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A deadbeat control method for a PWM converter applied to a superconducting magnet
Fusion Engineering and Design 58 – 59 (2001) 57 – 62 www.elsevier.com/locate/fusengdes
A deadbeat control method for a PWM converter applied to a superconducting magnet Y.M. Miura a,*, M. Matsukawa a, H. Nakano b a
b
Japan Atomic Energy Research Institute, Mukouyama, Naka-machi, Naka-gun, Ibaraki 311 -0193, Japan Department of System Engineering, Ibaraki Uni6ersity, Nakanarusawa, Hitachi-shi, Ibaraki 316 -8511, Japan
Abstract Current-type pulse-width-modulated (PWM) converter is regarded as one of the candidates for the superconducting magnet power supplies in next generation tokamak devices. However, this type PWM converter has a problem of the transient oscillation of the reactor current and capacitor voltage in an AC-side harmonic filter, which occurs owing to the LC resonance of the filter when the reference of the DC output current is changed rapidly. In this paper, the new PWM control method based on the deadbeat control theory is proposed to suppress the transient oscillation. The converter employing this method can control the reactor current and capacitor voltage in one sampling period under ideal conditions. In addition, the effect of dead time on the control is discussed. The performance of the newly proposed method is demonstrated through numerical circuit simulation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: PWM; Superconducting magnet; Tokamak
1. Introduction Next generation tokamak devices require superconducting poloidal field coils. In the case that conventional thyristor converters with phase control are applied to these coil power supplies, the power factor becomes very poor in average, and they require huge reactive-power compensators. From this viewpoint, current-type pulse-widthmodulated (PWM) converters are regarded as one of the candidates for the superconducting magnet power supplies in next generation tokamak * Corresponding author. Tel.: + 81-29-270-7439; fax: + 8129-270-7459. E-mail address: [email protected] (Y.M. Miura).
devices because they can work at the power factor of unity. In addition, the current-type PWM converters have an advantage over the voltage-type ones because it is easy to replace the conventional thyristor converters. However, the current-type PWM converters have a problem of the transient oscillation of the AC-side current and voltage. In general, it requires a harmonic filter located on the AC-side to reduce harmonics flowing out to a commercial power system. Since the harmonic filter consists of reactors and capacitors, the transient oscillation of the reactor current and capacitor voltage appears when the reference of the DC output current changes rapidly, owing to the LC resonance of the filter. This transient oscillation might continue in several ten sampling periods,
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Fig. 1. The schematic diagram of the current-type PWM converter.
and consequently degrade the control characteristic of the converter. In order to suppress this transient oscillation, Sato has proposed to employ the state-feedback control of the reactor current and capacitor voltage [1]. This method is based on the control theory in the continuous-time system, and the setting time is insufficiently long at low PWM frequency. Matsukawa investigated the application of the deadbeat control in the discrete-time system. In this method, the instantaneous values of the reactor current and capacitor voltage are controlled in two sampling periods [2]. Improving this method, we propose the new deadbeat control method in which they can be controlled in one sampling period. Therefore, a very short setting time is achieved even at low PWM frequency.
controllable switching devices such as insulated gate bipolar transistors (IGBTs). The single-phase equivalent circuit for the AC-side can be derived from Fig. 1 as shown in Fig. 2a. Moreover, it can be divided into two circuits: a voltage-source circuit (Fig. 2b) and a current-source circuit (Fig. 2c). The former represents the uncontrollable sinusoidal component depending on the voltage source, while the latter represents the component of the PWM control by the current source. Since it is easy to take the sinusoidal component into consideration by superimposing it on the PWMcontrolled component, we consider only the current-source circuit hereafter. With reference to
2. Deadbeat control method We first introduce the new method using a single-phase equivalent circuit for the AC-side of the current-type PWM converter. The typical configuration of the three-phase PWM converter shown in Fig. 1 consists of three parts: a threephase-bridge converter, an LC filter and an inductive load. In the bridge, S1 – S6 represent
Fig. 2. (a) The AC-side single-phase equivalent circuit of a current-type PWM converter, (b) a voltage-source circuit, (c) a current-source circuit.
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ing input quantities are necessary. In this case, they are the ON and OFF pulse widths of the converter input current, T2 and T3. The timings T2 and T3 to realize reference inputs [i*, *] s 6% c in one sampling period can be calculated from Eq. (2) by substituting [i*, *] for [is(t+T), 6%c(t+T)] as s 6% c follows: Fig. 3. The waveform of the converter input current.
Fig. 2c, the circuit equations for the PWM component are obtained as
n
0 d is = 1/C dt 6c
−1/L 0
n n
n
is 0 + i, − 1/C x 6c
(1)
where is is the reactor current, 6c is the capacitor voltage, ix is the converter input current, and L and C are the inductance and capacitance of the LC filter, respectively. Since the amplitude of ix is equal to one of the DC output current idc, ix can be treated as a constant in one sampling period. The current source generates a train of ix with respect to the switching states of S1 – S6, and the waveform of ix is illustrated in Fig. 3. Using the variables of the ON and OFF pulse widths, T2 and T3, defined in Fig. 3, the source current is(t+T) and capacitor voltage 6c(t + T) in one sampling period T can be derived from Eq. (1) as
is(t+ T) 6%c(t+T)
=
n
cos 0T sin 0T
+
− sin 0T cos 0T
n n is(t) 6%c(t)
n
cos 0T3 −cos 0(T2 +T3) i, sin 0T3 −sin 0(T2 +T3) x
T2 =
1 h2+i2 cos − 1 1− , 0 2
(3)
T3 =
1 h T2 tan − 1 − − , 0 i 2
(4)
where
n n h 1 = i ix
i*s cos 0T − 6%*c sin 0T
− sin 0T cos 0T
n n is(t) 6%c(t)
.
(5) The operational region of the converter is derived from the second term of Eq. (2) and the constraints of 05 T1 5 T, 05 T2 5 T, 05 T3 5T and T1 + T2 + T3 = T, and is shown in Fig. 4 as a shaded area on the state variable plane in the case of 0T= p/2. The boundary lines are defined by T1 = 0 and T3 = 0, and the cross-section of this region depends on the rotated angle 0T. When the state vector [h, i] (A) calculated from Eq. (5) exists within this region, the converter can realize the references in one sampling period. On the other hand, when the state vector [h%, i%] (B) is out of the region, namely, the references change largely and consequently the pulse width of ix is limited by the length of the sampling period T, the converter needs more than two sample periods to realize the references. For this case, we adopt
(2)
where 6%c is equal to 6c/w0L, and 0 is the filter resonant angular frequency (LC) − 1/2. The former term of this equation represents the free response component depending on the sample-and-hold values [is(t), 6%c(t)], and means that the state vector [is(t), 6%c(t)] rotates at the angle velocity of 0. The latter term represents the component controlled by the PWM process. In theory, to control two quantities in one sampling period, two manipulat-
Fig. 4. The operational region normalized by the converter input current on the state variable plane.
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another control algorithm in the simulation mentioned later; the converter controls the reactor current and capacitor voltage to realize the substituted state vector C on the boundary line, nearest to B. In this algorithm, the converter repeats this process while the state vector calculated in every sampling period is out of the operational region, and realizes the references in one sampling period when the state vector comes into this region. The evolution of this method to the three-phase circuit is considered. The control scheme of the converter shown in Fig. 1 is as follows: First, the references of active and reactive currents [i*, d i*] q are calculated from the DC output current idc, its reference i*dc and the power factor reference cos *. Second, using a two-phase to three-phase transform (dq transform) matrix, the references of the reactor currents [i*sa, i*sb, i*sc ] are obtained from [i*, d i*] q and the source voltage phase q. Third, the references of the capacitor voltages [6*sa, 6*sb, 6*sc ] are calculated from [i*sa, i*sb, i*sc ]. Finally, the pulse widths of the converter input current for each phase is calculated according to the way mentioned previously. However, for the three-phase current-type PWM converter, there are two constraints of the converter input currents: 1. The sum of the three-phase converter input currents [i*xa, i*xb, i*xc ] must be zero; ixa + ixb + ixc = 0. 2. All or one of the three-phase converter input currents must be zero. Namely, for two phases, the pulse widths of the converter input currents can be determined independently, while for the remaining phase, the pulse width must be calculated from results of the other two phases dependently. There are some methods to satisfy the constraints, and we adopt the following algorithm: 1. Calculate the pulse widths of the converter input current for each phase independently. 2. Compare the ON pulse widths of the three phases. 3. Determine again the pulse widths of the phase that has the second ON pulse width to satisfy the above constraints. The example of this process is illustrated in Fig. 5, and in this case, the pulse of the phase-b converter input current is dependently determined and conse-
Fig. 5. Evolution of the deadbeat control for the three-phase circuit.
quently split. For such a phase as the phase-b, however, the references are realized in one sampling period because of the three-wire circuit conditions of isa + isb + isc = 0 and 6ca + 6cb + 6cc = 0. 3. Simulation results To verify the newly proposed method, we carried out the numerical simulation. The circuit shown in Fig. 1 was modeled by substituting the three controllable current sources for the converter, and the constants used for simulation are summarized in Table 1. In this simulation, we assumed the DC output current is constant, that is, the amplitude of the converter input currents is constant. The resonance frequency of the LC filter is set to be relatively high (275 Hz) compared with the control frequency, similarly to ordinary largecapacity converters. The simulation results of a step response test are shown in Fig. 6. In this case, the converter operates at the power factor of unity, and the reference of the reactor current amplitude was suddenly reduced from 20 to 15 A and increased inversely when the phase of the phase-a reactor current was p/2 (the worst case for the transient oscillation). The result shows that the reactor current and capacitor voltage were controlled in Table 1 Constants of the PWM converter for the circuit simulation filter reactor L filter capacitor C source voltage es source voltage frequency fs sampling frequency DC output current idc
2 mH 167 mF 200 V 50 Hz 2 kHz 100 A constant
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Fig. 6. The simulation results of a step response test: reactor currents and capacitor voltages.
one sampling period without the transient oscillation against the both step-changes. Fig. 7 shows the simulation results of the step response in the case that the pulse width was limited owing to the large-step change of the reactor current references of 40 A. The references were realized in three sampling periods for the step-down, and in four sampling periods for the step-up. The both results indicate that the control algorithm functioned well, and the setting time
was sufficiently short even in the case that the pulse width was limited.
4. Discussions One of the problems of this method is that a long calculation time should be required because of a number of computations of trigonometric and its inverse functions for generating pulses. In
Fig. 7. The simulation results of the step response test in the case that the references are out of the operational region: reactor currents and capacitor voltages.
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practice, we plan to employ the table-look-up method; namely, the numerical data table enough to ensure the precision of the control such as trigonometric functions is previously provided on the hardware memory. Recent increase of the memory size has allowed the realization of this method. Another problem of this method is that the pulse width is limited by the dead time TD to calculate the pulse. It is impossible to generate the pulses of the converter input currents during the dead time TD, and thus this limitation reduces the operational region of the converter. However, the same algorithm as mentioned previously can be applied to the control; in the case that the pulse width is limited, the converter controls the reactor currents and capacitor voltages to the substituted point on the boundary line, which is newly defined by T1 =TD.
5. Conclusions We can conclude from the simulation results and discussions that the newly proposed method, which is able to realize the very short setting time and suppress the transient oscillation even at low PWM frequency, can be applied to the large-capacity current-type PWM converter for the superconducting magnets in future tokamaks.
References [1] Y. Sato, T. Kataoka, State feedback control of currenttype PWM AC-to-DC converters, IEEE Trans. Ind. Applicat. 29 (1993) 1090 – 1097. [2] M. Matsukawa et al., Dead beat control of current and voltage produced across an input filter in a current-source type three-phase PWM converter, Proc. of Int. Power Electronics Conf., Tokyo (2000) pp. 917 – 922.
A deadbeat control method for a PWM converter applied to a superconducting magnet Y.M. Miura a,*, M. Matsukawa a, H. Nakano b a
b
Japan Atomic Energy Research Institute, Mukouyama, Naka-machi, Naka-gun, Ibaraki 311 -0193, Japan Department of System Engineering, Ibaraki Uni6ersity, Nakanarusawa, Hitachi-shi, Ibaraki 316 -8511, Japan
Abstract Current-type pulse-width-modulated (PWM) converter is regarded as one of the candidates for the superconducting magnet power supplies in next generation tokamak devices. However, this type PWM converter has a problem of the transient oscillation of the reactor current and capacitor voltage in an AC-side harmonic filter, which occurs owing to the LC resonance of the filter when the reference of the DC output current is changed rapidly. In this paper, the new PWM control method based on the deadbeat control theory is proposed to suppress the transient oscillation. The converter employing this method can control the reactor current and capacitor voltage in one sampling period under ideal conditions. In addition, the effect of dead time on the control is discussed. The performance of the newly proposed method is demonstrated through numerical circuit simulation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: PWM; Superconducting magnet; Tokamak
1. Introduction Next generation tokamak devices require superconducting poloidal field coils. In the case that conventional thyristor converters with phase control are applied to these coil power supplies, the power factor becomes very poor in average, and they require huge reactive-power compensators. From this viewpoint, current-type pulse-widthmodulated (PWM) converters are regarded as one of the candidates for the superconducting magnet power supplies in next generation tokamak * Corresponding author. Tel.: + 81-29-270-7439; fax: + 8129-270-7459. E-mail address: [email protected] (Y.M. Miura).
devices because they can work at the power factor of unity. In addition, the current-type PWM converters have an advantage over the voltage-type ones because it is easy to replace the conventional thyristor converters. However, the current-type PWM converters have a problem of the transient oscillation of the AC-side current and voltage. In general, it requires a harmonic filter located on the AC-side to reduce harmonics flowing out to a commercial power system. Since the harmonic filter consists of reactors and capacitors, the transient oscillation of the reactor current and capacitor voltage appears when the reference of the DC output current changes rapidly, owing to the LC resonance of the filter. This transient oscillation might continue in several ten sampling periods,
0920-3796/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 3 7 9 6 ( 0 1 ) 0 0 2 8 4 - 8
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Y.M. Miura et al. / Fusion Engineering and Design 58–59 (2001) 57–62
Fig. 1. The schematic diagram of the current-type PWM converter.
and consequently degrade the control characteristic of the converter. In order to suppress this transient oscillation, Sato has proposed to employ the state-feedback control of the reactor current and capacitor voltage [1]. This method is based on the control theory in the continuous-time system, and the setting time is insufficiently long at low PWM frequency. Matsukawa investigated the application of the deadbeat control in the discrete-time system. In this method, the instantaneous values of the reactor current and capacitor voltage are controlled in two sampling periods [2]. Improving this method, we propose the new deadbeat control method in which they can be controlled in one sampling period. Therefore, a very short setting time is achieved even at low PWM frequency.
controllable switching devices such as insulated gate bipolar transistors (IGBTs). The single-phase equivalent circuit for the AC-side can be derived from Fig. 1 as shown in Fig. 2a. Moreover, it can be divided into two circuits: a voltage-source circuit (Fig. 2b) and a current-source circuit (Fig. 2c). The former represents the uncontrollable sinusoidal component depending on the voltage source, while the latter represents the component of the PWM control by the current source. Since it is easy to take the sinusoidal component into consideration by superimposing it on the PWMcontrolled component, we consider only the current-source circuit hereafter. With reference to
2. Deadbeat control method We first introduce the new method using a single-phase equivalent circuit for the AC-side of the current-type PWM converter. The typical configuration of the three-phase PWM converter shown in Fig. 1 consists of three parts: a threephase-bridge converter, an LC filter and an inductive load. In the bridge, S1 – S6 represent
Fig. 2. (a) The AC-side single-phase equivalent circuit of a current-type PWM converter, (b) a voltage-source circuit, (c) a current-source circuit.
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ing input quantities are necessary. In this case, they are the ON and OFF pulse widths of the converter input current, T2 and T3. The timings T2 and T3 to realize reference inputs [i*, *] s 6% c in one sampling period can be calculated from Eq. (2) by substituting [i*, *] for [is(t+T), 6%c(t+T)] as s 6% c follows: Fig. 3. The waveform of the converter input current.
Fig. 2c, the circuit equations for the PWM component are obtained as
n
0 d is = 1/C dt 6c
−1/L 0
n n
n
is 0 + i, − 1/C x 6c
(1)
where is is the reactor current, 6c is the capacitor voltage, ix is the converter input current, and L and C are the inductance and capacitance of the LC filter, respectively. Since the amplitude of ix is equal to one of the DC output current idc, ix can be treated as a constant in one sampling period. The current source generates a train of ix with respect to the switching states of S1 – S6, and the waveform of ix is illustrated in Fig. 3. Using the variables of the ON and OFF pulse widths, T2 and T3, defined in Fig. 3, the source current is(t+T) and capacitor voltage 6c(t + T) in one sampling period T can be derived from Eq. (1) as
is(t+ T) 6%c(t+T)
=
n
cos 0T sin 0T
+
− sin 0T cos 0T
n n is(t) 6%c(t)
n
cos 0T3 −cos 0(T2 +T3) i, sin 0T3 −sin 0(T2 +T3) x
T2 =
1 h2+i2 cos − 1 1− , 0 2
(3)
T3 =
1 h T2 tan − 1 − − , 0 i 2
(4)
where
n n h 1 = i ix
i*s cos 0T − 6%*c sin 0T
− sin 0T cos 0T
n n is(t) 6%c(t)
.
(5) The operational region of the converter is derived from the second term of Eq. (2) and the constraints of 05 T1 5 T, 05 T2 5 T, 05 T3 5T and T1 + T2 + T3 = T, and is shown in Fig. 4 as a shaded area on the state variable plane in the case of 0T= p/2. The boundary lines are defined by T1 = 0 and T3 = 0, and the cross-section of this region depends on the rotated angle 0T. When the state vector [h, i] (A) calculated from Eq. (5) exists within this region, the converter can realize the references in one sampling period. On the other hand, when the state vector [h%, i%] (B) is out of the region, namely, the references change largely and consequently the pulse width of ix is limited by the length of the sampling period T, the converter needs more than two sample periods to realize the references. For this case, we adopt
(2)
where 6%c is equal to 6c/w0L, and 0 is the filter resonant angular frequency (LC) − 1/2. The former term of this equation represents the free response component depending on the sample-and-hold values [is(t), 6%c(t)], and means that the state vector [is(t), 6%c(t)] rotates at the angle velocity of 0. The latter term represents the component controlled by the PWM process. In theory, to control two quantities in one sampling period, two manipulat-
Fig. 4. The operational region normalized by the converter input current on the state variable plane.
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another control algorithm in the simulation mentioned later; the converter controls the reactor current and capacitor voltage to realize the substituted state vector C on the boundary line, nearest to B. In this algorithm, the converter repeats this process while the state vector calculated in every sampling period is out of the operational region, and realizes the references in one sampling period when the state vector comes into this region. The evolution of this method to the three-phase circuit is considered. The control scheme of the converter shown in Fig. 1 is as follows: First, the references of active and reactive currents [i*, d i*] q are calculated from the DC output current idc, its reference i*dc and the power factor reference cos *. Second, using a two-phase to three-phase transform (dq transform) matrix, the references of the reactor currents [i*sa, i*sb, i*sc ] are obtained from [i*, d i*] q and the source voltage phase q. Third, the references of the capacitor voltages [6*sa, 6*sb, 6*sc ] are calculated from [i*sa, i*sb, i*sc ]. Finally, the pulse widths of the converter input current for each phase is calculated according to the way mentioned previously. However, for the three-phase current-type PWM converter, there are two constraints of the converter input currents: 1. The sum of the three-phase converter input currents [i*xa, i*xb, i*xc ] must be zero; ixa + ixb + ixc = 0. 2. All or one of the three-phase converter input currents must be zero. Namely, for two phases, the pulse widths of the converter input currents can be determined independently, while for the remaining phase, the pulse width must be calculated from results of the other two phases dependently. There are some methods to satisfy the constraints, and we adopt the following algorithm: 1. Calculate the pulse widths of the converter input current for each phase independently. 2. Compare the ON pulse widths of the three phases. 3. Determine again the pulse widths of the phase that has the second ON pulse width to satisfy the above constraints. The example of this process is illustrated in Fig. 5, and in this case, the pulse of the phase-b converter input current is dependently determined and conse-
Fig. 5. Evolution of the deadbeat control for the three-phase circuit.
quently split. For such a phase as the phase-b, however, the references are realized in one sampling period because of the three-wire circuit conditions of isa + isb + isc = 0 and 6ca + 6cb + 6cc = 0. 3. Simulation results To verify the newly proposed method, we carried out the numerical simulation. The circuit shown in Fig. 1 was modeled by substituting the three controllable current sources for the converter, and the constants used for simulation are summarized in Table 1. In this simulation, we assumed the DC output current is constant, that is, the amplitude of the converter input currents is constant. The resonance frequency of the LC filter is set to be relatively high (275 Hz) compared with the control frequency, similarly to ordinary largecapacity converters. The simulation results of a step response test are shown in Fig. 6. In this case, the converter operates at the power factor of unity, and the reference of the reactor current amplitude was suddenly reduced from 20 to 15 A and increased inversely when the phase of the phase-a reactor current was p/2 (the worst case for the transient oscillation). The result shows that the reactor current and capacitor voltage were controlled in Table 1 Constants of the PWM converter for the circuit simulation filter reactor L filter capacitor C source voltage es source voltage frequency fs sampling frequency DC output current idc
2 mH 167 mF 200 V 50 Hz 2 kHz 100 A constant
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Fig. 6. The simulation results of a step response test: reactor currents and capacitor voltages.
one sampling period without the transient oscillation against the both step-changes. Fig. 7 shows the simulation results of the step response in the case that the pulse width was limited owing to the large-step change of the reactor current references of 40 A. The references were realized in three sampling periods for the step-down, and in four sampling periods for the step-up. The both results indicate that the control algorithm functioned well, and the setting time
was sufficiently short even in the case that the pulse width was limited.
4. Discussions One of the problems of this method is that a long calculation time should be required because of a number of computations of trigonometric and its inverse functions for generating pulses. In
Fig. 7. The simulation results of the step response test in the case that the references are out of the operational region: reactor currents and capacitor voltages.
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practice, we plan to employ the table-look-up method; namely, the numerical data table enough to ensure the precision of the control such as trigonometric functions is previously provided on the hardware memory. Recent increase of the memory size has allowed the realization of this method. Another problem of this method is that the pulse width is limited by the dead time TD to calculate the pulse. It is impossible to generate the pulses of the converter input currents during the dead time TD, and thus this limitation reduces the operational region of the converter. However, the same algorithm as mentioned previously can be applied to the control; in the case that the pulse width is limited, the converter controls the reactor currents and capacitor voltages to the substituted point on the boundary line, which is newly defined by T1 =TD.
5. Conclusions We can conclude from the simulation results and discussions that the newly proposed method, which is able to realize the very short setting time and suppress the transient oscillation even at low PWM frequency, can be applied to the large-capacity current-type PWM converter for the superconducting magnets in future tokamaks.
References [1] Y. Sato, T. Kataoka, State feedback control of currenttype PWM AC-to-DC converters, IEEE Trans. Ind. Applicat. 29 (1993) 1090 – 1097. [2] M. Matsukawa et al., Dead beat control of current and voltage produced across an input filter in a current-source type three-phase PWM converter, Proc. of Int. Power Electronics Conf., Tokyo (2000) pp. 917 – 922.