Simplified Modelling of Oscillation Mode for Wind Power Systems

Hsiang Lin Chi

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OUTLINE

Abstract

Wind turbine generators can be operated in various types of system configurations. Once the configurations change, the system oscillation mode shapes change accordingly. The modelling of the full system is necessary for studying this issue, yet it is quite hard. In this paper, a simple approach is developed to study the mode shapes of wind power systems without the necessity of adopting the complex full-system models. The key is that the q-d axis model of electric power system is transformed into the single-axis model, so that it could integrate with the equivalent circuit model of drive train mechanism. After analyzing some of the system configurations organized by the well-known MOD-2 wind turbine generator unit, the proposed approach is found to be effective for analyzing various oscillation modes such as the local torsional oscillations, as well as the inter-unit and inter-area oscillations.

Keywords

Wind turbine generator (WTG); wind power system; oscillation mode; dynamics

I. Introduction

THERE have been many investigations on the dynamics and transient stability of wind power systems. One of the important issues is about the oscillations of the drive train. The modelling of wind turbine generator (WTG) units plays an important role in these studies. In [

1], a 2-mass model is used to study the effect of network faults. In [2], a 3-mass model is used to study the critical clearing time. In [3], a 6-mass model is used to analyze the transient stability affected by the inertia constant ratio of blades and hub. A comparative study of transient stability analysis by using different drive train models is presented in [4]. In [5], gear contact forces between wheels are modeled with linear spring acting in the plane of action along the contact line, leading to a 12-mass model. The model is thus able to investigate the influence of bearing stiffness on the internal dynamics of the drive train. In [6], the model derived by finite element is used to calculate the stress and deformation in the drive train components continuously with time. The effects of teeth defect and wear on the gear can be assessed. It can be seen obviously that every addition of the complexity to the model leads to additional information of the system.

However, microgrids have recently become more and more popular, especially for the wind power systems and hybrid systems. The attentions are no longer paid on the oscillations of mechanical parts alone. It becomes significant to study the electromechanical oscillations, the inter-unit oscillations and the inter-area oscillations [

7]-[11]. In [12], the difference in mode shapes is studied when a WTG is connected to a large utility or an isolated diesel generator unit. The approach could provide the basic information about both the mechanical and the electromechanical oscillations. Unfortunately, it cannot be further used in microgrids with more complex multi-machine configurations. The reason is that the electrical systems must be transformed and integrated to the mechanical side, yet only very simple electrical circuits can be treated in that way.

In order to study various oscillation modes of WTG systems in various system configurations, an approach opposed to the above-mentioned one is developed, where the mechanical systems are transformed and integrated into the electrical side. The key of the integration is that the generators (synchronous, induction or permanent magnet synchronous generators (PMSG)), converters (controlled rectifier and inverter) and circuits (transmission line, transformer, load and infinite bus) need to be transformed into the single-axis model, so that they can be combined with the wind turbine (WT) mechanical model. In this way, a WTG system, no matter how complex its configuration is, would turn to be an electrical circuit network. It will be possible to analyze such a network without difficulties, which is the merit of the proposed approach.

II. System Descriptions and Modelling

The WTG unit of MOD-2 system is adopted for investigation because complete studies of the unit are available [

13], [14]. It is a 2.5 MW unit and the generator is a 3-phase, 4.16 kV, 60 Hz, 4 pole, 1800 r/min synchronous generator rated 3.125 MVA. The blades adopt horizontal-axis propellers containing a 45-foot variable pitch tip section. Each blade is 150 feet long, and the blade speed is 17.55 r/min.

A. WTG Unit Model

For the WTG unit studied, the blade torque is transmitted to the generator via the drive train as shown in Fig. 1(a), where SG denotes the synchronous generator. For modelling the drive train, the mass-damping-spring model, as shown in Fig. 1(b), is adopted [

15], where J, K, D and

τ

are the inertia constant, stiffness coefficient, damping constant and torque, respectively; the subscripts B1, B2, H, G denote blade 1, blade 2, hub and generator rotor, respectively; and τ_EM is the electromagnetic (EM) torque. In the model, the inertias of low-speed shaft, gear box and high-speed shaft are included in the hub and generator rotor, respectively. The mechanical data are listed in Table I. By using the electromechanical analogy [16], the model can be transformed into the electrical circuit, as shown in the left block of Fig. 2(a), where ω_B₁, ω_B₂ and ω_G are the angular velocities of blade 1, blade 2 and generator rotor, respectively;

v_{q s}

and

i_{q s}

are the voltage and current referred to the q-axis, respectively;

I_{f d}^{'}

is the field current;

ω_{b}

is the rated system angular velocity; and

X_{q}

is the q-axis reactance..

Fig. 1 Drive train. (a) Scheme. (b) Mass-damping-spring model.

Table I Mechanical Parameters Referred to 3.125 MVA Base

Section	Inertia (p.u.)	Stiffness (p.u.)	Damping (p.u.)
J_B₁	11.3937500	-	-
K_HB₁/D_HB₁	-	1076.375	36.6375
J_B₂	11.3937500	-	-
K_HB₂/D_HB₂	-	1076.375	36.6375
J_H	0.5954375	-	-
K_HG/D_HG	-	35.500	2.2900
J_G	1.3068750	-	-

Fig. 2 Circuit model. (a) WTG unit. (b) Infinite/load bus.

For modelling a synchronous generator, the q-d axis model referred to the rotor reference frame is adopted, of which the electrical data are listed in Table II, where R_s, X_ls are the resistance and leakage reactance of the stator, respectively; R_fd, X_lfd are the resistance and leakage reactance of the field windings, respectively; X_mq and X_md are the magnetizing reactances referred to the q-axis and d-axis, respectively; and R_kq, X_lkq, R_kd, X_lkd are the resistances and leakage reactances referred to the q-axis and d-axis of damping windings, respectively. Based on the simplification process [

17], the generator circuit model can be simplified to a single-axis circuit model shown in the right block of Fig. 2(a), where an ideal transformer interfaces the electrical and mechanical parts.

Table II Generator Parameters Referred to 3.125 MVA Base

Parameter	Value (p.u.)	Parameter	Value (p.u.)
R_s	0.00750	R_fd	0.001063
X_ls	0.08100	X_lfd	0.188400
X_mq	0.92900	X_md	1.850000
R_kq	0.00988	R_kd	0.014230
X_lkq	0.06643	X_lkd	0.104900

By combining the circuit model of the drive train and the simplified circuit model of the synchronous generator, the whole circuit model of WTG unit is obtained in Fig. 2(a).

B. Infinite/Load Bus Model

A grid can be viewed as a group of turbine generators operating in parallel. Thus, many circuit models of the turbine generator unit are combined in parallel to represent the circuit model of a grid. The result is shown in Fig. 2(b), in which the shunt capacitance (equivalent to the inertia in mechanics) is very large and the series resistance and reactance are very small. In case of an ideal infinite bus, the model becomes a constant voltage source.

A load bus can be represented the same as a grid, because it is also a power unit which does not generate but dissipate power. Therefore, the power flows are in the opposite direction. In addition, the loads usually have very small or zero inertia, and the shunt capacitance of the model will be very small. In case of an ideal constant-torque load, the model becomes a constant current source.

III. Mode Shape Analysis for Single-machine Systems

The MATLAB/Simulink/SimPowerSystems is used to analyze the wind power systems. Since the whole circuit model is adopted for the WTG unit, the wind power system including both the mechanical and the electrical parts can be built easily by using the electrical components. By simply using the built-in instructions, the state space equations can be obtained, and the eigenvalues and eigenvectors of the system can be further calculated.

Assume that the WTG unit operates at 1.0 p.u. of output power with power factor of 0.8. The turn ratio of the electromechanical interface transformer is thus equivalent to 1.8 by adopting 2.5 MW power base and 4.16 kV voltage base. The detailed calculation can be accessed in [

17]. Moreover, it is assumed that the grid inertia is 100 times the WTG unit, and the load bus inertia is 1/1000 of the WTG unit.

A. Verification

In order to verify the proposed approach, the oscillation modes for the WTG unit connecting to infinite bus (i.e., the WTG-I system) are studied. Firstly, the undamped and damped eigenvalues are calculated by ignoring and considering both the mechanical damping and the electrical resistance, respectively. The results are shown in Table III.

Table III Undamped and Damped Eigenvalues for WTG-I System

Mode	Undamped eigenvalue	Damped eigenvalue
EM	$0 \pm 0.86$ i	$- 0.0234 \pm 0.86$ i
Generator	$0 \pm 27.13$ i	$- 0.4385 \pm 27.13$ i
Hub	$0 \pm 43.41$ i	$- 32.5251 \pm 28.75$ i
Blade	$0 \pm 6.87$ i	$- 0.8040 \pm 6.83$ i

Compared with [

13], both the mode frequency and the decrement factors (the real part of eigenvalues) are identical, which verifies the proposed approach. Based on the magnitude of eigenvectors, the mode shapes of the WTG-I system can be obtained. The WTG-L system represents the WTG unit supplying power to load directly. The mode shapes are shown in Figs. 3-5, where I and L denote the infinite bus and load, respectively. The modes are named according to the sensitive sections. Compared with [13], all mode shapes are almost the same. This further verifies the proposed approach.

Fig. 3 Mode shapes of drive train of WTG.

Fig. 4 Mode shapes of WTG-I system.

Fig. 5 Mode shapes of WTG-L system.

B. Comparison Between Isolated and Grid-connected Systems

In order to make a comparison between isolated and grid-connected systems, the case of the WTG-L system is further studied. It can be found that the generator mode and the EM mode present significant difference in shapes.

1)　Generator Mode

In Fig. 3, the oscillation modes of the drive train of WTG are shown. By comparing all mode shapes of the drive train of WTG with those of the WTG-L system and WTG-I system, it can be seen that the oscillation shapes of the mechanical parts are not changed under different electrical configurations, except for the generator mode. Under the WTG-I system configuration, the infinite bus forms a node due to its extremely large inertia. The generator rotor, hence, oscillates against the infinite bus. Under the WTG-L system configuration, however, the inertia of the load bus is small. Thus, the generator rotor and the load bus would oscillate to each other. In addition, the generator mode is found to present significant changes at mode frequency. For the drive train alone, the frequency is $ω = 3.76$ rad/s. Under the WTG-I and WTG-L configurations, it changes to $ω = 27.13$ rad/s and $ω = 3.75$ rad/s, respectively. It can be concluded that the electrical systems do affect the mechanical mode shape and frequency of a drive train of WTG, and the generator mode would be the most sensitive one.

2)　EM Mode

By comparing the WTG-L system with the WTG-I system, it can be found that the EM mode shapes of the two system configurations are totally different. For the WTG-I system configuration, the infinite bus ties together with the generator rotor, and the blades tie together with the hub. The combination of infinite bus and generator rotor forms a node, and the combination of blades and hub (i.e., WT) swings against it. Thus, the EM mode can also be named as the WT mode in the WTG-I system configuration. For the WTG-L system configuration, the combination of generator rotor, hub and blades forms a node, and the load bus swings against it. The EM mode can also be named as the load mode in the WTG-L system configuration. Therefore, the EM mode shape changes from the WT mode to the load mode when the configuration changes from the WTG-I system to the WTG-L system.

3)　Viewpoint of Dynamics

The mode shapes provide the basic information corresponding to mode behaviors. For the EM mode in the WTG-L system configuration, the combination of generator rotor, hub and blades forms a node, whereas a large magnitude of eigenvector appears at the load bus. This indicates that the EM mode is sensitive to electrical load disturbance, yet is totally insensitive to wind turbulence. From the viewpoint of control, it means that the dynamics of this mode can be improved by appropriate control of the system load, but cannot be improved by the pitch control or the generator automatic voltage regulator (AVR) control. However, for the EM mode in the WTG-I system configuration, a large magnitude of eigenvector appears at the combination of blades and hub. This indicates that the EM mode is sensitive to the turbulence of wind field, while the pitch control would be the effective way to provide damping for this mode. Therefore, there might be totally different characteristics under isolated and grid-connected wind power configurations.

To verify the different sensitivities subjected to wind turbulence for the WTG-L and WTG-I systems, the impulse responses excited by the blade torque disturbance for both systems are simulated. The hub-generator shaft torque and generator current are shown in Fig. 6. The results obviously corroborate the inference.

Fig. 6 Impulse responses excited by blade torque disturbance. (a) WTG-I system. (b) WTG-L system.

The same inference may be made for the generator mode. It can be easily judged from the shapes that both the generator AVR control and the electrical load have significant influences on the generator mode behaviors in the WTG-L system configuration. However, only the generator AVR control is important in the WTG-I system configuration.

Moreover, the changes in mode frequency can be a serious problem in some cases. For some WTG units with torsional vibration problems in main shaft, the frequency-based countermeasures may be adopted to alleviate the vibrations. The vibration suppression effect may be worse or even totally futile if the mode frequency is changed.

4)　Effect of Length of Transmission Line

Typically, a WTG unit is connected to network via a step-up transformer and a transmission line. The combined impedance of transformer and transmission line is found to significantly affect the mode shapes of the wind power system. This effect is studied for the WTG-I system. It is assumed that the combined impedances of short and long transmission lines are the same as and 3000 times that of the generator, respectively.

Based on the simulation results, it is found that there are obvious changes in the EM mode shapes. This is shown in Fig. 7. In case the transmission line is short, the generator rotor will be tied up by the much larger inertia of grid, which leads to that the WT swings against it. In contrast, in case the transmission line is long, the tie between the generator and the grid is broken. The generator rotor turns to unite with the WT, and they swing against the infinite bus.

Fig. 7 EM mode shapes for different lengths of transmission lines. (a) Long transmission line. (b) Short transmission line.

The shape variation process for the EM mode is analyzed and shown in Fig. 8(a). The transmission line impedance increases from 0 to 1000 times (defined as the K factor) the generator impedance. The frequency as a function of K factor for the generator mode is shown in Fig. 8(b). It can be observed from both figures that when the K factor exceeds 500, the tie between the generator rotor and grid is almost totally broken.

Fig. 8 Variation process of EM mode shapes. (a) Shape variation process. (b) Frequency variation with K factor.

IV. Mode Shape Analysis for Multi-machine Systems

In addition to the isolated and grid-connected system configurations, the WTG units may be clustered to form a microgrid. The proposed approach can be applied to such a multi-machine system without any difficulty.

A. Multi-WTG-to-load Systems

For the system of two paralleled WTG units supplying power to loads (i.e., the two-WTG-to-load system), the vibration modes are shown in Fig. 9. Compared with Figs. 3-5, it can be seen that both the modes in WTG-L and WTG-I systems are present. For the EM mode, both the load mode ( $ω = 875.26$ rad/s) and the WT mode ( $ω = 0.86$ rad/s) exist simultaneously. However, the influential control action is the pitch control for the WT mode, and the load control for the load mode. For the generator mode, both $ω = 3.76$ rad/s and $ω = 27.13$ rad/s modes exist simultaneously. The influential control actions are the generator AVR control for the $ω = 27.13$ rad/s mode, and the load control and the generator AVR control for the $ω = 3.76$ rad/s mode. For the system of three paralleled WTG units supplying power to loads (i.e., the three-WTG-to-load system), the oscillation modes are shown in Fig. A1 of Appendix A.

Fig. 9 Mode shapes of two-WTG-to-load system.

B. Multi-machine-interconnected Systems

Since the mechanical part shapes do not change with the electrical power system configurations, the blades and hub sections are combined and simplified as a WT section in order to simplify the mode shape diagrams of a multi-machine system. The mode shapes of a two-WTG-interconnected system for different lengths of transmission lines are shown in Fig. 10, where R_TX_-_LINE and K_TX_-_LINE are the resistance and inductance of the transmission line, respectively.

Fig. 10 Mode shapes of two-WTG-interconnected system.

In the short transmission line situation, it can be found that the mode shapes are the same as those of the two-WTG-to-load system, except that a new load mode is produced, in which two load buses swing to each other. When the transmission line becomes long, two changes happen. One is that for the WT mode and one of the generator modes, the originally united load buses become swing to each other. The other is that for the WT mode, the generator rotor separates from the node formed by the generator rotors and loads, and turns to tie together with the WT. For a three-WTG-interconnected system, the mode shapes are shown in Fig. A2 of Appendix A.

V. Mode Frequency Shifting

According to the equivalent circuit of WTG unit, the turn ratio of interface transformer depends on the generator operation conditions. Therefore, it is evident that some of the mode frequencies will vary following the variation in generator operation conditions. In case of the power output keeping at 1.0 p.u., the frequencies of generator mode and EM mode are calculated and listed in Table IV for the power factor decreasing from 1.0 to 0.8 with an interval of 0.05. In case of the power factor keeping at 0.8, the corresponding frequencies are listed in Table V for the different power outputs. It is found that the frequencies of both the generator mode and the EM mode are affected by the operation conditions. However, the generator mode is more sensitive to operation conditions than the EM mode.

Table IV Mode Frequencies for Different Power Factors with Power of 1.0 p.u.

Power factor	Ratio	Generator mode (rad/s)	EM mode (rad/s)
1.00	1.426	21.6094	0.8520
0.95	1.632	24.6477	0.8549
0.90	1.706	25.7412	0.8557
0.85	1.759	26.5265	0.8563
0.80	1.800	27.1334	0.8566

Table V Mode Frequencies for Different Power Outputs with Power Factor of 0.8

Power (p.u.)	Ratio	Generator mode (rad/s)	EM mode (rad/s)
1.00	1.800	27.1334	0.8566
0.95	1.755	26.4676	0.8562
0.90	1.710	25.8018	0.8558
0.85	1.666	25.1539	0.8553
0.80	1.622	24.5025	0.8548

VI. Pole Locus Analysis

One of the merits of the proposed WTG unit model is that the poles of a complex multi-machine system can be easily derived. Thus, the dynamics of each mode can be assessed. The WTG-I system is studied as an example, and three cases are analyzed.

A. Effect of Mechanical Damping

Referring to Figs. 3-5, it can be observed that the generator rotor swings against the hub for the EM, the generator and the hub modes. That implies a viscus damper installed at the hub-generator shaft section will significantly influence the behavior of those modes. This phenomenon is shown in Fig. 11(a), where the pole locus corresponding to various D_HG is depicted.

Fig. 11 Effects on pole locus. (a) Damping of hub-generator shaft. (b) Generator resistance. (c) Transmission line impedance.

For the generator mode ( $ω = 27.13$ rad/s), the locus is away from the vertical axis, which means that the mode gets more stable. For the hub mode ( $ω = 28.75$ rad/s) and the EM mode ( $ω = 0.86$ rad/s), their tendency is similar. Initially, the locus is toward the horizon axis and away from the vertical axis. Hence, the effect of D_HG is also positive. The dynamics of the agonizing EM mode, of which the decrement factor is only -0.0234, can also be improved by a damper installed at the hub-generator shaft section. When the damping keeps increasing, the complex number pole pair turns to become the real number poles. If the damping increases even larger, new complex number pole pair will be generated. However, the locus will move away from the horizon axis and toward the vertical axis. That means the increase in damping will have a negative effect in this situation.

B. Effect of Resistance

The generator resistance will affect the modes in relevance to electrical parts. This is shown in Fig. 11(b). Both the generator mode and the EM mode have improved in decrement factors, while the frequencies remain unchanged.

C. Effect of Transmission Line

The effect of transmission line impedance is shown in Fig. 11(c). It can be seen that the decrement factor of EM mode ( $ω = 0.86$ rad/s) decreases following the increase in transmission line length. Therefore, the effect is negative. The EM mode is the dominant one since it has small decrement factor and is sensitive to the turbulence of wind field, and therefore, a very long transmission line is not allowed. The same phenomenon is also found in other multi-machine systems.

VII. Analysis of Responses due to Disturbances

Another merit of the proposed WTG unit model is that the Bode plots and impulse responses can be easily obtained for a complex wind power system. The responses due to the interactions between the electrical and the mechanical parts can thus be assessed. As an example, the WTG-I system is studied. The results are shown in Fig. 12, which demonstrates the frequency responses of blade torques (T_B₂_H, T_B₁_H), the hub-generator shaft torque T_HG, and the generator current I_q.

Fig. 12 Frequency responses. (a) Excited by generator voltage disturbance. (b) Excited by blade torque disturbance.

The responses shown in Fig. 12(a) are induced by generator voltage disturbance. It shows that only the modes with $ω = 0.86$ rad/s and $ω = 27.13$ rad/s are noticeable. However, responses at all sections are below 0 dB, which means the disturbances will not be amplified.

The responses shown in Fig. 12(b) are induced by the torque disturbance of blades. It shows that the noticeable modes are different at various sections. The $ω = 0.86$ rad/s mode appears at all of the sections and presents the highest amplitude, which is the most important mode without doubt. Since the responses exceed 0 dB, the disturbances will be amplified. For the blade sections, the $ω = 6.87$ rad/s mode is also important. It presents the responses that exceed 0 dB. The reason is that this is the blade mode. For the hub-generator shaft, no other modes are noticeable except the $ω = 0.86$ rad/s mode. For the generator current, the $ω = 27.13$ rad/s mode is also noticeable. However, the response is far below 0 dB.

By the cross comparison between Fig. 12(a) and (b), it can be found that the blade torque disturbance on generator current is more significant than the generator voltage disturbance on torsional vibrations of turbine shafts and blades. To further validate this, the time domain simulations are conducted. The results are shown in Fig. 13(a) and (b), where the impulse responses induced by the two types of disturbance are presented. It can be observed that only small torsional vibrations are induced at the blades and shafts subject to voltage disturbance. On the contrary, significant oscillations are induced at the generator current subject to blade torque disturbance.

Fig. 13 Impulse responses. (a) Excited by generator voltage disturbance. (b) Excited by blade torque disturbance.

The proposed WTG unit model can also be used to assess the interactions between two WTG units in a multi-machine system. The two-WTG-to-load system is studied as an example. The resulting Bode plots are shown in Fig. 14, where the blades of unit 1 act as a shaker and the responses on the unit 2 are presented. It can be seen that the EM mode ( $ω = 0.86$ rad/s) are induced with a gain exceeding 0 dB. Therefore, disturbing on one unit could cause significant impact on the other unit.

Fig. 14 Frequency responses of two-WTG-to-load system.

VIII. Discussion

A. Induction Generator

Although the MOD-2 adopts a synchronous generator, most of commercial WTGs are designed using squirrel cage induction generators. It is interesting to know what will be different if the synchronous generator is replaced by an induction generator.

In [

12], based on the equivalent torsional stiffness and simplified linear model of induction generator dynamics, a mechanical model of WTG-I system is built. If the electromechanical analogy is applied, it can be found that the equivalent circuit will be in the same form as Fig. 2, except that the generator resistance is inversely proportional to the slip rate. According to the analysis in Section VI-B, it can be deduced that there will not be significant difference in mode shapes if an induction generator is adopted instead of a synchronous generator. However, more damping will be provided to some of the modes. Furthermore, the power factor correction (PFC) capacitors are usually installed with the induction generators, and therefore, the generator terminal should be treated as a load bus. The mode shapes would be resembled to the ones mentioned in Section IV. However, since the capacitance of the PFC capacitor is between the equivalent capacitance of a normal load and a grid, the frequencies of the corresponding modes would be between those of the two cases accordingly.

B. Direct Drive WTG Unit

A direct drive WTG unit includes converters as the grid interfaces. The typical scheme is shown in Fig. 15. The power train consists of blades, hub, main shaft, PMSG, rectifier, direct current (DC) link capacitor and inverter. The modelling of the PMSG, rectifier and inverter can be done as shown in Appendix B. By combining the circuit models of WT mechanism, PMSG, rectifier, DC lick capacitor and inverter, the complete model of direct drive WTG unit can be obtained, as shown in Fig. 16. Based on the unit model, much more complex multi-machine systems with different configurations can be developed and analyzed easily.

Fig. 15 Diagram of direct drive WTG system.

Fig. 16 Complete model of direct drive WTG unit.

However, the modelling of converters is based on the presumption of constant control instruction. When it comes to the emerging sub-synchronous oscillation phenomenon [

18], [19], which is usually called as the sub-synchronous torsional interaction, the control instruction such as the delay angle in a controlled rectifier would oscillate in a certain frequency. This results in the turn ratio variation of the equivalent transformer in a converter. The overall equivalent circuit of a direct drive wind power system turns to be a time varying circuit. The circuit can still be analyzed; nevertheless, it will not be that easy as a normal circuit. The mode frequencies will not be kept constant but spread in a range, and so as the mode shapes. Much more in-depth studies are needed to sufficiently present the complete profile.

C. Accuracy of Simplified Model

Compared with the full machine model, the simplified machine model has neglected the transients of stator and rotor windings, and the effects of damping windings and salient pole. These simplifications result in significant deviations in transient responses. The transient responses of the machine rotor angle are simulated for the WTG-I system configuration with the simplified model, full model, and full model plus power system stabilizer (PSS), respectively. A three-phase-to-ground fault occurs at 1.0 s and disappears at 1.05 s at the far end of transmission line. Different mechanical damping as well as different transmission line lengths have been considered in the simulations. The overall results are shown in Fig. 17. It is obvious that the major difference of adopting the simplified model is mainly on the damping behavior. The simplified model provides a more conservative results than the full model. If there is a power damping control such as the PSS, the errors increase even much more. However, these will not obstruct the applications of the simplified model, because the approach is not attempting to analyze the system stability or dynamics in detail. Instead, it focuses on acting as a preliminary preprocessing for stability and dynamics studies and providing the basic information.

Fig. 17 Rotor angle responses. (a) Decreased mechanical damping/short transmission line. (b) Decreased mechanical damping/long transmission line. (c) Increased mechanical damping/short transmission line. (d) Increased mechanical damping/long transmission line.

Furthermore, by comparing the effects of mechanical damping and transmission line length, it can be seen that the increased mechanical damping and the long transmission line will be helpful for decreasing the error. Therefore, the accuracy is dependent on the application situations.

IX. Conclusion

The main contribution of the proposed approach is that not only the local torsional oscillation modes, but also the electromechanical, the inter-unit and even the inter-area modes can be studied without the necessity to adopt the complex full model.

It is easily applied to solve the commonly encountered sub-synchronous resonance and oscillation problems to find the torsional interaction or transient amplification effects. However, the capability is limited because only a simplified machine model is adopted. The lack of control systems will make it hard to study the emerging sub-synchronous oscillation phenomenon. Nevertheless, this can be set as a research goal for future modification and improvement.

After using the proposed approach to study some of the different configurations of WTG systems, the following noteworthy results are summarized.

1) The electrical systems would affect the shapes of some of the oscillation modes, even the mechanical modes.

2) The EM mode shape is significantly dependent on system configurations.

3) The transmission line length is an important factor to influence the mode shapes of WTG systems.

Appendix

Appendix A

The mode shapes of three-WTG-to-load and three-WTG-interconnected systems are shown in Figs. A1 and A2, respectively.

Fig. A1 Mode shapes of three-WTG-to-load system.

Fig. A2 Mode shapes of three-WTG-interconnected system.

Appendix B

A. Single-axis Model of PMSG

For a PMSG, the typical dynamic equations in q-d reference frame are as the following formulas.

\{\begin{array}{l} [\begin{array}{l} v_{q s} \\ v_{d s} \end{array}] = [\begin{matrix} r_{s} + p L_{q} & ω_{r} L_{d} \\ - ω_{r} L_{q} & r_{s} + p L_{q} \end{matrix}] [\begin{array}{l} i_{q s} \\ i_{d s} \end{array}] + [\begin{matrix} ω_{r} λ_{m}^{'} \\ 0 \end{matrix}] \\ τ_{E M} = \frac{P}{2} [λ_{m}^{'} i_{q s} + (L_{d} - L_{q}) i_{q s} i_{d s}] \end{array}

(B1)

where P is the pole number; p is the differentiation operator; $v_{d s}$ is the d-axis voltage; $i_{d s}$ is the d-axis current; $v_{q s}$ is the q-axis voltage; $i_{q s}$ is the q-axis current; $r_{s}$ is the stator resistance; $L_{d}$ is the d-axis stator inductance; $L_{q}$ is the q-axis stator inductance; $λ_{m}^{'}$ is the q-d axis flux linkage; $ω_{r}$ is the rotor angular velocity (electrical); and $τ_{E M}$ is the EM torque.

Assume that the initial angle of rotor reference frame is zero degree, thus $v_{d s} = 0$ , and the relationship between the d-axis and q-axis currents becomes:

i_{d s} = \frac{ω_{r} L_{q}}{r_{s} + p L_{d}} i_{q s}

(B2)

Rewriting the q-axis equation by using the above formula, we have:

v_{q s} = [\frac{(r_{s} + p L_{q}) (r_{s} + p L_{d}) + ω_{r}^{2} L_{q} L_{d}}{r_{s} + p L_{d}}] i_{q s} + ω_{r} λ_{m}^{'}

(B3)

Usually, $L_{q} = L_{d} = L_{s}$ , and the term $ω_{r}^{2} L_{q} L_{d}$ can be neglected. Moreover, by changing the electrical angular velocity $ω_{r}$ to mechanical angular velocity $ω_{M}$ and the q-d axis flux linkage $λ_{m}^{'}$ to the abc axis flux linkage $λ_{m}$ , the equations can be obtained as the followings. Based on those equations, the single-axis circuit model can be obtained as shown in Fig. B1.

\{\begin{array}{l} v_{q s} = (r_{s} + p L_{s}) i_{q s} + \frac{P}{2} \sqrt[]{\frac{2}{3}} λ_{m} ω_{M} \\ τ_{E M} = \frac{P}{2} \sqrt[]{\frac{2}{3}} λ_{m} i_{q s} \end{array}

(B4)

Fig. B1 Single-axis model of PMSG.

B. Rectifier Model

The rectifier scheme is shown in Fig. B2(a). Assume that the input voltages of the rectifier are ideal sinusoidal waves with peak value $V_{p}$ and angular velocity $ω_{e}$ . The phase-a voltage can be expressed as:

v_{a s} = V_{p} c o s ω_{e}

(B5)

Fig. B2 Rectifier. (a) Scheme. (b) Ideal transformer model.

By using the Park’s transformation to the three-phase voltages $v_{a s}$ , $v_{b s}$ and $v_{c s}$ , we obtain:

\{\begin{array}{l} v_{q s} = \sqrt[]{\frac{3}{2}} V_{p} \\ v_{d s} = 0 \end{array}

(B6)

Assume the delay angle of the rectifier to be $α$ , and neglect the commutation loss. Then the output voltage of the rectifier will be:

v_{r} = \frac{3 \sqrt[]{3}}{π} V_{p} c o s α = (\frac{3 \sqrt[]{2}}{π} c o s α) v_{q s}

(B7)

The power invariant is applied to the rectifier assuming that no losses are considered. So, the rectifier can be modeled as an ideal transformer as shown in Fig. B2(b).

C. Inverter Model

The pulse width modulation (PWM) inverter scheme studied is shown in Fig. B3(a). Its operation can be analyzed by expressing the output voltage of the inverter as a product of the six-step continuous phase voltage and a pulse train with pulses of unit magnitude. For example, $v_{a s}$ , the voltage of phase a, may be expressed as the following formula.

Fig. B3 PWM inverter. (a) Scheme. (b) Ideal transformer model.

v_{a s} = \frac{2 v_{i}}{π} (c o s (ω_{e} t) + \frac{1}{5} c o s (5 ω_{e} t) - \frac{1}{7} (c o s 7 ω_{e} t) + \dots) P_{M}

(B8)

where $v_{i}$ is the input voltage of the inverter; and $P_{M}$ is the pulse train, which may be expressed in Fourier series as:

P_{M} = η + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n π} (s i n (12 n ω_{e} t) - s i n (12 n ω_{e} (t - T_{1})))

(B9)

where $η = T_{1} / T$ ; and $T = π / 6 ω_{e}$ .

By using the Park’s transformation to the voltages $v_{a s}$ , $v_{b s}$ and $v_{c s}$ , we obtain:

\{\begin{array}{l} v_{q s} = \sqrt[]{\frac{3}{2}} \frac{2 v_{i}}{π} P_{M} (1 + \frac{2}{35} c o s (6 ω_{e} t) - \frac{2}{143} c o s (12 ω_{e} t) + \dots) \\ v_{d s} = \sqrt[]{\frac{3}{2}} \frac{2 v_{i}}{π} P_{M} (\frac{12}{35} s i n (6 ω_{e} t) - \frac{24}{143} s i n (12 ω_{e} t) + \dots) \end{array}

(B10)

If the harmonics are neglected, the voltage formulas are further reduced to:

\{\begin{array}{l} v_{q s} = \frac{\sqrt[]{6} η}{π} v_{i} \\ v_{d s} = 0 \end{array}

(B11)

Again, the power invariant applies to the inverter if it is assumed that no losses are considered. So, the inverter can be modeled as an ideal transformer, as shown in Fig. B3(b).

REFERENCES

S. K. Kim and E. S. Kim, “PSCAD/EMTDC-based modeling and analysis of a gearless variable speed wind turbine,” IEEE Transactions on Energy Conversion, vol. 22, no. 2, pp. 421-430, Jun. 2007. [百度学术]

S. K. Salman and A. L. J. Teo, “Windmill modeling consideration and factors influencing the stability of a grid-connected wind power-based embedded generator,” IEEE Transactions Power Systems, vol. 18, no. 2, pp. 793-802, May 2003. [百度学术]

H. Li and Z. Chen, “Transient stability analysis of wind turbines with induction generators considering blades and shaft flexibility,” in Proceedings of the 33rd Annual Conference of the IEEE Industrial Electronics Society (IECON), Taipei, China, Nov. 2007, pp. 1604-1609. [百度学术]

S. M. Muyeen, M. H. Ali, R. Takahashi et al., “Comparative study on transient stability analysis of wind turbine generator system using different drive train models,” IET Renewable Power Generation, vol. 1, no. 2, pp. 131-141, Jun. 2007. [百度学术]

J. Peeters, D. Vandepitte, and P. Sas, “Analysis of internal drive train dynamics in a wind turbine,” Wind Energy, vol. 9, no. 1-2, pp. 141-161, Jun. 2006. [百度学术]

O. Lundvall, N. Stromberg, and A. Klarbring, “A flexible multi-body approach for frictional contact in spur gears,” Journal of Sound and Vibration, vol. 278, no. 3, pp. 479-499, Dec. 2004. [百度学术]

M. Jafarian and A. M. Ranjbar, “Interaction of the dynamics of doubly fed wind generators with power system electromechanical oscillations,” IET Renewable Power Generation, vol. 7, no. 2, pp. 89-97, Mar. 2013. [百度学术]

E. Rezaei, A. Tabesh, and M. Ebrahimi, “Dynamic model and control of DFIG wind energy systems based on power transfer matrix,” IEEE Transactions on Power Delivery, vol. 27, no. 3, pp. 1485-1493, Jul. 2012. [百度学术]

L. Fan, H. Yin, and Z. Miao, “On active/reactive power modulation of DFIG-based wind generation for interarea oscillation damping,” IEEE Transactions on Energy Conversion, vol. 26, no. 2, pp. 513-521, Jun. 2011. [百度学术]

L. Wang and K. H. Wang, “Dynamic stability analysis of a DFIG-based offshore wind farm connected to a power grid through an HVDC link,” IEEE Transactions on Power Systems, vol. 26, no. 3, pp. 1501-1510, Aug. 2011. [百度学术]

A. J. Laguna, N. F. B. Diepeveen, and J. W. V. Wingerden, “Analysis of dynamics of fluid power drive-trains for variable speed wind turbines: parameter study,” IET Renewable Power Generation, vol. 8, no. 4, pp. 398-410, May 2014. [百度学术]

J. A. M. Bleijs, “Wind turbine dynamic response-difference between connection to large utility network and isolated diesel micro-grid,” IET Renewable Power Generation, vol. 1, no. 2, pp. 95-106, Jun. 2007. [百度学术]

O. Wasynczuk, D. T. Man, and J. P. Sullivan, “Dynamic behavior of a class of wind turbine generators during random wind fluctuation,” IEEE Transactions on Power Apparatus and Systems, vol. 100, no. 6, pp. 2837-2845, Jun. 1981. [百度学术]

P. C. Krause and O. Wasynczuk, “Method of resynchronizing wind turbine generators,” IEEE Transactions on Power Apparatus and Systems, vol. 100, no. 10, pp. 4301-4308, Oct. 1981. [百度学术]

R. Wamkeue, Y. Chrourou, J. Song et al., “State modeling based prediction of torsional resonances for horizontal-axis drive train wind turbine,” in Proceedings of IET Conference on Renewable Power Generation (RPG 2011), Edinburgh, UK, Sept. 2011, pp. 1-5. [百度学术]

C. H. Lin, “Modeling and analysis of power trains for hybrid electric vehicles,” International Transactions on Electrical Energy Systems, vol. 28, no. 6, pp. 1-19, Jun. 2018. [百度学术]

W. M. Lin, C. C. Tsai, and C. H. Lin, “Analysing the linear equivalent circuits of electromechanical systems for steam turbine generator units,” IET Generation, Transmission & Distribution, vol. 5, no. 7, pp. 685-693, Jul. 2011. [百度学术]

D. Shu, X. Xie, H. Rao et al., “Sub- and super-synchronous interactions between STATCOMs and weak AC/DC transmissions with series compensations,” IEEE Transactions on Power Electronics, vol. 33, no. 9, pp. 7424-7437, Sept. 2018. [百度学术]

Y. Xu, H. Nian, T. Wang et al., “Frequency coupling characteristic modeling and stability analysis of doubly fed induction generator,” IEEE Transactions on Energy Conversion, vol. 33, no. 3, pp. 1475-1486, Jan. 2018. [百度学术]

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