Analysis of Torque Ripple Reduction in Induction Motor DTC Drive with Multiple Voltage Vectors.
Since the end of the 1980s [1-2] up to now, direct torque control (DTC) has been the subject of analyses of a vast number of scientific papers. Good dynamic characteristics, small sensitivity towards machine parameters variations, simplicity of algorithms without inverse transformations and complicated calculations make this type of motor control very attractive. However, the shortcomings of the classical DTC, such as big torque ripple and changing switching frequency impede its implementation in electric drives. These two major drawbacks of the classical DTC are the consequence of the hysteresis comparators for torque and flux that make the classical DTC simple and without complex calculations. The requirements for defining the constant switching frequency together with elimination of the torque ripple have provided a number of different DTC algorithms . By introducing the space vector modulation (SVM) or PWM, the direct torque control has permitted the significant torque ripple decrease [4-6]. This improvement has been made by introducing the PI regulator at the expense of the system's bandwidth reduction. The PI regulator output, as the continuous voltage references in the synchronous rotating dq reference frame, requires the inverse transformations that further increases the processor's calculating time. Introducing the fuzzy logic control may also provide good dynamic performances and reduce torque ripple .
A lot of authors propose different approaches with the aim of the torque ripple reduction. One of these approaches is finding out the optimal duty ratio of the PWM registers that corresponds to the necessary reference voltage in each switching cycle [8-10]. These algorithms require the longer calculating time due to the relatively more complex calculation with regard to the classical DTC. Also, the precise determination of the machine flux is of the essential importance in these algorithms, which requires more advanced estimation methods. This particularly applies to the Dead Beat Direct Toque Control (DBDTC) algorithms [11-12], based on reaching the torque reference in the smallest possible number of inverter switching cycles. DBDTC algorithms provide good results in terms of response time and torque ripple reduction, but to the detriment of the longer calculating time that requires more powerful processor.
If the point lies in the torque ripple reduction, as well as in defining the constant switching frequency in the classical DTC approach the most frequent proposals incline to the online adaptation of the torque hysteresis band [13-14]. The FPGA processor implementation and the possibility of considerable reduction of calculating and switching time [15-16] may attenuate the shortcomings of the classical DTC. The multilevel inverter implementation may also influence the torque ripple reduction, having at its disposal the larger number of voltage vectors [17-18]. Nevertheless, all these methods require the elements that raise the DTC drive's price, both due to the more complex inverter structures and the higher performance processors.
In contrast to the mentioned papers, this one presents a modified algorithm of the classical DTC. In addition to this, the proposed algorithm aims at retaining the good properties and simplicity of the classical DTC approach and introduces the changes that would fulfil the requirements for torque ripple reduction and the constant switching frequency. The algorithm is based on the variable intensity voltage vectors implementation and the torque hysteresis comparator modification. In , similar proposition was given for torque ripple reduction. The algorithm calculation time is divided in two or three switching periods with different active voltage vectors, thus forming higher number of available voltage vectors. In this way algorithm calculation time is two or three times longer than switching frequency. The last decade has shown significant increase of DSP calculation capabilities and clock frequency while inverter switching frequency basically stayed the same (rarely higher than 20kHz). The preferred reduction of the calculation time of the algorithm shown in  leads to unwanted increase in the switching frequency and switching losses. In contrast to this, the DTC algorithm proposed in this paper suggests combination of only one active and zero voltage vector during the switching cycle. In this way algorithm calculation time can be reduced to the switching cycle duration.
The paper seeks to demonstrate the experimental results of the proposed algorithm and the estimated torque analysis between the two sampling periods. Additionally, the torque ripple analysis has been performed for a larger number of the available voltage vectors.
II. THEORETICAL OVERVIEW OF CLASSICAL AND PROPOSED DTC ALGORITHM
One of the main direct torque control types of a voltage inverter powered induction machine is the stator flux direct control, by means of the optimal inverter switching state selection . The machine torque Te may be presented as the interdependence of stator and rotor flux in the following way:
[mathematical expression not reproducible] (1)
where, P - refers to pole pair number; Ls,Lr - refer to stator and rotor inductivity; Lm - mutual inductivity; [[psi].sub.s],[[psi].sub.r]-stator and rotor flux modules; [[psi].sub.] ,[[theta].sub.[psi]] - stator and rotor flux angles in relation to the reference axis.
In this way, the fast change of the stator flux in relation to the rotor one enables the direct torque control according to (1).
By selecting one of 6 active three phase inverter voltage vectors (Fig. 1) it is possible to influence the intensity and position of the stator flux in relation to the slowly changing rotor flux. In this way, the need for changing the instantaneous amplitude or angle of the flux vector, thus changing the instantaneous value of the motor torque may be provided by selecting the appropriate voltage vector. The change of the stator flux vector during the time At is illustrated in Fig. 1.
The selection of the necessary voltage vectors depends on demands for flux and torque change and may be presented by the switching Table I, where k is the number of section (Sk) shown in Fig. 1, within which the stator flux is located.
The demands for the flux [S.sub.w] and the torque Sm are obtained through the hysteresis comparators, shown in Fig. 2.
Depending on the estimated stator flux and torque values, demands are generated for their increase S=+1, reduction S=-1, or zero demand is generated S=0.
In contrast to the classical DTC, for which full voltage vector is applied during the complete switching cycle, in the proposed algorithm the variable intensity voltage vector is applied. Furthermore, in classical DTC, inverter's switching elements do not change their state during the whole cycle thus defining zero or one out of six active voltage vectors. During the application of any one of active voltage vectors, the full value of the DC bus voltage is applied on the motor during the period of the switching cycle At. Due to that, depending on the motor speed and torque and flux demands, large ripples of flux and torque occur. According to  and , taking as a starting point the equation (1) it is possible to arrive at the equations (2), (3), (4) and (5), which define the voltage vector impact on the torque increment during the time At.
[mathematical expression not reproducible] (2)
[mathematical expression not reproducible] (3)
[mathematical expression not reproducible] (4)
[mathematical expression not reproducible] (5)
where, [mathematical expression not reproducible] Rs- stator resistance, Rr - rotor resistance and [[omega].sub.m](t) - motor speed at the time instant t.
The first member of torque increment AT 1 has a negative sign and reduces the torque during the time At. It comprises the influence of stator resistance and rotor time constant on torque. The second member AT2 represents the influence of applied voltage vector on torque increment. This member has a significantly larger influence on torque in comparison to the first member and also depends on the motor speed. Equation (4) clearly shows that the voltage vector intensity has the decisive influence on torque increment during the inverter switching cycled.
With the introduction of PWM, it is possible to define several voltage vectors of the same direction but different intensity. In this way, we should have more voltage vectors at our disposal. Let us assume that stator flux vector is located in section 2 at the time instant t. If the demands for flux and torque increase are active, the voltage vector U3 would be applied according to the Table I. If we had three intensities of the voltage vector U3 at our disposal, we could choose between three corresponding flux vectors (and therefore three different torque increment values) in the following instant t+[DELTA]t, as shown in Fig. 3.
In this way, the application of the appropriate voltage vector will enable the torque ripple reduction during the cycle At. It is necessary to introduce the multilevel torque comparator in order to enable the application of different voltage vector intensities. Therefore, it is necessary to define 7-level hysteresis comparator for three different voltage vectors' intensities (Fig. 4).
According to equations (2) to (4), the machine torque decreases during the zero voltage vector application. For this reason, the zero voltage vector may be treated as the torque reducing voltage vector. Furthermore, the positive torque increment is always smaller than the negative torque increment with motor speed greater than zero. This effect is particularly noticeable at higher speeds. A detailed analysis of the speed impact on positive and negative torque slopes for the possible voltage vector scope in the stationary [alpha][beta] reference frame has been conducted in the literature . The analysis shows that the torque's positive increment intensity and the selection of the voltage vector that should enable the torque increase significantly decrease at higher rotor speed.
Torque hysteresis comparator in Fig. 4 can be modified in order to mitigate this effect so that the torque reference lies at the bottom of the zero vector sector. This leaves space for higher intensity vector application if the motor torque is smaller than the reference. In this way, the modified multilevel torque hysteresis will appear as shown in Fig. 5.
The inverter's constant switching frequency is defined by introducing the PWM, whereas the application of appropriate voltage vector is defined by the proposed algorithm with the aim of torque ripple reduction. The comparison between the classical DTC and the multilevel DTC method with variable intensity voltage vectors is given in , where the hysteresis' width influence and voltage vector intensity influence on torque ripple was analysed.
III. THE ANALYSIS OF EXPERIMENTAL RESULTS AND COMPARISON OF THE PROPOSED AND CLASSICAL DTC METHOD
The classical and proposed DTC algorithms have been implemented and tested on the DSP platform MSK2812 shown in Fig. 6 . The platform disposes of the TMS320F2812 fixed point processor with 150MHz (6.67ns) clock frequency. The two-level inverter module consists of 6 IGBT switches with 310V DC bus. The inverter switching frequency is set up at 20 kHz thereby defining the one-cycle duration of 50[micro]s.
Fig. 7 provides the layout of the key operations during the cycle. The interrupt service routine (ISR) which initiates the PWM update, AD current conversion and start of DTC calculation is carried out for each counter (cnt) underflow. All DSP data is obtained over serial communication with DSP F2812 and DMC Developer Pro software.
In classical DTC, the inverter branch switch is either open or closed in the course of the whole 50[micro]s cycle duration. By contrast, the voltage value (the PWM duty ratio) depends on the applied voltage vector intensity and varies from 0 to 100% for the proposed DTC method. The algorithms have been tested on the unloaded three-phase induction motor. The motor data are provided in the Appendix. Flux estimation is realized by open loop estimator based on machine current model , described by equations (6-9):
[mathematical expression not reproducible] (6)
[mathematical expression not reproducible] (7)
[mathematical expression not reproducible] (8)
[mathematical expression not reproducible] (9)
The reason for choosing this flux estimator is its simplicity, short calculation time and satisfactory results in low speed region. The illustration of the proposed modification of the DTC algorithm is not influenced by the flux estimator implementation. The flux estimator described in  is very well known, and is based on measured motor currents and motor speed. Such an estimator could easily be applied to both on-line and off-line processing of data, as described in section IV. Flux hysteresis comparator level was placed on 1% of the motor's nominal value [[psi].sub.bw] =0.01Wb). Torque hysteresis comparator level for the classical DTC was set for the value of 10% of the motor's nominal torque Tbw_c=0.129Nm. Since the classical DTC torque hysteresis disposed of 3 levels, the modified 7-level torque hysteresis width (according to Fig. 5) is defined by (10):
[mathematical expression not reproducible] (10)
Torque hysteresis comparator width, with higher or smaller level number, is defined in analogy with (10).
Fig. 8 shows the obtained results for the classical DTC and the proposed DTC with 7-level torque hysteresis comparator (4 voltage vector intensities - the total of 24 active vectors). The torque reference is given in the form of the periodical square wave with amplitude set to [+ or -]30% of the motor nominal torque, which equals to [+ or -]0.387Nm.
Fig. 8 shows the estimated torque, flux, velocity and motor current for the classical DTC (on the left hand side) and the proposed DTC algorithm (on the right hand side) for the duration interval of 0.2s. The machine magnetization was enabled along the [alpha] axis up to the reference flux of 0.7Wb (0.67p.u. due to the reduced DC bus voltage) at the beginning of the experiment, after which the torque reference is released. It is clearly noticeable that the proposed algorithm with the multilevel hysteresis gives a considerably reduced torque and current ripple. The reduced ripple is also noticeable in the stator flux, as a consequence of the application of voltage vectors of smaller intensity. Small divergence of the actual (estimated) torque from the reference (Fig. 8a) is caused by influence of motor speed, the resultant EMF and torque hysteresis limits. During the transient, as well as in the steady state, estimated torque has static error depending on motor EMF, i.e. the motor speed. Multilevel hysteresis comparator selects the voltage vector with appropriate intensity to overcome EMF in equation (4) according to the magnitude of the torque error. The error is bounded to the maximum level of torque comparator and can be moderated by adaptation of hysteresis limits, as well as by introduction of voltage feed-forward action which will cancel out the influence of the EMF. However, in many applications, the torque control loop is enclosed in the outer speed control loop, that would compensate the existing torque error.
Fig. 9 displays the data collected at the end of each calculation cycle, i.e. for 50[micro]s, in order to obtain a more precise insight in the torque ripple information. Figures 8 and 9 confirm a significant torque ripple reduction. The torque ripple in the proposed method is approximately 4-5 times smaller for only 4 defined intensities of the voltage vectors and the appropriate 7-level torque hysteresis. In the diagrams with smaller time frame (Fig. 9(b)) the considerably smaller torque increment for the proposed algorithm (on the right) in comparison to the classical DTC (on the left) is visible.
By adequate choice of number of levels in the multilevel hysteresis comparator, resulting in corresponding number of different voltage vector intensities, it is possible to significantly reduce the torque ripple or render it to the appropriate level. Also, the reduced stator current ripple is reflected in the harmonic losses reduction as well as the reduction of the acoustic noise level. The proposed algorithm enables also the retention of good dynamic characteristics of the classical DTC. Namely, the higher intensity vector is applied in case of the larger difference between the reference and estimated torque thereby enabling the fast response and the preserved drive dynamics with the high-bandwidth control. The exact position of the stator flux is not crucially significant for the voltage vector selection, but rather solely a sector in which flux is located. Instead of sophisticated flux estimation methods [23-25], current model estimator (6-9) was used. As the proposed algorithm does not depend on machine parameters and precise knowledge of stator flux position, simpler flux estimators that do not depend on speed can be utilized. For example, by using estimator based on voltage integration [26-27], proposed DTC algorithm (including flux estimator) will require only the knowledge of machine stator resistance Rs . This preserves the simplicity of the classical DTC and the enables the proposed algorithm implementation on the processors of lower computational performances.
Combination of three or more active voltage vectors in ,  requires a new, more complex switching table depending on available voltage vectors. On the other hand, the method proposed in this paper is based on standard DTC switching table (Table I) and uses 6 basic active voltage vectors with different intensities depending on the torque error level. Further increase in number of active voltage vectors does not require the creation of new switching tables. Necessary hysteresis level settings can be easily automatized and related with the desired number of voltage vector intensities.
IV. OFFLINE PARALLEL TORQUE ESTIMATION WITH STATOR CURRENT OVERSAMPLING
The previous analysis of the data, collected with sampling interval of 50[micro]s, clearly indicate the advantages of the proposed algorithm in terms of torque ripple reduction. In the given graphs, the displayed values have been acquired at the end of the calculation algorithm on DSP and they retain the constant values during the whole computation and the switching cycle (the stepped graph in Fig. 9(b)). However, it is of great interest to analyse the current and torque waveforms within the time interval of 50[micro]s, that is, between two sampling periods (ISR).
The analysis of results, collected with sampling frequency of 6.25MHz, has been performed. This acquired 312 points of motor current and voltage in the one calculation cycle interval comprising 50[micro]s. Current and voltage measurement has been performed independently of and in parallel with the DSP performing the presented DTC algorithms. Current measurement has been performed by means of current probes (Table IV) in two motor phases. The signal acquisition has been carried out by the mixed signal oscilloscope with recording length of 1Mpoints per channel. The results have been collected for 5 channels: 2 currents (a and b motor phase currents) and 3 PWM signals (upper inverter leg switches).
Fig. 10 displays the acquired current values with DSP (50[micro]s) and the same current values collected by oscilloscope - OSC (0.16[micro]s) in time interval of 0.2s. Fig. 10(a) shows the stator currents in [alpha] and [beta] axes for classical DTC (on the left hand side) and the proposed DTC algorithm (on the right hand side). At first, the pronounced current noise is noticeable, which is the consequence of the inverter's transistors commutation. This noise is considerably pronounced for the proposed DTC algorithm, which is logical given the PWM modulation nature. In classical DTC the inverter changes its switching simultaneously (at the beginning of the cycle) at all three inverters' phases.
For symmetrical PWM, which is utilized in the proposed algorithms, the switching state change is not simultaneous and depends on the applied voltage intensity. Hence, the switching elements commutation impact of each inverter branch can occur individually during one switching cycle of 50[micro]s.
The delay in motor current signals measured by DSP, also evident in Fig. 10(c), is introduced by the low pass anti-aliasing filter of the current-measuring circuit on DSP board. The RC time constant of the filter gives the delay in the signal of 30ns. Nevertheless, it is evident from Fig. 10(b) that the current ripple, which is the consequence of the DTC algorithm itself, is significantly reduced by the proposed algorithm. Due to its high switching frequency, the switching noise exceeds the audible range and influences neither the drive's dynamics nor performance.
In order to compare the differences in current waveforms with respect to the applied algorithm, it was necessary to reduce the commutation noise by averaging the obtained signals with zero phase shift [29-30]. The signals are filtered offline by the moving average (12 samples) filter, in both the forward and reverse direction, attenuating spectral components higher than 500 kHz. The time diagrams of the obtained filtered current signals are shown in Fig. 11.
The filtered stator current values are further used for offline flux and torque estimation. The estimation results are compared with estimated values simultaneously recorded in DSP. In this way, insight into the behaviour of the estimated torque between two DSP samples (50[micro]s) is provided. The results are shown in Fig.12.
The online estimated torque values on DSP (50[micro]s) and offline estimated torque values (0.16[micro]s) are given in Fig. 12. Flux and torque estimation algorithm, utilized for offline estimation, is identical to the utilized algorithm on DSP. The delay due to anti-aliasing filtration of current values used by DSP is mapped to the estimated torque as well. The estimated torque change in is almost linear between consequent DSP sampling points. Yet, it is possible to notice in Fig. 12(c) that the torque values (OSC) for classical DTC may exceed the torque values that are "seen" by the DSP. These deviations are noticeable both at increase and decrease of torque, and are marked in Fig. 12c (on the left hand side) with "*". In this way, the DSP may register somewhat smaller torque ripple values than the actual. If the switching commutations noise is disregarded, this effect almost does not exist for the proposed DTC algorithm. The torque increments are smaller during one cycle due to the application of the smaller intensity vector, therefore, the actual torque does not deviate significantly from the DSP values sampled at 50[micro]s.
Fig. 13(a) displays the online and offline estimated torque values of the classical and proposed DTS, in smaller time range. In addition to this, the upper transistor gate command have been shown for all three inverter phases (T1, T3 and T5), as well as the voltage vector selection and its percentage intensity in relation to the full vector. It is noticeable in Fig. 13(c) (on the left hand side) that the classical DTC takes voltage vectors of different direction and full intensity to fulfil the demands per flux and torque. In case of the proposed DTC algorithm (on the right hand side), the torque increments are smaller due to application of voltage vectors with smaller intensity, as well as the smaller variation in the direction of the voltage vector.
V. TORQUE RIPPLE ANALYSIS FOR VARIOUS VOLTAGE VECTOR INTENSITIES
The results obtained by offline processing of the acquired current and transistor commands are congruent with values calculated by DSP with sampling interval of 50[micro]s, to a great extent. Small differences in these signals are the consequence of the fixed point arithmetic
of DSP, the offsets of the current probes and LEM sensors at DSP MSK2812 setup, dead time, as well as the low pass filter at the DSP measurement unit. Figures 12 and 13 show that the current and torque ripple are significantly reduced due to the application of variable intensity voltage vectors.
The following analysis of torque ripple in the offline estimated torque has been given depending on the number of the available voltage vectors in the proposed DTC algorithm. The ripple analysis is performed by measuring the estimated torque value deviation from its mean value. For the purpose of the analysis the range with positive torque is taken, in the experiments with torque reference as shown in Fig. 12. The selected time interval disposed of 600000 estimated torque values, which is quite sufficient for the analysis of this type. The proposed DTC algorithm has been analysed with 3, 4 and 5 different intensities of the active voltage vectors and the corresponding hysteresis comparators with 5, 7 and 9 levels. The obtained results are presented both graphically and numerically.
Fig. 14 displays the torque ripple for the classical and proposed DTC with 3, 4 and 5 voltage vector intensities. The average torque value shown on the left hand side in Fig. 14 is obtained by numerically fitting exponential curve to the recorded estimated torque values. The torque ripple, calculated as the absolute value of the deviation of the estimated torque from its average value, is shown in Fig. 14 (on the right hand side). The mean value of torque ripple, as well as the ripple ratio for the classical and proposed DTC methods, is given numerically in Table II.
The results shown in Table II indicate that the torque ripple is reduced drastically by increasing the number of different voltage vector intensities. The torque ripple is reduced by a factor of 2.72 compared to classic DTC in case of 3 voltage vector intensities. With 4 voltage vector intensities the torque ripple reduction factor is 4.68, and 6.59 in case of 5 vector intensities.
The analysis shows that it is possible to select an appropriate number of voltage vectors in order to meet demands of maximum torque ripple. Namely, this simple DTC algorithm allows placing torque ripple within defined limits with only a few changes in the code of the classical DTC algorithm, without the modification of the switching table.
Certainly, torque ripple cannot be quite minimized in this way as in case of PWM-DTC or DBDTC algorithms, due to finite number of comparator levels and use of the classic DTC switching table. Nevertheless, the proposed algorithm shows its possibility of significant torque ripple reduction without utilization of complex flux estimators, coordinate transformation, current controllers or PI regulators.
This paper proposes a modified DTC algorithm with multilevel torque hysteresis comparator based on classical DTC with two-level inverter. The proposed algorithm has retained both the simplicity and the advantages of the classical DTC algorithm. The torque ripple with only 4 predefined voltage vectors have been reduced more than 4 times without complicated calculations and PI regulators. The proposed DTC algorithm is convenient for processors with PWM unit, which is nowadays common even in low performance processors. The calculation duration for the multilevel hysteresis comparator based DTC algorithm with TMS320F2812 DSP is only 0.5[micro]s (1.2%-2%) longer than the calculation duration for the classical DTC. This negligibly small prolongation of the calculation time is a consequence of the simple commands at defining the comparator condition and voltage vector selection. This paper provides the comparative survey of currents and estimated torque sampled at 50[micro]s and oversampled at 0.16[micro]s. Good agreement of these data has been confirmed, and the torque behaviour has been analysed between two DSP samples (50[micro]s) for both classical and proposed DTC. Taking into consideration the torque values obtained with the sample of 0.16[micro]s, the torque ripple analysis for classical and proposed DTC has been performed. Graphical and numerical analysis for the torque ripple has been provided for the classical DTC and three proposed DTC methods with 3, 4, and 5 voltage vectors intensities. It has been shown that the ripple value significantly decreases with the introduction of a larger number of active voltage vectors with different intensities. This allows selection of an appropriate number of voltage vectors necessary to keep torque ripple within desired limits.
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Marko M. ROSIC, Milan Z. BEBIC
Faculty of Electrical Engineering, University of Belgrade, 11000 Belgrade, Serbia email@example.com, firstname.lastname@example.org
This work was supported in part by Ministry of Education, Science and Technological Development of Republic of Serbia within the project TR33016.
Digital Object Identifier 10.4316/AECE.2015.01015
TABLE I. SELECTION OF VOLTAGE VECTORS [S.sub.[psi]] [S.sub.m] 1 0 -1 1 [U.sub.k+1] [U.sub.7] or [U.sub.8] [U.sub.k-1] -1 [U.sub.k+2] [U.sub.7] or [U.sub.8] [U.sub.k-2] TABLE II. NUMERICAL TORQUE RIPPLE ANALYSIS IN CLASSICAL DTCAND PROPOSED DTC METHOD mean Torque classical/proposed torque ripple/nominal torque ripple ripple torque [%] ratio [Nm] Classical DTC 0.1311 10.16 1 (1 voltage intensity) Proposed 1 0.0482 3.74 2.72 (3 voltage intensities) Proposed 2 0.0280 2.17 4.68 (4 voltage intensities) Proposed 3 0.0199 1.54 6.59 (5 voltage intensities) TABLE III. SIEBER LS71 MOTOR PARAMETERS Un [V] 400 [R.sub.s] [[OMEGA]] 24.6 [I.sub.n] [A] 0.95 [R.sub.r] [[OMEGA]] 16.1 [P.sub.n] [W] 370 [L.sub.m] [H] 1.46 [n.sub.n] [[min.sup.-1]] 2860 [L.sub.s] [H] 1.48 p [pole num.] 2 [L.sub.r] [H] 1.48 TABLE IV. OSCILLOSCOPE, AMPLIFIER AND CURRENT PROBES Oscilloscope Tektronix MSO2014B Current probes Tektronix A6302,10A Amplifier Tektronix TM502A
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|Author:||Rosic, Marko M.; Bebic, Milan Z.|
|Publication:||Advances in Electrical and Computer Engineering|
|Date:||Feb 1, 2015|
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