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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012

361

Voltage Disturbance Rejection for Matrix Converter-Based PMSM Drive System Using Internal Model Control

Changliang Xia, Member, IEEE, Yan Yan, Peng Song, and Tingna Shi

Abstract—A strategy based on internal model control (IMC) is proposed for a matrix converter-based permanent magnet synchronous machine (PMSM) drive system to reduce the adverse impact on drive performance caused by nonlinear output characteristics of matrix converter in the case of input voltage disturbance. Based on the duty-cycle space vectors and small-signal model, the relationship between output and input disturbances is obtained in the synchronous reference frame. Output characteristics of matrix converter are analyzed, and practical considerations are discussed for the purpose of controller design. A general design procedure of the robust IMC controller is described, and parameters of the controller are determined. Numerical simulations and experiments with a 10-kW prototype are carried out. The results show that good dynamic and steady-state performance on PMSM speed regulation is achieved under the unbalanced and distorted input voltage conditions, and the immunity of the drive system is veri?ed to be improved. Index Terms—Input voltage disturbance, internal model control (IMC), matrix converter, permanent magnet synchronous machine (PMSM).

I. I NTRODUCTION HE matrix converter is a forced commutated direct acto-ac converter. It is known for power regeneration, highquality input/output waveform, controllable input power factor, and compact design of power circuit [1], [2]. These features are very advantageous to the modern electrical drives and have been further applied in such areas as wind power generation [3], [4], power supply unit [5], and dynamic voltage restorer [6]. In the topology of matrix converter, output and input lines are directly connected by an array of controlled bidirectional switches, without any intermediate energy storage stage. One

T

Manuscript received June 20, 2010; revised October 29, 2010 and January 17, 2011; accepted March 7, 2011. Date of publication March 28, 2011; date of current version October 4, 2011. This work was supported in part by the National Science Fund for Distinguished Young Scholars under Grant 50825701, in part by the Key Program of National Natural Science Foundation of China under Grant 51037004, in part by the National Natural Science Foundation of China under Grant 51077097, and in part by the Key Technologies Research and Development Program of Tianjin under Grant 10ZCKFGX02800. C. Xia is with the School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China, and also with the School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300160, China (e-mail: motor@tju.edu.cn). Y. Yan, P. Song, and T. Shi are with the School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China (e-mail: yanyan@tju.edu.cn; song_peng@tju.edu.cn; tnshi@tju.edu.cn). Digital Object Identi?er 10.1109/TIE.2011.2134058

practical problem of such topology is low immunity to power grid disturbances, such as unbalance, harmonics, and voltage sags [7]. Considerable research has been done on the analysis of matrix converter operation in the case of input voltage disturbance. [8] proposes a novel analysis method based on symmetrical component transformation to determine the matrix converter performance in terms of output voltage quality and points out that the disturbances in power supply may generate undesirable low-frequency harmonics in output voltage. In literature [9]–[14], several control techniques have been proposed to solve the problem. Among those techniques, the feedforward compensation method based on input voltage measurement has the advantage of simplicity, but the presence of L–C input ?lter may cause unstable operation of matrix converter under certain conditions [15]. In such a case, oscillations may be superimposed on both input voltage and line current, and the operation of the converter may be highly disturbed. In [16]– [19], stability issues of matrix converter were addressed using a small-signal analysis. It is shown that the stability can be sensibly improved when the duty cycles of switching con?gurations are calculated using the digital ?ltering input voltages, but the compensation capability of matrix converter against input voltage disturbances is deteriorated to some extent [18]. In this paper, a strategy based on internal model control (IMC) is proposed for a matrix converter-based permanent magnet synchronous machine (PMSM) drive system to overcome the unfavored impact introduced by digital ?lter and guarantee the drive performance under input disturbance conditions. Contents of this paper are organized as follows. Section II describes the modeling of a matrix converter-based PMSM drive system, based on the modulation strategy proposed in [18], and transfer characteristics of matrix converter and output behaviors in terms of input voltage disturbances are analyzed in detail. Practical considerations are presented in Section III, a robust control scheme based on IMC is developed in Section IV, and its performance is evaluated by simulation in Section V. Then, the experimental setup and results are analyzed in Section VI, showing the feasibility and effectiveness of the proposed strategy. Finally, in Section VII, some conclusions are established. II. M ODELING AND A NALYSIS A matrix converter-based PMSM drive system is shown in Fig. 1. It consists of a nonideal supply, a second-order L–C

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012

where mlh are the duty cycles of the switches Slh , l ∈ {A, B, C }, h ∈ {a, b, c}. Introducing the duty-cycle space vectors and the coordinate transformation, (5) and (6) can be written in [20]

? vo = 1.5 (vi m? i + vi m d )

(7) (8)

ii = 1.5 (io mi + i? o md )

Fig. 1. Scheme of matrix converter-based PMSM drive system.

?lter, a matrix converter, and a PMSM. For the following analysis, the variables for the input side of the matrix converter are transformed into a reference frame rotating at a supply angular frequency ωi . The output side quantities are transformed into a direct and quadrature reference frame rotating at the electrical angular speed of the PMSM rotor with the d-axis aligned with the rotor ?ux vector. A. Mathematical Model of PMSM In the d–q reference frame, the dynamics of a PMSM can be represented by the following nonlinear differential equations: diod vod R Lq = ? iod + pωr ioq dt Ld Ld Ld dioq voq R Ld ψm pωr = ? ioq ? pωr iod ? dt Lq Lq Lq Lq dωr 1 = (Te ? TL ). dt J The torque equation can be expressed as Te = 1.5p [ψm ioq + (Ld ? Lq )iod ioq ] (4) (1) (2) (3)

where the duty-cycle space vector md is de?ned in a reference frame rotating at the angular speed ωi + pωr ; the duty-cycle space vector mi is de?ned in a reference frame rotating at the angular speed ωi ? pωr ; and vi , vo , ii , io are the voltage and current space vectors for input and output sides of matrix converter. The symbol ? denotes the complex conjugate. The input current modulation of matrix converter can be represented by introducing an arbitrary vector ψ ref . Here, ψ ref is called modulation vector, and it de?nes the direction along which the input current is modulated. (ii · j ψ ref ) = 0. (9)

A general modulation strategy related to the output voltage and input current control requirement can be solved from (7)–(9) md = mi = vo,ref ψ ref k + ?? 3(vi · ψ ref ) vi io

? ψ ref vo,ref k ? ? 3(vi · ψ ref ) vi io

(10) (11)

where vo,ref is the reference output voltage vector; the parameter k is arbitrary; and (·) denotes the scalar product. In [18], with the aim of improving the stability of matrix converter drive system, the input voltages are measured and ?ltered in the synchronous reference frame, given by dviLf /dt = vi /τ ? viLf /τ (12)

where iod , ioq , vod , voq are the stator current and voltage components; R is the stator resistance; Ld and Lq are d- and q -axis inductances; ψm is the amplitude of the ?ux induced by the permanent magnets; ωr is the rotor angular speed; Te and TL are electromagnetic torque and load torque; J is the moment of inertia; and p is the number of pole pairs. B. Transfer Characteristics of Matrix Converter Neglecting the effects of switching harmonics, considering the average values of voltages and currents over a switching interval, the input–output relationships of matrix converter can be expressed as ? ? ? ?? ? voA mAa mAb mAc via ? voB ? = ? mBa mBb mBc ? ? vib ? (5) voC mCa mCb mCc vic ?? ? ? ? ? mAa mBa mCa ioA iia ? iib ? = ? mAb mBb mCb ? ? ioB ? (6) iic mAc mBc mCc ioC

with τ as the time constant of the digital ?lter. Furthermore, the duty cycles of switching con?gurations are calculated by using the digital ?ltering input voltages. Letting k = 0, vi = viLf , and ψ ref = viLf , the duty-cycle space vectors in (10) and (11) result in

? md = vo,ref / 3viLf ? ? mi = vo,ref . / 3viLf

(13) (14)

Applying the aforementioned modulation strategy, from (7), the transfer characteristics of matrix converter can be obtained vo = (vi · viLf ) vo,ref = KM C vo,ref (viLf · viLf ) (15)

where KM C is the ampli?er gain of matrix converter. Equation (15) indicates the in?uence of input voltages on actual outputs for a given reference. Ideally, the power supply is balanced and sinusoidal, and the steady-state solution of (12) is viLf = vi , which leads to vo = vo,ref . It means that the ampli?er gain of matrix converter KM C equals 1. However,

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Fig. 2. Normalized loci of matrix converter voltage vectors with typical disturbances in the ac distribution system. (a) Input voltage with 10% unbalanced degree. (b) Input voltage with 10% ?fth harmonic component. (c) Input voltage with 10% seventh harmonic component.

if the supply voltages are unbalanced or have some typical distortions, according to (12) and (15), the actual output voltage will be in?uenced by input voltages. With τ = 0.8 ms and the reference output voltage being 0.6 p.u./25 Hz, Fig. 2 shows the normalized loci of matrix converter voltage vectors with typical disturbances in ac distribution system. In these cases, the ampli?er gain KM C becomes nonlinear and ?uctuates around 1, as seen in Fig. 2, where the output voltages apparently oscillate around the reference including speci?c harmonics. C. Analysis of Output Characteristics The output voltage references are assumed to be balanced and sinusoidal, and Vo,ref = Vod,ref + jVoq,ref . Around the steady-state operating point ? Vi = ViLf = Vi ? ? ?V = V o o,ref (16) M = V d o,ref /(3Vi ) ? ? ? M = V? /(3V )

i o,ref i

where ViH indicates the amplitude and phase angle of the disturbance harmonic component; and ωH is the corresponding angular frequency with respect to the reference frame rotating at ωi . Calculating (18) and substituting (22) into (21), the output voltage disturbances can be expressed in d- and q -axis components Δvod = Vod,ref ViH 2Vi jωH τ 1 + jωH τ ejωH t

?

? + ViH

jωH τ 1 + jωH τ

e?jωH t

(23)

Δvoq =

Voq,ref ViH 2Vi

jωH τ 1 + jωH τ

ejωH t

?

? + ViH

jωH τ 1 + jωH τ

e?jωH t .

(24)

linearizing (7), (12), (13) and (14) leads to following smallsignal equations:

? ? Δvo = 1.5 (Δvi M? i + Δvi Md + Vi Δmi + Vi Δmd )

(17) dΔviLf /dt = Δvi /τ ? ΔviLf /τ Δmd =

? ? Vo,ref ΔviLf /

(18) (19) (20)

3Vi2

? ? Δmi = ? Vo,ref ΔviLf / 3Vi2 .

Combining (17) with (20) yields

? ? Δvo = Vo,ref (Δvi ? ΔviLf ) + Δvi ? ΔviLf

/(2Vi ). (21)

To predict the behavior of matrix converters in the case of unbalanced or nonsinusoidal supply voltages, the input voltage vector is assumed to consist of two terms, one representing the ideal supply voltages, the other representing the disturbance. Generally, the disturbance can be expressed by complex Fourier series. Here, to facilitate the analysis, the disturbance is given by Δvi = ViH ejωH t (22)

Equations (23) and (24) express output voltage disturbances as a function of output reference Vo,ref , disturbance harmonic component ViH , angular frequency ωH , and time constant τ . A similar formula for output and input disturbances when matrix converter feeds an R–L passive load is presented in [18]. According to the aforementioned expressions, it can be concluded that (1) the d- and q -axis output disturbances consist of two components, which rotate in opposite directions at speed of ωH within the reference frame. As a result, the synthesized output disturbance is characterized by oscillation; (2) the amplitude of output disturbances are proportional to the reference values for given input disturbance Δvi and time constant τ , which means that the output distortion gets worse with higher reference value, and the input disturbance causes different d- and q -axis components of output voltage disturbances according to the distinct references; (3) the d- and q -axis components of output voltage disturbance depend on the amplitude and phase angle of the input disturbance for given reference value Vo,ref , angular frequency ωH , and time constant τ . III. P RACTICAL C ONSIDERATIONS FOR C ONTROLLER D ESIGN In the presence of input voltage disturbance, under the applied modulation strategy, the actual fundamental output voltages do not agree with the reference voltages due to the

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012

Fig. 3. IMC structure. Fig. 4. IMC structure for current controller design.

nonlinear transfer characteristics of matrix converter. The undesired output voltage disturbances result in stator harmonics current, speed ?uctuation, torque ripple, and machine parameter variation, as far as a matrix converter-based PMSM drive system is concerned. According to the analysis presented in Section II, extra considerations for controller design to suppress disturbances involve the following aspects. 1) The output voltage distortion becomes worse with higher reference value. This characteristic has signi?cant in?uence on the drive system performance, particularly on the occasions of acceleration, deceleration, and heavy load. 2) The impact of input disturbance on output voltage varies with the machine operating condition. Normally, the d- and q -axis components of output disturbances are diversely affected by the input perturbation. IV. P ROPOSED C ONTROLLER D ESIGN A control scheme based on IMC is developed for the matrix converter-based PMSM drive system to guarantee the drive performance under input disturbance conditions. IMC is selected in the proposed strategy due to its two major advantages. First, the dynamic and steady errors caused by unmeasurable disturbances and uncertainty of control process can be reduced by IMC controller since the difference between the plant and the reference model is used as feedback. Second, by de?ning a suitable reference model, the IMC theory provides a systematic approach in the controller design with the assurance of robustness, and it is convenient to regulate parameters online. A. Internal Model Control Fig. 3 shows the basic con?guration of internal model con? (s), trol, which consists of the plant G(s), a reference model G an internal model controller CIM C (s), and the disturbance D(s). The advantage of using IMC is partly due to the special form of the feedback signal, i.e., E (s) = ? (s) D(s) + R(s)CIM C (s) G(s) ? G ? (s) I + CIM C (s) G(s) ? G (25)

Note that the IMC control scheme can be mathematically equivalent to a conventional feedback controller by de?ning F (s) = CIM C (s) ? (s) I ? CIM C (s)G . (26)

With respect to the IMC structure, the general properties are summarized as follows. 1) When the reference model is accurate, the stability of the overall system depends on the stability of both controller and plant. 2) According to the optimal control theory [21], the system control error can be minimized by making ? (s) = I , under the assumption that G ? (s) = CIM C (s)G G(s) and that G(s) is stable. ? (0) = I , a 3) When the IMC controller satis?es CIM C (0)G unit step change in R(s) will yield zero steady-state error for any asymptotically constant disturbance D(s). B. Current Controller Design Combining the PMSM plant with the dynamic behaviors of matrix converter, the IMC structure for the current controller design can be shown by the block diagram in Fig. 4. Considering the voltage transfer radio of the matrix converter, voltage limiters are imposed on the control process. According to (1) and (2), the transfer function matrix between the stator current and voltage is GP M (s) = io (s) R + Ld s ?pωr Lq = pωr Ld R + Lq s vo (s)

?1

(27)

where vo = [vod voq ? pωr ψm ]T and io = [iod ioq ]T . From (15), the transfer characteristics of matrix converter can be described by a proportional element. Given the dead-time lag due to the PWM sampling delay, a simpli?ed model of matrix converter is given by GM C (s) = = vo (s) vo,ref (s) KM C /(1 + Ts s) 0 0 KM C /(1 + Ts s) (28)

where I is the identity matrix. ? (s) = G(s), the feedFor a perfect reference model, i.e., G back signal is just the disturbance D(s). When there are differences between the plant and the model, E (s) provides the information of disturbance and model–plant mismatch, and the robustness can be obtained by compensating for the derivation appropriately.

where vo,ref = [vod,ref voq,ref ]T and Ts represent the sampling delay. In most cases, variation of the converter gain can be merged with the output voltage disturbance signal Δvo ; thus, the gain KM C in (28) can be replaced by 1.

XIA et al.: VOLTAGE DISTURBANCE REJECTION FOR MATRIX CONVERTER-BASED PMSM DRIVE SYSTEM

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TABLE I R ATED PARAMETERS OF PMSM

Fig. 5.

IMC structure for speed controller design.

TABLE II PARAMETER VARIATIONS

Fig. 6.

Control scheme of matrix converter-based PMSM drive system.

With (27) and (28) combined and possible variation and measurement error of machine parameters considered, an uncertainty set Πc of the converter–machine model is de?ned by Πc = Pc (s)|Pc (s) = 1 R + Ld s ?pωr Lq 1 + Ts s pωr Ld R + Lq s

?1

, (29)

R = Rn (1+ ζ1 ), Ld = Ldn (1+ ζ2 ), Lq = Lqn (1+ ζ3 )

where the actual value of machine parameter is written as a perturbation from its nominal value, with Rn , Ldn , and Lqn being the nominal values of machine parameters and ζ1 , ζ2 , ζ3 being uncertainty bounds. The reference model in Fig. 4 is de?ned by ?c (s) = P 1 Rn + Ldn s ?pωr Lqn pωr Ldn Rn + Lqn s 1 + Ts s

?1

.

(30)

Fig. 7. Current responses to independent step command at t = 0. (a) Unit step response to iod,ref . (b) Unit step response to ioq,ref .

According to the second property of IMC structure, the controller is ?rst designed regardless of the uncertainty of converter–machine system, which yields

?1 ?c CCIM C (s) = P (s).

(31)

where the positive integer n is chosen to be suf?ciently large to guarantee that the IMC controller is proper; and α is a design parameter. According to the order of the reference model de?ned in (30), letting n = 2, the IMC controller becomes CCIM C (s) = Lc (s)CCIM C (s) α2 (1 + Ts s) Rn + Ldn s ?pωr Lqn = . (33) pωr Ldn Rn + Lqn s (s + α)2 In accordance with (26), put the IMC control scheme into conventional feedback controller, which results in Fc (s) = α(1 + Ts s) Rn + Ldn s ?pωr Lqn . (34) pωr Ldn Rn + Lqn s 2s(s/2α + 1)

Second, the controller is augmented by a ?lter to make the system robust [22]. The ?lter is usually a low-pass ?lter since the uncertainty of the plant generally increases with frequency. According to the third property of IMC, one of the simplest forms of the ?lter which makes offset-free tracking can be represented by Lc (s) = α s+α

n

(32)

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012

Fig. 10.

Speed responses to unit step command at t = 0.

Fig. 8. Current responses for PI controller to independent step disturbance at t = 0. (a) Unit step responses to Δvod . (b) Unit step response to Δvoq .

Fig. 11.

Speed responses to unit step load disturbance at t = 0.

TABLE III E LECTRICAL PARAMETERS AND C ONTROL DATA OF M ATRIX C ONVERTER -BASED PMSM D RIVE S YSTEM

C. Speed Controller Design Setting iod,ref = 0, together with the mechanical dynamics (3) and (4), the IMC structure for the speed controller design can be represented by the block diagram in Fig. 5, where Gcc (s) represents the equivalent model of the current loop. Given the limitation of maximum torque, the current limits are imposed on the controlled process. Given the possible uncertainty of the controlled plant in Fig. 5, the uncertainty set Πs is de?ned as

Fig. 9. Current responses for IMC controller to independent step disturbance at t = 0. (a) Unit step responses to Δvod . (b) Unit step response to Δvoq .

Πs = Ps (s)|Ps (s) =

1.5pψm , ψm = ψmn (1 + θ1 ), Js(1 + τc s) (36)

By using the aformentioned controller, the transfer function of the closed-loop system in Fig. 4 can be written as follows: Gcc (s) = io (s) L (s) = c 0 io,ref (s) 0 ≈ Lc (s)

1 2s/α+1

J = Jn (1 + θ2 ), τc = τcn (1 + θ3 )

0

1 2s/α+1

0

.

(35) Further, by neglecting s2 term, the nominal time constant of current loop can be obtained.

where τc is the time constant of current loop, τcn (= 2/α), ψmn , and Jn are the nominal values of time constant, ?ux linkage, and inertia, respectively, and θ1 , θ2 , and θ3 are the uncertainty bounds. The reference model of speed loop is de?ned by ?s (s) = P 1.5pψmn . Jn s(1 + τcn s) (37)

XIA et al.: VOLTAGE DISTURBANCE REJECTION FOR MATRIX CONVERTER-BASED PMSM DRIVE SYSTEM

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Fig. 12. Responses of the proposed control system under different input voltage conditions.

Similar with the design procedure of current controller, the IMC controller in speed loop is of the form ? ?1 (s). CSIM C (s) = Ls (s)P s (38)

where Ls (s) is the transfer function of the ?lter. According to (26), transform the IMC controller (38) into conventional feedback controller, which results in Fs (s) = Ls (s) ? ?1 P (s). I ? Ls (s) s (39)

It is known that introducing the integrator in control loop can avoid offset. Here, a three-order ?lter is constructed according to (39) Ls (s) = (3λs + 1) (λs + 1)3 (40)

speed loop generates the reference current ioq,ref through the controller Fs (s) presented in (43), while the reference current iod,ref is kept as zero. The differences between the reference currents and the feedback currents generate the voltage references of matrix converter through the controller Fc (s) presented in (34). Under the modulation strategy de?ned in (13) and (14), the input voltage measurement is performed and a digital phase-locked loop described in [23] is adopted to realize the input voltage synchronization. The calculation of duty cycles is then carried out by the output reference and the ?ltered input voltage described in (12). V. N UMERICAL S IMULATIONS To investigate the performance of the proposed controllers, uncertain plants and controllers shown in Figs. 4 and 5 are developed by using Matlab/Robust Control Toolbox. The characteristics, such as reference tracking, disturbance rejection, and sensitivity to parameter variations, are preliminarily evaluated by comparing the independent step responses to individual control channels. The rated parameters of PMSM used in the simulation are given in Table I, and the uncertainty bounds of the corresponding parameters are listed in Table II. Based on the model in Fig. 4, the decoupling characteristics of the proposed current controller are tested by simulation. Taking machine parameter variation into account, ?ve plants are selected at random from the uncertainty set Πc , and the corresponding responses to independent step commands are shown in Fig. 7. The transient responses to a step change in d-axis reference current are shown in Fig. 7(a). With the design parameter α being 1320 Hz, d-axis feedback signal approaches the reference value with the rise time of around 3 ms. Meanwhile, q -axis current remains to zero. Similar results can be obtained from Fig. 7(b) as far as the step change of q -axis reference current is concerned.

where λ is a design parameter. Substituting (40) and (37) into (38), the IMC controller in speed loop can be expressed by CSIM C (s) = (3λs + 1) Jn s(1 + τcn s) . (λs + 1)3 1.5pψmn (41)

For the conventional feedback structure, substituting (41) into (26), yields Fs (s) = Jn (1 + τcn s)(3λs + 1) . 4.5pψmn λ2 s(λs/3 + 1) (42)

Furthermore, with λ = 3τcn , the controller Fs (s) can be simpli?ed to a typical PI controller Fs (s) = Jn (3λs + 1) . 4.5pψmn λ2 s (43)

Fig. 6 shows the overall control scheme of matrix converterbased PMSM drive system. Under the cascade control, the

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012

Fig. 13. Photograph of the experimental setup.

Fig. 15. Experimental waveforms for the step change in the load torque, with ωr,ref being 30 r/min, TL increasing from 0 to 120 N · m. (a) Input line-to-neural voltage (64 V/div, 20 ms/div) and input line current (2.5 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (6.67 r/min/div, 500 ms/div).

Fig. 14. Experimental waveforms for the step change of speed reference, with ωr,ref increasing from 20 to 40 r/min, TL being 120 N · m. (a) Input line-toneural voltage (64 V/div, 20 ms/div) and input current (2.5 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (6.67 r/min/div, 500 ms/div).

Subjected to the voltage disturbances, performance of the current controllers shown in Fig. 4 is analyzed and compared with that of PI controller. Five plants are selected from the uncertainty set Πc , and the corresponding responses to independent step changes of voltage disturbances are shown in Figs. 8 and 9. As can be seen, the maximal oscillation amplitudes of

IMC strategy are quite similar to those of PI control. It is shown that the IMC strategy regulates much faster in the case of voltage disturbance. In Fig. 8, the settling time of PI control is longer than 0.3 s, and its effect on voltage disturbance rejection is signi?cantly dependent on plant parameters. By applying the IMC strategy, the current oscillations caused by the disturbance step change can be suppressed to zero within 40 ms. Based on the model in Fig. 5, the reference tracking of the proposed speed controller is tested by applying a step change of speed reference. In the simulation, ?ve different plants are randomly sampled from the uncertainty set Πs , and the design parameter λ is 5 ms. The obtained results are shown in Fig. 10. As can be seen, the feedback signals approach the reference value with a fast dynamic response. The maximal overshoot level is less than 8% in spite of parameter variations. Subjected to a load disturbance, transient responses of the proposed speed control are evaluated. Again, the controller performance is checked by ?ve different plants, which are selected randomly from the uncertainty set Πs . The step responses to disturbance are shown in Fig. 11. As can be seen, the speed

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Fig. 16. Experimental waveforms for a triangular speed reference, with ωr,ref varying between 20 and 40 r/min, TL being 120 N · m. (a) Input lineto-neural voltage (64 V/div, 20 ms/div) and input current (2.5 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (6.67 r/min/div, 500 ms/div).

Fig. 17. Experimental waveforms during a speed reversal command, with ωr,ref step changing from 30 to ?30 r/min. (a) Input line-to-neural voltages (64 V/div, 20 ms/div) and input line current (2.5 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (13.34 r/min/div, 500 ms/div).

waveforms restore quickly after a slight drop due to the sudden increase of load torque. Taking into account the switching behaviors of matrix converter and the delay related to the digital implementation of the control algorithm, a further test on voltage disturbance rejection is applied to the overall system shown in Fig. 6. In the simulation, the matrix converter is controlled by space vector modulation technology [24]. The system parameters are shown in Tables I–III. The following power supply conditions have been considered: 1) ideal inputs, 2) unbalanced inputs, and 3) inputs with ?fth harmonic components. The ideal input voltage is 380 V and 50 Hz. A negative sequence component and a ?fth harmonic, both with the amplitude of 31.1 V, is added to the ideal power supply to form the unbalanced and distorted input voltage, respectively. In each case, the load torque steps between 50 and 95% rated torque. Responses of the system are shown in Fig. 12, with the speed reference of 100 r/min. As can be seen from Fig. 12, the proposed strategy works well in the disturbed power supply conditions. The speed and

torque in the case of the unbalanced and distorted supply respond as fast and accurately as those in the case of the ideal supply. Such performance results from the desired stator currents. According to the analysis in Section II, there are two main harmonics in the output currents under the unbalanced input voltage condition with the frequencies of 80 and 120 Hz, respectively. These speci?c harmonics are proved to be extremely reduced, and the contents of them are only 0.16 and 0.18% in the simulation by a spectrum analysis of stator currents. The good effect on stator current control is also supported by the exactly sinusoidal current in Fig. 12. A similar conclusion can be drawn from Fig. 12 as far as the distorted power supply is concerned. Hence, it can be seen that the proposed IMC controller compensates the nonlinear characteristics of matrix converter very well, and restrains the unfavored impact of input disturbance effectively. VI. E XPERIMENTAL R ESULTS An experimental setup was built and tested on a 10-kW PMSM to verify the feasibility and effectiveness of the

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Fig. 18. Experimental waveforms for the step change in speed reference, with ωr,ref increasing from 20 to 40 r/min, TL being 120 N · m. (a) Input line-to-line voltages (106 V/div, 20 ms/div) and input line current (4.2 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (6.67 r/min/div, 500 ms/div).

Fig. 19. Experimental waveforms for the step change in the load torque, with ωr,ref being 30 r/min, TL increasing from 0 to 120 N · m. (a) Input line-to-line voltages (106 V/div, 20 ms/div) and input line current (4.2 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (6.67 r/min/div, 500 ms/div).

proposed controller, as shown in Fig. 13. The complete system was connected to the utility grid through the variacs. An input ?lter was adopted to attenuate switching harmonics. A clamp circuit was designed to protect the matrix converter against overcurrent and overvoltage that occur on input side and/or output side of matrix converter. The controller proposed in Fig. 6 was implemented in a digital platform, which used a TMS320F28335 digital signal processor for the proposed control strategy and an EP1C6 FPGA for switch commutation [25]. The parameters of the experimental system are shown in Tables I–III. The experiments were performed under normal and unbalanced input voltage conditions, respectively. The results were recorded by a YOKOGAWA digital storage oscilloscope DL1640. A. Performance Under Normal Voltage Conditions The normal input voltages of the matrix converter are set to 200 V/50 Hz. Fig. 14 shows the measured input current, input voltage, stator current, and speed waveforms, with speed reference value stepping up from 20 to 40 r/min and the load

torque being 120 N · m. As can be seen in Fig. 14, the duration for speed acceleration is about 100 ms. During this transient period, the acceleration of rotor remains constant until its speed approaches the reference value. Moreover, the input current is in phase with the input line-to-neural voltage and is sinusoidal in steady-state conditions. Fig. 15 shows the transient responses of the proposed controller with the load torque stepping up from zero to 120 N · m and speed reference maintaining at 30 r/min. It can be seen that the stator current increases sharply at the moment of load step, and the speed of the PMSM appears only very slightly pit. In Fig. 16, a triangular speed command is used to check the offset-free tracking capability of the IMC controller. The reference value varies between 20 and 40 r/min with a period of 2.6 s, and the load torque is 120 N · m. Fig. 16(a) shows the sinusoidal and inphase dynamical currents and voltages at the input side of matrix converter, and Fig. 16(b) shows an offsetfree tracking to command. Fig. 17 shows the experimental results of the transient state when the speed reference value steps from 30 to ?30 r/min.

XIA et al.: VOLTAGE DISTURBANCE REJECTION FOR MATRIX CONVERTER-BASED PMSM DRIVE SYSTEM

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Fig. 20. Experimental waveforms for a triangular speed reference, with ωr,ref varying between 20 and 40 r/min, TL being 120 N · m. (a) Input line-to-line voltages (106 V/div, 20 ms/div) and input line current (4.2 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (6.67 r/min/div, 500 ms/div).

Fig. 21. Experimental waveforms during a speed reversal command, with ωr,ref step changing from 30 to ?30 r/min. (a) Input line-to-line voltages (106 V/div, 20 ms/div) and input line current (4.2 A/div, 20 ms/div). (b) Stator current (2.3 A/div, 500 ms/div) and speed response (13.34 r/min/div, 500 ms/div).

A permanent magnet synchronous generator feeding resistances is used to load the PMSM. In Fig. 17(a), the input current of the drive is reverse in phase with the input voltage during the deceleration of the PMSM, which means the drive operates in regenerative mode during the deceleration transient. Such results prove that the matrix converter fed PMSM can be manipulated in two or four quadrants given a proper load. B. Performance Under Input Voltage Disturbances Experimental results in the case of input voltage unbalance are shown in Figs. 18–21. The unbalanced voltages are set to 220 V/200 V/200 V (50 Hz), supplied by two variacs. Operating modes of the PMSM are identical to those in Figs. 14–17, respectively. From Figs. 18–21, behavior of the PMSM under unbalanced input voltage is very similar with that under normal input voltage. Stator current and speed of the PMSM responds fast and accurately to the step change in speed command and load torque. The drive system exhibits offset-free tracking to the triangular speed command. Transient of reversal speed regulating

indicates a short-time two-quadrant operation of the PMSM. As can be seen in Figs. 18–21, unbalanced input voltage causes distorted input current, which is the most marked difference between the two groups of experimental results. It should be pointed out that the noticeable ripple present in speed both under normal and unbalanced input voltage condition is caused by high cogging torque of the PMSM. Feedback control on torque or optimized structure design for the PMSM can suppress such ripples, which is the content of followup research of this paper. VII. C ONCLUSION Input voltage disturbances cause apparent harmonics in output voltage of matrix converter and deteriorate speed regulation performance of the matrix converter-based PMSM drive system. According to the analysis of transfer characteristic, the amplitude of output voltage oscillations is proportional to the output references. Therefore, the impact of input voltage disturbances on drive performance varies with machine operating conditions. The worse cases occur under the conditions

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of acceleration, deceleration, and heavy loading. A practical control scheme based on IMC has been proposed in this paper to improve the performance of matrix converter-based PMSM drive system in such circumstances. The effect of the IMC controller on voltage disturbance rejection has been evaluated by simulation and experiment. The results prove that the proposed strategy has good performance under the input voltage disturbance conditions. R EFERENCES

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Changliang Xia (M’08) was born in Tianjin, China, in 1968. He received the B.S. degree in electrical engineering from Tianjin University, Tianjin, in 1990 and the M.S. and Ph.D. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 1993 and 1995, respectively. He is currently a Professor with the School of Electrical Engineering and Automation, Tianjin University, and also with the School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin. In 2008, he became “Yangtze Fund Scholar” Distinguished Professor and is currently supported by the National Science Fund for Distinguished Young Scholars. His research interests include electrical machines and their control systems, power electronics, and control of wind generators.

Yan Yan was born in Tianjin, China, in 1981. She received the B.S. and M.S. degrees in electrical engineering from the Tianjin University of Science and Technology, Tianjin, in 2004 and 2007, respectively, and Ph.D. degree in electrical engineering from Tianjin University, Tianjin, China, in 2010. She is currently a Lecturer with the School of Electrical Engineering and Automation, Tianjin University, Tianjin. Her research interests include the design and control of matrix converter for electric drive applications and power converters for wind power generation.

Peng Song was born in Gansu, China, in 1980. He received the B.S. and M.S. degrees in electrical engineering from Tianjin University, Tianjin, China, in 2003 and 2006, respectively, where he is currently working toward the Ph.D. degree in electrical engineering with the School of Electrical Engineering and Automation. His research interests include control of permanent magnet synchronous motor and highperformance motor drive fed by matrix converter.

Tingna Shi was born in Zhejiang, China, in 1969. She received the B.S. and M.S. degrees from Zhejiang University, Hangzhou, China, in 1991 and 1996, respectively, and the Ph.D. degree from Tianjin University, Tianjin, China, in 2008. She is currently an Associate Professor with the School of Electrical Engineering and Automation, Tianjin University. Her research interests include electrical machines and their control systems, power electronics, and electric drives.

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