An Online VSI Error Parameter Identification Method for Multiphase IM With Non-Sinusoidal Power Supply

Voltage-source-inverter (VSI) nonlinearity compensation is an important scheme for improve motor performance. The traditional off-line method is cumbersome and the compensation effect is poor because it does not simulate the real operation of the motor. To solve this problem, an on-line identification method for compensating multiphase machine based on non-sinusoidal power supply is proposed in this paper. First, a full order air-gap flux observer on harmonic plane was established. Then the voltage error is estimated based on a model reference adaptive system (MRAS) and compensated to the given voltage. In the process of identification, the adaptive law of dead-time voltage is designed, and a feedback gain matrix satisfying Lyapunov stability is also designed. Finally, the feasibility and the effectiveness for harmonic suppression of the proposed method are verified on an experimental platform of a seven-phase induction motor platform.


NOMENCLATURE
Rotation angle of the magnetic fields.  The off-line compensation method consists of look-up-table 53 (LUT) [16] and fitting curve [8], [17]. This kind of skills 54 is easy to understand but cumbersome in operation and not 55 universal. When the bus voltage or dead time was changed, 56 the off-line compensation method needs to be re-measured. 57 Online compensation methods also mainly include two kinds.

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One of them is approximates off-line fitting curve compensa-59 tion [18], [19], [20]. It can adjust the parameters of the fitting 60 function online to achieve that actual voltage is equal to given 61 voltage. However, these method did not take into account 62 voltage fluctuations at neutral points. The other approach is 63 to treat the VSI nonlinearity as a constant times the polar-64 ity of the current under synchronous coordinate frame [21], 65 [22], [23]. Then, the constant can be estimated by Low-Pass 66 Filter, Kalman-Filter and current injection [24], [25], [26]. 67 The method proposed in [24]  Lyapunov stability is designed in this paper.

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The outline of this paper is as follows. In section II, 90 we explained how the nonlinearity of VSI produce and estab-91 lish the voltage error model of traditional method. Then, 92 we analyzed the shortcomings of the traditional model and 93 establish the VSI model under synchronous coordinate frame. 94 In section III, we built a full-order flux observer for third har-95 monic plane based on AFOC and identified the voltage error 96 by MRAS method. Moreover, in the process of error voltage 97 adaptive law calculation, we designed a set of feedback gain 98 matrix satisfying Lyapunov's stability. Next, the algorithm 99 is validated on a self-designed seven-phase induction motor 100 experimental platform. Finally, the conclusion is drawn.

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A. NONLINEARITY OF VSI MODEL 103 As shown in Fig.1, the switch signal S 1 is 0 or 1. Take phase 104 A as an example, if i sa 0 and S 1 = 1, the voltage between 105 A and ground is u ao = V dc 2 − V sat , where V sat is the saturation 106 voltage of the active switch S 1 . If i sa 0 and S 1 = 0, u ao = 107 − V dc 2 −V d where, V d is the forward voltage of the antiparallel 108 diode. In case of i sa < 0, the terminal voltage u ao = V dc 2 + V d 109 when S 1 = 1. The voltage become − V dc 2 + V sat when S 1 turn 110 to 0. In general, the final form of the voltage can be expressed 111 as following [23]: where sgn (i sa ) is the sign function and defined as: In addition, to prevent the short-circuit caused by the upper 117 and lower IGBT in the same leg being turned on at the same 118 time, a dead time delay is usually applied to ensure that the 119 IGBT is completely turned off before the another IGBT in 120 the same leg conduct. Fig.2 (a) displays the reference signal 121 VOLUME 10, 2022

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In practice, the inverter on and off time needs to be taken into   Fig.3 shows the voltammetry curve after injecting direct 155 current into a single phase. The other curve is the voltammetry 156 curve of error voltage derived from (5). Generally, there are 157 two kinds of off-line feed-forward compensation methods: 158 LUT and curve fitting.

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For the multiphase induction motors, the third harmonic cur-161 rent injection can improve the distribution of the motor air 162 gap flux, so that it could achieve increment of the torque den-163 sity [31]. Furthermore, there are two mutually independent 164 coordinate planes are formed after PARK transformation with 165 third harmonic current injection. Fortunately, the extra plane 166 can also be used to identity voltage error.

168
According to the principle of linear superposition, the airgap 169 flux density can be expressed as: and k 3 = 1 6 are the optimized proportionality 172 coefficients using genetic algorithm [14], [32].

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In order to obtain better flux distribution, this paper adopt 174 a non-sinusoidal power supply control technique which is 175 based on airgap flux orientation [13]. The rms current is 176 reduced by 10.5% when the motor operates at rated load. The 177 given currents on the third harmonic plane are shown in (7a) 178 and (7b): Where: Because of the injection of the third harmonic current, the 186 third harmonic plane is formed which is independent of the 187 fundamental plane. As the harmonic plane is based on airgap 188 flux field orientation, x − y reference frame, its state equation 189 variables should be the currents and the airgap flux, which is 190 shown in: The offline measurement method is simple, but when the bus 222 voltage or dead zone parameters need to be adjusted, the 223 compensation curve needs to be manually measured off-line 224 again. Moreover, manual measurement method injects direct 225 current into only one phase and assumes that the potential at 226 neutral point N , in Fig.1, is the same as point O. However, 227 when the motor is running normally, the N point potential 228 is affected by the current of each phase [23] and it is not 229 equal to 0.

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As shown in Fig.1, after considering the voltage fluctuation 231 of N points, u no = 0, the actual phase voltage is: The phase voltage of the other phases is expressed in the 234 same way as phase A, which is expressed as following: In the case of symmetrical and stable operation of the 237 motor, the voltage fluctuation u no can be derived from 238 (10) and (11).  10) and (13), the voltage error of phase A 245 inverter can be obtained as (14).
Due to V dc is much larger than (V sat − V d ), the second 250 term in (14) can be neglected normally [25]. So (14) can be 251 simplified as: where: 257 VOLUME 10, 2022   It can be seen that u err is mainly influenced by V dc 258 and t dead .

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According to (15), when seven-phase sinusoidal current 260 is injected into the seven-phase induction motor, the volt-  where: Usually, IMs is analyzed under synchronous rotation 279 frame. Therefore, the VSI nonlinearity model can be calcu-280 lated from (16).
and T (θ r1 , θ m3 ) is represents the Park transformation. It is 285 noticed that fifth harmonic variables are not listed, because 286 fifth harmonic current was controlled to zero.

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Once VSI nonlinearity is taken into consideration, the actual 289 machine model in rotating reference frame on third harmonic 290 plane can be written as: where G 3 is the feedback gain matrix.
where: P is a symmetric positive definite matrix.

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The derivative of the Lyapunov function (21) becomes: According to the Lyapunov stability theorem, a sufficient 313 condition for the asymptotic stability of the estimator is that L r3 e isx3 f x3 + e isy3 f y3 λ u (L m3 L rσ 3 + L r3 L sσ 3 ) We assumed that: Obviously, is a fourth order real symmetric matrix. Q n 342 represents leading principal minor of Q, where n = 1, 2, 3, 4. 343 Substituting (25) into (26) can we obtain: In order to the satisfy Lyapunov stability theorem, Q 3 must 351 be a negative definite matrix. G 3 , a necessary not sufficient 352 condition for the stability is easy to be designed as:

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The proposed algorithm is validated on a seven-phase induc-361 tion motor experimental platform. Fig.7 displays the com-362 plete experimental platform, which is composed of DC power 363 supply, dSPACE-1005 controller, multiphase VSI and alter-364 nating current servo load. The multi-phase VSI is constructed 365 by several groups of IGBT device, filter capacitors, current 366 and voltage sensors, etc. In particular, the IGBT model is 367 the INFINEON FF150R12ME3G. The sampling frequency 368 of the platform is 10 kHz, and the dead time is set to 3 µs to 369 ensure the safety of the system. Table 1 and Table 2 exhibit 370 all parameters of the self-designed squirrel-cage seven-phase 371 induction motor. 372 VOLUME 10, 2022

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If the motor is running at high speed, the phase voltage of described in detail in the Appendix. The seven-phase machine 383 runs steadily at 30 rpm and rated-load condition. The algo-384 rithm starts at 0.5 s. Fig.8 (a) exhibits that u err estimated 385 by traditional method can converge to 6.34 V and 2.49 V 386 when the bus voltage is set 200 V and 100 V respectively. 387 When the bus voltage up to 400 V, traditional method has 388 poor performance. It can not converge to a constant. At V dc = 389 400 V, when the deadtime is set to 6 µs, the system crashed 390 after 1 second, which is illustrated in Fig.8 (b).

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However, the proposed method can operate stably under all 392 the above conditions and can converge stably to a constant. 393 In Fig.9 (a), the estimated error voltage converges to 3.95 V 394 when the bus voltage is 100 V. When V dc increase to 200 V, 395 400 V respectively,û err become 6.54 V and 11.52 V after 1 s. 396 In Fig.9 (b), it can be seen that when t dead is adjusted to 4 µs 397 and 6 µs under V dc = 400 V, estimated value of voltage error 398 converge to 15.25 V and 22.72 V correspondingly.  According to the IGBT parameters from data sheet, the 400 reference value of u err can be calculated, which is list in 401 Table 3. It can be seen that, under above conditions, the  As we all know, the long-term operation will result in the 418 temperature increasing of the motor and the winding resis-419 tance will change. Therefore, this paper also shows the perfor-420 mance of the proposed algorithm when the actual parameters 421 of the motor deviate from the control parameters. Fig.11 422 shows the VSI error identification curve when resistance 423 deviation occurs in the proposed method. In Fig.11, R sref is 424 the reference in the control system, and R sact is the actual 425 value. When the reference resistance is 20% greater than the 426 actual resistance, the estimate of u err will be an about 6.5% 427 reduction. Otherwise, when the reference resistance is 20% 428 smaller than the actual resistance, the estimate of u err will 429 rise about 6.7%. It can be seen that the proposed method is 430 affected by the variation of motor parameters. The reason is 431 that the proposed method is based on the model of the air-gap 432 flux observer. When the model itself has errors, the parameter 433 identification results will also produce errors.  As for Fig.13 (a), when the offline feedforward method 459 was adopted, the current distortion has been little improved 460 on the zero crossing point. The value of SHD decrease to 461 3.22%, which is shown as Fig.13 (b). Among them, there has 462 been a noticeable decrease in 9th and 11th order harmonics.

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In Fig.14  phenomenon is almost eliminated which is shown as 468 Fig.15 (a). Moreover, the harmonic content decreases more 469 significantly, SHD = 2.43%. These experiments are enough 470 to conclude that the proposed method not only can adapt to 471 various operating conditions, but also has obvious advantages 472 in harmonic suppression.

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Non-sinusoidal power supply is a significant application for 475 multiphase motors. In this paper, a VSI nonlinearity identifi-476 cation method based on third harmonic injection is proposed. 477 In addition, this method utilizes third harmonic plane only, 478 and the fundamental plane can be used for other parameter 479 estimation purposes. In order to make the adaptive law of 480 voltage error stable, we designed a feedback gain matrix 481 of a full-order air-gap flux observer. Based on the above 482 algorithm, a series of experiments are carried out on a seven-483 phase induction motor. By comparing the estimated values 484 with the reference values which is calculated from IGBT data 485 sheet, the errors are satisfactory under different conditions. 486 It is important to note that the accuracy of the estimates is 487 affected by parameters of the flux observer, especially the 488 stator resistance. Finally, the proposed method can not only 489 estimate the reliable voltage error of the inverter, but also 490 effectively reduce the SHD.

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In this Appendix, the traditional VSI nonlinearity estimation 493 method mentioned in the experiment is given [22], [23]. 494 In [22], the authors obtained the VSI error by direct calcu-495 lation which is expressed as: The denominator of (A2) contains f y , and f y is close to 503 zero, so that the perturbation of f y is going to be amplified 504 in the estimation. A low pass filter is used to reduce this 505 disturbance [22], but the effect is not obvious. Therefore, after 506 two years, they optimized the perturbation of the estimated 507 value by using MRAS [23]. As for the induction machine