High frequency resonance in DFIG-based wind farm with variable power capacity

: As wind power penetration has been gaining in the power grid for decades, a large number of the doubly fed induction generator(DFIG) based wind farms are being established around the globe. The power capacities of these wind farms may vary around hundreds of MW, and most of the wind farms are connected to long transmission cables whose impedances can not be ignored and require careful attention. Several works have investigated the impedance interaction between the DFIG based wind farm and long transmission cables which may unfortunately cause high frequency resonance (HFR). The main contribution of this paper is to investigate the influence of the variable wind farm capacity on the behavior of the HFR when certain transmission cables are provided. It is found out that the potential HFR may happen in certain wind farms, and the larger wind farm capacity causes more severe HFR due to the relatively weaker grid transmission capability. Simulation results based on Matlab/Simulink are given to validate the analysis of HFR.


Introduction
The recent rapid growth of renewable power generation has been reported all over the world. Among the several different renewable power sources, wind power generation is one of the most promising solutions [1][2][3] . From the early year of 21st century, the doubly fed induction generator(DFIG) based wind farm has drawn increasing attention from commercial companies as well as the scientific researchers.
Often the grid voltage of the wind farm is not ideal, but is subject to detrimental operational conditions such as grid voltage three-phase unbalance and harmonics distortion, three-/single-phase voltage fault, as well as frequency deviation. In order to deal with these problems, several enhanced control strategies have been implemented regarding grid voltage fault [4] , three-phase unbalance [5][6] , harmonic distortion [7][8] , as well as for the special topology of connection to the dc voltage grid [9] .
Normally, the DFIG based wind farm delivers renewable power to the main grid via the long transmission cables. As reported previously, the cable impedance of the long transmission cable is relatively larger than the conventional stiff grid [10][11][12] , thus the wind farm is often under the operational condition of weak grid due to the large cable impedance.
In the discussion of this paper, the impedance modeling of the long transmission cable is assumed to be a series connection of several Π units [12] , where there is a cable resistance and a cable inductance in series connection, and two cable shunt capacitances at both ends [10][11][12] in each single Π unit. On the other hand, the DFIG based wind farm behave mostly inductive in the control frequency range [13][14][15][16] . Due to their respective inductive and capacitive behaviors, a resonance may happen as a consequence of the impedance interaction between the wind farm and the transmission cable, which will be explained in detail.
HFR analysis has been conducted in [13][14][15][16], where several influencing factors are considered, i.e., different parameters of the long transmission cable, different parameters of single DFIG system unit, variable rotor speed, variable output power. Moreover, the active damping strategies were also proposed in [14][15][16] to mitigate the HFR and ensure safe and reliable operation of the DFIG system. Nevertheless, the power capacity of the DFIG based wind farm is always set constant in the previous discussions [13][14][15][16] . Practically, the power capacity of the commercial wind farms may vary within a wide range around the globe due to different climate conditions, local grid transmission capability and economic concerns. Thus, it is necessary to discuss the HFR under the condition of variable power capacity of the DFIG based wind farm. This paper is organized as following: the impedance modeling of a single DFIG system unit as well as the wind farm is established in Section 2. Then, the impedance modeling of the Π units based long transmission cable is established in Section 3. Thereafter, the HFR can be analyzed on the basis of the obtained impedance modeling of both the wind farm and the cable in Section 4. Note that, several case studies with different power capacity of the wind farm will be analyzed in this section, and the relationship between the occurrence of HFR and the wind farm power capacity can be obtained. Finally, the simulation results by Matlab/Simulink are provided in Section V in order to verify the analysis of the HFR.

Impedance modeling of the wind farm
This section firstly introduces the diagram of the DFIG based wind farm connected to the transmission cables in Fig.1, where then the impedance modeling of the wind farm can be deduced. Fig.1 shows the diagram of the DFIG based wind farm connected to the long transmission cable. Each single DFIG unit includes two parts: 1) the RSC and the DFIG generator; 2) the GSC and the output LCL filter. The parameters of the DFIG unit are shown in Table 1. The three-terminal transformer T 1 is adopted to adjust the output voltage to 690 V.

Introduction to the wind farm and the cable
The transformer T 2 is implemented to increase the voltage level up to 33kV in order to connect each single DFIG unit to the 33kV common AC bus. The impedance of the 33kV transmission cable is neglected due to its relatively shorter length compared with the 150kV cable.
After collecting the wind power at 33kV AC bus, the voltage level is again increased up to 150kV through the transformer T 3 . The 150kV long transmission cable is assumed to be modeled as multiple Π units in series. Each Π unit includes the cable resistor R 1 and the cable inductor L 1 in series connection and the cable shunt capacitor C 1 /2 at both ends.

Impedance modeling of the DFIG based wind farm
Before deducing the impedance modeling of the DFIG based wind farm, it is necessary to first obtain the impedance modeling of the single DFIG unit, which includes both the RSC and the DFIG generator, as well as the GSC and the output LCL filter [13][14][15][16] as following.
The grid part of the DFIG system contains the GSC and the LCL filter, and its impedance modeling in the stationary frame shown in Fig.2 can be obtained [13][14][15][16], where, Z Cf = 1/sC f , Z Lf = sL f , Z L g = sLg. C f is the LCL-filter capacitance, L f is the converter side inductance, and Lg is the LCL grid side inductance. K 1 is the voltage ratio between V G and V PCC defined as is the PI current controller containing the proportional part K pgsc and the integral part K igsc /(sjω 0 ), the parameters of K pgsc and K igsc are given in Table 1. G d (sjω 0 ) is the digital control delay of 1.5 sample period caused by the sampling and PWM update [13][14][15][16] . ω 0 is the grid fundamental angular speed of 100π rad/s, it is introduced to represent the rotation from the stationary to the synchronous frame. The control of the DC-link voltage and the grid synchronization in the GSC are neglected due to the slower dynamic response [13][14][15][16] .
The impedance of the RSC and DFIG generator in the stationary frame can be obtained in Fig.3 as [13][14][15][16], where H=Z Lσr +(R r +Z RSC )/slip; Z RSC =G c (sjω 0 )G d (sjω 0 ), Z Lm =sL m ; Z Lσr =sL σr ; Z Lσs =sL σs . R r is the rotor resistance, L m is the mutual inductance, L σr is the rotor leakage inductance, and L σs is the stator leakage inductance. K 2 is the voltage ratio between V S and V PCC as defined The rotor current control is implemented in the synchronous reference frame, and needs to be rotated to the rotor stationary frame using the slip angular speed expressed as [13][14][15][16], where, ω r is the rotor angular speed.
Since the RSC and DFIG generator Z SR and the GSC and LCL-filter Z G are connected in parallel, the impedance of the single DFIG unit Z single can be obtained based on (1) and (2) as, where, Since it is appropriate and reasonable to assume that all the DFIG units in the wind farm are working in the same conditions, it is assumed in this paper that all the single DFIG units are identical with same parameters, and thus the impedance of the wind farm can be calculated by dividing the impedance of the single DFIG unit with the number of units n DFIG as following.
where, n DFIG is the number of DFIG units.

Impedance modeling of the cable
The modeling of the transmission cable has been investigated in [12] and the Π unit based cable modeling is a common solution which is shown in Fig.4.
As can be seen, the Π unit contains the cable resistance R 1 and inductance L 1 in series connection, and the shunt capacitance C 1 /2 between the cable and the ground. Note that R 1 , L 1 and C 1 are the cable parameters of each single Π unit, and the length of each single Π unit l single needs to be taken into consideration as following. 1 single 0 1 single 0 1 single 0 where, R 0 , L 0 and C 0 are the per km resistance, inductance and capacitance as given in Table 1 where, n is the number of Π units. K 4 = V HV /V MV is the voltage ratio between the high voltage V HV = 150 kV and the medium voltage V MV = 33 kV. Fig.5 shows the Bode diagram of the impedance modeling of the transmission cable with multiple Π units n = 8, 10, 12, lsingle =5km (total length = 40, 50, 60km). According to [10], the length of the single unit lsingle is appropriately chosen as 5km, as this value makes a reasonable compromise between the cable modeling complexity and accuracy. As shown in Fig.5, as the transmission cable becomes longer, the magnitude peak moves towards lower frequency. Moreover, the potential resonance frequency range, where the phase response of the cable is -90°, also moves towards lower frequency range as the cable becomes longer. For instance, for the case of n = 10 in green, the potential resonance frequency (phase = -90°) is from 2000Hz to 2600Hz; while for the case of n =12 unit in red, the potential resonance frequency (phase = -90°) shifts to lower as from 1700Hz to 2200Hz.

Analysis of HFR in wind farms
Based on the impedance modeling of both the DFIG based wind farm and the long cable obtained above, the HFR can be analyzed on the basis of the Bode diagram analysis method.
Note that due to the limited space in this paper, the following discussion of HFR will be conducted with a constant cable length of 60km (n = 12 Π units), while the power capacity of the wind farm may vary as 300MW (n DFIG = 150), 400MW (n DFIG = 200) and 500MW (n DFIG = 250) respectively, which is possible in the practical commercial wind farm around the globe. Fig.6 shows the Bode diagram of the impedance of the 60km 150kV transmission cable and the impedance of the DFIG based wind farm with the capacity of (a) 300MW (n DFIG = 150); (b) 400MW (n DFIG = 200); (c) 500MW (n DFIG = 250). By comparing the impedance of the wind farm with three different power capacities (red curves in the three Bode diagrams), it can be found out that its magnitude response drops as the capacity increases, which is due to the increasing number of DFIG units working in parallel, while the phase response remains constant regardless of the different power capacity.
On the other hand, in each of these three cases, the magnitude intersection point exists at 1775Hz with 300MW wind farm in Fig.6(a), 1810Hz with 400MW wind farm in Fig.6(b), and 1840Hz with 500MW wind farm in Fig.6(c). The phase differences between the cable and the wind farm are all 180° in these three cases, thus the HFRs at 1775Hz, 1810Hz and 1840Hz occur consequently for the three case studies. Note that although additional magnitude intersection points exist at other frequencies, the phase difference at these frequencies are smaller than 180°, thus no HFR will occur at these frequencies.

Simulation setup
In order to validate the HFR in the wind farm, a simulation model is built where the control block is shown in Fig.7 and the DFIG system parameters can be found in Table 1. The transmission cables are simulated as shown in Fig.1 with their parameters listed in Table 1. The rotor speed is set to 1200r/min (0.8pu), with the synchronous speed of 1500r/min(1.0pu). The DC-link voltage is 1200V. The switching frequency f sw for both RSC and GSC is 2.5kHz, the sampling frequency f s for both RSC and GSC is 5kHz. The output wind power is set as 1.0pu active power and 0.0pu reactive power.
Clearly it is impossible to run the simulation based on Mablab/Simulink with 200 single DFIG units whose power capacity is 2MW each, two single DFIG units are adopted here, while the impedance of the transmission cables are multiplied with 75, 100, 125 in simulation in order to represent the total number of n DFIG = 150, 200, 250 DFIG units, which finally build up a 300MW, 400MW, 500 MW wind farm.  Fig.7 shows the control block diagram of a single DFIG unit connecting to the grid through long transmission cables. As can be seen, for the RSC control, an Enhanced Phase Locked Loop (EPLL) is used to obtain the grid voltage fundamental synchronous angular speed ω 1 and angle θ 1 , while an encoder gives out the DFIG rotor position θ r and speed ω r . The rotor current I+ rdq is first sampled and then controlled based on the reference value I+* rdq with a PI controller to output the harvested wind energy. The output of the rotor current PI closed-loop control V + rdqPI are added together with the decoupling compensation, giving out the rotor control voltage V + rdq, it is then transformed to the rotor stationary frame as the input to the Space Vector Pulse Width Modulation (SVPWM).

Control block diagram
As for the GSC control, the dc-link voltage V dc is well regulated by a PI controller, and its output is delivered as the converter side inductance filter current reference I+* fdq, which is used to regulate the actual converter side inductance filter current I+ fdq by a PI controller. Similarly, the GSC control voltage V+ gdq can be obtained by the PI current controller output and the decoupling compensation. Fig.8~Fig.10 give the simulation results of the DFIG based wind farm with different power capacity of 300MW in Fig.8, 400MW in Fig.9 and 500 MW in Fig.10. The transmission cable is set constant as 60km with n = 12.

Simulation results
As can be observed from the simulation results in Fig.8~Fig.10, the three cases all contain the HFR components in the DFIG system performance. That is, 1) For the case of 300MW wind farm, analysis HFR = 1775Hz in Fig.6(a), simulation HFR=1785Hz in Fig.8; 2) For the case of 400MW wind farm, analysis HFR = 1810 Hz in Fig.6(b), simulation HFR=1820Hz in Fig.9; 3) For the case of 500MW wind farm, analysis HFR = 1840Hz in Fig.6(c), simulation HFR=1850Hz in Fig.10; Table 2 summarizes the analysis and simulation results of the HFR, and it can be found that the analysis

Conclusion
This paper has investigated the HFR phenomenon in the DFIG based wind farm connected to the long transmission cable. The long cable is modeled as multiple Π units in series, including the cable resistor and inductor in series connection and the shunt capacitor at both ends. Several conclusions can be obtained.
(1) The Bode diagram based analysis method can be adopted to theoretically explain the principle of the HFR, and simulation results are provided to validate the HFR.
(2) The larger power capacity of the DFIG based wind farm results in slightly higher HFR frequency due to its smaller magnitude caused by the larger number of DFIG units.
(3) From the perspective of grid impedance weakness, it can be found that the weaker grid transmission capability (i.e., larger impedance of the long cable) results in the severer HFR phenomenon. For instance, the relatively small capacity 300MW of the wind farm causes small amount of HFR, while the relatively large capacity 500MW of the wind farm causes significant HFR. damping strategy for DFIG system high frequency resonance,"