fluctuation using modulation-based strategies have been proposed [2], [3], followed by analytical approaches and methods to overcome it [4]. A direct torque control (DTC) [6] scheme was proposed in [7] and a modification was introduced in [8]. Despite the extensive research, some problems still remain unsolved. The presented need for higher dc voltage input leads to high torque and flux ripples in direct control approaches, such as DTC. One way to overcome this problem would be to decrease the sampling interval, but that would lead to higher hardware requirements and costs for the drive system, which are inconsistent with the cost minimization FSTPI is chosen to guarantee. Given the fact that a FSTPI topology is more suitable for low-power applications, where torque and flux quality are of utmost importance, a way to reduce torque and flux ripples without increasing the sampling frequency and within reasonable switching frequency is necessary. Over the last decade, model predictive control (MPC) [9] has been gaining more popularity due to its numerous advantages, such as fast dynamics, inherent robustness, and explicit inclusion of constraints. MPC-based approaches have proven to be promising candidates for controlling electric drives [10]– [17], as well. One very famous approach for LV ac drives targeting on flux and torque control is predictive torque control (PTC) [18], [19]. In order to take advantage of the fast and flexible control of torque and stator flux that PTC offers, a PTC approach

C1 S1

S2 is,acb

iC1 Vdc

n

IM iC2 C2 S3

S4

Fig. 1: Four-switch three-phase inverter driving an IM.

for the FSTPI is proposed. Since even within one sampling interval torque and flux ripple could become worse, the fact that PTC penalizes the predicted torque and flux deviations for every switching possibility is expected to reduce the ripple. An additional advantage of PTC for the FSTPI is that by penalizing the capacitor voltage imbalance in the optimization procedure, the voltage fluctuation problem can be directly restrained. In order to further reduce torque and flux ripples, a variable switching point predictive torque control (VSP2 TC) [20], [21] approach for the FSTPI is presented; in [22] a similar approach, called variable switching point predictive current control (VSP2 CC), is introduced in order to directly control the current and minimize its ripple. In the aforementioned works the switching is allowed to take place in between the sampling interval so as to minimize the physical quantity of concern without the need of decreasing the sampling interval. A modification to the original algorithm is presented, aiming to reduce both torque and flux ripples. A trade-off factor between torque and flux ripple is introduced for the mathematical calculation of the switching point. The proposed method is tested via simulations and a comparative investigation between DTC, PTC and VSP2 TC is presented. This paper is organized as follows. In Section II the physical system, comprised of a FSTPI and an IM, is described. In Section III, the control objectives are mentioned and the control algorithm is presented. Simulation results under steady-state and transient conditions are provided in Section IV. Finally, the paper is summarized in Section V where conclusions are drawn. II. P HYSICAL S YSTEM A. Four-Switch Three-Phase Inverter The conduction state of the FSTPI (Fig. 1) is associated with the binary variables u1 to u4 . A binary “1” indicates a closed switch. Switches S1 − S3 and S2 − S4 are complementary. Let Vdc be the dc-link voltage, vC1 , vC2 the capacitor voltages and ia , ib , ic the three-phase load currents. The assumption of stiff capacitor voltages, equal to Vdc /2, has proven to be unfortunate in previous works [2], [3]. In this paper, the state-variable vn = vC1 − vC2 is introduced in order to model the capacitor voltage imbalance. Using circuit analysis on the premise that a constant dc-voltage Vdc is guaranteed and that the capacitors C1 , C2 are ideal and utilizing the relationships Vdc = vC1 + vC2 , iC1 − iC2 = ic , we derive: 2 dvn = ic . (1) dt C1 + C2 Moreover, with respect to the assumed symmetric, wye connected load neutral point, the phase voltages are given by [3] vC vC (2a) va = 1 (2u1 − u2 ) + 2 (2u1 − u2 − 1) 3 3 vC vC (2b) vb = 1 (2u2 − u1 ) + 2 (2u2 − u1 − 1) 3 3 vC vC vc = 1 (−u1 − u2 ) + 2 (2 − u1 − u2 ) . (2c) 3 3

jβ

01

11

α

10

00

Fig. 2: Four-switch three-phase inverter voltage vectors in the αβ plane, if vC1 = vC2 . TABLE I: Four-switch three-phase inverter vectors in the αβ plane S1

S2

vi

0

0

v1

1

0

v2

1

1

v3

0

1

v4

vα − 61 Vdc + 16 vn 1 V + 16 vn 2 dc 1 V + 16 vn 6 dc 1 − 2 Vdc + 16 vn

vβ √ 3 − 6 Vdc + 63 vn √ √ − 63 Vdc + 63 vn √ √ 3 Vdc + 63 vn 6 √ √ 3 Vdc + 63 vn 6 √

Subsequently, the Clarke transformation iαβ = Kiabc

(3a)

v αβ = Kv abc # " 2 1 −1/2 −1/2 K= √ √ 3 0 3/2 − 3/2

(3b) (3c)

is used to transform the three phase currents iabc and voltages v abc to their stationary αβ plane values iαβ and v αβ , respectively. Finally, exploiting the fact that ia + ib + ic = 0 so as to express the current ic appearing in (1) in the αβ plane, the acquired FSTPI model is 1 1 vα = vn + (4u1 − 2u2 − 1)Vdc (4a) 6√ 6√ 3 3 vn + (2u2 − 1)Vdc (4b) vβ = 6 6 √ dvn 1 3 =− iα − iβ . (4c) dt C1 + C2 C1 + C2 Note that (4c) describes the capacitor voltage imbalance. Moreover, (4a) and (4b) indicate that the inverter utilizes four active vectors in the αβ plane for the four permutations of the binary variables u1 , u2 . These vectors, which also depend on the voltage imbalance, are presented in Table I. For the case of vn = 0 the vectors are shown in the αβ plane (Fig. 2). Even in balanced conditions, the topology has an inherent asymmetry, utilizing two big and two small vectors. As a consequence, the output voltage that the topology can deliver is limited (without overmodulation) by a maximum amplitude of [2] 1 (5) Vsmax = √ min(vC1 , vC2 ) , 3 a relationship that illustrates the decreased voltage gain of the FSTPI as an important disadvantage, since it is at best half the corresponding voltage of a SSTPI.

PTC

B. Induction Machine

Te

The dynamics of the IM are modeled in the stator αβ reference frame. The complex stator current is , the complex stator flux ψ s = ψsα + jψsβ and the angular velocity of the rotor ωr are the state variables, and the continuous-time state equations are [23]: dψ s = v s − rs is dt τsr

1 dis + is = jωr τsr is + dt rsr J

(6a)

1 1 − jωr ψ s + us τr rsr (6b)

dωr = Te − Tℓ dt 3 Te = p(ψ s × is ) 2 q Ψs =

2 + ψ2 , ψsα sβ

(6c) (6d) (6e)

where the complex stator voltage us is in the stator αβ plane. Based on the model parameters, i.e. the stator rs and the rotor rr resistances and the stator ls , the rotor lr and the mutual lm inductances, the coefficients in (6) are τsr = σls /rsr , 2 σ = 1 − lm /(ls lr ), rsr = rs + rr ls /lr and τr = lr /rr . Variable J stands for the inertia, Te for the electromagnetic torque, Tℓ for the mechanical load torque, p is the number of pole pairs and Ψs is the stator flux magnitude. C. Complete Continuous- and Discrete-Time State-Space Model As (4) implies, the four-switch output voltage depends not only on the switching state of the inverter, but also on the capacitor voltages, which are significantly affected by the stator currents. This intricacy makes the FSTPI modeling more complicated than the SSTPI case. The state vector is selected to be: h iT x = isα isβ ψsα ψsβ vn . (7)

The switching states serve as the input u = [u1 u2 ]T and the output is y = [Te Ψs vn ]T . In a subsequent step (4), (6a) and (6b) are discretized using Euler forward approximation. The discrete-time state-space model of the machine is of the form

Te,ref VSP2 TC

k

k+1 k+2

Time (Sampling instants)

k+N

Fig. 3: Principle of VSP2 TC for torque ripple reduction.

namely the minimization of the voltage imbalance, previous works have shown that the torque and flux quality can be adequately satisfying despite the voltage imbalance [2], if the control algorithm is adjusted accordingly. However, it is desirable for the control algorithm to keep the capacitor voltages close to each other, because if it fails to achieve this the voltage gain will be further diminished as dictated by (5). One of the main problems of the four-switch three-phase topology is, as mentioned before, the high torque and flux ripples. Even in PTC, each switching state is applied for at least one sampling interval Ts , so an active state with high torque slope, for example, could lead to high torque ripple during that interval. If this state was applied for a time interval less than Ts , the torque ripple could be reduced. This can be achieved by allowing the switching to take place in between the sampling interval Ts . Fig. 3 shows a simplified example, where only two active states exist (one with high torque slope and one with low torque slope). In PTC the switching state can change only at specific instants (k, k + 1, . . . , k + N ). If a variable switching point is introduced and the switching state can change anytime, the torque ripple can be effectively reduced. The switching frequency however, as concluded also from Fig. 3, might increase. The sampling interval though remains the same; each IGBT can switch only once within every Ts . The same method can be applied in order to reduce the flux ripple, as well. B. Control Algorithm

A. Control Objectives and Motivation

Let k denote the k th step and Ts be the sampling interval. The change of switching state occurs at time tz , where kTs ≤ kTs + tz ≤ (k + 1)Ts , where the subscript z ∈ {1, 2, 3, 4} denotes the switching state applied at time (k) tz among the four alternatives. Let nint ∈ N denote the normalized time instant of the switching within the interval, (k) so that tz = nint Ts . Finally, u(t) represents the input, i.e. the switching state, applied at time instant t. The procedure for the calculation of the variable switching point comprises the following steps, executed at step k.

The control objective of the introduced approach is to regulate the torque and the stator flux magnitude to their reference values, as well as to equally share the dc-link voltage across the two capacitors. The importance of the first objective has already been clarified in the introduction of this work (see Section I). With regards to the second control objective,

Step 1: At step k, the switching state calculated at the (k−1) previous sampling interval u(k − 1 + nint ) is applied. According to (8b), the electromagnetic torque Te (k) and the stator flux magnitude Ψs (k) at instant kTs are estimated from measurements of the stator currents, rotor speed and capacitor voltage.

x(k + 1) = (I + ATs )x(k) + BTs u(k) ! y(k) = C x(k) ,

(8a) (8b)

where Ts is the sampling interval, I is the identity matrix, and matrices A, B and C (state-dependent) are calculated using elementary algebra. III. C ONTROL S CHEME

Te

constant within Ts , no matter what particular time in the interval Ts the corresponding switching state will be ap(k) plied, is made, i.e. mz,Te (k) = mz,Te (k + nint ) = mz,Te and (k) mz,Ψs (k) = mz,Ψs (k + nint ) = mz,Ψs .

(k)

Tez (k + nint ) mTe

d1,Te

mz,Te

Te,ref d2,Te

Te (k) Tez (k + 1)

(k + 1)Ts kTs kTs + tz Time (sampling instants) (a) The torque at kTs and kTs + tz is calculated based on the previous (k−1) switching state u(k − 1 + nint ), hence Tez (k + nkint ) = Te (k) + mTe tz . For every possible switching state z ∈ {1, 2, 3, 4} applied at instant kTs + tz , the torque at instant (k + 1)Ts is recalculated Tez (k + 1) = Te (k) + mTe tz + mz,Te (Ts − tz ). Ψs Ψs (k) Ψsz (k + 1)

(k)

Ψsz (k + nint ) Ψs,ref

mz,Ψs

m Ψs d1,Ψs

d2,Ψs

(k + 1)Ts kTs kTs + tz Time (sampling instants) (b) The flux at kTs and kTs + tz is calculated based on the previous (k−1) switching state u(k − 1 + nint ), hence Ψsz (k + nkint ) = Ψs (k) + mΨs tz . For every possible switching state z ∈ {1, 2, 3, 4} applied at instant kTs + tz , the flux at instant (k + 1)Ts is recalculated Ψsz (k + 1) = Ψs (k) + mΨs tz + mz,Ψs (Ts − tz ).

Step 4: In this step, the variable switching point is calculated for every switching alternative that changes the switching state. The goal is to reduce both torque and stator flux ripples. Fig. 4(a) shows the torque time evolution for an interval Ts , calculated according to the assumptions of steps 3, and 4 using elementary mathematical analysis. Minimizing the deviations d1,Te and d2,Te , i.e. d21,Te + d22,Te , with respect to tz , would give a value for the switching time that directly reduces the torque ripple. The same strategy can be applied in order to minimize the stator flux ripple. Fig. 4(b) shows the flux for an interval Ts and the corresponding deviations d1,Ψs , d2,Ψs . However, the switching time tz is the same for both torque and flux, so the following objective function is defined: Etot (tz ) = d21,Te (tz ) + d21,Te (tz ) + γ(d21,Ψs (tz ) + d22,Ψs (tz )) , (11) where γ > 0 is a trade-off factor between the minimization of torque ripple and flux ripple. Demanding dEtot =0 dtz we obtain the following value for tz : tz =

Fig. 4: Calculation of torque and flux deviations. (In this example the torque and flux evolutions are shown for one value of z.)

Step 2: Using (8a), and by applying the previous switching (k−1) state u(k) = u(k−1+nint ) for the entire sampling interval, the predicted values of the stator current and stator flux are calculated. Based on these values, the torque Te0 (k + 1) and stator flux magnitude Ψs0 (k + 1) at k + 1 are computed. Considering a constant torque slope mTe and flux slope mΨs , for one Ts , the following expressions describe the relationship between the torque at steps k and k + 1, and between the flux at steps k and k + 1: Te0 (k + 1) = Te (k) + mTe Ts

(9a)

Ψs0 (k + 1) = Ψs (k) + mΨs Ts .

(9b)

Step 3: The predicted state and output variables at step k + 1 are recomputed, assuming this time that the switching state applied at step k for a time interval Ts is one out of the four possible switching states of the four-switch inverter. The corresponding torque Tez and flux magnitude Ψsz slopes are calculated now according to the relationships: Tez (k + 1) = Te (k) + mz,Te (k)Ts

(10a)

Ψsz (k + 1) = Ψs (k) + mz,Ψs (k)Ts ,

(10b)

where mz,Te (k), mz,Ψs (k) are the slopes if the switching state z ∈ {1, 2, 3, 4} is applied at step k. At this point the assumption that the torque and flux slopes mz,Te , mz,Ψs of the four possible switching states will approximately remain

mz,Te (mz,Te − mTe ) Ts + ∆ (Te,ref − Te (k))(2mTe − mz,Te ) + + ∆ mz,Ψs (mz,Ψs − mΨs ) Ts + +γ ∆ (Ψs,ref − Ψs (k))(2mΨs − mz,Ψs ) +γ , ∆

(12)

(13)

with ∆ =2m2Te − 2mTe mz,Te + m2z,Te +

+ γ(2m2Ψs − 2mΨs mz,Ψs + m2z,Ψs ).

Note that for values of the formula tz < 0 or tz > Ts , tz is restricted to tz = 0 and tz = Ts , respectively. Step 5: By taking into account the variable switching point for each switching alternative, the predicted values of the state (k) variables are calculated at step k + nint by applying (8a) for a time interval tz instead of Ts . Following, the predictions of torque Te (k+1), stator flux magnitude Ψs (k+1) and capacitor voltage imbalance vn are computed for each switching state, in a similar manner as before using now the time interval Ts − tz instead of tz . Step 6: In a last step, an objective function is formulated and minimized in real-time. The chosen function is X ! J(k) = ||Te,ref − Te (k + ξ|k)||2 + ξ∈{nk int ,1}

vn (k + ξ|k)||2 +λ1 ||Ψs,ref − Ψs (k + ξ|k)||2 + λ2 ||¯ (14)

minimize

J(k)

subject to eq. (8) .

(15)

Function (14) is minimized every Ts . The switching state (k) u(k + nint ) that results to the minimum associated cost is (k) applied to the inverter at time instant (k + nint )Ts . At the next time-step, the whole procedure is repeated with the new measurements and estimates. C. Capacitor Voltage Imbalance Penalization In order to regulate the voltage imbalance, a corresponding term has to be included in the objective function. The state variable vn contains the information of the capacitor voltage deviation. However, due to the nature of this particular topology, instead of directly penalizing vn , a modification is proposed. As (1) shows, the derivative of vn is mathematically related to the phase current ic . The phase current is indirectly defined by the torque and flux requirements expressed in the objective function and is ideally sinusoidal. Therefore, vn also has a sinusoidal component that should be exempted from penalization, since its presence is inevitable and thus can influence the total cost of the objective function. The magnitude of this component can be reduced by increasing the capacitance of the dc-link up to the point that the associated cost remains reasonable. The goal to isolate and remove the fundamental sinusoidal component can be accomplished in many ways. A band-reject filter can be used, or alternatively the procedure described below can be adopted. As concluded from (1), the fundamental ac component of vn lags the phase current ic by 90◦ . If an ideal, three phase system is assumed, we can approach this behavior by a suitable linear combination of the threephase currents, √ i.e. ia − ib . More specifically, the relationship d(ia −ib ) = 3ωic can be easily verified and thus the ac dt component of vn , i.e. vñ can be isolated √ 2 3 (ia − ib ) , (16) vñ = 3ω(C1 + C2 ) where ω = ωr and ia , ib are calculated at step k of the algorithm. (16) also provides a way to estimate the necessary capacitance, if the maximum current demand from the ac-drive and the tolerance in capacitor voltage fluctuation are known at nominal conditions. Finally, we define v¯n = vn − vñ

(17)

and v¯n is the term penalized in the objective function. Fig. 5 shows the dc-link imbalance before and after the removal of the fundamental sinusoidal component.

150 100

vn [V]

where the weighting factors λ1 , λ2 > 0 set the trade-off between the electromagnetic torque error, the stator flux magnitude error and the capacitor voltage imbalance. Notice that the variable v¯n is used instead of vn to penalize the voltage imbalance. The reason for this choice will be explained in Section III-C. Subsequently, the following optimization problem is formulated:

50 0 −50

−100 −150

0

10

20

30

40

50

60

70

80

Time [ms] Fig. 5: vn before (blue dashed line) and after (black solid line) the removal of the fundamental sinusoidal component. TABLE II: Parameters of the simulation setup Parameter dc-link voltage Vdc capacitance C1 , C2 number of pole pairs p inertia J stator resistance rs rotor resistance rr stator inductance ls rotor inductance lr mutual inductance lm

Value 1300 V 1 mF 1 0.062 kg·m2 1.2 Ω 1Ω 175 mH 175 mH 170 mH

D. Delay Compensation In digital control systems a delay exists between the time instant that measurements are made and the time instant that these values are delivered to the controller, owing to the fact that the algorithm requires some time to be executed. For the simulations of VSP2 TC in the ac-drive system, a delay of one sample was taken into account. This means that one more prediction step was taken into consideration in the algorithm, in order to compensate for the delay. IV. P ERFORMANCE E VALUATION The proposed algorithm was tested via simulations in Matlab/Simulink. The simulation setup consists of a FSTPI driving a laboratory squirrel-cage IM. The drive operates at fundamental frequency f1 = 50 Hz. The torque reference is Te,ref = 18 Nm and the stator flux magnitude reference Ψs,ref = 0.71 Wb. The sampling interval is Ts = 66.7µs. A delay time Ts for the algorithm execution was taken into account. Table II shows the remaining parameters of the simulation setup. A. Steady-State Operation To begin with, the steady-state performance of the system was tested. The adequate dc-link voltage enables increased attention to the torque and flux quality. For comparison reasons, the improved DTC algorithm presented in [8] and the PTC algorithm [18], [19] were also implemented. In Fig. 6 the DTC results are presented. The switching frequency is around 4.1 kHz. For the implementation of PTC the following objective function was formulated in order to penalize the torque

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is [A]

is [A]

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0 −10

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0

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Time [ms] (a) Three-phase stator currents.

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Amplitude %

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(b) Harmonic spectrum of the stator currents. The THD is 4.91%.


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Te [Nm]

Te [Nm]

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0

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19 18 17 16

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Time [ms] (c) Electromagnetic torque.

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1

ψsβ [Wb]

ψsβ [Wb]

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ψsα [Wb]

−1 −1

−0.5

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ψsα [Wb]

(d) Flux in αβ plane.

(d) Flux in αβ plane.

Fig. 6: Simulation results of the direct torque control (DTC) for steady-state operation at nominal speed (f1 = 50 Hz—fsw ≈ 4.1 kHz).

Fig. 7: Simulation results of the predictive torque control (PTC) for steadystate operation at nominal speed (f1 = 50 Hz—fsw ≈ 4.1 kHz).

and flux deviations and the capacitor voltage fluctuation:

tortion (THD) than DTC, as well (THDDTC = 4.91% whereas J(k) = ||Te,ref − Te (k + 1)||2 + λ1 ||Ψs,ref − Ψs (k + 1)||2 + THDPTC = 3.53%, see Figs. 6(b) and 7(b), respectively). Finally the algorithm for VSP2 TC described in the previous + λ2 ||¯ vn (k + 1)||2 (18) section was implemented. The results are shown in Fig. 8. Torque (Fig. 8(c)) and flux (Fig. 8(d)) ripples are significantly Fig. 7 presents the results of PTC. As can be seen, for the reduced and the current THD is lower (THDVSP2 TC = 2.66%, same switching frequency, i.e. fsw ≈ 4.1 kHz, the ripple of the Figs. 8(a) and 8(b)). It should be mentioned, though, that torque (Fig. 7(c)) and flux (Fig. 7(d)) that PTC delivers is re- the switching frequency is increased (5.2 kHz), something duced; the currents (Fig. 7(a)) are of lower total harmonic dis- expected from the analysis given in Section III. However, the

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Te [Nm]

is [A]

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6

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Time [ms] (a) Electromagnetic torque (magenta solid line) and its reference value (black dashed line).

5

0.78

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0.76

3

Ψs [Wb]

Amplitude %

14 12 10

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2 1

0.74 0.72 0.7 0.68

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Fig. 9: Simulation results with DTC for a step change in the electromagnetic torque reference at t = 15 ms.

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Ψs [Wb]

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ψsα [Wb] (d) Flux in αβ plane.

0.74 0.72 0.7 0.68 0.66

(VSP2 TC)

for Fig. 8: Simulation results of the proposed control strategy steady-state operation at nominal speed (f1 = 50 Hz—fsw ≈ 5.2 kHz).

sampling interval remains the same.

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Fig. 10: Simulation results with PTC for a step change in the electromagnetic torque reference at t = 15 ms.

B. Torque Step Change Response The performance of the control algorithms was also tested during transients; a step-down change in the torque reference was examined. The torque reference steps from Te,ref = 18 Nm to Te,ref = 10 Nm at time t ≈ 15 msec. The reaction of the controller in all three control methods is very fast, as depicted in Figs. 9, 10 and 11, where the response of the drive is shown for DTC, PTC and VSP2 TC, respectively. Such a behavior is

expected from DTC and PTC, since these control algorithms provide fast dynamics, but the simulation results prove that VSP2 TC as well preserves the fast responses of PTC. Moreover, as it can be observed, the flux remains unaffected by this change in the torque reference, see Figs. 9(b), 10(b) and 11(b).

22 20

Te [Nm]

18 16 14 12 10 8 6

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Time [ms] (a) Electromagnetic torque (magenta solid line) and its reference value (black dashed line). 0.78

Ψs [Wb]

0.76 0.74 0.72 0.7 0.68 0.66 0.64

0

5

10

15

20

25


30

35

40

Fig. 11: Simulation results with VSP2 TC for a step change in the electromagnetic torque reference at t = 15 ms.

V. C ONCLUSIONS

AND

F UTURE W ORK

In this paper a model predictive control (MPC) approach was adopted in order to drive an induction machine (IM) using a four-switch three-phase inverter (FSTPI). Predictive torque control (PTC) improves the performance of the topology in terms of torque, flux and output current quality compared to a direct torque control (DTC) approach. In order to further decrease torque and flux ripples, a modification to the original variable switching point predictive torque control (VSP2 TC) algorithm was introduced and tested. The results prove that the proposed method improves the performance of the system. There are many topics deriving from the present paper that require further research and are mostly associated with alleviating the aforementioned inherent disadvantageous characteristics of the topology. For example, the dc-link voltage can be reduced and the effect on the torque and flux quality can be presented. Moreover, stability and robustness issues can be examined. Finally, on an experimental basis, the drive system can be implemented to verify the simulations. R EFERENCES [1] H. W. van der Broeck and J. D. van Wyk, “A comparative investigation of a three-phase induction machine drive with a component minimized voltage-fed inverter under different control options,” IEEE Trans. Ind. Appl., vol. IA-20, no. 2, pp. 309–320, Mar./Apr. 1984. [2] F. Blaabjerg, D. O. Neacsu, and J. K. Pedersen, “Adaptive SVM to compensate dc-link voltage ripple for four-switch three-phase voltagesource inverters,” IEEE Trans. Power Electron., vol. 14, no. 4, pp. 743– 752, Jul. 1999. [3] M. B. R. Correa, C. B. Jacobina, E. R. C. da Silva, and A. M. N. Lima, “A general PWM strategy for four-switch three-phase inverters,” IEEE Trans. Power Electron., vol. 21, no. 6, pp. 1618–1627, Nov. 2006.

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