NOMENCLATURE
Rotor voltage in the rotor and stator reference frames
Stator voltage in the stator reference frame
Stator and rotor current in the stator reference frame
Stator and rotor current in the rotor reference frame
Stator and rotor fluxes in the stator reference frame
Rs, Rr Resistances of the stator and rotor
Lsσ, Lrσ Leakage inductances of the stator and rotor
Lm, Ls, Lr Mutual inductances, stator and rotor inductances
ω1, ωr Synchronous and rotor electrical angular speeds
Udc DC-LINK voltage of the back to back converter
pn Number of pole-pairs
k Winding conversion coefficient
Superscripts
→ Space vectors
s, r Stator and rotor reference frames
' Values after the winding conversion
Subscripts
α, β Stationary α-axis and β-axis
s, r Stator and rotor
cal Calculated result
sim Simulated result
ex Experimental result
Boldface Variables in the frequency domain
I. INTRODUCTION
Many wind farms employ wind turbines based on DFIG systems, where the converter is typically rated at around 30% of the system’s rated capacity. However, the rated slip power of the converter results in damage to the back-to-back converter when a grid voltage sag happens. The induced high voltage at the rotor side and the transient attenuating current through the rotor winding result in imbalances of the energy stream, which leads to voltage fluctuations of the DC-LINK, the high electromagnetic torque ripples, etc. Due to the existence of the transient flux components in the winding, it needs higher output voltage with different frequency to keep the rotor current under control, which exceeds the maximum voltage output of the RSC attributing to the DC-LINK.
Several types of LVRT technologies for DFIG wind turbine systems are summarized and evaluated respectively in [1]. The application of a crowbar circuit is a prevalent way to assimilate transient energy at the rotor side. The maximum short circuit current is analyzed in the time domain for a crowbar-protected DFIG during LVRT [2]. In [3], a short circuit of the rotor winding is simulated when the DC-link clamp effect occurs. A flowchart of the crowbar circuit control is proposed in [4]. To avoid over-voltage after a crowbar circuit is triggered, a principle of selecting the crowbar resistance is given by [5]. In [6], the flux response of a DFIG is researched with the crowbar as a linear circuit. One kind of adjustable resistance crowbar structure is proposed in reference [7]. The turn-on and turn-off strategy of the crowbar circuit is achieved depending on the DC-LINK voltage and the time constant of the damping of the DC component in the stator [8]. The rotor current and DC-LINK voltage are applied as criteria to optimize the crowbar resistance in [9]. Optimizing the resistance value of a crowbar circuit based on the Analytic Hierarchy Process (AHP) while taking into account its effects on the reactive power, rotor current, voltage and electromagnetic torque is proposed in [10].
Although many important articles have been published to investigate the bypass effect of crowbar circuits and the control logic, some difficulties still exist in the analytic expression of the transient current and flux linkage. In [11], the transient characteristics are analyzed and calculated under voltage dips when the rotor is assumed to be an open circuit, whereas the resolution of the transient components is not realized when the rotor current is not zero.
In [2], [6] and [12]-[15], the transient time constants of the stator and rotor are applied to resolve the transient behavior of the current and flux. However, a theoretical analysis by using the transient time constant as the attenuation time constant is not given, and the results of the resolutions reported in these papers are not identical. A short circuit current analysis of the DFIG is realized by solving the state space equation [16]. However, due to the convenience of resolving the matrix function, approximations and simplifications are made.
The physical transient behavior of the DFIG when there are grid voltage sags or swells in specific proportions is a hotspot and difficulty. The engineering design of the crowbar circuit is still based on experience. Furthermore, researchers are addressing this issue by assuming a linear crowbar circuit, whereas a nonlinear rectifier is utilized. Consequently the parameters of the crowbar need a lot of adjustment in engineering applications based on the trial method or simulations.
Due to the indefinite physical meaning of the transient course, simulations runs short of significance for investigating the effect of varying the crowbar resistance and DFIG parameters on the dynamic performance.
Concerning the resolving method based on the references, all of the models are approximate. In order to get an accurate expression of the LVRT response when the crowbar circuit is fired while keeping the conciseness of the results, a modeling method based on the Laplace transformation is proposed in this paper. It can be regarded as a rigorous mathematical method neglecting only the non-linear factors including the nonlinear excitation, skin effect, slot effect, etc.
The present paper extends the previous studies. Hereafter, the transient current and flux linkage are calculated by Laplace transformations and inverse transformations in both the frequency and time domains. As demonstrated in this paper, the attenuation characteristics of the transient components after the crowbar is fired depend on the crowbar resistance and rotation speed of the rotor. In an effort to satisfy grid code requirements, the selection strategy of the crowbar resistance is applied according to the proposed approach to guarantee the attenuation rate of the transient components, the reserved response time of the reactive power compensation and the controllability when excitation of the RSC is resumed. Concerning the setting of the crowbar resistance, an approximate linear calculation approach for nonlinear circuits is proposed.
Rigorous mathematical derivation is proposed in this paper, which is of great importance to the design of the crowbar resistance, and the dynamic behavior of a DFIG is analyzed during grid disturbances using active crowbar protection. This paper is organized as follows. In Section II, the analytic expressions of the transient current are proposed. In section III, the principle of selecting the crowbar resistor is discussed. The impedance of each transient fundamental component is derived for the nonlinear circuit in Section IV. A validated model is used in Section V to evaluate the crowbar performance with respect to the calculated crowbar resistance. Experimental results are shown in Section VI, and some conclusions are summarized in Section VII.
II. RESOLVING OF THE TRANSIENT CURRENT DURING VOLTAGE SAGS
The rotor voltage vector equation in the static reference frame of the rotor is expressed as:
where is the mutual-induction of the rotor phase winding.
After winding conversion, is equal to Then (1) can be calculated as:
In the stationary reference frame of the stator, (2) can be represented as:
where θr is the angle between phase A of the stator winding and the rotor winding. Due to heavy moment inertia, the rotating speed during a voltage sag is assumed to be constant, i.e., dθr/dt is equal to ωr which is constant during a voltage sag.
Considering the initial values, the Laplace transformation of (3) can be derived as follows:
The following equation is obtained by substituting (4) into
Considering the initial value of the flux, the Laplace transformation of the stator voltage vector equation in the stator reference frame is:
The expression of the stator voltage in the frequency domain is:
where θsp0 and θsn0 are the initial phase angles of the voltage in the positive and negative sequences, respectively.
Considering a short circuit of the rotor through the crowbar circuit the stator current, stator flux linkage, rotor current and rotor flux linkage can be composed of the zero-state response and the zero-input response whether the grid voltage fault is symmetrical or asymmetrical [17].
A. Solution of the Stator Current
Substituting (5) and (7) into (6) can calculate in the frequency domain as follows:
where is the zero-state response and is the zero-input response, which can be expressed as (9) and (10):
Where:
The four poles of NUM1(s)/DEN1(s) are:
where and
Furthermore the two poles of NUM2(s)/DEN2(s) are:
According to (11) and (12), the attenuation time constants of the zero-state and the zero-input responses are equal to each other for the stator current. With the above poles and the derivatives of DEN1(s) and DEN2(s), which are dDEN1(an)/ds and dDEN2(bn)/ds, the stator current in the time domain is calculated through the inverse Laplace transformation :
Consider that Re(a1) = Re(a2) =0. Meanwhile, the values of Re(a3), Re(a4), Re(b1) and Re(b2) are not zero. Therefore, the steady state and transient components of the stator current are described as (14) and (15):
In (15), the value of -1/Re(a3), -1/Re(a4), -1/Re(b1) and -1/Re(b2) is the attenuation time constant of each transient component[17].
B. Solution of the Rotor Current
The rotor current in the frequency domain can be expressed by substituting into (4):
where is the zero-state response and is the zero-input response of the rotor current. The corresponding expressions are given in detail in Appendix A.
The four poles of NUM3(s)/DEN3(s) and the three poles of NUM4(s)/DEN4(s) are expressed by (18) and (19):
With the poles and the derivatives of DEN3(s) and DEN4(s), the rotor current in the time domain is calculated by the inverse Laplace transformation:
On account of the zero value of Re(c1) and Re(c2) , the steady state current and the transient component of the rotor are described by (21) and (22):
In (22), the value of -1/Re(c3), -1/Re(c4), -1/Re(d1), -1/Re(d2) and -1/Re(d3) is the time constant of the attenuation for each transient component.
III. SELECTION OF CROWBAR RESISTOR FOR THE LINEAR CIRCUIT
The characteristics of the transient current are researched in this section based on a 690V/30kW laboratory DFIG test rig. The attenuation rate of the transient current depends on the real parts of a3, a4 and d1, which are the function of ωr and shown in Fig.1 and Fig.2. The resolving is based on the parameters listed in Appendix B. The resistance of the rotor winding (after the winding conversion) is equal to 0.3226Ω, and the resistance (after the winding conversion) of the linear crowbar circuit is adjusted from 0Ω to 1.2774Ω. With the expression of given in Appendix A, approaches zero while 0.7ω1 ≤ ωr ≤ 1.3ω1. Therefore, it is clear that the attenuation of can be omitted.
Fig. 1Curves of Re(a3).
Fig. 2Curves of Re(a4).
Since 0 > Re(a3) >> Re(a4), it can be concluded that the attenuation rates of the transient components of either the stator or the rotor current mainly lie on Re(a3). From Fig.1, it can be seen that as long as the attenuation rate can be guaranteed when the DFIG runs at the highest positive slip, the crowbar circuit can insure a higher attenuation rate for the transient components at a higher rotating speed.
According to the response time of the reactive power compensation required by grid codes in China, which is 75ms after voltage sags, the time duration of the crowbar conducting state is set up to 40ms in this paper. It can be seen from Fig. 1 that if ωr is equal to the lowest bound of the slip. The attenuation ratio in 40ms when is given by:
According to the rotor resistance 0.3226 Ω after the winding conversion listed in Appendix B, the optimal resistor in the linear crowbar circuit is calculated by the winding conversion as follows:
IV. ANALYSIS OF THE FUNDAMENTAL WAVE COMPONENT FOR A NONLINEAR CROWBAR
Assuming that the crowbar resistor in the linear circuit was applied, the voltage drop of the crowbar resistance can be expressed as (25):
Considering the application of a nonlinear crowbar circuit based on the uncontrolled rectifier and the rotor current can be written as:
where fn and gn are functions of the extra harmonic current caused by the zero-state and the zero-input responses, and h is the harmonic order.
Due to the impedance of the “fundamental” components whose angle speeds depend on Im(cn) and Im(dn), once the value of is worked out through a linear calculation, can be given with the winding conversion applied in the nonlinear circuit:
Based on this principle, all of the extra harmonic components caused by the nonlinear circuit, whose angle speed is different with Im(cn) and Im(dn), are omitted.
V. SIMULATION RESULTS
For verifying the above theoretical analysis, the performance simulation of the DFIG for the nonlinear crowbar circuit is conducted with a sag depth of 0.2 p.u..The generator parameters are given in Appendix B. According to (24) and the equivalent conversion between the linear and nonlinear circuits, the optimal resistance applied in the nonlinear circuit to realize the fastest attenuation rate can be expressed as:
For comparing the calculated result with those of the simulation and the experiment, the crowbar resistors applied in this paper are uniform. Due to the limitations imposed by the experimental condition in the laboratory, a 0.25 Ω power resister which approximates the calculation result of (28) is applied instead. This substitution leads to a larger time constant -1/Re(a3). Through an equivalent conversion from a linear to a nonlinear circuit and a winding conversion, can be calculated including the rotor resistor and the crowbar resister:
As a result, is set to 0.25 ×(1/k2)Ω in the MATLAB/SIMPOWER model of a nonlinear crowbar circuit. The attenuation ratio in 40ms is if which is acceptable when compared with σopt . The attenuation time constants of the transient components are equal in the positive and negative sequence as concluded from (9), (10), (16) and (17). Without losing generality and for simplifying the model, simulations and experiments are carried out under the symmetrical fault condition.
A grid fault occurred at t=0.402s when the space vector of the grid voltage was and grid voltage space vector was with a phase jump after the occurrence of a sag. The electrical angular speed of the rotor is set to be constant which is 80π(rad/s), viz. , the mechanical rotational speed is 126(rad/s).
Fig. 3 and Fig. 4 show a comparison of the simulated and calculated transient stator current which are isim and ical in the stator stationary reference frame assuming that the crowbar resistor is fired permanently once a grid fault occurs. The horizontal axes indicate the time t in both the calculation and the simulation. For lining up the time bases of the simulated and calculated values, the calculated value is set to be zero before a fault occurs.
Fig. 3.Simulated and calculated stator current in α axis of stator.
Fig. 4.Simulated and calculated stator current in β axis of stator.
Fig. 5 and Fig. 6 give the simulated and calculated transient rotor currents which are isim and ical in the stator reference frame.
Fig. 5.Simulated and calculated rotor current in α axis of stator.
Fig. 6.Simulated and calculated rotor current in β axis of stator.
As shown by Fig. 5 and Fig. 6, the calculated current of the linear circuit is almost identical to the simulated result carried out using the nonlinear crowbar model. The extra harmonic current due to nonlinear rectifying has little influence on the results.
The amplitude of the crowbar current and the line voltage at the rotor side converter should be guaranteed to be within the safe operating area for the generator, the crowbar and the rotor side converter. The above result shows that When the crowbar is fired, it can be calculated that:
Thereafter, icrowbar_max can be calculated by (30) and (31) with a range of icrowbar_max ≤ 366A. Then the maximum value of the line voltage at the rotor side Uline_max is given by:
The above analysis shows that the rotor side converter is bypassed by the crowbar. However, the rotor current impact is inevitable.
VI. EXPERIMENTAL VALIDATION OF THE PROPOSED CALCULATION METHOD
In order to validate the analysis in the previous sections, tests are performed using the system shown in Fig. 7(a), while the theory of equivalent conversion between the linear and nonlinear crowbar is illustrated by Fig. 7(b). Pictures of experimental setup are shown in Fig. 7(c)-(e). The experimental system includes the following items:
Fig. 7.Experimental system: (a) 30kW test rig, (b) Equivalent conversion between linear and nonlinear crowbar, (c) The picture of the 30kW driving setup, (d) The power circuit of the converter for DFIG, (e) The 2MW converter for simulating the grid.
(1) The parameters of a DFIG are listed in Appendix B.
(2) A 2MW voltage source converter (VSC) operating at 2kHz switching frequency and a RLC filter are connected to the stator of the DFIG, which is controlled to simulate a voltage sag of the grid.
(3) A wind turbine simulation system that outputs a signal of the rotation speed and a DC motor is applied as the actuating mechanism.
(4) The resistance of which is equal to 0.0157Ω calculated by is converted to Rc_nonlinear = 0.25Ω and is used in the nonlinear crowbar circuit.
The identities of the calculated and the experimental transient current can be made out from Fig. 8 and Fig. 10. The time period while the crowbar is being fired is expressed by the rectangular box in Fig. 9 and Fig. 11. After the crowbar is switched off, the rotor current controlled by the RSC is resumed to supply reactive current (the RMS value of the phase current is equal to 35A) at the stator side.
Fig. 8.Calculated and experimental stator current when crowbar is turned on.
Fig. 9.Experimental stator current during crowbar is operating and the compensated reactive current of stator.
Fig. 10.Calculated and experimental rotor current when crowbar is turned on.
Fig. 11.Experimental rotor current during crowbar is operating and the reactive AC excitation after the control of RSC is resumed.
Fig. 9 and Fig. 11 show waveforms obtained by scope capture. Due to the scope design of the waveform recording function, the sign of the time axis is shown as negative in this time span. For lining up the time bases of the experimental and calculated values, the divisions of the horizontal axes in Fig. 8 and Fig. 10 are specified corresponding to those obtained by the scope capture, hence tsag is equal to -5.4225s. As can be seen from Fig. 12, the rotation speed of the DFIG changes in the experiment. The speed (ωr/pn) is 126rad/s before a grid fault. The time interval of the crowbar being fired is given in the rectangular box of Fig. 12, during which time the speed variation is small. The calculated values given by Fig. 8 and Fig. 10 are provided and ωr/pn is approximately equal to 126rad/s. This waveform is exported from the RAM on the DSP board. For lining up the time base of Fig. 12 and that of the scope capture, the division of the horizontal axis is specified corresponding to that obtained by the scope capture.
Fig. 12.Rotor speed of the DFIG.
In the experiment, the steady-state control algorithm realizes the controllability of the rotor current after the crowbar is switched off. The waves indicate the correctness of the proposed calculation method and the accuracy of the attenuating characteristics figured out.
VII. CONCLUSIONS
The proposed approach is worthwhile in three aspects: a) the DFIG’s influence on the grid of the transient current during LVRT is determined; b) the selection of the crowbar resistance is realized by a mathematical method; c) a theoretical basis is proposed for optimizing the parameters of the DFIG to improve the performance of LVRT.
Simulation and experimental studies on a 30kW wind power DFIG system are carried out to validate the proposed theoretical method. As a result, the following conclusions can be drawn:
1) On the basis of the linear resolution, the crowbar resistance can be selected and converted to a nonlinear circuit with the coefficient
2) The time constant of the exponential attenuation of transient components depends on the parameters of the DFIG, the rotating speed and the crowbar resistance, while being irrelevant to the depth of the voltage sag and the angle of the phase jump.
3) When the crowbar resistor is set to guarantee the attenuation rate if the DFIG is running with the highest positive slip, the crowbar circuit insures a higher attenuation rate of the transient components at a higher rotating speed.
4) The maximum negative real part of the exponents that is in the expressions of the transient components is pivotal for deciding the attenuation rate.
5) The resolution of the time constant realizes the selection of the crowbar resistance and the duration time set up of the crowbar protection, which reserves a response time for the reactive power support.
6) The crowbar resistor can be selected both to set the attenuation rate and to decrease the transient current of the DFIG, and the latter will be further studied in another paper.
7) Although it has been proven in this paper that Uline_max << Udc, Uline_max depends on the conditions before a sag, the asymmetrical characteristics of the voltage sag, the depth of the voltage sag and the parameters of the generator. The possibility that the design of the crowbar would be unable to insure Uline_max < Udc and a proper attenuation rate simultaneously will be researched in a future work.
References
- Y. He and P. Zhou, "Overview of the low voltage ride-through technology for variable speed constant frequency doubly fed wind power generation systems," Transactions of China Electrotechnical Society, Vol. 24, No. 9, pp. 140-146, Sep. 2009.
- J. Morren, and S. W. H. De Haan, "Short-circuit current of wind turbines with doubly-fed induction generator," IEEE Trans. Energy Convers., Vol. 22, No.1, pp. 174-180, Mar. 2007. https://doi.org/10.1109/TEC.2006.889615
- J. Zhai, B. Zhang, K. Wang, and W. Shao, "Three-phase symmetrical short circuit current characteristic analysis of doubly fed induction generator with crowbar protection," IEEE PES Innovative Smart Grid Technologies, pp. 1-5, May 2012.
- H. Soliman, M. I. Marei, R. M. El-Sharkawy, K. M. El-Bahrawy, "Analysis of the dynamic behavior of a DFIG during grid disturbances using active crowbar protection," in 12th International Conference on Environment and Electrical Engineering (EEEIC), pp. 160-163, May 2013.
- M. Ding, T. Hu, P. Han, and D. Wang, "Analysis of low voltage ride through ability of variable speed constant frequency wind turbine based on crowbar protection," in Asia-Pacific Power and Energy Engineering Conference (APPEEC), pp. 1-5, Mar. 2012.
- Z. Zheng, G. Yang, and H. Geng, "Short circuit current analysis of DFIG-type WG with crowbar protection under grid fault," in IEEE International Symposium on Industrial Electronics (ISIE), pp. 1072-1079, May 2012.
- Y. Ren and W. Zhang, "A novel control strategy of an active crowbar for DFIG-based wind turbine during grid fault," in IEEE International Electric Machines & Drives Conference (IEMDC), pp. 1137-1142, May 2011.
- L. Zhang, X. Jin, and L. Zhan, "A novel LVRT control strategy of DFIG based rotor active crowbar," in Asia-Pacific Power and Energy Engineering Conference (APPEEC), pp. 1-6, Mar. 2011.
- M. Wang, W. Xu, H. Jia, and X. Yu, "A novel method to optimizing the active crowbar resistance for low voltage ride through operation of doubly-fed induction generator based on wind energy," in IEEE International Symposium on Industrial Electronics (ISIE), pp. 957-962, May 2012.
- A. Zohoori, J. S. Moghani, S. H. Fathi, and V. Mortezapour, "An optimized crowbar for DFIG with AHP algorithm under single phase fault," in 10th International Conference on Environment and Electrical Engineering (EEEIC), pp. 1-4, May 2011.
- J. Lopez, E. Gubia, P. Sanchis, and X. Roboam, "Wind turbine based on doubly-fed induction generator under asymmetrical voltage dips," IEEE Trans. Energy Convers., Vol. 23, No. 1, pp. 321-330, Mar. 2008． https://doi.org/10.1109/TEC.2007.914317
- G. Pannell, D. J. Atkinson, and B. Zahawi, "Analytical study of grid-fault response of wind turbine doubly fed induction generator," IEEE Trans. Energy Convers., Vol. 25, No. 4, pp. 1081-1091, Dec. 2010. https://doi.org/10.1109/TEC.2010.2049494
- L. Zhang, X. Cai, and J. Guo. "Dynamic responses of DFIG fault currents under constant AC excitation condition," in Asia-Pacific Power and Energy Engineering Conference, pp. 1-4, Mar. 2009.
- Z. Wu, G. Wang, H. Li, T. Ding, and X. Gao, "Fault current contribution of DFIG under unsymmetrical fault," in 4th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT), pp. 1687-1692, Jul. 2011.
- X. Gao, G. Wang, Z. Wu, and H. Li, "Fault current contribution of doubly fed induction generator wind turbines under different control strategies," in International Conference on Advanced Power System Automation and Protection (APAP), Vol. 2, pp. 1209-1214, Oct. 2011.
- H. L. Zhou, G. Yang, and D. Y. Li, "Short circuit current analysis of DFIG wind turbine with crowbar protection," in International Conference on Electric Machine and Systems (ICEMS), pp. 1-6, Nov. 2009.
- Y. Tang, Y. Zhang, and Y. Fan, Dynamic analysis of AC machine, Chapter. 3, pp. 67-75, China Machine Press, 2004.
Cited by
- Crowbar resistance value-switching scheme conjoint analysis based on statistical sampling for LVRT of DFIG pp.2196-5420, 2018, https://doi.org/10.1007/s40565-018-0444-y