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Pulsating Torques in PWM Multi-Megawatt Drives for Torsional Analysis of Large Shafts Joseph Song-Manguelle, Member, IEEE, Jean-Maurice Nyobe-Yome, and Gabriel Ekemb

Abstract—This paper presents new, simple, and linear mathematical relationships to predict pulsating torque frequencies in industrial multi-megawatt pulse width-modulated (PWM) drives. The proposed relationships provide direct definition of possible drive operating points where the torsional resonance modes of the mechanical shaft assembly might be excited. The calculation accuracy of natural frequencies and the induction motor slip have been taken into account in the definition of a critical excitation band. Variable switching frequency PWM strategies have also been analyzed. Pulsating torque frequencies of complex load-commutated-inverter configurations have been proposed as complementary results. Several tests have been performed on an integrated-motor-compressor system operating at high-speed (3.3 kV, 6 MW at 10 000 r/min). Intensive simulation results of a higher power PWM drive for compressor applications are shown to validate the suggested approach. This paper is a key tool for electrical engineers involved in the design of multi-megawatt drive systems such as in liquefied natural gas compression and transportation, mining, or cements. Index Terms—High power, liquefied natural gas, loadcommutated inverter (LCI), mechanical resonance, multimegawatt, pulsating torque, pulsewidth modulation (PWM), torsional vibration.

Fig. 1. Typical O&G shaft assembly. (a) Responsibility of the mechanical load designer. (b) Responsibility of the VSDS designer. (c) Shared responsibility area.

I. I NTRODUCTION

T

HIS PAPER is the second contribution to improve communication and to clarify terminology between mechanical and electrical engineers involved in the design of complex drive trains such as in oil and gas (O&G) applications, mining, or cement industries. The first contribution is focused on a comprehensive modeling of torsional resonances for multimegawatt drives design [1]. Rotating machine designers generally perform torsional analysis of their mechanical systems in the design phase. That analysis evaluates the behavior of the load in case of torsional stress on the mass assembly system. This activity is carried out most of the time by mechanical engineers. A key data during

Paper 2008-IDC-218, presented at the 2008 Industry Applications Society Annual Meeting, Edmonton, AB, Canada, October 5–9, and approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. Manuscript submitted for review November 21, 2008 and released for publication June 11, 2009. First published November 17, 2009; current version published January 20, 2010. J. Song-Manguelle was with the Advanced Technology Group, Baldor Electric, Montreal, QC H3J 1R5, Canada. He is now with the GE Global Research Center, Niskayuna, NY 12309 USA (e-mail: [email protected]). J.-M. Nyobe-Yome and G. Ekemb are with the Industrial Electronics and Systems Laboratory, University of Douala-ENSET, Douala, Cameroon. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2009.2036515

Fig. 2. Possible configurations of the O&G drives. (a) Full mechanical system. (b) Hybrid electromechanical system.

that analysis is the set of externally applied torque on the shaft assembly, provided by electrical engineers. Fig. 1 shows a typical configuration of drive trains in O&G applications such as liquid natural gas compression and transportation. Its simplified version is shown Fig. 2. A gas turbine is mounted in the same shaft assembly as a compressor and an electric motor. A gear box is used to adapt the compressor high speed (more than 6000 r/min) and the motor speed (1500–3600 r/min). In the case of direct drive, Fig. 2(a), the motor is a starter. Operating at constant speed, the motor rating can exceed 12 MW. An autotransformer is sometimes needed to reduce the voltage drop during start-up. Dynamic air-gap torque equations of the motor, provided by the motor manufacturer are usually enough to analyze the torsional behavior of the shaft assembly.

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SONG-MANGUELLE et al.: PULSATING TORQUES IN PWM MULTI-MEGAWATT DRIVES FOR TORSIONAL ANALYSIS

When the process required more than one operating point, Fig. 2(b), a variable-speed drive system (VSDS: transformer, variable-frequency converter, motor) is used. The VSDS generates a set of torque components depending on the converter topology: thyristors-based current source converter, insulated-gate-bipolar-transistor- or integrated-gatecommutated-thyristor-based voltage source inverter, etc., the control strategy (U/F , vector control, direct torque control) and on the modulation strategy (symmetrical, asymmetrical pulse width modulation, etc.). Those parameters are out of the scope of rotating load manufacturers. The converter topology depends on the process needs, international standards and on the final customer requirements. Generally, the final customer requirements depend on the existing power system characteristics, sometimes on the interchangeability requirements of the existing loads. For a given VSDS configuration, the control and modulation strategies depend only on the VSDS manufacturer. For the success of the project, electrical and mechanical engineers need to communicate with the same terminology. Unfortunately, electrical engineers are most of the time showing a lack of knowledge in the interpretation of the concern and the requirements of mechanical engineers: critical speed, torsional vibration of the mechanical load are notions which usually create some frictions. Mechanical engineers not used to VSDS operation show also a lack of knowledge and confusion in the formulation of their needs. The torque produced by the VSDS is the key data from drive manufacturers to be provided to rotating load manufacturers for torsional analysis purposes. However, VSDS manufacturers generally provide the motor air-gap torque harmonics, based on simulation results. The simulations are performed at the nominal operating points, and some time at the system critical speeds, if they are situated within the VSDS operating limits. This paper shows that this approach induced an incorrect torsional analysis on the shaft assembly. For rotating machinery designers, this paper is an aid in the definition of their requirements, and in the interpretation of the electrical engineer simulation results. For multi-megawatt drive manufacturers, this paper helps in the prediction of possible operating points where the VSDS might excite the shaft assembly. Previous references on the evaluation of pulsating torque frequencies are limited to loadcommutated inverters (LCIs) [4]. The presented investigation is focused on the voltage source inverter-based pulse width modulation (PWM). The possible drift of shaft natural frequencies, as well as the variability of switching frequency, are taken into account. Principles described in this paper can be successfully applied in existing industrial three-level neutral-point-clamped converters, including integrated motocompressor systems [5]. Intensive simulation results confirm that the suggested approach can be extended to other industrial PWM drives. II. S HAFT A SSEMBLY M ODEL FOR T ORSIONAL A NALYSIS A. Simplified Model Fig. 3 shows a simplified flexible shaft with two rotors.

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Fig. 3. Simplified flexible shaft with two rotors.

If an external torque is applied to the shaft, the decoupled Newton’s law can be written, as shown in (1) and solved1 as (2). Assuming the particular case where the externally applied torque has the same pulsation as the natural pulsation of the shaft. Rotor position trajectories have a component at the same pulsation as the natural frequency of the system. Its magnitude also depends on the natural frequency, on the moments of inertia of the two masses and on the external torque magnitude. However, its magnitude is proportional to time. Therefore, even for a small external torque magnitude, the amplitude of this component is increasing over time [1] ⎧ ⎨ J2 d2 θ21 = K12 (θ1 − θ2 ) + Ta ∗ cos(ωnat t) dt (1) d2 θ K 2 ⎩ J1 dt22 = K12 (θ2 − θ1 ), with : ωnat = J112 J2 J1 +J2 ⎧ θ (t) = λ1 (t ⎪ sin(ωnat t) + cos(ωnat t)) ⎨ 1 θ2 (t) = λ1 JJ12 t sin(ωnat t) − cos(ωnat t) , (2) ⎪ ⎩ Ta with : λ1 = 2(J1 +J2 )ωnat . The risk that the positions can swing with a magnitude increasing over time is the main issue in all rotating equipment, [2]. The corresponding torque component on the shaft will also increase over time, leading to a shutdown if the system has enough protection or damping, otherwise the consequence would be a mechanical failure in the rotating system which may lead to the destruction of the whole train. These conclusions can be extended to a more complex system, having more than two mass rotors. As the number of elements increases, the number of natural frequencies also increases. Each natural frequency corresponds to the so-called torsional mode of the system. If n-masses are connected to the same shaft assembly by (n − 1) flexible shaft sections, there will be (n − 1) torsional modes [6]. These deductions can also be extended to smaller systems like hard disk drives, where the external forces which might lead to excitation are generated by form errors of ball bearings, [2], [3]. B. Simulation Results Assuming a system with one resonance mode located at 20 Hz. In addition, external torque have been applied to the model with the following characteristics: Text = 1000 + 300 ∗ cos(2π ∗ 55 ∗ t) N · m.

(3)

The simulation results, Fig. 4, show that, even if an external torque harmonic has a magnitude high as 30% of the 1 A generalized model have been presented in [1], as well as other solutions of Newton’s law, considering different initial conditions.

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the process requirements, and at the same time, they need to avoid torque harmonics located near the shaft assembly natural frequencies. III. P ULSATING T ORQUE F REQUENCIES IN M ULTI -M EGAWATT-BASED PWM D RIVES A. Preliminary Considerations

Fig. 4. Externally applied torque with pulsation different to the shaft natural frequency.

Fig. 5. Externally applied torque pulsating at the shaft natural.

dc-component, the resultant torque on the shaft assembly stays in acceptable limits. The following torque have been applied to the same system ⎧ ⎨ Text = 1000+T1 ∗cos(2π∗20∗t) N · m if t < 0.5 s And T = 0 N · m, ⎩ 1 if t ≥ 0.5 s. T1 = 50 N · m, (4) The externally applied torque has an harmonic component at the natural frequency of the shaft assembly, for t ≥ 0.5 s. Fig. 5 shows the time domain evolution of the external torque as well as the torque on the shaft. These results show that, even if the harmonic component of the externally applied torque has a magnitude as low as 5% of its dc-component, the resultant torque on the shaft increases and reaches a value ten times bigger that the dc torque in only 4 s. These results are the evidence that the concern in torsional analysis if not the magnitudes of the externally applied torque. Their locations in the frequency domain are the critical parameters. In any large train shaft, the variable frequency drive (VFD) is the main root cause of externally applied torque. Theoretically, the VSDS can generate an infinity number of torque harmonics. The main challenge for VSDS manufacturers is to control the dc-component of the motor torque according to

Pulsating torques are the set of torque harmonics in the motor air gap. They correspond to the difference between the electromagnetic torque and the load torque fluctuation. For a given machine, neglecting the torque linked to the machine construction, the frequencies of pulsating torques are totally independent to the machine. They depend on the control and the topology of the VFD. However, their magnitudes depend on the machine parameters. The electromagnetic torque is created with the set of voltage applied to the machine. The machine equivalent model is based on inductors and resistors of the windings. The machine inductors depend on the permeability and the topology of the winding [7]. The air-gap torque is created according to the machine model and the applied voltage. The applied voltage combined to the machine model generates stator and rotor current harmonics, and they depend on the chosen PWM strategy. Voltage harmonics of several PWM strategies are formulated in [8]. Therefore, its possible to make detailed mathematical developments in order to get the analytical expression of the torque, based on the forward and backward approach described in [9] and [10]. However, pulsating torque frequencies have a much more significant impact on the shaft eigenmode excitation than their magnitudes. For that reason, the next section is focused only on the analytical form of torque frequencies. This approach is suitable for torsional analysis purposes, assuming that the natural frequencies of the shaft assembly are known. The air-gap torque in alternating current machines supplied by a PWM inverter can be summarized as follows: Te = TDC +

∞

TAmph cos(ωh t + θh )

(5)

h=1

where TDC is the dc-component of the torque, and corresponds to the desired torque needed to drive the shaft assembly. The existing control strategies are focused on the control of that parameter. The torque harmonic frequencies Fh depend on the network frequency and on the operating points for LCIs [4]. For voltagesource inverters, Fh depends on the semiconductor switching frequency and on the operating point, characterized by the frequency of the fundamental component of the stator current. In the next section, a generalized characteristic of Fh for VSI is proposed. B. Generalized Campbell Diagram for VSI-Based Asymmetrical PWM Strategy Assuming a carrier frequency fsw and a fundamental stator current frequency f0 , the pulsating torque frequencies can be

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written as follows: Fmn = |mfsw ± nf0 |.

(6)

The torque components with dominant magnitudes are located at the following frequencies, where m and n are integers specified below. • For m = 0, F0n = nf0 ,

n = 6l

∀ l = 1, 2, 3 . . . .

(7)

Example: F06 = 6f0 , F0,12 = 12f0 , F0,18 = 18f0 F0,24 = 24f0 , F0,30 = 30f0 , F0,36 = 36f0 . • For m even integer, m and n are given as follows: m = 2j, ∀ j = 1, 2, 3, . . . n = 3(2i), ∀ i = 0, 1, 2, . . . .

(8)

(9)

Fig. 6. Generalized Campbell diagram for induction motor supplied by a VSIbased asymmetric PWM.

Example: F20 F26 F46 F2,18

= 2fsw , F40 = 4fsw , F60 = 6fsw = |2fsw ± 6f0 |, F2,12 = |2fsw ± 12f0 | = |4fsw ± 6f0 |, F4,12 = |4fsw ± 12f0 | = |2fsw ± 18f0 |, F4,18 = |4fsw ± 18f0 |.

• For m odd integer, m and n are given as follows: m = 2j + 1, ∀ j = 0, 1, 2, . . . n = 3(2i + 1) ∀ i = 0, 1, 2, 3, . . . .

(10)

(11)

Examples: F13 = |fsw ± 3f0 |, F19 = |fsw ± 9f0 | F33 = |3fsw ± 3f0 |, F39 = |3fsw ± 9f0 | F1,15 = |fsw ± 15f0 |, F3,15 = |3fsw ± 15f0 |.

(12)

Equations (6)–(12) are a set of frequencies where the pulsating torque magnitudes could reach nonnegligible values. The graphical representation of that frequencies are shown in Fig. 6. This representation corresponds to a generalized Campbell diagram of pulsating torque components in an induction motor supplied with an asymmetrical PWM strategy.

Fig. 7. Simulated air-gap torque with 825-Hz carrier frequency.

C. Validation A VSI-based asymmetrical PWM strategy have been simulated. The 35-MW VSDS is driving a compressor as a load with a three-level neutral-point-clamped converter. Torque waveform of each operating point was recorded after all the control variables have reached their steady states. Then, a fast Fourier transform (FFT) of the resultant torque has been calculated under a window such that the torque harmonic resolution is equal to 5 Hz. Illustrative cases of the time domain air-gap torque at 65and 75-Hz motor operating frequencies are shown in Fig. 7, with a carrier frequency of 825 Hz. Corresponding FFT of the time domain waveforms are shown in Figs. 8 and 9. Exact value of air-gap torque frequencies based on the principle described

Fig. 8. Torque frequency response. Fsw = 825 Hz. Fmot = 65 Hz.

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Fig. 9. Torque frequency response. Fsw = 825 Hz. Fmot = 75 Hz.

Fig. 11. Campbell diagram of a VSI with 750-Hz switching frequency. “o” are simulation results.

Fig. 10. Campbell diagram of a VSI with 825-Hz carrier frequency. “o” are simulation results.

Fig. 12.

in previous sections are also shown, with a good correlation between prediction and simulations. The same principle is applied to 12 operating points, corresponding to the following stator current frequencies: 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 100, and 105 Hz. The carrier frequency was chosen constant at 825 Hz. The simulated results shown in Fig. 10 are superposed to the predicted results. Torque harmonic components whose magnitudes are at least equal to 0.1% of the nominal torque are shown. This principle is also applied to the same system with a carrier frequency reduced to 750 Hz, and eight operating points located at 10, 20, 40, 50, 60, 80, 90, 100, and 110 Hz. Fig. 11 shows that simulation results correlated to predictions. These results with different carrier frequencies and different motor operating frequencies validate the proposed relationships. Some torque frequencies have redundant combinations of m and n, which correspond to a superposition of current harmonics effects in the machine. It can be seen that some

Graphical solution to localize the possible resonance excitation area.

simulated points do not match the predicted lines. This can happen because the parameters m and n have been limited to certain values. D. Exciting Shaft Assembly Torsional Resonance With a VSDS Assuming a shaft assembly with k rotor masses. There are (k − 1) shaft sections and (k − 1) resonance modes: fnat0 , fnat1 , . . . , fnatk−1 . Solving (13) will give a set of solutions to predict possible operating points where high risks of excitating the torsional resonance modes of the shaft assembly are located [fnat0 , . . . , fnatk−1 ] = |mfsw ± nf0 |.

(13)

Values of m and n are given according to (6)–(12). Theoretically, there are many solutions in all the VSDS operating range. A graphical solution of (13) is shown in Fig. 12.

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Assuming a carrier frequency of 1 kHz, the intersection between the natural frequencies fnat0 = 45 Hz, fnat1 = 308 Hz, fnat2 = 530 Hz and the Campbell diagram predicted lines corresponds to the graphical solution of (13). IV. S YSTEM S ENSITIVITY In the proposed modeling approach of torsional resonances for multi-megawatt drives design described in [1], and summarized in the previous sections, we assumed that natural frequencies of any shaft assembly are constants in all the process operating points. This assumption might be wrong in highspeed motor applications, where magnetic bearings are used to lift the rotor. Therefore, the suggested prediction of pulsating torque frequencies might not be enough to assess the risk of shaft excitation. A. Flexible Shaft Critical Bands: Calculation Accuracy Effects Natural frequencies of shaft assembly depends on stiffness constants and mass moments of inertia [1]. Assuming each rotor is a rigid body with j rotating interdependent points, with a mass Mj , the mass moment of inertia can be approximated as follows, assuming each point is located at a distance Rj to the rotation axis Δ: Mj Rj2 . (14) JΔ =

Fig. 13. Example of critical bands of a shaft assembly.

j

With that approximation, the natural frequencies are calculated with an accuracy %. A sensitivity analysis of torsional modes to uncertainty in shaft mechanical parameters have been carried out [11]. The natural frequencies are not constant. They can slightly drift depending on mechanical system parameters. Rotating load speed defined by mechanical engineers are seen by the motor designers as motor rotor speeds. Drive engineers need to transform the rotor speed into stator current frequencies, taking into account the motor slip. Assuming the motor slip has a maximum value, s, and considering the accuracy of the system natural frequencies, a “critical band” of the mechanical system can be defined from the frequency point of view (1 − − s)fnat ≤ fnat ≤ (1 + + s)fnat .

(15)

Critical bands can be plotted on a Campbell diagram. Any intersection between the predicted lines and the critical bands increases the probability of exciting the torsional resonance modes of the shaft assembly. Assuming power switches switching frequency of 1 kHz and three natural frequencies fnat1 = 45 Hz, fnat2 = 380 Hz, fnat3 = 530 Hz, a motor maximum slip of s = 0.04 and the mass moments of inertia are calculated with = 0.01 accuracy. Fig. 13 shows the critical resonance bands and the predicted Campbell diagram of a VSDS. B. Pulsating Torques of Variable-Frequency PWM Strategies To reduce the switching losses in VSI-based PWM strategy, variable switching frequency can be implemented. The switch-

Fig. 14. Principle of the variable switching frequency PWM strategy.

ing frequency of the power semiconductors are synchronously defined depending on the motor operating points. This approach is suitable for high-speed applications, where the stator current frequency can be as high as 300 Hz [12] for induction motors. The principle of the variable switching frequency gear change is shown Fig. 14. The predicted Campbell diagram lines of a variable switching frequency PWM strategy are linearly

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Fig. 16.

Fig. 15. Campbell diagram of a variable switching frequency PWM strategy.

moving with the stator frequency. Assuming a fundamental frequency fnom = 200 Hz, and the gear change defined in 0 ≤ f0 < x1 ,

fsw = δ0 ,

x1 ≤ f0 < x2 ;

fsw = δ1 f0

x2 ≤ f0 < x3 ,

fsw = δ2 , f0

x3 ≤ f0 < x4 ;

fsw = δ3 . f0

Example of complex LCI configurations.

Any complex LCI configuration can be reduced to the uniform connection (see Fig. 16(a)). Therefore, the pulsating torques are located at the following frequencies. The pulsating torque frequencies of Fig. 16(b) (12/12-pulse LCI) can be written as follows: fmn = |m ∗ fN ± n ∗ fM |, where m = p ∗ l, n = q ∗ l; l = 0, 1, 2, 3 . . . (18) ⎧ F0,12 = 12fM , F0,24 = 24fM , F0,36 = 36fM . . . ⎪ ⎨ F12,0 = 12fN , F24,0 = 24fN , F36,0 = 36fN . . . F ⎪ ⎩ 24,12 = |24fN ± 12fM |, F24,24 = |24fsw ± 24fM | . . . F36,12 = |36fN ± 12f0 |, F36,24 = |36fN ± 24fM | . . .

(16)

(19)

With x1 = 45%fnom , x2 = 65%fnom , and δ2 = 6, Fig. 15 shows the Campbell diagram of a variable switching frequency PWM strategy. This diagram is valid for 90 ≤ fnom < 130 Hz.

In addition, the pulsating torque frequencies of Fig. 16(c) (24/12-pulse LCI) can be written as follows:

V. P ULSATING T ORQUES OF C OMPLEX LCI C ONFIGURATIONS The pulsating torque frequencies of uniform LCI configuration [Fig. 16(a)] are well known. Assuming a constant network frequency fN (or slightly shifting within the industrial and international standard margins), and a drive output frequency fM . Assuming a p-pulse transformer configuration, p = 6 for 6-pulse LCI system, p = 12 or p = 24 for 12-pulse and 24-pulse LCI systems. The pulsating torques are located at the following frequencies [1], [4]: fmn = |m ∗ fN ± n ∗ fM |, where m = p ∗ l, n = p ∗ l; l = 0, 1, 2, 3 . . .. (17) Complex configurations of LCI are used to reduce the current harmonics in the grid side, as well as in the motor side. From the motor side point of view, the current harmonic reduction is directly reducing the air-gap torque ripple. The number of pulses p in the grid side can be different to the number of pulses q in the motor side. Fig. 16(b) shows a possible 12/12 pulse LCI configuration, where p = 12 in the grid side and q = 12 in the motor side. An other example is shown Fig. 16(c) for a 24/ 12-pulse, where p = 24 and q = 12.

⎧ F0,12 = 12fM , F0,24 = 24fM , F0,36 = 36fM . . . ⎪ ⎨ F24,0 = 24fN , F48,0 = 48fN , F72,0 = 72fN . . . F ⎪ ⎩ 24,12 = |24fN ± 12fM |, F24,24 = |24fsw ± 24fM | . . . F48,12 = |48fN ± 12f0 |, F48,24 = |48fN ± 24fM | . . . . (20) As shown in Fig. 17(a), increasing the number of pulses helps clean the Campbell diagram. The first torsional mode is usually located below 50 Hz. Fig. 17(d) shows a zoom on the Campbell diagram of the three LCI configurations, with a maximum pulsating torque frequency of 100 Hz. These figures will help to define in which operating points the predicted lines of the Campbell diagram will cross the torsional mode frequencies. Therefore, multi-megawatt drive manufacturers should perform a few simulations around each area, taking into account the critical bands as defined in (15). VI. C ONCLUSION The investigations presented in this paper were focused on multi-megawatt voltage source inverter-based PWM strategies. A new modeling approach as well as simple and linear analytical relationships of pulsating torque frequencies have been proposed. The mathematical formulation of pulsating torque frequencies is a helpful tool for multi-megawatt drive manufacturers involved in the integration of large shafts.

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Fig. 17. Campbell diagram of complex LCI configurations. (a) p = 6; q = 6. (b) p = 12; q = 12. (c) p = 24; q = 12. (d) Low-frequency range.

Using the mechanical and electrical similarities defined in [1], the complete drive train can be simulated by multimegawatt drive manufacturers, in any electrical engineering simulation software. The simulated motor torque signal should be provided to the equivalent electrical circuit of the mechanical load as a controllable voltage source. This approach will help to reduce the risk of torsional resonance excitation early in the design phase. Critical speed calculation accuracy, as well as induction motor slip, have been taken into account to define the critical bands of possible torsional resonance excitation. Variable frequency PWM strategies have been also investigated. Pulsating torque frequencies of complex LCI configuration have been included in this investigation, in order to provide a complete design tool to multi-megawatt drive manufacturers. The Campbell diagram has been chosen as the main graphical representation of possible torque harmonics frequency locations in order to be easily understood by rotating load manufacturers. ACKNOWLEDGMENT The authors would like to thank Dr. S. Schröder, Research Scientist at GE Global Research, Munich, Germany, for his contribution to this paper.

R EFERENCES [1] J. Song-Manguelle, C. Sihler, and J. M. Nyobe-Yome, “Modeling of torsional resonances for multi-megawatt drives design,” in Conf. Rec. IEEE IAS 43rd Annu. Meeting, Edmonton, AB, Canada, Oct. 2008, pp. 1–8. [2] K. Ono and K. Takahasi, “Theoretical analysis of shaft vibration supported by a ball bearing with small sinusoidal waviness,” IEEE Trans. Magn., vol. 32, pt. 2, no. 3, pp. 1709–1774, May 1996. [3] K. Deeyiengyang and K. Ono, “Suppression of resonance amplitude of disk vibrations by squeeze air bearing plate,” IEEE Trans. Magn., vol. 37, pt. 1, no. 2, pp. 820–825, Mar. 2001. [4] J. J. Simond, A. Sapin, T. Xuan, R. Wetter, and P. Burmeiter, “12-pulse LCI synchronous drive for a 20 MW compressor: Modeling, simulation and measurements,” in Proc. 40th IEEE IAS Annu. Meeting, Hong Kong, Oct. 2005, vol. 4, pp. 2302–2308. [5] Integrated Compressor Line-ICL: An Integrated High-Speed Electric Motor Driven Compressor, GE Oil and Gas, Florence, Italy, 2007. [Online]. Availabe: www.ge.com/oilandgas [6] C. M. Ong, Dynamic Simulation of Electric Machinery using Matlab/ Simulink. Englewood Cliffs, NJ: Prentice-Hall, 1999. [7] J. Song-Manguelle, “Synthesis of AC machine modeling for digital control purposes,” M.S. thesis, ENSET, Univ. Douala, Douala, Cameroon, 1997. [8] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principle and Practice. New York: Wiley-Interscience, 2003, ser. IEEE Press Series on Power Eng.. [9] S. M. Abdulrahman, J. G. Kettleborough, and I. R. Smith, “Fast calculation of harmonic torque pulsations in a VSI/induction motor drive,” IEEE Trans. Ind. Electron., vol. 40, no. 6, pp. 561–569, Dec. 1993. [10] T. A. Lipo, P. C. Krause, and H. E. Jordan, “Harmonic torque and speed pulsations in a rectifier-inverter induction motor drive,” IEEE Trans. Power App. Syst., vol. PAS-88, no. 5, pp. 569–587, May 1969.

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[11] J. V. Milanovic, C. P. Fu, R. Radosavljecic, and Z. Lazarevic, “Sensitivity of torsional modes and torques to uncertainty in shaft mechanical parameters,” Elect. Power Compon. Syst., vol. 29, no. 10, pp. 867–881, Oct. 2001. [12] MV7000 Brochure: Entering a New Dimension for Reliability and Performance in Medium-Voltage AC Drives, Converteam, Massy, France, 2007. [Online]. Available: www.converteam.com

Joseph Song-Manguelle (M’07) was born in Makak, Cameroon. He received the B.S. and M.S. degrees in pedagogical sciences and electrical engineering from the University of Douala, Douala, Cameroon, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, in 2004. From 1997 to 1999, he was a Lecturer at the University of Douala. In 1999, he joined the Industrial Electronics Laboratory, EPFL, as a Postgraduate Assistant, where, from 2001 to 2004, he was a Research Associate. In 2004, he joined the High-Power Electronics Laboratory, GE Global Research Center, Munich, Germany, dealing with high-power drives topologies and control for various applications such as oil and gas. From 2006 to 2007, he was with Thermodyn, a GE oil and gas business in Le Creusot, France, as a Senior Electrical Engineer and Team Leader, dealing with high-power drives acquisition and torsional issues in compressor drive trains, including highspeed applications. From 2007 to 2008, he was with Baldor Electric, Montreal, QC, Canada, as a Senior Design Engineer. He spent part of his time at the Baldor Drives Center, Fort Smith, AR, focusing on low-voltage high-power regenerative drives. Since 2008, he has been with the Power Conversion Systems organization at the GE Global Research Center, Niskayuna, NY. His areas of interest are topologies and control of high-power electronics, as well as large-scale integrated electromechanical systems.

Jean-Maurice Nyobe-Yome was born in Yaoundé, Cameroon. He received the Diploma degree in electrical engineering of technical high-school teaching from the University of Douala-ENSET, Douala, Cameroon, in 1984; the M.S. degree in electrical engineering from the Ecole Normale Supérieure de Cachan, Cachan, France, and the Université de Paris VI, Paris, France, in 1986; and the Ph.D. degree from the Université de Monpellier II, Montpellier, France, in 1993. In 1985, he joined the University of DoualaENSET as a Lecturer, where he is currently an Associate Professor. He is also Head of the Electrical Engineering Department and Head of the Industrial Electronics and Systems Laboratory. Since 1994, he has supervised more than 200 Master’s theses in electrical engineering sponsored by local companies and the Cameroonian government. He is a Senior Member of the Cameroonian Commission for Technical Education. His areas of interest are resonant power conversion, high-power variable-speed drives, windmill/diesel twinning, and applied pedagogical sciences to electrical engineering.

Gabriel Ekemb was born in Matomb, Cameroon. He received the B.S. and M.S. degrees in pedagogical sciences and electrical engineering and the D.E.A. degree in electrical engineering from the University of Douala-ENSET, Douala, Cameroon, in 1995, 1997, and 2006, respectively, where he is currently working toward the Ph.D. degree, in collaboration with the University of Quebec, Rouyn-Noranda, QC, Canada. From 1997 to 2003, he was a Teacher at the Technical High-School of Nkongsamba, Cameroon. Since 2003, he has been with the Industrial Electronics and Systems Laboratory, University of Douala-ENSET, as a Senior Research Associate. His areas of interest are high-power conversion, electromechanical intereactions, and renewable energy power conversion.

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 1, JANUARY/FEBRUARY 2010

Pulsating Torques in PWM Multi-Megawatt Drives for Torsional Analysis of Large Shafts Joseph Song-Manguelle, Member, IEEE, Jean-Maurice Nyobe-Yome, and Gabriel Ekemb

Abstract—This paper presents new, simple, and linear mathematical relationships to predict pulsating torque frequencies in industrial multi-megawatt pulse width-modulated (PWM) drives. The proposed relationships provide direct definition of possible drive operating points where the torsional resonance modes of the mechanical shaft assembly might be excited. The calculation accuracy of natural frequencies and the induction motor slip have been taken into account in the definition of a critical excitation band. Variable switching frequency PWM strategies have also been analyzed. Pulsating torque frequencies of complex load-commutated-inverter configurations have been proposed as complementary results. Several tests have been performed on an integrated-motor-compressor system operating at high-speed (3.3 kV, 6 MW at 10 000 r/min). Intensive simulation results of a higher power PWM drive for compressor applications are shown to validate the suggested approach. This paper is a key tool for electrical engineers involved in the design of multi-megawatt drive systems such as in liquefied natural gas compression and transportation, mining, or cements. Index Terms—High power, liquefied natural gas, loadcommutated inverter (LCI), mechanical resonance, multimegawatt, pulsating torque, pulsewidth modulation (PWM), torsional vibration.

Fig. 1. Typical O&G shaft assembly. (a) Responsibility of the mechanical load designer. (b) Responsibility of the VSDS designer. (c) Shared responsibility area.

I. I NTRODUCTION

T

HIS PAPER is the second contribution to improve communication and to clarify terminology between mechanical and electrical engineers involved in the design of complex drive trains such as in oil and gas (O&G) applications, mining, or cement industries. The first contribution is focused on a comprehensive modeling of torsional resonances for multimegawatt drives design [1]. Rotating machine designers generally perform torsional analysis of their mechanical systems in the design phase. That analysis evaluates the behavior of the load in case of torsional stress on the mass assembly system. This activity is carried out most of the time by mechanical engineers. A key data during

Paper 2008-IDC-218, presented at the 2008 Industry Applications Society Annual Meeting, Edmonton, AB, Canada, October 5–9, and approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. Manuscript submitted for review November 21, 2008 and released for publication June 11, 2009. First published November 17, 2009; current version published January 20, 2010. J. Song-Manguelle was with the Advanced Technology Group, Baldor Electric, Montreal, QC H3J 1R5, Canada. He is now with the GE Global Research Center, Niskayuna, NY 12309 USA (e-mail: [email protected]). J.-M. Nyobe-Yome and G. Ekemb are with the Industrial Electronics and Systems Laboratory, University of Douala-ENSET, Douala, Cameroon. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2009.2036515

Fig. 2. Possible configurations of the O&G drives. (a) Full mechanical system. (b) Hybrid electromechanical system.

that analysis is the set of externally applied torque on the shaft assembly, provided by electrical engineers. Fig. 1 shows a typical configuration of drive trains in O&G applications such as liquid natural gas compression and transportation. Its simplified version is shown Fig. 2. A gas turbine is mounted in the same shaft assembly as a compressor and an electric motor. A gear box is used to adapt the compressor high speed (more than 6000 r/min) and the motor speed (1500–3600 r/min). In the case of direct drive, Fig. 2(a), the motor is a starter. Operating at constant speed, the motor rating can exceed 12 MW. An autotransformer is sometimes needed to reduce the voltage drop during start-up. Dynamic air-gap torque equations of the motor, provided by the motor manufacturer are usually enough to analyze the torsional behavior of the shaft assembly.

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SONG-MANGUELLE et al.: PULSATING TORQUES IN PWM MULTI-MEGAWATT DRIVES FOR TORSIONAL ANALYSIS

When the process required more than one operating point, Fig. 2(b), a variable-speed drive system (VSDS: transformer, variable-frequency converter, motor) is used. The VSDS generates a set of torque components depending on the converter topology: thyristors-based current source converter, insulated-gate-bipolar-transistor- or integrated-gatecommutated-thyristor-based voltage source inverter, etc., the control strategy (U/F , vector control, direct torque control) and on the modulation strategy (symmetrical, asymmetrical pulse width modulation, etc.). Those parameters are out of the scope of rotating load manufacturers. The converter topology depends on the process needs, international standards and on the final customer requirements. Generally, the final customer requirements depend on the existing power system characteristics, sometimes on the interchangeability requirements of the existing loads. For a given VSDS configuration, the control and modulation strategies depend only on the VSDS manufacturer. For the success of the project, electrical and mechanical engineers need to communicate with the same terminology. Unfortunately, electrical engineers are most of the time showing a lack of knowledge in the interpretation of the concern and the requirements of mechanical engineers: critical speed, torsional vibration of the mechanical load are notions which usually create some frictions. Mechanical engineers not used to VSDS operation show also a lack of knowledge and confusion in the formulation of their needs. The torque produced by the VSDS is the key data from drive manufacturers to be provided to rotating load manufacturers for torsional analysis purposes. However, VSDS manufacturers generally provide the motor air-gap torque harmonics, based on simulation results. The simulations are performed at the nominal operating points, and some time at the system critical speeds, if they are situated within the VSDS operating limits. This paper shows that this approach induced an incorrect torsional analysis on the shaft assembly. For rotating machinery designers, this paper is an aid in the definition of their requirements, and in the interpretation of the electrical engineer simulation results. For multi-megawatt drive manufacturers, this paper helps in the prediction of possible operating points where the VSDS might excite the shaft assembly. Previous references on the evaluation of pulsating torque frequencies are limited to loadcommutated inverters (LCIs) [4]. The presented investigation is focused on the voltage source inverter-based pulse width modulation (PWM). The possible drift of shaft natural frequencies, as well as the variability of switching frequency, are taken into account. Principles described in this paper can be successfully applied in existing industrial three-level neutral-point-clamped converters, including integrated motocompressor systems [5]. Intensive simulation results confirm that the suggested approach can be extended to other industrial PWM drives. II. S HAFT A SSEMBLY M ODEL FOR T ORSIONAL A NALYSIS A. Simplified Model Fig. 3 shows a simplified flexible shaft with two rotors.

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Fig. 3. Simplified flexible shaft with two rotors.

If an external torque is applied to the shaft, the decoupled Newton’s law can be written, as shown in (1) and solved1 as (2). Assuming the particular case where the externally applied torque has the same pulsation as the natural pulsation of the shaft. Rotor position trajectories have a component at the same pulsation as the natural frequency of the system. Its magnitude also depends on the natural frequency, on the moments of inertia of the two masses and on the external torque magnitude. However, its magnitude is proportional to time. Therefore, even for a small external torque magnitude, the amplitude of this component is increasing over time [1] ⎧ ⎨ J2 d2 θ21 = K12 (θ1 − θ2 ) + Ta ∗ cos(ωnat t) dt (1) d2 θ K 2 ⎩ J1 dt22 = K12 (θ2 − θ1 ), with : ωnat = J112 J2 J1 +J2 ⎧ θ (t) = λ1 (t ⎪ sin(ωnat t) + cos(ωnat t)) ⎨ 1 θ2 (t) = λ1 JJ12 t sin(ωnat t) − cos(ωnat t) , (2) ⎪ ⎩ Ta with : λ1 = 2(J1 +J2 )ωnat . The risk that the positions can swing with a magnitude increasing over time is the main issue in all rotating equipment, [2]. The corresponding torque component on the shaft will also increase over time, leading to a shutdown if the system has enough protection or damping, otherwise the consequence would be a mechanical failure in the rotating system which may lead to the destruction of the whole train. These conclusions can be extended to a more complex system, having more than two mass rotors. As the number of elements increases, the number of natural frequencies also increases. Each natural frequency corresponds to the so-called torsional mode of the system. If n-masses are connected to the same shaft assembly by (n − 1) flexible shaft sections, there will be (n − 1) torsional modes [6]. These deductions can also be extended to smaller systems like hard disk drives, where the external forces which might lead to excitation are generated by form errors of ball bearings, [2], [3]. B. Simulation Results Assuming a system with one resonance mode located at 20 Hz. In addition, external torque have been applied to the model with the following characteristics: Text = 1000 + 300 ∗ cos(2π ∗ 55 ∗ t) N · m.

(3)

The simulation results, Fig. 4, show that, even if an external torque harmonic has a magnitude high as 30% of the 1 A generalized model have been presented in [1], as well as other solutions of Newton’s law, considering different initial conditions.

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the process requirements, and at the same time, they need to avoid torque harmonics located near the shaft assembly natural frequencies. III. P ULSATING T ORQUE F REQUENCIES IN M ULTI -M EGAWATT-BASED PWM D RIVES A. Preliminary Considerations

Fig. 4. Externally applied torque with pulsation different to the shaft natural frequency.

Fig. 5. Externally applied torque pulsating at the shaft natural.

dc-component, the resultant torque on the shaft assembly stays in acceptable limits. The following torque have been applied to the same system ⎧ ⎨ Text = 1000+T1 ∗cos(2π∗20∗t) N · m if t < 0.5 s And T = 0 N · m, ⎩ 1 if t ≥ 0.5 s. T1 = 50 N · m, (4) The externally applied torque has an harmonic component at the natural frequency of the shaft assembly, for t ≥ 0.5 s. Fig. 5 shows the time domain evolution of the external torque as well as the torque on the shaft. These results show that, even if the harmonic component of the externally applied torque has a magnitude as low as 5% of its dc-component, the resultant torque on the shaft increases and reaches a value ten times bigger that the dc torque in only 4 s. These results are the evidence that the concern in torsional analysis if not the magnitudes of the externally applied torque. Their locations in the frequency domain are the critical parameters. In any large train shaft, the variable frequency drive (VFD) is the main root cause of externally applied torque. Theoretically, the VSDS can generate an infinity number of torque harmonics. The main challenge for VSDS manufacturers is to control the dc-component of the motor torque according to

Pulsating torques are the set of torque harmonics in the motor air gap. They correspond to the difference between the electromagnetic torque and the load torque fluctuation. For a given machine, neglecting the torque linked to the machine construction, the frequencies of pulsating torques are totally independent to the machine. They depend on the control and the topology of the VFD. However, their magnitudes depend on the machine parameters. The electromagnetic torque is created with the set of voltage applied to the machine. The machine equivalent model is based on inductors and resistors of the windings. The machine inductors depend on the permeability and the topology of the winding [7]. The air-gap torque is created according to the machine model and the applied voltage. The applied voltage combined to the machine model generates stator and rotor current harmonics, and they depend on the chosen PWM strategy. Voltage harmonics of several PWM strategies are formulated in [8]. Therefore, its possible to make detailed mathematical developments in order to get the analytical expression of the torque, based on the forward and backward approach described in [9] and [10]. However, pulsating torque frequencies have a much more significant impact on the shaft eigenmode excitation than their magnitudes. For that reason, the next section is focused only on the analytical form of torque frequencies. This approach is suitable for torsional analysis purposes, assuming that the natural frequencies of the shaft assembly are known. The air-gap torque in alternating current machines supplied by a PWM inverter can be summarized as follows: Te = TDC +

∞

TAmph cos(ωh t + θh )

(5)

h=1

where TDC is the dc-component of the torque, and corresponds to the desired torque needed to drive the shaft assembly. The existing control strategies are focused on the control of that parameter. The torque harmonic frequencies Fh depend on the network frequency and on the operating points for LCIs [4]. For voltagesource inverters, Fh depends on the semiconductor switching frequency and on the operating point, characterized by the frequency of the fundamental component of the stator current. In the next section, a generalized characteristic of Fh for VSI is proposed. B. Generalized Campbell Diagram for VSI-Based Asymmetrical PWM Strategy Assuming a carrier frequency fsw and a fundamental stator current frequency f0 , the pulsating torque frequencies can be

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written as follows: Fmn = |mfsw ± nf0 |.

(6)

The torque components with dominant magnitudes are located at the following frequencies, where m and n are integers specified below. • For m = 0, F0n = nf0 ,

n = 6l

∀ l = 1, 2, 3 . . . .

(7)

Example: F06 = 6f0 , F0,12 = 12f0 , F0,18 = 18f0 F0,24 = 24f0 , F0,30 = 30f0 , F0,36 = 36f0 . • For m even integer, m and n are given as follows: m = 2j, ∀ j = 1, 2, 3, . . . n = 3(2i), ∀ i = 0, 1, 2, . . . .

(8)

(9)

Fig. 6. Generalized Campbell diagram for induction motor supplied by a VSIbased asymmetric PWM.

Example: F20 F26 F46 F2,18

= 2fsw , F40 = 4fsw , F60 = 6fsw = |2fsw ± 6f0 |, F2,12 = |2fsw ± 12f0 | = |4fsw ± 6f0 |, F4,12 = |4fsw ± 12f0 | = |2fsw ± 18f0 |, F4,18 = |4fsw ± 18f0 |.

• For m odd integer, m and n are given as follows: m = 2j + 1, ∀ j = 0, 1, 2, . . . n = 3(2i + 1) ∀ i = 0, 1, 2, 3, . . . .

(10)

(11)

Examples: F13 = |fsw ± 3f0 |, F19 = |fsw ± 9f0 | F33 = |3fsw ± 3f0 |, F39 = |3fsw ± 9f0 | F1,15 = |fsw ± 15f0 |, F3,15 = |3fsw ± 15f0 |.

(12)

Equations (6)–(12) are a set of frequencies where the pulsating torque magnitudes could reach nonnegligible values. The graphical representation of that frequencies are shown in Fig. 6. This representation corresponds to a generalized Campbell diagram of pulsating torque components in an induction motor supplied with an asymmetrical PWM strategy.

Fig. 7. Simulated air-gap torque with 825-Hz carrier frequency.

C. Validation A VSI-based asymmetrical PWM strategy have been simulated. The 35-MW VSDS is driving a compressor as a load with a three-level neutral-point-clamped converter. Torque waveform of each operating point was recorded after all the control variables have reached their steady states. Then, a fast Fourier transform (FFT) of the resultant torque has been calculated under a window such that the torque harmonic resolution is equal to 5 Hz. Illustrative cases of the time domain air-gap torque at 65and 75-Hz motor operating frequencies are shown in Fig. 7, with a carrier frequency of 825 Hz. Corresponding FFT of the time domain waveforms are shown in Figs. 8 and 9. Exact value of air-gap torque frequencies based on the principle described

Fig. 8. Torque frequency response. Fsw = 825 Hz. Fmot = 65 Hz.

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Fig. 9. Torque frequency response. Fsw = 825 Hz. Fmot = 75 Hz.

Fig. 11. Campbell diagram of a VSI with 750-Hz switching frequency. “o” are simulation results.

Fig. 10. Campbell diagram of a VSI with 825-Hz carrier frequency. “o” are simulation results.

Fig. 12.

in previous sections are also shown, with a good correlation between prediction and simulations. The same principle is applied to 12 operating points, corresponding to the following stator current frequencies: 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 100, and 105 Hz. The carrier frequency was chosen constant at 825 Hz. The simulated results shown in Fig. 10 are superposed to the predicted results. Torque harmonic components whose magnitudes are at least equal to 0.1% of the nominal torque are shown. This principle is also applied to the same system with a carrier frequency reduced to 750 Hz, and eight operating points located at 10, 20, 40, 50, 60, 80, 90, 100, and 110 Hz. Fig. 11 shows that simulation results correlated to predictions. These results with different carrier frequencies and different motor operating frequencies validate the proposed relationships. Some torque frequencies have redundant combinations of m and n, which correspond to a superposition of current harmonics effects in the machine. It can be seen that some

Graphical solution to localize the possible resonance excitation area.

simulated points do not match the predicted lines. This can happen because the parameters m and n have been limited to certain values. D. Exciting Shaft Assembly Torsional Resonance With a VSDS Assuming a shaft assembly with k rotor masses. There are (k − 1) shaft sections and (k − 1) resonance modes: fnat0 , fnat1 , . . . , fnatk−1 . Solving (13) will give a set of solutions to predict possible operating points where high risks of excitating the torsional resonance modes of the shaft assembly are located [fnat0 , . . . , fnatk−1 ] = |mfsw ± nf0 |.

(13)

Values of m and n are given according to (6)–(12). Theoretically, there are many solutions in all the VSDS operating range. A graphical solution of (13) is shown in Fig. 12.

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Assuming a carrier frequency of 1 kHz, the intersection between the natural frequencies fnat0 = 45 Hz, fnat1 = 308 Hz, fnat2 = 530 Hz and the Campbell diagram predicted lines corresponds to the graphical solution of (13). IV. S YSTEM S ENSITIVITY In the proposed modeling approach of torsional resonances for multi-megawatt drives design described in [1], and summarized in the previous sections, we assumed that natural frequencies of any shaft assembly are constants in all the process operating points. This assumption might be wrong in highspeed motor applications, where magnetic bearings are used to lift the rotor. Therefore, the suggested prediction of pulsating torque frequencies might not be enough to assess the risk of shaft excitation. A. Flexible Shaft Critical Bands: Calculation Accuracy Effects Natural frequencies of shaft assembly depends on stiffness constants and mass moments of inertia [1]. Assuming each rotor is a rigid body with j rotating interdependent points, with a mass Mj , the mass moment of inertia can be approximated as follows, assuming each point is located at a distance Rj to the rotation axis Δ: Mj Rj2 . (14) JΔ =

Fig. 13. Example of critical bands of a shaft assembly.

j

With that approximation, the natural frequencies are calculated with an accuracy %. A sensitivity analysis of torsional modes to uncertainty in shaft mechanical parameters have been carried out [11]. The natural frequencies are not constant. They can slightly drift depending on mechanical system parameters. Rotating load speed defined by mechanical engineers are seen by the motor designers as motor rotor speeds. Drive engineers need to transform the rotor speed into stator current frequencies, taking into account the motor slip. Assuming the motor slip has a maximum value, s, and considering the accuracy of the system natural frequencies, a “critical band” of the mechanical system can be defined from the frequency point of view (1 − − s)fnat ≤ fnat ≤ (1 + + s)fnat .

(15)

Critical bands can be plotted on a Campbell diagram. Any intersection between the predicted lines and the critical bands increases the probability of exciting the torsional resonance modes of the shaft assembly. Assuming power switches switching frequency of 1 kHz and three natural frequencies fnat1 = 45 Hz, fnat2 = 380 Hz, fnat3 = 530 Hz, a motor maximum slip of s = 0.04 and the mass moments of inertia are calculated with = 0.01 accuracy. Fig. 13 shows the critical resonance bands and the predicted Campbell diagram of a VSDS. B. Pulsating Torques of Variable-Frequency PWM Strategies To reduce the switching losses in VSI-based PWM strategy, variable switching frequency can be implemented. The switch-

Fig. 14. Principle of the variable switching frequency PWM strategy.

ing frequency of the power semiconductors are synchronously defined depending on the motor operating points. This approach is suitable for high-speed applications, where the stator current frequency can be as high as 300 Hz [12] for induction motors. The principle of the variable switching frequency gear change is shown Fig. 14. The predicted Campbell diagram lines of a variable switching frequency PWM strategy are linearly

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Fig. 16.

Fig. 15. Campbell diagram of a variable switching frequency PWM strategy.

moving with the stator frequency. Assuming a fundamental frequency fnom = 200 Hz, and the gear change defined in 0 ≤ f0 < x1 ,

fsw = δ0 ,

x1 ≤ f0 < x2 ;

fsw = δ1 f0

x2 ≤ f0 < x3 ,

fsw = δ2 , f0

x3 ≤ f0 < x4 ;

fsw = δ3 . f0

Example of complex LCI configurations.

Any complex LCI configuration can be reduced to the uniform connection (see Fig. 16(a)). Therefore, the pulsating torques are located at the following frequencies. The pulsating torque frequencies of Fig. 16(b) (12/12-pulse LCI) can be written as follows: fmn = |m ∗ fN ± n ∗ fM |, where m = p ∗ l, n = q ∗ l; l = 0, 1, 2, 3 . . . (18) ⎧ F0,12 = 12fM , F0,24 = 24fM , F0,36 = 36fM . . . ⎪ ⎨ F12,0 = 12fN , F24,0 = 24fN , F36,0 = 36fN . . . F ⎪ ⎩ 24,12 = |24fN ± 12fM |, F24,24 = |24fsw ± 24fM | . . . F36,12 = |36fN ± 12f0 |, F36,24 = |36fN ± 24fM | . . .

(16)

(19)

With x1 = 45%fnom , x2 = 65%fnom , and δ2 = 6, Fig. 15 shows the Campbell diagram of a variable switching frequency PWM strategy. This diagram is valid for 90 ≤ fnom < 130 Hz.

In addition, the pulsating torque frequencies of Fig. 16(c) (24/12-pulse LCI) can be written as follows:

V. P ULSATING T ORQUES OF C OMPLEX LCI C ONFIGURATIONS The pulsating torque frequencies of uniform LCI configuration [Fig. 16(a)] are well known. Assuming a constant network frequency fN (or slightly shifting within the industrial and international standard margins), and a drive output frequency fM . Assuming a p-pulse transformer configuration, p = 6 for 6-pulse LCI system, p = 12 or p = 24 for 12-pulse and 24-pulse LCI systems. The pulsating torques are located at the following frequencies [1], [4]: fmn = |m ∗ fN ± n ∗ fM |, where m = p ∗ l, n = p ∗ l; l = 0, 1, 2, 3 . . .. (17) Complex configurations of LCI are used to reduce the current harmonics in the grid side, as well as in the motor side. From the motor side point of view, the current harmonic reduction is directly reducing the air-gap torque ripple. The number of pulses p in the grid side can be different to the number of pulses q in the motor side. Fig. 16(b) shows a possible 12/12 pulse LCI configuration, where p = 12 in the grid side and q = 12 in the motor side. An other example is shown Fig. 16(c) for a 24/ 12-pulse, where p = 24 and q = 12.

⎧ F0,12 = 12fM , F0,24 = 24fM , F0,36 = 36fM . . . ⎪ ⎨ F24,0 = 24fN , F48,0 = 48fN , F72,0 = 72fN . . . F ⎪ ⎩ 24,12 = |24fN ± 12fM |, F24,24 = |24fsw ± 24fM | . . . F48,12 = |48fN ± 12f0 |, F48,24 = |48fN ± 24fM | . . . . (20) As shown in Fig. 17(a), increasing the number of pulses helps clean the Campbell diagram. The first torsional mode is usually located below 50 Hz. Fig. 17(d) shows a zoom on the Campbell diagram of the three LCI configurations, with a maximum pulsating torque frequency of 100 Hz. These figures will help to define in which operating points the predicted lines of the Campbell diagram will cross the torsional mode frequencies. Therefore, multi-megawatt drive manufacturers should perform a few simulations around each area, taking into account the critical bands as defined in (15). VI. C ONCLUSION The investigations presented in this paper were focused on multi-megawatt voltage source inverter-based PWM strategies. A new modeling approach as well as simple and linear analytical relationships of pulsating torque frequencies have been proposed. The mathematical formulation of pulsating torque frequencies is a helpful tool for multi-megawatt drive manufacturers involved in the integration of large shafts.

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Fig. 17. Campbell diagram of complex LCI configurations. (a) p = 6; q = 6. (b) p = 12; q = 12. (c) p = 24; q = 12. (d) Low-frequency range.

Using the mechanical and electrical similarities defined in [1], the complete drive train can be simulated by multimegawatt drive manufacturers, in any electrical engineering simulation software. The simulated motor torque signal should be provided to the equivalent electrical circuit of the mechanical load as a controllable voltage source. This approach will help to reduce the risk of torsional resonance excitation early in the design phase. Critical speed calculation accuracy, as well as induction motor slip, have been taken into account to define the critical bands of possible torsional resonance excitation. Variable frequency PWM strategies have been also investigated. Pulsating torque frequencies of complex LCI configuration have been included in this investigation, in order to provide a complete design tool to multi-megawatt drive manufacturers. The Campbell diagram has been chosen as the main graphical representation of possible torque harmonics frequency locations in order to be easily understood by rotating load manufacturers. ACKNOWLEDGMENT The authors would like to thank Dr. S. Schröder, Research Scientist at GE Global Research, Munich, Germany, for his contribution to this paper.

R EFERENCES [1] J. Song-Manguelle, C. Sihler, and J. M. Nyobe-Yome, “Modeling of torsional resonances for multi-megawatt drives design,” in Conf. Rec. IEEE IAS 43rd Annu. Meeting, Edmonton, AB, Canada, Oct. 2008, pp. 1–8. [2] K. Ono and K. Takahasi, “Theoretical analysis of shaft vibration supported by a ball bearing with small sinusoidal waviness,” IEEE Trans. Magn., vol. 32, pt. 2, no. 3, pp. 1709–1774, May 1996. [3] K. Deeyiengyang and K. Ono, “Suppression of resonance amplitude of disk vibrations by squeeze air bearing plate,” IEEE Trans. Magn., vol. 37, pt. 1, no. 2, pp. 820–825, Mar. 2001. [4] J. J. Simond, A. Sapin, T. Xuan, R. Wetter, and P. Burmeiter, “12-pulse LCI synchronous drive for a 20 MW compressor: Modeling, simulation and measurements,” in Proc. 40th IEEE IAS Annu. Meeting, Hong Kong, Oct. 2005, vol. 4, pp. 2302–2308. [5] Integrated Compressor Line-ICL: An Integrated High-Speed Electric Motor Driven Compressor, GE Oil and Gas, Florence, Italy, 2007. [Online]. Availabe: www.ge.com/oilandgas [6] C. M. Ong, Dynamic Simulation of Electric Machinery using Matlab/ Simulink. Englewood Cliffs, NJ: Prentice-Hall, 1999. [7] J. Song-Manguelle, “Synthesis of AC machine modeling for digital control purposes,” M.S. thesis, ENSET, Univ. Douala, Douala, Cameroon, 1997. [8] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principle and Practice. New York: Wiley-Interscience, 2003, ser. IEEE Press Series on Power Eng.. [9] S. M. Abdulrahman, J. G. Kettleborough, and I. R. Smith, “Fast calculation of harmonic torque pulsations in a VSI/induction motor drive,” IEEE Trans. Ind. Electron., vol. 40, no. 6, pp. 561–569, Dec. 1993. [10] T. A. Lipo, P. C. Krause, and H. E. Jordan, “Harmonic torque and speed pulsations in a rectifier-inverter induction motor drive,” IEEE Trans. Power App. Syst., vol. PAS-88, no. 5, pp. 569–587, May 1969.

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[11] J. V. Milanovic, C. P. Fu, R. Radosavljecic, and Z. Lazarevic, “Sensitivity of torsional modes and torques to uncertainty in shaft mechanical parameters,” Elect. Power Compon. Syst., vol. 29, no. 10, pp. 867–881, Oct. 2001. [12] MV7000 Brochure: Entering a New Dimension for Reliability and Performance in Medium-Voltage AC Drives, Converteam, Massy, France, 2007. [Online]. Available: www.converteam.com

Joseph Song-Manguelle (M’07) was born in Makak, Cameroon. He received the B.S. and M.S. degrees in pedagogical sciences and electrical engineering from the University of Douala, Douala, Cameroon, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, in 2004. From 1997 to 1999, he was a Lecturer at the University of Douala. In 1999, he joined the Industrial Electronics Laboratory, EPFL, as a Postgraduate Assistant, where, from 2001 to 2004, he was a Research Associate. In 2004, he joined the High-Power Electronics Laboratory, GE Global Research Center, Munich, Germany, dealing with high-power drives topologies and control for various applications such as oil and gas. From 2006 to 2007, he was with Thermodyn, a GE oil and gas business in Le Creusot, France, as a Senior Electrical Engineer and Team Leader, dealing with high-power drives acquisition and torsional issues in compressor drive trains, including highspeed applications. From 2007 to 2008, he was with Baldor Electric, Montreal, QC, Canada, as a Senior Design Engineer. He spent part of his time at the Baldor Drives Center, Fort Smith, AR, focusing on low-voltage high-power regenerative drives. Since 2008, he has been with the Power Conversion Systems organization at the GE Global Research Center, Niskayuna, NY. His areas of interest are topologies and control of high-power electronics, as well as large-scale integrated electromechanical systems.

Jean-Maurice Nyobe-Yome was born in Yaoundé, Cameroon. He received the Diploma degree in electrical engineering of technical high-school teaching from the University of Douala-ENSET, Douala, Cameroon, in 1984; the M.S. degree in electrical engineering from the Ecole Normale Supérieure de Cachan, Cachan, France, and the Université de Paris VI, Paris, France, in 1986; and the Ph.D. degree from the Université de Monpellier II, Montpellier, France, in 1993. In 1985, he joined the University of DoualaENSET as a Lecturer, where he is currently an Associate Professor. He is also Head of the Electrical Engineering Department and Head of the Industrial Electronics and Systems Laboratory. Since 1994, he has supervised more than 200 Master’s theses in electrical engineering sponsored by local companies and the Cameroonian government. He is a Senior Member of the Cameroonian Commission for Technical Education. His areas of interest are resonant power conversion, high-power variable-speed drives, windmill/diesel twinning, and applied pedagogical sciences to electrical engineering.

Gabriel Ekemb was born in Matomb, Cameroon. He received the B.S. and M.S. degrees in pedagogical sciences and electrical engineering and the D.E.A. degree in electrical engineering from the University of Douala-ENSET, Douala, Cameroon, in 1995, 1997, and 2006, respectively, where he is currently working toward the Ph.D. degree, in collaboration with the University of Quebec, Rouyn-Noranda, QC, Canada. From 1997 to 2003, he was a Teacher at the Technical High-School of Nkongsamba, Cameroon. Since 2003, he has been with the Industrial Electronics and Systems Laboratory, University of Douala-ENSET, as a Senior Research Associate. His areas of interest are high-power conversion, electromechanical intereactions, and renewable energy power conversion.