Predictive Torque Control of an Induction Machine

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Predictive Torque Control of an Induction Machine Fed by a Matrix Converter With Reactive Input Power Control Ren´e Vargas, Member, IEEE, Ulrich Ammann, Member, IEEE, Boris Hudoffsky, Jose Rodriguez, Senior Member, IEEE, and Patrick Wheeler, Member, IEEE

Abstract—This paper presents a new control method for a matrix-converter-based induction machine drive. A discrete model of the converter, motor, and input filter is used to predict the behavior of torque, flux, and input power to the drive. The switching state that optimizes the value of a quality function, used as the evaluation criterion, is selected and applied during the next discrete-time interval. Experimental results confirm that the proposed strategy gives high-quality control of the torque, flux, and power factor with a fast dynamic control response. The key implementation issues are analyzed in depth to give an overview of the realization aspects of the proposed algorithm. Index Terms—AC–AC power conversion, matrix converter (MC), motor drives, predictive control, torque and flux control.

I. INTRODUCTION HE MATRIX converter (MC) is a single-stage power converter, capable of feeding an m-phase load from an n-phase source without using energy storage components [1]. The MC represents an alternative to the back-to-back converter in applications where size and weight are important. The absence of large capacitors or inductances allows the MC to give a compact solution [2], [3]. Several modulation techniques have been developed for MCs. These can be classified into two main groups: scalar and space vector methods [4]–[7]. The higher number of switching states and the direct interaction between the source and load introduces a certain amount of complexity into the analysis and implementation of an MC-based induction motor drive [8]–[12]. Predictive control is a control theory that was developed at the end of the 1970s [13]. Variants of this type of control strategy, associated with modulation techniques, have been used for power

T

Manuscript received May 5, 2009; revised July 27, 2009 and October 28, 2009. Current version published May 7, 2010. This work was supported by the Chilean Research Fund CONICYT under Grant 1100404, by the Universidad T´ecnica Federico Santa Mar´ıa, and by the Institute of Power Electronics and Electrical Drives, Universit¨at Stuttgart. Recommended for publication by Associate Editor K.-B. Lee. R. Vargas is with ABB Switzerland, R&D Traction Converters, Austrasse, 5300 Turgi, Switzerland (e-mail: [email protected]). U. Ammann and B. Hudoffsky are with the Institute of Power Electronics and Electrical Drives, Universit¨at Stuttgart, Stuttgart 70550, Germany (e-mail: [email protected]; [email protected]). J. Rodriguez is with the Electronics Department, Universidad T´ecnica Federico Santa Mar´ıa, Valparaiso 2390123, Chile (e-mail: [email protected]). P. Wheeler is with the School of Electrical and Electronic Engineering, University of Nottingham, Nottingham, NG10 1NX, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2010.2040839

conversion and motor drive control [14]–[18]. The application of this family of nonlinear control techniques for torque and flux control in induction machines (IMs) has received attention from researchers due to the techniques’ qualities of fast dynamic torque response, low torque ripple, and reduced switching frequency [19]–[23]. Model-based predictive control (MPC) has been introduced for motor current control [24], [25] and implemented on a variety of converter topologies [26]–[32]. An alternative technique for controlling the torque and flux of an IM has also been investigated [33]. The method has been considered for MCs through simulation studies [34], [35]. Both approaches share a common element: a quality function, which is evaluated for every valid switching state of the converter based on predictions obtained from a model of the system. The objective of this paper is to develop and experimentally validate an MC-based IM drive control method using MPC. This method features fast dynamic response, low torque ripple, and reactive input power control. The simple approach is based on the evaluation of an objective function through a unified switching-state selection criteria. This use of quality functions allows further attributes to be added to the method [24], such as reduction of switching losses, common-mode voltage control, spectrum regulation, etc. The method does not require additional modulation stages and can utilize all the allowable space vectors generated by the MC, including the rotating vectors. II. POWER CIRCUIT AND BASIC CONCEPTS The MC is a single-stage power converter based on an array of controlled semiconductor switches. For a 3 × 3 scheme, a three-phase source feeds, through the MC, a three-phase load, as shown in Fig. 1. The input filter attenuates the high-frequency switching components in the input current. Three-phase variables are characterized as complex vectors by means of a 2-D representation. Throughout this paper, this representation is considered when vectors are used. From Fig. 1, the output, or load voltage space, vector is defined as vo =

2 (va + a · vb + a2 · vc ) 3

(1)

where a = ej /2π 3 , vx , with x ∈ {a, b, c}, are output phase voltages. A similar definition can be applied to obtain the input current vector ie , the source current vector is , the source voltage vector vs , and the load or output current vector io . The MC topology has nine bidirectional switches, each with a corresponding switching function Sxy , with x ∈ {u, v, w}

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VARGAS et al.: PREDICTIVE TORQUE CONTROL OF AN INDUCTION MACHINE FED BY A MATRIX CONVERTER

Fig. 1.

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Power circuit of the system.

and y ∈ {a, b, c}, as shown in Fig. 1. Considering that the load should not be in an open circuit, due to its inductive nature, and that phases of the source should not be short circuited, switching functions should fulfil, at all times, the following equation: Su y + Sv y + Sw y = 1

∀ y ∈ {a, b, c}

Fig. 2.

Output voltage space vectors generated by the MC.

Fig. 3.

Block diagram of the control strategy.

(2)

where Sxy = 0 represents switch xy open and Sxy = 1 represents switch xy closed. This restriction allows the topology to have 27 valid switching states and implies the requirement for a commutation strategy in the switching process [8]. These 27 switching states can be classified into three groups according to the kind of output voltage and input current vector that each switching state generates, which are as follows. 1) All three output phases are connected to the same input phase. Space vectors from this group of switching states have zero amplitude. 2) Two output phases connected to a common input phase, and the third output phase connected to a different input phase. This group generates stationary space vectors with varying amplitude and fixed direction. 3) Each output phase is connected to a different input phase. Space vectors have a constant amplitude, but its angle varies at the supply angular frequency. The trajectories of the output voltage space vectors, assuming balanced three-phase input voltages of 230 Vrm s per phase, are shown in Fig. 2. Vectors from group 1) can be identified by the symbol , while groups 2) and 3) are represented by ◦ and
, respectively. Most control techniques (including most space vector modulation (SVM) based methods and direct torque control (DTC) strategies [4], [6], [10]) use only zero and stationary vectors. The method proposed in this paper considers all valid switching states, including the rotating vectors that have the benefit of producing lower common-mode voltage.

is performed using a quality function minimization technique. The quality function g represents the evaluation criteria in order to select the best switching state for the next sampling interval. For the computation of g, the input current vector is , the electric torque Te , and the stator flux ψs on the next sampling interval are predicted, assuming the application of each valid switching state, by means of a mathematical model of the input filter and IM. These predicted values are indicated by the superscript “p” and are compared with their reference values denoted by the superscript “∗” within g. A propotional–integral (PI) controller is used to generate the reference torque Te∗ to the predictive algorithm. It also compensates deviations of the prediction model caused by possible variations of the parameters of the IM.

III. PREDICTIVE TORQUE CONTROL Predictive torque control (PTC) consists of choosing, at fixed sampling intervals, one of the 27 feasible switching states of the MC. A diagram of the PTC strategy is shown in Fig. 3. The selection of the switching state for the following time interval

A. Evaluation Criterion: Quality Function g The quality function represents the evaluation criteria used to decide which switching state is the best to apply. The function is composed of the absolute error of the predicted torque, the

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absolute error of the predicted flux magnitude, and the absolute error of the predicted reactive input power, resulting in g=

λT |Te∗

−

Tep |

+

λψ |ψs∗ 

−

ψsp |

+

λQ |Q∗in

−

Qpin |

IM can be presented as dψs (6) dt dψr vr = Rr ir + − jpωψr = 0 (7) dt where Rs and Rr are the stator and rotor resistances, ψs and ψr are the stator and rotor fluxes, ω is the mechanical rotor speed, and p is the number of pole pairs of the IM. The stator and rotor fluxes are related to the stator and rotor currents by vo = R s i o +

(3)

where λT , λψ , and λQ are weighting factors that handle the relationship between reactive input power, torque, and flux conditions. The units of each weighting factor are defined to maintain g as a magnitude without a physic interpretation. In this sense, λT is measured in Newton·meter inverse, λψ in webers inverse, and λQ in voltampere inverse. In practice, only two of these factors are required to adjust the method. The third can be kept constant. The reason is that the weighting factors must handle the relative importance of the errors in g and this task can be accomplished simply by changing two ratio magnitudes. For example, λT can be set as constant, using λψ and λQ to change the values of the weighting ratios λψ /λT and λQ /λT . Through the development of this study, the value of λT is kept constant at λT = 1(N·m)−1 . The values of λψ and λQ are modified to evaluate the performance of the method and find the optimum set of factors. The quality function g must be calculated for each of the 27 feasible switching states. The state that generates the optimum value, in this case a minimum, will be chosen and applied during the next sampling period. The states that generate the higher predictions of torque error, flux error, and reactive input power error will be penalized with higher values of g, and thus, will not be selected. In this sense, the technique assigns costs to the objectives reflected in g, weighted by λT , λQ , and λψ , and then chooses the switching state that presents the lowest cost. B. Models Used to Obtain Predictions 1) MC Model: The equation that relates load or output voltages with input voltages of the MC can be expressed as      va Su a Sv a Sw a veu  vb  =  Su b Sv b Sw b   vev  . (4) vc Su c Sv c Sw c vew    T

Output voltages applied to the load can then be considered as dependant variables of the switching functions, reflected in matrix T , and the input voltages. The relation between input currents and output or load currents is expressed as      S u a Su b Su c ia ieu  iev  =  Sv a Sv b Sv c   ib  . (5) iew Sw a Sw b Sw c ic    TT

Input currents depend on output currents and the switching state of the MC, due to the inductive nature of the load. 2) Load Model: In this section, a mathematical discretetime model is derived to predict the behavior of the system under a given switching state, based on well-known [19]–[23], [33]–[35] dynamic equations for an IM. The stator and rotor voltage equations in fixed stator coordinates for a squirrel-cage

ψs = Ls io + Lm ir

(8)

ψr = Lm io + Lr ir

(9)

where Ls , Lr , and Lm are the self- and mutual inductances. Finally, the electric torque produced by the machine can be obtained by Te = =

Lm 3 p Im{ψ¯r ψs } 2 Lr Ls − L2m Lm 3 p (ψr α ψsβ − ψr β ψsα ) 2 Lr Ls − L2m

(10)

where ψ¯r is the complex conjugate of vector ψr , and the subscripts α and β represent real and imaginary components of the associated vector. Equations (6) and (7) can be rewritten, solving the stator and rotor currents in terms of the stator and rotor fluxes from (8) and (9), as −Rs Lr Rs Lm dψs = ψs + ψ r + vo dt Ls Lr − L2m Ls Lr − L2m

(11)

dψr Rr Lm Rr Ls = ψs − ψr − jpωψr . (12) 2 dt Ls Lr − Lm Ls Lr − L2m The next step is to define a discrete-time model based on these continuous-time equations. Using a forward Euler approximation [19], the following discrete equations are computed from (11) and (12) as

Ts Rs Lr ψs (k) ψsp (k + 1) = 1 − Ls Lr − L2m + ψrp (k + 1) =

Ts Rs Lm ψr (k) + Ts vo (k) Ls Lr − L2m

(13)

Ts Rr Lm ψs (k) Ls Lr − L2m

Ts Rr Ls + 1− ψr (k) − jpTs ω(k)ψr (k) Ls Lr − L2m (14)

where Ts is the sampling period. An alternative approach is to compute the state-space representation of (11) and (12), and apply a discretization process similar to the one presented for the input filter in the next section [31]–[33]. This approach must deal with the fact that ω in (14) changes in time, causing the state-space representation to be a linear time-varying system. Although under certain assumptions, this method produces a more accurate representation [33], the time-varying nature of the

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VARGAS et al.: PREDICTIVE TORQUE CONTROL OF AN INDUCTION MACHINE FED BY A MATRIX CONVERTER

system implies more computational requirements. This forces the control platform to use a higher sampling time, which could produce a deterioration in the performance of the drive [24]. Equations (10), (13), and (14) are used by the proposed method to predict the stator flux and the electric torque produced by the IM during the next sampling interval if a certain voltage vector vo (k) is applied from the MC. 3) Input Filter Model: The input filter’s dynamic can be described by the following continuous-time equations [27], [31] as dis + ve (15) vs = Rf is + Lf dt dve (16) is = ie + Cf dt where Lf and Rf are the inductance and resistance from the line and filter, and Cf is the filter capacitance. This continuous-time system can be rewritten as   1  −1  0 0 Cf    Cf    u(t)  x(t) ˙ = (17)   −1 −R  x(t) +  1 f 0 Lf L Lf     f   Bc

Ac

with

x(t) =

ve



is

u(t) =

and

vs ie

 .

(18)

A discrete state-space model can be derived when a zero-order hold input is applied to a continuous-time system described in state space form as (17). Considering a sampling period, Ts , the discrete-time system derived from (17) is x(k + 1) = Aq x(k) + Bq u(k) with

 Ac Ts

Aq = e

and

Ts

Bq =

(19)

eA c (T s −τ ) Bc dτ. (20)

0

Discrete-time variables will match continuous-time variables at the sampling instants [31]. To predict the grid’s current, it is possible to solve is (k + 1) from (19) as ips (k + 1) = Aq (2, 1)ve (k) + Aq (2, 2)is (k) + Bq (2, 1)vs (k) + Bq (2, 2)ie (k)

(21)

where Λ(m, n) is the (m, n) element of matrix Λ. To analyze the resulting effect on the reactive input power, it is necessary to consider the instantaneous power theory [36]. The instantaneous reactive input power can be predicted, based on predictions of the input current, as Qp (k + 1) = Im{vs (k + 1)¯ips (k + 1)} = vsβ (k + 1)ipsα (k + 1) − vsα (k + 1)ipsβ (k + 1). (22) Line voltages are low-frequency signals. Based on this, the method considers vs (k + 1) ≈ vs (k).

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Fig. 4. Time diagram of the calculations performed during the sampling interval between time k and k + 1.

C. Time Frame for the Calculations Equations (10), (13), (14), and (22) are used to obtain predictions of torque, flux, and reactive power for each of the 27 valid switching combinations. The cost of a switching state is evaluated applying these predictions in (3). Nevertheless, it is important to evaluate the predictions at the appropriate sampling instant, taking into consideration the computational requirements of a real implementation of the method. An explanation of the tasks that the control processor must perform is presented in Fig. 4. The switching state that was selected at the preceding sampling interval will be applied between time (k) and (k + 1) (shadowed sampling interval in Fig. 4). The algorithm begins by acquiring measurements of the system variables. Then, the model is used to update, at time (k + 1), the variables that will be required for the prediction. This process implies a one-step-forward estimation using the equations described in Section III-B, considering that the switching state that was previously selected at the preceding sampling interval is held during the present interval (k) to (k + 1). Consequently, this process implies just one application or use of the prediction model, i.e., only for the previously selected switching state, as indicated in Fig. 4 as actualization at time (k + 1). This step, previous to the exhaustive search algorithm based on predictions, is usually not explained when presenting similar MPC strategies. However, this delay affects the performance of the control technique and has been considered in previous reports [17], [28], [29]. After updating the variables at time (k + 1), it is possible to start with the predictive algorithm to analyze the effect that every feasible switching state would produce if applied during the next sampling interval. It is important to notice that this process, presented in Fig 4 as predictions for each switching state, is also computed within the shadowed sampling interval (k) to (k + 1). It implies calculating predictions for each of the 27 switching states, in order to evaluate these predictions by means of g and select the best state for the period (k + 1) to (k + 2). The effect of the switching state under evaluation, to be applied at time (k + 1), must be assessed at time (k + 2), since this is when its effects will be noticeable in the system. A flow diagram of the steps required to implement the strategy on a digital control platform is presented in Fig. 5. It is worth noting that the prediction equation used from time (k + 1) to (k + 2) are the same as equations used from time (k) to (k + 1) (see Section III-B) shifting one-step forward in time. D. Adjusting the Weighting Factors The PTC method presented in this paper uses three weighting factors that the designer must adjust: λT or torque control

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TABLE 1 PARAMETERS OF THE MC SETUP AND CONTROL METHOD

Fig. 5.

Flow diagram of the implemented code on the control platform.

parameter, λψ or flux control parameter, and λQ or reactive input power control parameter. These weighting factors handle the cost assigned to each specific objective within the quality function g. As mentioned, only two of these factors are required to adjust the method, the third can be kept constant. In this sense, the value of λT is set constant at λT = 1 (N·m)−1 . The proposed control technique can be classified as finitestate model predictive control (FS-MPC). FS-MPC has emerged as a promising control tool for power converters and drives [24]. One of the major advantages is the possibility to control several system variables with a single control law, by including them with appropriate weighting factors, as PTC does with torque, flux, and reactive input power (it would be possible to add further variables in order to improve the performance of the drive in specific topics). However, these coefficients are determined empirically. The design problem is to select appropriate values for these weighting factors. The topic has been treated in [24] and [37] and the reported method implies testing feasible values until reaching to an appropriate result. The method exposed in these publications gives guidelines to starting the search process and sets boundaries for the exploration. In the specific case of PTC, an appropriate starting point for the search process is to consider the three objectives with an equal relative importance. This implies considering the nominal values (indicated by a subscript N ) of the variables that are being balanced in the equation. In this sense λT TN = λψ ψs N = λQ (is N vs N ) = K

(23)

where K is an arbitrary constant. As an example, for K = 73 and considering the parameters of the experimental setup (see Table I), the result is λT = 1 (N·m)−1 , λψ = 72 Wb−1 , and λQ = 0.02 (VA)−1 . The value of K was chosen in order to

obtain the required value of λT = 1 (N·m)−1 . Note that this is only a suggested starting point for a search process over λψ and λQ for finding the optimum values [37]. A series of 6161 simulations were carried out in order to analyze the performance of the method under a wide range of parameter values and to define the optimum set of weighting factors. This exhaustive search process was performed setting λT = 1(N·m)−1 and sweeping λQ in 101 equidistant values between 0 and 0.025 (VA)−1 , and λψ in 61 equidistant intervals between 1 and 301 Wb−1 . Each value of λQ is tested with each value of λψ , resulting in the total of 6161 simulations. The total harmonic distortion (THD) of the input current is , and the standard deviation of the flux and torque signals were considered as merit functions, defined to evaluate the performance of the system working under each set of parameters. The following definition for the standard deviation of a data vector x was considered:   n n  1  1 (xi − x ¯)2 where x ¯= xi (24) σx =  n − 1 i=1 n i=1 and n is the number of elements in the sample. The result of the exhaustive search process, in terms of the standard deviation of the flux and torque, and the THD of the input current, is shown in Fig. 6. In the graphics, a logarithmic function is applied to the merit functions, to help in the visualization of the plotted information. The darker areas represent the regions where the merit functions reach low values, indicating a good performance of the system in terms of the specific objective (flux, torque, and input current, respectively). These regions are highlighted in the images by a white scattered line, defined as acceptable regions on each case. It is possible to observe, based on Fig. 6 (left), that there is a minimum acceptable value of λψ , and that for greater values of λψ , higher values of λQ are admissible. From Fig. 6 (center),

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Fig. 6. Simulation results obtained from an exhaustive search process over λQ and λψ (λT = 1 (N·m)−1 constant). (Left) Standard deviation of ψ s . (Center) Standard deviation of T e . (Right) is THD (all three merit functions in a logarithmic scale).

Evidently, the selection of λψ affects the allowable range of λQ , and the center of the optimum area is close to λψ = 180 Wb−1 and λQ = 0.003 (VA)−1 . This suggested set of optimum values (based on simulations) are far from the starting point proposed based by (23), but at least this starting point lies in the area where the system can operate. Equation (23) suggests only a set of parameters to begin with the search algorithm. The suggested set of optimum values and acceptable regions, obtained from the search process based on simulations, will still have to undertake an experimental validation in order to find the optimum set of parameters. This task is completed in the next sections.

Fig. 7. Superposition of the three acceptable regions in order to find the area of suitable set of parameters.

it can be inferred that for high values of λψ and λQ , the torque performance is penalize and a worse performance of σTe is obtained. This make sense, since λT is kept constant, therefore the relative importance of the torque error is reduced in g as the other weighting factor increase. Also from Fig. 6 (center), it can be observed that there is a minimum value of λψ from which an appropriate torque response can be obtained, in concordance with the physics fact that the presence of flux is required in order to generate electric torque from the IM. Finally, the input current THD presents an acceptable region near the center of the analyzed range for λψ and for a narrow set of values for λQ [see Fig. 6 (right)]. The explanation for that phenomenon lies in the fact that the method requires an appropriate output performance in order to achieve an acceptable control of the input variables. To find the best set of weighting factors, it is necessary to consider the three objectives analyzed in Fig. 6. This task can be graphically shown by superposing the three curves that indicate acceptable regions on each case. The result is shown in Fig. 7. The optimum region, according to this procedure, is a range of values of λψ between 80 and 250 Wb−1 . For λQ , the set of suggested values lies between 0.002 and 0.005 (VA)−1 .

IV. EXPERIMENTAL RESULTS The objective of this section is to present the performance of the method on an experimental platform, analyzing the main implementation issues. Throughout this section, the procedure used to adjust the method’s parameters is analyzed and contrasted based on experimental results. Similar control strategies have been analyzed in simulations. An IM is controlled by predicting its torque and flux in [34]. No reactive input power control is considered, being equivalent to the presented method if λQ = 0. The reactive power control feature is added in [35], but only to work with unity power factor (PF). That means that the method is equivalent to the presented technique if Q∗in = 0, directly minimizing the absolute reactive input power [see (3)]. The addition of Q∗in allows the method to work with a PF different to the unity if required [31] (in most applications, it is desired to have cosφ = 1). Despite these differences, which can be overcome for certain values of λQ and Q∗in , and the load’s parameters, the presented results can be contrasted to simulation results included in [34] and [35]. Most of the conclusions over the performance of the method and the behavior of the MC drive can also be observed, and are independent of these differences. Simulations were used in this paper only for the exhaustive search process described in the previous section. The analysis included in this section is based

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Fig. 8. Experimental setup. (Upper) Converter. (Lower right) Machine. (Lower left) interface circuits.

solely on experimental results, as a solid basis to demonstrate the effectiveness of the method. A. Setup Description and Implementation Issues An 11 kW, 4-pole IM is connected to an MC with 50 A peak output current capability running at 400 V grid voltage. The MC consists of an arrangement of 18 insulated-gate bipolar transistors (IGBTs) (IXDN 55N120D1) in common emitter configuration, and additional filter and protection components. Control of the commutation and overcurrent protection is provided by an field programmable gate array (FPGA) circuit, following a current-controlled four-step commutation scheme. The dc load machine is fed from a line-commutated thyristor rectifier with an appropriate armature current control to apply torque within the range from −50 to 50 N·m, whereas the IM control system contains a speed controller, according to Fig. 3. The experimental setup converter, IM, and necessary interface boards are shown in Fig. 8. A powerful dSPACE 1103 rapid prototyping platform is used to implement the predictive control strategy. Its 1 GHz clock frequency allows for a sampling time of 14.5 µs for the predictive torque and reactive input power control. The superimposed speed control loop is handled at 290 µs sampling time, using a standard PI controller. The predictive torque and reactive power control algorithm is implemented in C code, following the two-step approach previously explained to compensate the calculation delay. Field weakening is easily achieved by reducing the flux reference value for a speed rage over 1330 r/min. A complete list of parameters of the system and control strategy are included in Table I. A key aspect of the experimental implementation of the control method is the flux observer (see Fig. 3). Since the proposed method is based on the calculation of the actual flux in the IM, this estimation has to be as accurate and reliable as possible. Equations (13) and (14) are used to predict the future behavior of the IM, but there might be differences in the result of the calculation and the real value, depending on the knowledge

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Fig. 9.

Block diagram of the electrical part of an IM (discrete-time equations).

of the machines parameters. To reduce the error, a simple P (proportional) controller is applied in the predictive formulas, specifically in the stator voltage equation, to implement the flux observer. In this flux observer, the P controller corrects the predicted value based on the measurement of the stator current. By correcting the calculated stator current, both the stator and rotor flux estimations are corrected to be closer to their actual values. The procedure is explained in the following paragraph and is based on the previously presented IM model. As stated in Section III-B.2, (13) and (14) are discrete-time versions of the stator and rotor voltage equations, respectively, reached after solving the stator and rotor currents in terms of the stator and rotor fluxes. It is possible to express (13) to explicitly show the stator current as ψs (k + 1) = ψs (k) − Ts Rs 



  Lr Lm   × ψ (k) − ψ (k)  + Ts vo (k). s r Ls Lr − L2m  Ls Lr − L2m     i o (k )

(25) By means of (13) and (14), it is possible to build an equivalent block diagram for the electrical part of the IM, as shown in Fig. 9, with κ = 1/Ls Lr − L2m . As presented in the figure, (13) and (14) allow access to the stator and rotor flux from a onestep prediction. Note that this model includes both the stator and rotor voltage equations. If the machines parameters were exactly known, no more effort would be necessary to match the observed to the real values, and these equations could be used as they are for the flux observer. Since there is always a mismatch in the parameters—as well as variations caused by temperature, etc.—the flux observing equations can be improved by adding an additional input based on measurements to correct the result considering the real behavior of the machine.

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The effect of errors and uncertainties in the load’s parameters using predictive control is treated in [24] and [26]. It is worth mentioning that previous knowledge of the load’s parameters is a requirement on any control method in order to tune linear controllers and select the modulation frequency. The implementation described for the flux observer helps the method to mitigate the effect of these uncertainties in the predictions. The feasibility of the proposed control algorithm is backed up with experimental results. Through the subsequent experimental tests, the torque weighting factor is set constant at λT = 1 (N·m)−1 (as stated previously), and λQ and λψ are modified to find the best set of parameters. B. Adjusting the Flux Control Parameter

Fig. 10.

Block diagram of the flux observer.

The flux observer, utilized in the experimental validation, uses both the stator and rotor voltage equations (13) and (14). The stator equation (13), or more explicit (25), is modified to use the measured stator current and to include a P controller to correct the estimated stator current calculated by the model based on the observed stator and rotor fluxes [38]. The implemented stator equation is expressed as ψsp (k + 1) = ψsp (k) − Ts Rs io (k) + P Ts [io (k) − ipo (k)] +Ts vo (k)   

(26)

Correction Term

where io (k) is the measured stator current and ipo (k) is the stator current calculated from the estimated fluxes, as a result from the observer equations. Note that the correction term is proportional to the error between the measured and the estimated stator currents. The concept of adding an error compensator to the observer model, comparing the model current ipo (k) and the machine current io (k) to generate a correction term in the estimation, has been studied and analyzed in the literature [39]. The rotor equation is used without modifications in the model. The resulting block diagram of the flux observer is shown in Fig. 10. From the figure, it is possible to observe that the fluxes’ estimations are obtained based on both the stator and rotor voltage equations, using the machine equations presented in Fig. 9. The model’s calculation is enhanced by including information obtained from measuring the stator current of the IM, as stated in (26). The value of the electric torque Tep (k + 1) is obtained from (10), using ψsp (k + 1) and ψrp (k + 1) from the flux observer. For the second prediction step, there is no additional correction term, i.e., (13) and (14) are directly used to get the values of ψsp (k + 2) and ψrp (k + 2) based on ψsp (k + 1), ψrp (k + 1), and vo (k + 1), or voltage vector under evaluation. Once more, Tep (k + 2) is determined by applying ψsp (k + 2) and ψrp (k + 2) into (10). For more details regarding flux estimation, see [38] and [39].

The flux control parameter λψ is vital for an appropriate performance of the drive. On the other hand, the λQ factor mostly affects the input power performance. Thus, the system can work with λQ = 0 (VA)−1 maintaining an appropriate torque and flux control, but diminishing the interaction with the mains (see Figs. 6 and 7). Three experimental results for the proposed method under a speed reversal are presented in Fig. 11, each one with a representative value of λψ . In the first case, λψ = 50 Wb−1 , close to the lowest value admitted by the experimental setup. The method is sensitive to low values of λψ —to maintain an appropriate flux control is critical in every control method for IMs—and the proposed technique is unable to control torque or flux with lower values. This fact is noticeable during the speed reversal, with an increased ripple on the flux signal. On the other hand, with λψ = 600 Wb−1 , the priority for the method is placed on controlling flux, assigning a lower relative cost to the torque objective. The result is an excellent flux signal, but a considerably deteriorated torque response. An adequate tradeoff is reached with λψ = 300 Wb−1 , in which flux and torque are properly controlled. This supposes a selected value slightly above the suggested range based on simulations (see Fig. 7), but still close to the upper limit of 250 Wb−1 . C. Adding Reactive Input Power Control The MC must not only act as drive for the machine, but it must also be able to control input currents in order to regulate the input PF. This objective is achieved by PTC through the last term in g(3), weighted by λQ . The λQ parameter must balance the importance of reactive input power control with other objectives reflected in g, i.e., torque and flux control. By assigning cost to the reactive input power error in the quality function g, switching states that produce higher differences between Q∗in and Qpin will be penalized and not selected to be applied during the next sampling interval. The requirement in most cases is to work with unity PF. For this reason, the reference reactive power is zero, i.e., Q∗in = 0. Hence, the method will directly try to minimize the predicted reactive power Qpin by synchronizing input currents with input voltages, achieving a close to unity PF. The use of a value of Q∗in different to zero enables the method to work with inductive or capacitive PF [31].

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Fig. 11. Speed reversal with a value of λψ lower than the optimum, λψ = 50 Wb−1 , with the selected value, λψ = 600 Wb−1 : speed, electric and reference torque and flux (real component).

λψ = 300 Wb−1 , and with a higher value,

Fig. 12. Steady-state performance of the drive without reactive input power control λQ = 0 (VA)−1 , with control using the selected value λQ = 0.003 (VA)−1 , and with a higher value λQ = 0.007 (VA)−1 : electric and reference torque, input current, and voltage’s phase and reactive input power.

Setting the appropriate value of λQ enables the method for a correct interaction with the mains. As for λψ , the analysis is based on testing the performance of the system with several values, to evaluate the conclusions obtained from the simulated search process. Three representative cases are presented in Fig. 12. The system is tested in steady state at 1000 r/min with a 15-N·m load. To study the performance of the method on each case, the standard deviation of the torque signal σT and the measured PF are included in the figure. The first case [see Fig. 12 (left)] presents the performance of the converter with λQ = 0 (VA)−1 , i.e., assigning no cost to reactive input power. Evidently, the efforts of the method are focused on maintaining torque and flux control, reaching a low standard deviation of Te , but disregarding the input performance of the MC. This is reflected on a low PF, a significant ripple in the input current, and a considerable amount of reactive power delivered to the mains. The opposite occurs with λQ = 0.007 (VA)−1 [see Fig. 12 (right)]. In this case, a high importance is placed on controlling reactive power. As a result, input currents present lower distortion and are synchronized with the respective input phase voltages, generating almost no reactive input power and close to unity PF. The problem is that the machine’s torque is not properly controlled [see Fig. 12 (right)] due to the exaggerated importance given to reactive power control, reaching the highest

deviation of Te with σT = 0.77 N·m. An appropriate balance is reached with λQ = 0.003(VA)−1 in Fig. 12 (center). Both torque and reactive input power are suitably controlled, with σT = 0.3 N·m and a close to unity PF. The selected value of λQ = 0.003 (VA)−1 agrees with the optimum value delivered by the search process based on simulations. On the other hand, it was expected that the standard deviation of Te resulted higher on the experimental tests than in simulations, due to the ideal conditions assumed in the last case. For instance, in simulations, the load torque is considered to be constant, while in the experimental setup, a dc load machine is fed from a line-commutated thyristor rectifier. Nevertheless, the validity of the information provided from simulations was corroborated with the experimental results in Figs. 11 and 12. The performance of the complete drive-control strategy is shown, during a speed reversal, in Figs. 13 and 14. Results presented in Fig. 13 were obtained considering only torque and flux control, λψ = 300 Wb−1 and λQ = 0 (VA)−1 , while in Fig. 14, reactive input power control is included, using the optimum set of weighting factors λψ = 300 Wb−1 and λQ = 0.003 (VA)−1 . Note that λT = 1 (N·m)−1 at all times. Both cases present similar output behavior, with a fast torque response, smooth frequency transition on the output current, and sinusoidal flux. In Fig. 14, PTC is controlling not only machine’s variables, but

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VARGAS et al.: PREDICTIVE TORQUE CONTROL OF AN INDUCTION MACHINE FED BY A MATRIX CONVERTER

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Fig. 13. Speed reversal without the term to control reactive input power λQ = 0 (VA)−1 : speed, electric and reference torque, output current, flux (real and imaginary components), input current, and voltage’s phase and reactive input power.

Fig. 14. Speed reversal including the term to control reactive input power λQ = 0.003 (VA)−1 (optimum set of weighting factors): speed, electric and reference torque, output current, flux (real and imaginary components), input current, and voltage’s phase and reactive input power.

also the input performance of the converter. This fact causes a tradeoff noticeable while approaching to final speed and reaching steady state in the torque signal, with a slight amount of ripple in Fig. 14. The main difference, as expected, can be observed in the input current and reactive input power. Fig. 13 presents a distorted current that redounds in a significant amount of reactive power to the mains. In contrast, a more sinusoidal input current—with an appropriate amount of ripple, expected for the topology—and in synchrony with the input phase voltage is obtained considering reactive power control in Fig. 14. As a result, the reactive input power is relatively low.

The behavior of input and output electric variables without and including reactive input power control is shown in Figs. 15 and 16. Both cases were measured in steady state at 1000 r/min and with a 25-N·m load. It can be observed how the addition of the term to control reactive input power causes the current to match in phase with the input voltage, considerably reducing its ripple and the reactive input power delivered to the grid by the converter. The input current distortion observed in Fig. 15 causes an alteration on the envelope waveform of the output voltage, vab , which corresponds to the maximum line-to-line voltage at the input capacitors (vex , where x = u, v, w, in Fig. 1). This slight distortion has no effect on the load current, as shown in

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Fig. 15. Input and output electric variables in steady state without reactive input power control, λQ = 0 (VA)−1 : output current and voltage, input current (voltage’s phase in dotted line), and reactive input power.

Fig. 16. Input and output electric variables in steady state including reactive input power control, λQ = 0.003 (VA)−1 (optimum set of weighting factors): output current and voltage, input current (voltage’s phase in dotted line), and reactive input power.

Figs. 15 and 16 for two reasons: the low-pass filtering nature of the load and the fact that the predictive method considers the instantaneous (updated) value of the input voltages, considering this variation in the strategy. In Figs. 17 and 18, the harmonic content of the input current and output voltage for λQ = 0 (VA)−1 and λQ = 0.003 (VA)−1 are shown. Figs. 17 and 18 represent the spectrum analysis of variables shown in Figs. 15 and 16 in the time domain, respectively. It is possible to observe a drastic reduction on the input current’s distortion by including the strategy to control reactive power. The THD of iu was reduced from 75.1% to 17.6%, also adding the correct phase to synchronize it with the input voltage, as shown in Fig. 16. A high amount of energy can  be seen at the input filter’s resonance frequency fr = 1/2π Lf Cf near 2 kHz. The method effectively avoids exciting this resonance in order to reduce the total distortion in Fig. 18. The output voltage presents, in both cases, a spread spectrum, with significant energy in a wide range of frequencies. This is an attribute of many nonlinear or nonmodulated control methods. For the presented predictive approach, a concentrated spectrum can be reached by applying the strategy described

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 6, JUNE 2010

Fig. 17. Spectrum analysis without the strategy to control reactive power, λQ = 0 (VA)−1 . (Top) Input current iu . (Bottom) Output voltage va b .

Fig. 18. Spectrum analysis including the strategy to control reactive power, λQ = 0.003 (VA)−1 (optimum set of weighting factors). (Top) Input current iu . (Bottom) Output voltage v a b .

in [28]. The MC’s switched output pattern and the method’s fixed time period for the switching state’s transitions is noticeable by a higher amplitude harmonic at the sampling frequency fs = 1/Ts = 69 kHz. Nevertheless, the average switching frequency per IGBT fsf is considerably lower. The highest reachable value is fsf = fs /6 = 11.5 kHz, assuming the maximum number of transitions as each Ts [27], [31]. In practice, the measured average switching frequency per IGBT was fsf = 6 kHz. A last test of the performance of the drive is shown in Fig. 19, applying a step change on the reference torque using the optimum set of weighting factors. The IM starts from standstill, delivering only magnetizing current from the MC. No torque is applied from the drive. At time t = 94 ms, the reference torque Te∗ is modified externally (the speed controller is bypassed) from zero to maximum torque. It is possible to observe in Fig. 19 how the machine’s torque reaches its reference value in around 1 ms, the motor’s current is fast forced to a sinusoidal waveform, and the speed increases with a rate proportional to Te . The torque measurement is obtained based on the result delivered by the flux observer and (10).

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VARGAS et al.: PREDICTIVE TORQUE CONTROL OF AN INDUCTION MACHINE FED BY A MATRIX CONVERTER

Fig. 19.

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Behavior of the system under a step change on the reference torque (0 to 80 N·m): speed, output current, and electric and reference torque.

V. CONCLUSION The proposed control method for an MC-based IM drive has been shown to give excellent performance, effectively controlling both the machine and the input current. The method has a fast torque response, unity PF with low reactive power delivered to the mains, and low torque and flux ripple. PTC considers a fictional cost assigned to specific objectives, balanced by weighting factors. These factors are determined by analyzing the performance of the system for different values. The presented control approach represents a basic frame in which other features can be added [24]–[31], for example, to improve the efficiency of the drive [32]. The objectives of this paper are to introduce the method and its theoretical background, analyzing, in depth, the most relevant issues related to its implementation and showing its excellent performance based on experimental results. An assessment with conventional control and modulation methods was not included in the scope of this paper and is a topic that will be faced in the next step of this research. PTC takes advantage of the discrete nature of the MC’s switching states and the control processor. The method also utilizes the rotating vectors, usually discarded by MC modulation techniques. The high sampling frequency required should not be a problem nowadays, opening interesting possibilities with a conceptually different approach to optimization in the control of power converters and drives.

[6]

[7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17]

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[22] K. Drobnic, M. Nemec, D. Nedeljkovic, and V. Ambrozic, “Predictive direct control applied to AC drives and active power filter,” IEEE Trans. Power Electron., vol. 56, no. 6, pp. 1884–1892, Jun. 2009. [23] M. Nemec, D. Nedeljkovic, and V. Ambrozic, “Predictive torque control of induction machines using immediate flux control,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 2009–2017, Aug. 2007. [24] S. Kouro, P. Cortes, R. Vargas, U. Ammann, and J. Rodr´ıguez, “Model predictive control—A simple and powerful method to control power converters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1826–1838, Jun. 2009. [25] S. Bolognani, S. Bolognani, L. Peretti, and M. Zigliotto, “Design and implementation of model predictive control for electrical motor drives,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1925–1936, Jun. 2009. [26] J. Rodr´ıguez, J. Pontt, C. Silva, P. Correa, P. Lezana, P. Cort´es, and U. Ammann, “Predictive current control of a voltage source inverter,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 495–503, Feb. 2007. [27] S. M¨uller, U. Ammann, and S. Rees, “New time-discrete modulation scheme for matrix converters,” IEEE Trans. Ind. Electron., vol. 52, no. 6, pp. 1607–1615, Dec. 2005. [28] P. Cort´es, J. Rodr´ıguez, D. E. Quevedo, and C. Silva, “Predictive current control strategy with imposed load current spectrum,” IEEE Trans. Power Electron., vol. 23, no. 2, pp. 612–618, Mar. 2008. [29] P. Cort´es, J. Rodr´ıguez, P. Antoniewicz, and M. Kazmierkowski, “Direct power control of an AFE using predictive control,” IEEE Trans. Power Electron., vol. 23, no. 5, pp. 2516–2523, Sep. 2008. [30] R. Vargas, P. Cort´es, U. Ammann, J. Rodr´ıguez, and J. Pontt, “Predictive control of a three-phase neutral-point-clamped inverter,” IEEE Trans. Ind. Electron., vol. 54, no. 5, pp. 2697–2705, Oct. 2007. [31] R. Vargas, J. Rodr´ıguez, U. Ammann, and P. Wheeler, “Predictive current control of an induction machine fed by a matrix converter with reactive power control,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4362– 4371, Dec. 2008. [32] R. Vargas, U. Ammann, and J. Rodr´ıguez, “Predictive approach to increase efficiency and reduce switching losses on matrix converters,” IEEE Trans. Power Electron., vol. 24, no. 4, pp. 894–902, Apr. 2009. [33] H. Miranda, P. Cort´es, J. I. Yuz, and J. Rodr´ıguez, “Predictive torque control of induction machines based on state space models,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1916–1924, Jun. 2009. [34] J. Rodr´ıguez, J. Pontt, R. Vargas, P. Lezana, U. Ammann, P. Wheeler, and F. Garc´ıa, “Predictive direct torque control of an induction motor fed by a matrix converter,” Proc. Int. Power Electron. Appl. Conf. (EPE 2007), 2–5 Sep., pp. 1–10. [35] R. Vargas, M. Rivera, J. Rodr´ıguez, J. Espinoza, and P. Wheeler, “Predictive torque control with input PF correction applied to an induction machine fed by a matrix converter,” in Proc. Conf. Rec. IEEE (Power Electron. Spec. Conf.), Rhodes, Greece, Jun. 2008, pp. 9–14. [36] H. Akagi, Y. Kanagawa, and A. Nabae, “Generalized theory of the instantaneous reactive power in three phase circuits,” in IEEE J. Int. Power Electron. Conf. (IPEC), Tokyo, Japan, 1983, pp. 1375–1386. [37] P. Cort´es, S. Kouro, B. La Rocca, R. Vargas, J. Rodr´ıguez, J. I. Le´on, S. Vazquez, and L. G. Franquelo, “Guidelines for weighting factors adjustment in finite state model predictive control of power converters and drives,” in Proc. IEEE Int. Conf. Ind. Tech. (ICIT), Melbourne, Australia, Feb. 2009, pp. 1–7. [38] U. Baader, “Die direkte-selbstregelung (dsr) ein verfahren zur hochdynamischen regelung von drehfeldmaschinen,” Fortschritt-Berichte, VDIVerlag GmbH, D¨usseldorf, Germany, 1988. [39] J. Holtz, “Sensorless control of induction motor drives,” Proc. IEEE, vol. 90, no. 8, pp. 1359–1394, Aug. 2002. Ren´e Vargas (S’05–M’09) received the Engineer and M.Sc. degrees in electronics engineering, in 2005, and the Ph.D. degree for his work on predictive control applied to matrix converters, in 2009, from the Universidad T´ecnica Federico Santa Mar´ıa (UTFSM), Valpara´ıso, Chile. For a total period of eight months within 2006–2008, he was with the Institute of Power Electronics and Electrical Drives, University of Stuttgart, Germany. In 2009, he was a Research Assistant at the Power Electronics Research Group, UTFSM. In 2010, he joined ABB Switzerland, R&D Traction Converters, as Development Engineer. He has authored or coauthored over 20 papers in leading international conferences and journals, mainly on the topic of new control techniques applied to power conversion and drives.

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Ulrich Ammann (M’06) received the Dipl.-Ing. degree in electrical engineering in 2002 from the University of Stuttgart, Stuttgart, Germany, where he is currently working toward the Ph.D. degree in discrete-time modulation schemes, including predictive techniques. In 2002, he joined the Institute of Power Electronics and Electrical Drives, University of Stuttgart, as a Research Assistant. His current research interests include electric drives, high-power current sources, and automotive power electronics.

Boris Hudoffsky received the Dipl.-Ing. (F.H.) degree in mechanical engineering and automation in 2001 from the University of Applied Science Furtwangen, Furtwangen, Germany, and the Dipl.Ing. degree in electrical engineering in 2007 from the University of Stuttgart, Stuttgart, Germany, where he is currently working toward the Ph.D. degree in current measurement at the Institute of Power Electronics and Electrical Drives. Since 2001, he has been with TR Electronic GmbH, Trossingen, Germany for three years.

Jose Rodriguez (M’81–SM’94) received the Engineer degree in electrical engineering from the Universidad T´ecnica Federico Santa Mar´ıa (UTFSM), Valpara´ıso, Chile, in 1977, and the Dr.-Ing. degree in electrical engineering from the University of Erlangen, Erlangen, Germany, in 1985. Since 1977, he has been a Professor with the UTFSM, where he was the Director of the Electronics Engineering Department from 2001 to 2004, the Vice Rector of academic affairs from 2004 to 2005, and has been a Rector since 2005. During his sabbatical leave in 1996, he was with Siemens Corporation, Santiago, Chile, where he was responsible for the mining division. He has extensive consulting experience in the mining industry, particularly in the application of large drives, such as cycloconverter-fed synchronous motors for semiautogenous grinding mills, high-power conveyors, controlled ac drives for shovels, and power quality issues. He was the Director of more than 40 R&D projects in the field of industrial electronics. He has coauthored more than 250 journal and conference papers, and has contributed one book chapter. His research group has been recognized as one of the two centers of excellence in engineering in Chile in 2005 and 2006. His current research interests include multilevel inverters, new converter topologies, and adjustable-speed drives. Prof. Rodriguez is an active Associate Editor of the IEEE Power Electronics and the IEEE Industrial Electronics Societies since 2002. He was the Guest Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS in five opportunities [Special Sections on Matrix Converters (2002), Multilevel Inverters (2002), Modern Rectifiers (2005), High-Power Drives (2007), and Predictive Control of Power Electronic Drives (2008)].

Patrick Wheeler (M’00) received the B.Eng. (Hons.) degree and the Ph.D. degree in electrical engineering from the University of Bristol, Bristol, U.K., in 1990 and 1994, respectively. Since 1993, he has been with the University of Nottingham, Nottingham, U.K., where he was a Research Assistant in the Department of Electrical and Electronic Engineering, a Lecturer in the Power Electronics, Machines and Control Group during 1996, and has been a Full Professor in the same research group since January 2008. He has authored or coauthored more than 200 papers in leading international conferences and journals. His research interests include power conversion and more electric aircraft technology.

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