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Improvement of energy conversion efficiency and damping of wind turbine response in grid connected DFIG based wind turbines
Electrical Power and Energy Systems 95 (2018) 11–25
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improvement of energy conversion efficiency and damping of wind turbine response in grid connected DFIG based wind turbines Mohsen Rahimi Department of Electrical and Computer Engineering, University of Kashan, Kashan, Iran
a r t i c l e
i n f o
Article history: Received 11 January 2017 Received in revised form 21 July 2017 Accepted 6 August 2017
Keywords: DFIG Drive train dynamics Torsional modes Power-speed curve Wind turbine stabilizer
a b s t r a c t This paper examines dynamic performance of DFIG based wind turbine regarding torsional electromechanical oscillations when the wind turbine is controlled in power control mode. In the power control mode, the stator active power is set to its reference value, where the reference active power is determined based on a predefined power-speed curve. Torsional oscillations associated with the poorly torsional modes may reduce the aerodynamic efficiency, deteriorate the power quality and increase stresses on the drive train system. The paper indeed analytically investigates the drive train dynamics for different operating regions of the power-speed curve. It is shown that under some operating points, torsional oscillations are well damped and the DFIG control improves the damping of torsional modes. On the other hand, at some regions of the power-speed curve, the DFIG control does not provide significant damping action on the torsoinal modes. Then a wind turbine stabilizer (WTS) is implemented to suppress torsional vibrations in the drive train system. The role of the WTS is to add positive damping to the generator rotor oscillations by controlling the active power and providing a component of electrical torque proportional to the rotor speed deviations. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction DFIGs are commonly used variable speed generators in wind turbine (WT) applications. Wind turbines based on DFIG systems result in lower converter cost and lower power losses compared to full converter based wind turbines (WTs) [1]. As shown in Fig. 1, the DFIG-based WT consists of drive train system (turbine blades, gear box, shafts, and other mechanical parts), wound rotor induction generator, back-to-back voltage source converters linking the rotor to the grid, pitch control and converters control systems. The rotor-side converter (RSC) is used to control the generator real and reactive power, while the grid-side converter (GSC) is used to control the dc-link voltage and reactive power exchanged with the grid [2]. The wind turbine equivalent shaft is relatively softer than the typical turbine shaft in conventional power plants [3]. Hence, the drive train of the wind turbine should be described with at least a two-mass model [4], or three-mass model [5], since a single-mass shaft model cannot properly represent mechanical oscillations. Mechanical oscillations of the multi mass drive train dynamics are known as torsional oscillations related to the torsional oscillatory modes. The electromechanical oscillations in a E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijepes.2017.08.005 0142-0615/Ó 2017 Elsevier Ltd. All rights reserved.
conventional power system are related to the local and inter-area modes associated with the tendency of the interconnected synchronous generators to remain in synchronism [6]. However, in a mixed power system with high penetration of wind power generation, there are two types of oscillatory electromechanical modes: local and inter-area oscillatory modes related to the synchronous generators, and torsional oscillatory modes associated with the wind turbines [7,8]. From the DFIG control point of view, there are two modes for the RSC control using field oriented control (FOC): power control mode and speed control mode. The reference control value is stator power for the power control mode [9,10], and the rotational speed for the speed control mode [11,12]. In the power control mode, the active power control is carries out based on a predefined piecewise power-speed curve generating the active power reference signal for the power control loop. In most practical cases, the DFIGs are controlled in the power control mode, in which the generator active power is set to its reference value. In literatures, little work has been published regarding the effect of power-speed curve on the torsional oscillations. Low frequency torsional oscillations associated with the wind turbine generators have been investigated by several papers in the literatures. In [13–15] some approaches are proposed to mitigate the torsional oscillations of the drive train when the wind turbine is subjected to a severe grid
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 1. Schematic diagram for the DFIG system under study.
fault. Without the damping controller, the torsional oscillations excited by the grid fault are very strong after the grid fault incident. In [14], the damping control for the torsional oscillations is realized by using STATCOM/BESS topology. In [16–18] the SSR phenomenon caused by induction generator effects as well as torsional interactions in a series compensated wind park are discussed. In [16] by using the STATCOM an auxiliary subsynchronous damping control is developed to meet the damping torque in the range of critical torsional frequencies. In [19] a detailed model of the wind turbine with complex drive train is proposed which is the starting point in the research of torsional vibrations. In [20], It is shown that the pitch angle control may excite the torsional oscillations in DFIG based wind turbines. Torsional oscillations occur as the pitch bandwidth increases where the controller bandwidth begins to interfere with the shaft natural frequency. In [21], an activedamping strategy is proposed for the suppression of speed and torsional oscillations in direct-driven permanent-magnet synchronous generator wind turbine. In [22], two mitigation methods for torsional resonance are proposed: an active vibration absorber and the other method uses rotor angular acceleration measurements to provide virtual inertia for resonance detuning. In [23], the interactions of the torsional dynamics of the wind turbines and power modulation for oscillations in DFIG-based wind generators are investigated. Active power modulation for inter area oscillation damping is effective. However, it may interact with the torsional dynamics. According to the above references, the torsional oscillations in the wind turbine generators are excited due to four main reasons: excitation by the grid fault, torsional interactions in a series compensated wind turbine due to SSR phenomenon, interference between the bandwidth of pitch angle control and shaft torsional frequency, and interactions of torsional dynamics and power modulation. As stated above, the control of DFIG in the power control mode is realized based on a predefined piecewise power-speed curve generating the active power reference signal for the power control loop. This paper deals with the stability analysis of the drive train dynamics, torsional oscillatory modes, and damping improvement of torsional oscillations in DFIG based wind turbine at different regions of the power-speed curve when the wind turbine is controlled in the power control mode. As the main contribution, it is shown that the torsional oscillations may be excited due to characteristic of the power speed curve at some operating points. In other words, at some regions of the power-speed curve, the DFIG control does not provide significant damping torque on the torsional modes. Hence, unlike the related literatures, in this paper the torsional oscillations may be raised due to nature of the power speed curve which has not been investigated before. The paper first analytically examines the dynamics of the drive train and related torsional oscillations for different operating regions of the powerspeed curve. The torsional oscillatory modes with low damping ratio affect the dynamic performance of the wind turbine. This is because torsional oscillations are transmitted to the generator
speed, electrical torque and output active power deteriorating the power quality and inducing fatigue stresses on the mechanical components. Then, an auxiliary active damping control known as wind turbine stabilizer (WTS) is proposed improving the damping ratio of the torsional oscillatory modes. Consequently the impact of the WTS is to introduce a damping torque component suppressing the torsional oscillations in the drive train components. As an advantage, in this paper, a more accurate solution is formulated and more comprehensive analysis is made for examining the issue of torsional oscillations. This paper is organized as follows: Section 2 deals with the wind turbine modeling and control. Modeling of linearized drive train dynamics is presented in Section 3. Dynamic stability analyses of the drive train dynamics, with and without WTS, are given in Sections 4 and 5. Simulation results are presented in Section 6, and finally Section 7 draws the conclusions. 2. Wind turbine modeling and control The purpose of this section is to present the dynamic model and power controller design of DFIG based WT. The following assumptions for the system model are considered: (a) positive direction for the stator and rotor currents is assumed into the generator, (b) all system parameters and variables are in per unit and referred to the stator side in generator, (c) the equations are given in dqsynchronous reference frame with the stator voltage orientation (SVO). The stator and rotor voltages and fluxes, electromagnetic torque, and stator output active power in reference frame rotating at angular speed of x are given by [2]:
v sdq ¼ Rs isdq þ jxwsdq þ
v rdq ¼ Rr irdq þ jx2 wrdq þ ws ¼ Ls is þ Lm ir wr ¼ Lm is þ Lr ir Te ¼
Lm ðw irq wsq ird Þ Ls sd
Ps ¼ Re v sdq isdq
1 dwsdq
xb dt
1 dwrdq
xb dt
ð1Þ
ð2Þ
ð3Þ
ð4Þ ð5Þ
where w, v and i represent the flux, voltage and current. Subscripts s and r denote the stator and rotor quantities, respectively. Ls and Lr are the stator and rotor self inductances, Lm is the mutual inductance. x2 is the rotor slip frequency, xb is the base angular frequency, and x is the speed of d-q reference frame. Also, Rs and Rr
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
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are the stator and rotor resistances. The superscript ⁄ in (5) represents the conjugate operator. 2.1. Active power flow in DFIG In the dq reference frame with the stator voltage orientation (SVO), the d-axis is aligned with the stator voltage vector, and consequently v sd ¼ V s and v sq ¼ 0. With stator and rotor resistances neglected, the steady state stator/rotor voltage equations in SVO frame are obtained as follows: V s ¼ v sd ffi xs wsq , V sq ffi xs wsq ¼ 0 and V r ffi LLms sV s , where xs , equal to 1 pu, is the stator angular fre-
quency, s is the generator slip, and V s is the stator voltage amplitude. This results in the approximate expressions of wsd 0 and wsq ws . Considering (5) and according to the above explanations, the stator output active power, P s , active power delivered to the rotor, P r , and electromagnetic torque as functions of the stator voltage and rotor current can be written as
Ps
Lm V s ird Ls
ð6Þ
Pr
Lm sV s ird Ls
ð7Þ
Te ¼
Lm ðw ird Þ Ls sq
ð8Þ
Hence, in the SVO frame, isd and ird represent the active power components of the stator and rotor currents, respectively. On the other hand, isq and irq stand for the reactive power components of the stator and rotor currents, respectively. xs ¼ 1 pu and According to (8), and considering V s ¼ v sd ffi xs wsq , the electromagnetic torque in per unit is approximately equal to the stator power. Therefore,
T e ffi Ps ffi
Lm V s ird Ls
ð9Þ
Considering (6) and (7), we have: P r ffi sP s . Fig. 2 depicts the active power flow direction of DFIG at sub-synchronous and super-synchronous operation modes. According to Fig. 2 the power delivered to the grid by the DFIG is given by
Po ffi Ps Pr ¼ ð1 sÞP s
ð10Þ
2.2. Active power control loop According to (6), there is a direct relation between d-axis rotor current and stator output active power. Therefore, in DFIG based WT, stator active power injected to the grid can be controlled by the d-axis rotor current control through rotor voltage regulation. Fig. 3 depicts the output active power control loop, where P s and P sref are the active power and reference active power of the stator. The transfer function sþaaI I in Fig. 3, is the closed loop transfer
function of inner rotor current control loop, and aI is the closed loop bandwidth of the rotor current control. Hence, the d-axis rotor reference current is determined by the outer active power controller, as shown in Fig. 3. Assuming the power controller to be PI, K P ðsÞ ¼ kPP þ kIP , and by canceling the s pole aI with the zero of the controller, we have
(
aP ¼ LLms kPP aI V s kIP ¼ kPP aI
Fig. 2. Active power flow direction in DFIG.
ð11Þ
where aP is the bandwidth of stator active power control loop. Then, the transfer function from P sref to P s is given by
Fig. 3. Stator active power control loop.
Ps aP ¼ Psref s þ aP
ð12Þ
Indeed, the rotor current control is the inner control loop, and the stator active power control represents the outer control loop. The active power reference in Fig. 3 is determined based on a predefined power-speed characteristic curve, given by
Psref ¼ f ðxg Þ
ð13Þ
where f ðÞ represents the power-speed curve function. Fig. 4 depicts a typical power-speed curve for the DFIG wind turbine system. Considering Fig. 4, the power-speed curve is divided into four regions: 1. Low speed operating region: this region, i.e. section 1 of Fig. 4, corresponds to low wind speed operation. At this region the generator speed is relatively low, which is usually around 30% below synchronous speed. 2. Middle-speed operating region: in this region, i.e. section 2 of Fig. 4, the speed of the turbine is adjusted to yield a maximum power capture at a given wind speed. 3. Constant speed operation region: this region corresponds to section 3 of Fig. 4, in which the generator speed is approximately constant up to rated power. At high wind speed, generator speed is not able to continuously follow optimum
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 5. The electromagnetic torque-speed block diagram.
According to (9), (12) and (13), the electromagnetic torque, T e , as a function of the generator speed, xg , can be given by Fig. 5. As stated in Section 2, the stator active power reference, P sref , is determined based on a predefined power-speed curve, given by Psref ¼ f ðxg Þ, where f ðxg Þ represents the power-speed curve function. Linearizing T e around an operating point, xg0 , yields
DT e ffi DP s ¼
Fig. 4. A typical power-speed curve for the DFIG wind turbine system.
df ðxg Þ aP Dxg dxg xg0 s þ aP
ð19Þ
operation, and consequently the controller attempts to maintain generator speed constant. 4. Constant power operation region: this region, i.e. section 4 of Fig. 4, stands for a constant power characteristic beyond the speed limit followed by a blade pitch control for high wind speed. In this region the generator speed is limited by reducing the turbine mechanical power through pitching the blade angle.
According to (19), DT e depends on the sensitivity of the f ðxg Þ to the generator speed. The turbine torque, T t , is extracted from aerodynamic model of the wind turbine. The inputs to the aerodynamic model are the wind speed, V w , the blade pitch angle, b, and the turbine speed, xt . The output of the aerodynamic model is the turbine mechanical power, Pt. The following relationship is used to obtain Pt from the wind power [3]:
3. Modeling of linearized drive train dynamics
Pt ¼ Pwind C P ðk; bÞ 1 Pwind ¼ qpR2 V 3w 2
The drive train of the wind turbine is usually represented by a two-mass model. In the two-mass model, the low speed mass of the turbine is connected to the high speed mass of the generator through a flexible shaft modeled as a spring and damper. The linearized model of the two-mass drive train system is given by
ð20Þ
where Pwind is the power contained in the wind, C P is the power coefficient, q is the air density, and R is the radius of the blades. C P is a function of tip-speed ratio, k, and blade pitch angle, b. k is the ratio of the rotor blade tip speed and the wind speed. The turbine torque is given by T t ¼ Pt =xt . Linearizing T t around the turbine operating speed xt0 yields
dDxg DT e þ ks Dh þ Dtg ðDxt Dxg Þ ¼ 2Hg dt
ð14Þ
dDh ¼ xb ðDxt Dxg Þ dt
ð15Þ
DT t
dDxt DT t ks Dh Dtg ðDxt Dxg Þ ¼ 2Ht dt
ð16Þ
Replacing (19) and (21) into (14) and (16) results in the linearized drive train dynamic as depicted in Fig. 6. Block diagram of Fig. 6 represents the small signal model of the drive train dynamics.
where xt and xg are the turbine and generator speeds in (pu), h is the shaft twist angle in (rad), Hg and Ht are the inertia constants of the turbine and generator in (sec), respectively, ks is the shaft stiffness coefficient in (pu/elec. rad), Dtg is the damping coefficient of the shaft in (pu), Te and Tt are the generator electrical torque and the turbine mechanical torque, respectively, in (pu). Also, the expression ks h þ Dtg ðxt xg Þ in (19) and (20) is known as shaft torsional torque and is depicted by T sh . Considering (14)–(16), the open loop transfer function from the electromagnetic torque, DT e , to the generator speed, xg , is given by
Dx g 1 2Hg s2 þ Dtg s þ ks xb ¼ : 2Hg Ht 2ðHg þ Ht Þs DT e s2 þ Dtg s þ ks xb
ð17Þ
Hg þHt
The open loop transfer function of (17) contains oscillating pole and zero with undamped natural frequency given by (18). The oscillating pole with frequency of x01 (in (18)) is known as torsional mode. Since the damping coefficient Dtg is relatively small, the torsional mode may results in low frequency oscillations if no damping control is provided by the controllers.
x01 x02
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ks xb ðHg þ Ht Þ ¼ 2Hg Ht sffiffiffiffiffiffiffiffiffiffiffi ks xb ¼ 2Ht
ð18Þ
P t0
x2t0
Dxt þ Pwind0
@C P ðk; bÞ dk : Dx t @k dxt
ð21Þ
4. Stability analysis of the drive train dynamics Considering (14)–(16), (19) and (21), the linearized drive train dynamic may be written as
Dx ¼ A Dx þ B Du Dy ¼ C Dx þ D Du where
Dx
is
the
ð22Þ vector
of
state
variables,
i.e.
Dx ¼
½Dxg ; Dh; Dxt ; DT e T , Du is the input variable, i.e. Du ¼ ½DV w and Dy is the output variable, i.e. Dy ¼ ½Dxg . Also, A is the 4 4 state matrix, B is the 4 1 input vector, C is the 1 4 output vector, and D is a scalar value. The matrix A and vectors B, C and D are given in C. In this section small signal stability analysis of the drive train dynamics is presented. The studies are carried out on a DFIG test system with parameters of Appendix A. The system under study is a 4-pole, 710 KW, 690 v, 50 Hz DFIG system consisting of 660 KW wound rotor induction generator with rated generator speed of 1620 rpm. In the studies the bandwidth of the stator active power control loop, aP , is selected equal to 31.4 rad/sec, i.e. 5 Hz.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 6. Linearized block diagram of the drive train dynamics.
The drive train of the turbine-generator has the following parameters: Hg ¼ 0:55 sec, Ht ¼ 3:5 sec, ks ¼ 0:5 pu/elec. rad, Dtg ¼ 1:2 pu, and xb ¼ 314 rad=sec. Hence, according to (18), the undamped natural frequency of the torsional modes is obtained equal to x01 ¼ 12:85 rad=sec (f 01 ¼ 2:045 Hz) with damping ratio of n ¼ 0:049. The element a41 of the state matrix A in (22) is dependent on the sensitivity of the power-speed curve function to the generator speed, i.e.
df ðxg Þ , d xg
and consequently f ðxg Þ plays an important role
on the drive train dynamics. Fig. 18, given in Appendix B, shows the power-speed curve of the DFIG test system with parameters of Appendix A. According to Fig. 18, the power-speed curve is a piecewise function with four operating regions. In the following, we examine the dynamics of the drive train for different operating regions.
4.1. Dynamic analysis at operating region 1 Simulation results show that for wind speeds of 5 and 6 m/s, the DFIG system operates at region 1 of the power-speed curve. Table 1 shows the operating points, system modes, and dominant state variables for wind speed of 6 m/s. According to Table 1, the modes K1;2 ¼ 5:75 j14 are relatively damped oscillatory modes corresponding to the turbine speed Dxt and shaft twist angle Dh. The mode K3 ¼ 20:1 is a damped simple mode corresponding to the state variables DT e and Dxg . This mode is highly dependent on the bandwidth of active power control loop, aP . The mode K4 ¼ 1:45 is a non-oscillatory mode corresponding to the turbine speed Dxt . At region 1, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , dxg
According to Table 2, the modes K1;2 ¼ 1 j120:9 are poorly damped oscillatory modes corresponding to the generator speed Dxg and shaft twist angle Dh. The corresponding damping ratio and natural frequency are n ¼ 0:077 and xn ¼ 12:9 rad/sec (2.05 Hz). Indeed, these modes correspond to open loop torsional oscillating modes with low damping ratio. This is because at region 2, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , dxg
in pu is approximately equal to 0.65 providing a low
damping action on the generator speed. The mode K3 ¼ 30:9 is a damped simple mode corresponding to the state variable DT e . This mode is approximately equal to the bandwidth of active power control loop, aP . The mode K4 ¼ 0:1221 is a simple mode corresponding to the turbine speed Dxt . 4.3. Dynamic analysis at operating region 3 For wind speeds of 9–12 m/sec, the DFIG operates at region 3 of the power speed curve. Table 3 shows the operating points, system modes, and dominant state variables for the wind speed of 12 m/sec. According to Table 3, the modes K1;2 ¼ 13:75 j260:1 are relatively damped oscillatory modes corresponding to the generator speed Dxg and DT e . Also, the modes K3;4 ¼ 2:78 j30:72 are relatively damped oscillatory modes corresponding to the Dxt and Dh. At region 3, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , dxg
in pu is equal to 29 introducing damping
action by making a component of electrical torque proportional to the generator speed. Hence, the mechanical dynamics are well damped and the open loop torsional oscillatory modes are not appeared on the dynamic mechanical response.
in pu is equal to 10 introducing damping
action by making a component of electrical torque proportional to the generator speed.
4.2. Dynamic analysis at operating region 2 For wind speeds of 7 and 8 m/sec, the DFIG operates at region 2 of the power speed curve. Table 2 shows the operating points, system modes, and dominant state variables for the wind speed of 7 m/sec.
4.4. Dynamic analysis at operating region 4 For wind speeds higher than 13 m/sec, the DFIG operates at region 4 of the power speed curve. Table 4 shows the operating points, system modes, and dominant state variables for the wind speed of 14 m/sec. According to Table 4, the modes K1;2 ¼ 0:79 j120:83 are poorly damped oscillatory modes corresponding to the generator speed Dxg and shaft twist angle Dh. The corresponding damping ratio and natural frequency are n ¼ 0:061 and xn ¼ 12:85 rad/sec (2.04 Hz), respectively. Indeed,
Table 1 Operating conditions, system modes, and dominant state variables for the wind speed of 6 m/sec. Operating Conditions System Modes Dominant State variables
C P ¼ 0:3871, P t ¼ 88:5 KW, P o ¼ 79:5 KW, xg ¼ 1236 rpm K3 ¼ 20:1 K1;2 ¼ 5:75 j14 Dxt Dh DT e
Dxg
K4 ¼ 1:45 Dxt
–
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Table 2 Operating conditions, system modes, and dominant state variables for the wind speed of 7 m/sec. C P ¼ 0:425 P t ¼ 155 KW P o ¼ 144:7 KW xg ¼ 1301 rpm K3 ¼ 30:9 K1;2 ¼ 1 j12:9 Dxg Dh DT e
Operating Conditions System Modes Dominant State variables
K4 ¼ 0:1221 Dxt
–
–
Table 3 Operating conditions, system modes, and dominant state variables for the wind speed of 12 m/sec. C P ¼ 0:3525 P t ¼ 647:3 KW P o ¼ 627:3 KW xg ¼ 1585:7 rpm K1;2 ¼ 13:75 j26:1 Dxg DT e
Operating Conditions System Modes Dominant State variables
K3;4 ¼ 2:78 j3:72 Dxt
Dh
Table 4 Operating conditions, system modes, and dominant state variables for the wind speed of 14 m/sec. C P ¼ 0:25 P t ¼ 730 KW P o ¼ 710 KW xg ¼ 1626 rpm K3 ¼ 31 K1;2 ¼ 0:79 j12:83 Dxg Dh DT e
Operating Conditions System Modes Dominant State variables
–
K4 ¼ 0:125 Dxt
–
Fig. 7. Structure of the DFIG active power control considering auxiliary damping control known as wind turbine stabilizer.
Fig. 8. Linearized drive train dynamics with auxiliary damping control known as wind turbine stabilizer.
these modes correspond to open loop torsional oscillating modes with low damping ratio. This is because at region 4, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , d xg
is
equal to zero without damping action on the generator speed. The mode K3 ¼ 31:4 is a damped simple mode corresponding to the state variable DT e . This mode is approximately equal to the bandwidth of active power control loop, aP . The mode K4 ¼ 0:125 is a simple mode corresponding to the turbine speed Dxt . 5. Wind turbine stabilizer (WTS) According to the explanations given in Section 4, at operating regions 2 and 4, the generator control does not provide damping action on the mechanical dynamics and consequently open loop torsional oscillation modes with low damping ratio appear as the
closed loop system modes. These modes play an important role on the dynamic response when wind turbine is subjected to a disturbance. In the following, an auxiliary active damping control known as wind turbine stabilizer (WTS), is proposed improving the damping Table 5 system modes and dominant state variables with the proposed WTS for the wind speed of 14 m/sec. System Modes
Dominant State Variables
K1
302
K2;3 K4 K5
4 j10:4 0.12
K6
33
State variable corresponding to the phase compensation term Dh Dxg Dxt – 5.8 State variable corresponding to the high-pass filter dynamics DT e –
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
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Fig. 9. Time response of the wind turbine for the wind speed of 8 m/sec with and without the wind turbine stabilizer (WTS), (a) generator speed, (b) generator output power, (c) electromagnetic torque, (d) shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 10. Time response of the wind turbine for the wind speed of 12 m/sec with and without the wind turbine stabilizer (WTS), (a) generator speed, (b) generator output power, (c) electromagnetic torque, (d) shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
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Fig. 11. Time response of the wind turbine for the wind speed of 14 m/sec with and without the wind turbine stabilizer (WTS), (a) generator speed, (b) generator output power, (c) electromagnetic torque, (d) shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
ratio of the drive train oscillatory modes at operating regions 2 and 4. Thus the effect of the WTS is to increase the damping torque component suppressing the torsional oscillations in the drive train dynamics. To provide positive damping, the WTS must provide a component of electrical torque proportional to the rotor speed deviations. Fig. 7 shows the structure of the DFIG active power control considering auxiliary damping control known as wind turbine stabilizer. The main control signal in Fig. 7 is extracted based on the power-speed curve for the generator power control at normal operations. Also, the auxiliary control signal is used for damping improvement of torsional modes when torsional oscillations are excited and appeared on the generator speed. Fig. 8 shows the linearized drive train dynamics of the DFIG system by taking the effect of auxiliary damping control, i.e. wind turbine stabilizer, into account. Fig. 8 is an extension of the block diagram of Fig. 6 including the WTS effect represented by GWTS ðsÞ. The transfer function GWTS ðsÞ in Fig. 8 represents the WTS. Considering Fig. 8, the electrical torque can be given by
DT e ffi
df ðxg Þ aP aP Dxg þ GWTS ðsÞ Dx g s þ aP dxg xg0 s þ aP
ð23Þ
where the second term in (23) is the electrical torque due to WTS and is given by
DT eWTS ffi GWTS ðsÞ
aP Dxg s þ aP
ð24Þ
For producing the damping torque by the WTS, it is required that DT eWTS be proportional to Dxg . Therefore, we select GWTS ðsÞ in the following form given by (25).
GWTS ðsÞ ¼ K WTS
1 þ sT 1 sT H 1 þ sT 2 1 þ sT H
ð25Þ
According to (25), GWTS ðsÞ contains two terms: a gain-phase compensator, and a high-pass filter block. The gian K WTS determines the amount of damping provided by the WTS. The phase compensation term,
1þsT 1 , 1þsT 2
produces the appropriate phase-lead to
compensate for the phase-lag of
aP
sþaP
.
The frequency of interest for reaction of WTS is the torsional oscillation frequency which is approximately 2.05 Hz (12.9 rad/sec) for the test system. Hence, the phase compensation term should provide compensation at this frequency. The high-pass filter with time constant of T H allows only the torsional oscillations in the generator speed to pass, and blocks the steady state changes in the speed to pass. Hence, the WTS responds only when low frequency torsional oscillations appear on the generator speed. The first term of electrical torque in (23) is highly dependent on the power-speed curve and operating conditions. As stated before, for the operating conditions correspond df ðx Þ ing to the region 4 of the power-speed curve, dxgg is zero and generator control does not participate to improve the damping of torsional oscillations. Hence, for the worst case conditions, the first term of (23) is zero, and DT e is equal to DT eWTS .
Fig. 12. Wind turbine time responses against voltage dip, under two different values of the grid short circuit power, at wind speed of 13 m/sec when the WTS is inactive, (a) Generator speed, (b) Shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
For the DFIG test system, the closed loop bandwidth of active power has been designed as ap ¼ 2p 5 rad=sec. Hence, at the torsional frequency of s ¼ j12:9, we have
DT eWTS ffi GWTS ðsÞjs¼j12:9
31:4 Dx g j12:9 þ 31:4
¼ ðGWTS ðj12:9ÞÞ ð0:925\ 220 ÞDxg
ð26Þ
According to (26), if Dxg through the phase compensation term, 1þsT 1 , 1þsT 2
is advanced by 220 at the frequency of 12.9 rad/sec, the
DT eWTS will be in phase with Dxg and thus will improve the damping of torsional oscillations. The amount of damping provided by the GWTS ðsÞ depends on the gain value K WTS . Hence, with the exact compensation of the phase lag of sþaaP P , the damping torque due to
WTS at x ¼ 12:9 rad=sec is DT eWTS ffi K WTS 0:925 Dxg . According to the above explanations, we select the parameters of GWTS ðsÞ for the DFIG test system as follows: K WTS ¼ 10, T 1 ¼ 0:035, T 2 ¼ 0:0032, and T H ¼ 0:177. Table 5 shows the system modes, and dominant state variables with the proposed WTS for the wind speed of 14 m/sec. Considering Table 5, the mode K1 ¼ 302 is a well damped mode corresponding to phase compensation term of WTS. The modes K2;3 ¼ 4 j10:4 are oscillatory modes corresponding to the generator speed Dxg and shaft twist angle Dh. The corresponding damping ratio and natural frequency of these modes are n ¼ 0:35 and xn ¼ 11:14 rad/sec (2.04 Hz). The modes K4 ¼ 0:12 and
21
K6 ¼ 33 are the simple modes associated with the state variables Dxt and DT e , respectively. Also, the mode K5 ¼ 5:8 is the mode corresponding to the high-pass filter of WTS. Comparing Tables 4 and 5 shows that by using the WTS, the poorly damped torsional modes (0:79 j12:83 in Table 4) with n ¼ 0:061 are changed to damped oscillatory modes (4 j100:4 in Table 5) with n ¼ 0:35. In other words, the WTS has increased the damping ratio of the torsional modes by 0.289 and reduced the modes natural frequency by 2.4 rad/sec. Hence, as will be seen in Section 6, the WTS improves the wind turbine dynamic response regarding the torsional oscillations when the wind turbine is subjected to a disturbance. 6. Time domain simulations In this section the effect of proposed wind turbine stabilizer (WTS) on the torsional oscillations of the DFIG based wind turbine is examined by time domain simulations. The system under study, shown in Fig. 1, consists of 710 KW, 690 V, 50 Hz DFIG based WT connected to a 20 kV distribution system. It exports power to the grid through a 10 km-20 kV cable and transformer T1. Transformer T1 is rated 900 KVA with the equivalent series reactance of 5%. The DFIG based WT parameters used in this simulation are given in Appendix A. In the simulations, two cases are considered: the case without the wind turbine stabilizer (WTS), and the case with the proposed WTS.
Fig. 13. Wind turbine time responses against voltage dip, under two different values of the grid short circuit power, at wind speed of 13 m/sec when the WTS is active, (a) Generator speed, (b) Shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 14. DFIG wind turbine connected to the IEEE-39 bus test system [24].
Fig. 15. Wind turbine responses for the step change of the wind speed from 9 to 14 m/sec at t = 4 sec, where DFIG connected at bus 14 of the IEEE 39 bus test system, (a) Generator speed, (b) Shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 9 depicts the time responses of the generator speed, output active power, electromagnetic torque, and shaft torsional torque for the wind speed of 8 m/sec. The DFIG average output active power and generator speed at V w ¼ 8 m=sec are Po ¼ 210 kW and xg ¼ 1470 rpm (0.982 pu), respectively, and the DFIG operates at the region 2 of the power-speed curve. Considering Fig. 9, in the case without the WTS, damping of torsional oscillations is lower than that in the case with the proposed WTS. This is because for V w ¼ 8 m=sec, and in the case without the WTS, generator active power control does not provide sufficient damping action on the generator speed. Fig. 10 shows the time responses of the wind turbine for the wind speed of 12 m/sec. The DFIG output active power and generator speed at V w ¼ 12 m=sec before the voltage dip are P o ¼ 627:3 kW (0.88 pu) and xg ¼ 1587:7 rpm (1.058 pu), respectively, and the DFIG operates at the region 3 of the power-speed curve. According to Fig. 10, the torsional oscillations at the both cases of with and without WTS, are well damped. This is because for V w ¼ 12 m=sec the DFIG operates at the region 3 of the power-speed curve, and thus sensitivity of the power-speed curve function to the generator speed is relatively high. Fig. 11 depicts the time responses of the generator speed, output active power, electromagnetic torque, and shaft torsional torque for the wind speed of 14 m/sec. The DFIG output active power and generator speed at V w ¼ 14 m=sec are P o ¼ 710 kW (1 pu) and xg ¼ 1626 rpm (1.084 pu), respectively, and the DFIG operates at the region 4 of the power-speed curve. According to Fig. 11, the frequency of the oscillations appeared after clearing the voltage dip is approximately 2 Hz corresponding to the
23
undamped natural frequency of the torsional modes. This is in agreement with the frequency of torsional modes computed at Section 4. Considering Fig. 11, in the case without the WTS, damping of torsional oscillations is lower than that in the case with the proposed WTS. This is because for V w ¼ 14 m=sec, sensitivity of the power-speed curve function to the generator speed is equal to zero, and thus in the case without the WTS, generator active power control does not provide any damping action on the generator speed. On the other hand, in the case with the proposed WTS, positive damping torque component is provided suppressing the torsional oscillations in the drive train dynamics. According to Fig. 11(d), in the case without the WTS, the shaft torsional torque contains large torsional oscillations with frequency of 2 Hz that may result in fatigue and stress on the wind turbine shaft. Figs. 12 and 13 examine the dynamic performance of the wind turbine under two different values of the grid short circuit power, i.e. Ssc = 4 MVA and 10 MVA. Fig. 12 corresponds to the case without the wind turbine stabilizer, and Fig. 13 corresponds to the case with the wind turbine stabilizer. Figs. 12 and 13 depict the time responses of the generator speed and shaft torsional torque at the wind speed of 13 m/sec when the wind turbine is subjected to a voltage dip. In Fig. 12, the WTS is inactive and after the voltage dip, weakly damped oscillations appear on the wind turbine response. However, according to Fig. 12, in the case with Ssc = 4 MVA, the effect of the grid short circuit power on the wind turbine torsional oscillations is more severe than that in the case with Ssc = 10 MVA. Considering Fig. 12(a) and (b), in the case with Ssc = 4 MVA, the
Fig. 16. Wind turbine responses against 95% voltage dip at t = 10 sec, where DFIG connected at bus 14 of the IEEE 39 bus test system, (a) Generator speed, (b) Shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
response of the generator speed is unstable and relatively large oscillations appear on the shaft torsional torque. In Fig. 13, the WTS is active and after the voltage dip, well damped oscillations appear on the wind turbine response. Unlike Fig. 12, in Fig. 13 due to WTS, at both of the Ssc = 4 MVA and Ssc = 10 MVA, the response of the generator speed is stable and well damped oscillations appear on the shaft torsional torque.
6.1. Time domain simulations using IEEE 39-bus New England test system Fig. 14 depicts the IEEE 39-bus New England test system [24], in which the DFIG wind turbine system is connected at bus 14. Figs. 15–17 depict the times responses of the DFIG wind turbine connected at bus 14 of the IEEE 39-bus test system. Fig. 15 compare time responses of the generator speed and shaft torsional torque for the cases with and without the WTS, under the step change of the wind speed from 9 to 14 m/sec at t = 4 sec. Before the wind step change, V w ¼ 9 m/sec, xg ¼ 1:043 pu, and wind turbine operates in region 3 of the power speed curve. After the wind step change, the wind turbine operates in region 4 of the power speed curve. According to Fig. 15(a) and (b), in the case without the wind turbine stabilizer and after the wind step change, torsional oscillations appear on the responses of the generator speed and shaft torsional. This is because the rotor side converter control cannot provide adequate action for damping of torsional oscillations at region 4. On the other hand, in the case with the wind turbine stabilizer, the added auxiliary control (WTS) can
provide positive damping action, and thus the torsional oscillations are well damped. Fig. 16 depicts the time responses of the generator speed and shaft torsional torque under 95% voltage dip at the stator terminal, at the wind speed of 13 m/sec, and for the cases with and without the WTS. Also, Fig. 17 depicts the times responses of the DFIG wind turbine connected at bus 11 of the IEEE 39-bus test system. It shows the time responses of the generator speed and shaft torsional torque under 80% voltage dip, at the wind speed of 13 m/sec, and for the cases with and without the WTS. According to Figs. 16 and 17, in the cased without the WTS, weakly damped oscillations appear on the wind turbine responses after the voltage dip. However, in the cased with the WTS, the torsional oscillations after the voltage dip are well damped.
Fig. 18. Power-speed curve for the DFIG system under study.
Fig. 17. Wind turbine responses against 80% voltage dip at t = 10 sec, where DFIG connected at bus 11 of the IEEE 39 bus test system, (a) Generator speed, (b) Shaft torsional torque.
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7. Conclusion
References
This paper first analytically examines the dynamics of the drive train for different operating regions of the power-speed curve when the wind turbine is controlled in power control mode. It is shown that at operating regions 1 and 3 of the power-speed curve, torsional oscillations are well damped and the DFIG control improves the damping of torsional modes. On the other hand, at operating regions 2 and 4 of the power-speed curve, the generator control does not provide damping action on the poorly damped torsional oscillatory modes. Then an auxiliary active damping control known as wind turbine stabilizer (WTS), is proposed improving the damping ratio of the drive train oscillatory modes at operating regions 2 and 4. The effect of the WTS is to produce a damping torque component suppressing the torsional oscillations in the drive train dynamics. As the future research, it may be possible to propose a comprehensive auxiliary control system for damping of torsional oscillations, subsynchronous oscillations, and power system electromechanical oscillations.
[1] Rahimi M, Parniani M. Low voltage ride-through capability improvement of DFIG-based wind turbines under unbalanced voltage dips. Electric Power Energy Syst 2014;60:82–95. [2] Rahimi M, Parniani M. Coordinated control approaches for low-voltage ridethrough enhancement in wind turbines with doubly fed induction generators. IEEE Trans Energy Convers 2010;25(3):873–83. [3] Ackerman T. Wind power in power systems. 1st ed. Wiley; 2005. [4] Tabesh A, Iravani R. Transient ehaviour of a fixed-speed grid-connected wind farm. In: Proc. int. conf. power systems transients, Montreal, Canada, June 2005, p. 1–5. [5] Ramtharan G, Jenkins N, Anaya-Lara O, Bossanyi E. Influence of rotor structural dynamics representations on the electrical transient performance of FSIG and DFIG wind turbines. Wind Energy 2007;10:293–301. [6] Kundur P. Power system stability and control. McGraw-Hill; 1994. [7] Savulescu Savu C. Real-time stability in power systems. 2nd ed. SpringerVerlag; 2014. [8] Hovd AB. Modal analysis of weak networks with the integration of wind power [Master of Science Thesis]. NTNU University; 2008. [9] Gayen PK, Chatterjee D, Goswami SK. Stator side active and reactive power control with improved rotor position and speed estimator of a grid connected DFIG (doubly-fed induction generator). Energy 2015;89:461–72. [10] Li H, Chen Z, Pedersen JK. Optimal power control strategy of maximizing wind energy tracking and conversion for VSCF doubly fed induction generator system. In: IEEE 5th international power electronics and motion control conference, vol. 3, August 2006, p. 1–6. [11] Pena R, Clare JC, Asher GM. Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation. IEE Electric Power Appl 1996;143:231–41. [12] Hilal M, Maaroufi M, Ouassaid M. Doubly fed induction generator wind turbine control for a maximum power extraction. Int Conf Multimedia Comput Syst 2011:1–7. [13] Chen L, Xu H, Wenske J. Active damping of torsional vibrations in the drive train of a DFIG wind turbine. In: Proc. int. conf. renewable energies and power quality, Cordoba, Spain, April 2014. [14] Muyeen SM, Ali MH, Takahashi R, Murata T. Blade-shaft torsional oscillation minimization of wind turbine generator system by using STATCOM/ESS. In: Proc. power tech. conf., Lausanne, July 2007, p. 184–9. [15] Xing Z, Liang L, Guo H, Wang X. Damping control study of the drive train of DFIG wind turbine. In: Proc. int. conf. energy and environment technology, Guilin, Guangxi, Oct. 2009, p. 576–9. [16] El-Moursi MS, Bak-Jensen B, Abdel-Rahman MH. Novel STATCOM controller for mitigating SSR and damping power system oscillations in a series compensated wind park. IEEE Trans Power Elec 2010;25(2):429–41. [17] El Moursi MS, Khadkikar V. Novel control strategies for SSR mitigation and damping power system oscillations in a series compensated wind park. In: Proc. 38th annual conf. industrial elec. society, Montreal, QC, Oct. 2012, p. 5335–42. [18] Abdou AF, Abu-Siada A, Pota HR. Damping of subsynchronous oscillations and improve transient stability for wind farms. In: IEEE PES Conf. Innovative Smart Grid Technologies Asia (ISGT), Perth, WA, Nov. 2011, p. 1–6. [19] Todorov M, Dobrev I, Massouh F. Analysis of torsional oscillation of the drive train in horizontal-axis wind turbine. In: Proc. electromotion joint symposium, Lille, France, July 2009, p. 1–7. [20] Lei T, Bames M, Ozakturk M. Doubly-fed induction generator wind turbine modelling for detailed electromagnetic system studies. IET Renew Power Gener 2013;7(2):180–9. [21] Geng H, Xu D, Wu B, Yang G. Active damping for PMSG-based WECS with DClink current estimation. IEEE Trans Indus Elec 2011;58(4):1110–9. [22] White WN, Yu Z, Lucero C. Active damping of torsional resonance in wind turbine drivetrains. In: Proc. ind. elec. society conf., Dallas, TX, Oct. 2014, p. 1957–63. [23] Fan L, Yin H, Miao Z. On active/reactive power modulation of DFIG-based wind generation for interarea oscillation damping. IEEE Trans Energy Conv 2011;26 (2):513–21. [24] Surinkaew T, Ngamroo I. Coordinated robust control of DFIG wind turbine and PSS for stabilization of power oscillations considering system uncertainties. IEEE Trans Sustain Energy 2014;5(3):823–33.
Appendix A Parameters of the 710 KW, 690 V, 50 Hz, DFIG-WT:
V base ¼ 690 V Sbase ¼ 710 KVA f base ¼ 50 Hz
xb ¼ 2pf b ¼ 314 rad=sec
Hg ¼ 0:55 sec Ht ¼ 3:5 sec Dtg ¼ 1:2 pu ks ¼ 0:5 pu=elec rad xs ¼ 1 pu Appendix B Power-speed curve for the DFIG system under study is shown in Fig. 16. In Fig. 16, the generator speed and stator power are in pu, where the base values of the generator speed and power are 1500 rpm and 710 KVA, respectively. Appendix C Matrix A and vectors B, C and D related to linearized drive train dynamics:
h B¼ 0 0
1 2Ht
3P t0
1
xt0 V w0
wind0 @C P ðk;bÞ Px @k t0
0
k0 V w0
0
iT
C ¼ ½ 1 0 0 0 D ¼ ½0 2
Dtg 2H
ks 2H
Dtg 2H
1 2H
3
g g g g 6 7 6 xb 0 xb 0 7 1 P t0 @C P ðk;bÞ dk 6 7 A¼6 2 þ P wind0 Dtg 7 and N ¼ ks N 0 6 7 x 2H @k d xt0 t t 2Ht 0 4 2Ht 5 df ðxg Þ :aP 0 0 aP dxg
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improvement of energy conversion efficiency and damping of wind turbine response in grid connected DFIG based wind turbines Mohsen Rahimi Department of Electrical and Computer Engineering, University of Kashan, Kashan, Iran
a r t i c l e
i n f o
Article history: Received 11 January 2017 Received in revised form 21 July 2017 Accepted 6 August 2017
Keywords: DFIG Drive train dynamics Torsional modes Power-speed curve Wind turbine stabilizer
a b s t r a c t This paper examines dynamic performance of DFIG based wind turbine regarding torsional electromechanical oscillations when the wind turbine is controlled in power control mode. In the power control mode, the stator active power is set to its reference value, where the reference active power is determined based on a predefined power-speed curve. Torsional oscillations associated with the poorly torsional modes may reduce the aerodynamic efficiency, deteriorate the power quality and increase stresses on the drive train system. The paper indeed analytically investigates the drive train dynamics for different operating regions of the power-speed curve. It is shown that under some operating points, torsional oscillations are well damped and the DFIG control improves the damping of torsional modes. On the other hand, at some regions of the power-speed curve, the DFIG control does not provide significant damping action on the torsoinal modes. Then a wind turbine stabilizer (WTS) is implemented to suppress torsional vibrations in the drive train system. The role of the WTS is to add positive damping to the generator rotor oscillations by controlling the active power and providing a component of electrical torque proportional to the rotor speed deviations. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction DFIGs are commonly used variable speed generators in wind turbine (WT) applications. Wind turbines based on DFIG systems result in lower converter cost and lower power losses compared to full converter based wind turbines (WTs) [1]. As shown in Fig. 1, the DFIG-based WT consists of drive train system (turbine blades, gear box, shafts, and other mechanical parts), wound rotor induction generator, back-to-back voltage source converters linking the rotor to the grid, pitch control and converters control systems. The rotor-side converter (RSC) is used to control the generator real and reactive power, while the grid-side converter (GSC) is used to control the dc-link voltage and reactive power exchanged with the grid [2]. The wind turbine equivalent shaft is relatively softer than the typical turbine shaft in conventional power plants [3]. Hence, the drive train of the wind turbine should be described with at least a two-mass model [4], or three-mass model [5], since a single-mass shaft model cannot properly represent mechanical oscillations. Mechanical oscillations of the multi mass drive train dynamics are known as torsional oscillations related to the torsional oscillatory modes. The electromechanical oscillations in a E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijepes.2017.08.005 0142-0615/Ó 2017 Elsevier Ltd. All rights reserved.
conventional power system are related to the local and inter-area modes associated with the tendency of the interconnected synchronous generators to remain in synchronism [6]. However, in a mixed power system with high penetration of wind power generation, there are two types of oscillatory electromechanical modes: local and inter-area oscillatory modes related to the synchronous generators, and torsional oscillatory modes associated with the wind turbines [7,8]. From the DFIG control point of view, there are two modes for the RSC control using field oriented control (FOC): power control mode and speed control mode. The reference control value is stator power for the power control mode [9,10], and the rotational speed for the speed control mode [11,12]. In the power control mode, the active power control is carries out based on a predefined piecewise power-speed curve generating the active power reference signal for the power control loop. In most practical cases, the DFIGs are controlled in the power control mode, in which the generator active power is set to its reference value. In literatures, little work has been published regarding the effect of power-speed curve on the torsional oscillations. Low frequency torsional oscillations associated with the wind turbine generators have been investigated by several papers in the literatures. In [13–15] some approaches are proposed to mitigate the torsional oscillations of the drive train when the wind turbine is subjected to a severe grid
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 1. Schematic diagram for the DFIG system under study.
fault. Without the damping controller, the torsional oscillations excited by the grid fault are very strong after the grid fault incident. In [14], the damping control for the torsional oscillations is realized by using STATCOM/BESS topology. In [16–18] the SSR phenomenon caused by induction generator effects as well as torsional interactions in a series compensated wind park are discussed. In [16] by using the STATCOM an auxiliary subsynchronous damping control is developed to meet the damping torque in the range of critical torsional frequencies. In [19] a detailed model of the wind turbine with complex drive train is proposed which is the starting point in the research of torsional vibrations. In [20], It is shown that the pitch angle control may excite the torsional oscillations in DFIG based wind turbines. Torsional oscillations occur as the pitch bandwidth increases where the controller bandwidth begins to interfere with the shaft natural frequency. In [21], an activedamping strategy is proposed for the suppression of speed and torsional oscillations in direct-driven permanent-magnet synchronous generator wind turbine. In [22], two mitigation methods for torsional resonance are proposed: an active vibration absorber and the other method uses rotor angular acceleration measurements to provide virtual inertia for resonance detuning. In [23], the interactions of the torsional dynamics of the wind turbines and power modulation for oscillations in DFIG-based wind generators are investigated. Active power modulation for inter area oscillation damping is effective. However, it may interact with the torsional dynamics. According to the above references, the torsional oscillations in the wind turbine generators are excited due to four main reasons: excitation by the grid fault, torsional interactions in a series compensated wind turbine due to SSR phenomenon, interference between the bandwidth of pitch angle control and shaft torsional frequency, and interactions of torsional dynamics and power modulation. As stated above, the control of DFIG in the power control mode is realized based on a predefined piecewise power-speed curve generating the active power reference signal for the power control loop. This paper deals with the stability analysis of the drive train dynamics, torsional oscillatory modes, and damping improvement of torsional oscillations in DFIG based wind turbine at different regions of the power-speed curve when the wind turbine is controlled in the power control mode. As the main contribution, it is shown that the torsional oscillations may be excited due to characteristic of the power speed curve at some operating points. In other words, at some regions of the power-speed curve, the DFIG control does not provide significant damping torque on the torsional modes. Hence, unlike the related literatures, in this paper the torsional oscillations may be raised due to nature of the power speed curve which has not been investigated before. The paper first analytically examines the dynamics of the drive train and related torsional oscillations for different operating regions of the powerspeed curve. The torsional oscillatory modes with low damping ratio affect the dynamic performance of the wind turbine. This is because torsional oscillations are transmitted to the generator
speed, electrical torque and output active power deteriorating the power quality and inducing fatigue stresses on the mechanical components. Then, an auxiliary active damping control known as wind turbine stabilizer (WTS) is proposed improving the damping ratio of the torsional oscillatory modes. Consequently the impact of the WTS is to introduce a damping torque component suppressing the torsional oscillations in the drive train components. As an advantage, in this paper, a more accurate solution is formulated and more comprehensive analysis is made for examining the issue of torsional oscillations. This paper is organized as follows: Section 2 deals with the wind turbine modeling and control. Modeling of linearized drive train dynamics is presented in Section 3. Dynamic stability analyses of the drive train dynamics, with and without WTS, are given in Sections 4 and 5. Simulation results are presented in Section 6, and finally Section 7 draws the conclusions. 2. Wind turbine modeling and control The purpose of this section is to present the dynamic model and power controller design of DFIG based WT. The following assumptions for the system model are considered: (a) positive direction for the stator and rotor currents is assumed into the generator, (b) all system parameters and variables are in per unit and referred to the stator side in generator, (c) the equations are given in dqsynchronous reference frame with the stator voltage orientation (SVO). The stator and rotor voltages and fluxes, electromagnetic torque, and stator output active power in reference frame rotating at angular speed of x are given by [2]:
v sdq ¼ Rs isdq þ jxwsdq þ
v rdq ¼ Rr irdq þ jx2 wrdq þ ws ¼ Ls is þ Lm ir wr ¼ Lm is þ Lr ir Te ¼
Lm ðw irq wsq ird Þ Ls sd
Ps ¼ Re v sdq isdq
1 dwsdq
xb dt
1 dwrdq
xb dt
ð1Þ
ð2Þ
ð3Þ
ð4Þ ð5Þ
where w, v and i represent the flux, voltage and current. Subscripts s and r denote the stator and rotor quantities, respectively. Ls and Lr are the stator and rotor self inductances, Lm is the mutual inductance. x2 is the rotor slip frequency, xb is the base angular frequency, and x is the speed of d-q reference frame. Also, Rs and Rr
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are the stator and rotor resistances. The superscript ⁄ in (5) represents the conjugate operator. 2.1. Active power flow in DFIG In the dq reference frame with the stator voltage orientation (SVO), the d-axis is aligned with the stator voltage vector, and consequently v sd ¼ V s and v sq ¼ 0. With stator and rotor resistances neglected, the steady state stator/rotor voltage equations in SVO frame are obtained as follows: V s ¼ v sd ffi xs wsq , V sq ffi xs wsq ¼ 0 and V r ffi LLms sV s , where xs , equal to 1 pu, is the stator angular fre-
quency, s is the generator slip, and V s is the stator voltage amplitude. This results in the approximate expressions of wsd 0 and wsq ws . Considering (5) and according to the above explanations, the stator output active power, P s , active power delivered to the rotor, P r , and electromagnetic torque as functions of the stator voltage and rotor current can be written as
Ps
Lm V s ird Ls
ð6Þ
Pr
Lm sV s ird Ls
ð7Þ
Te ¼
Lm ðw ird Þ Ls sq
ð8Þ
Hence, in the SVO frame, isd and ird represent the active power components of the stator and rotor currents, respectively. On the other hand, isq and irq stand for the reactive power components of the stator and rotor currents, respectively. xs ¼ 1 pu and According to (8), and considering V s ¼ v sd ffi xs wsq , the electromagnetic torque in per unit is approximately equal to the stator power. Therefore,
T e ffi Ps ffi
Lm V s ird Ls
ð9Þ
Considering (6) and (7), we have: P r ffi sP s . Fig. 2 depicts the active power flow direction of DFIG at sub-synchronous and super-synchronous operation modes. According to Fig. 2 the power delivered to the grid by the DFIG is given by
Po ffi Ps Pr ¼ ð1 sÞP s
ð10Þ
2.2. Active power control loop According to (6), there is a direct relation between d-axis rotor current and stator output active power. Therefore, in DFIG based WT, stator active power injected to the grid can be controlled by the d-axis rotor current control through rotor voltage regulation. Fig. 3 depicts the output active power control loop, where P s and P sref are the active power and reference active power of the stator. The transfer function sþaaI I in Fig. 3, is the closed loop transfer
function of inner rotor current control loop, and aI is the closed loop bandwidth of the rotor current control. Hence, the d-axis rotor reference current is determined by the outer active power controller, as shown in Fig. 3. Assuming the power controller to be PI, K P ðsÞ ¼ kPP þ kIP , and by canceling the s pole aI with the zero of the controller, we have
(
aP ¼ LLms kPP aI V s kIP ¼ kPP aI
Fig. 2. Active power flow direction in DFIG.
ð11Þ
where aP is the bandwidth of stator active power control loop. Then, the transfer function from P sref to P s is given by
Fig. 3. Stator active power control loop.
Ps aP ¼ Psref s þ aP
ð12Þ
Indeed, the rotor current control is the inner control loop, and the stator active power control represents the outer control loop. The active power reference in Fig. 3 is determined based on a predefined power-speed characteristic curve, given by
Psref ¼ f ðxg Þ
ð13Þ
where f ðÞ represents the power-speed curve function. Fig. 4 depicts a typical power-speed curve for the DFIG wind turbine system. Considering Fig. 4, the power-speed curve is divided into four regions: 1. Low speed operating region: this region, i.e. section 1 of Fig. 4, corresponds to low wind speed operation. At this region the generator speed is relatively low, which is usually around 30% below synchronous speed. 2. Middle-speed operating region: in this region, i.e. section 2 of Fig. 4, the speed of the turbine is adjusted to yield a maximum power capture at a given wind speed. 3. Constant speed operation region: this region corresponds to section 3 of Fig. 4, in which the generator speed is approximately constant up to rated power. At high wind speed, generator speed is not able to continuously follow optimum
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 5. The electromagnetic torque-speed block diagram.
According to (9), (12) and (13), the electromagnetic torque, T e , as a function of the generator speed, xg , can be given by Fig. 5. As stated in Section 2, the stator active power reference, P sref , is determined based on a predefined power-speed curve, given by Psref ¼ f ðxg Þ, where f ðxg Þ represents the power-speed curve function. Linearizing T e around an operating point, xg0 , yields
DT e ffi DP s ¼
Fig. 4. A typical power-speed curve for the DFIG wind turbine system.
df ðxg Þ aP Dxg dxg xg0 s þ aP
ð19Þ
operation, and consequently the controller attempts to maintain generator speed constant. 4. Constant power operation region: this region, i.e. section 4 of Fig. 4, stands for a constant power characteristic beyond the speed limit followed by a blade pitch control for high wind speed. In this region the generator speed is limited by reducing the turbine mechanical power through pitching the blade angle.
According to (19), DT e depends on the sensitivity of the f ðxg Þ to the generator speed. The turbine torque, T t , is extracted from aerodynamic model of the wind turbine. The inputs to the aerodynamic model are the wind speed, V w , the blade pitch angle, b, and the turbine speed, xt . The output of the aerodynamic model is the turbine mechanical power, Pt. The following relationship is used to obtain Pt from the wind power [3]:
3. Modeling of linearized drive train dynamics
Pt ¼ Pwind C P ðk; bÞ 1 Pwind ¼ qpR2 V 3w 2
The drive train of the wind turbine is usually represented by a two-mass model. In the two-mass model, the low speed mass of the turbine is connected to the high speed mass of the generator through a flexible shaft modeled as a spring and damper. The linearized model of the two-mass drive train system is given by
ð20Þ
where Pwind is the power contained in the wind, C P is the power coefficient, q is the air density, and R is the radius of the blades. C P is a function of tip-speed ratio, k, and blade pitch angle, b. k is the ratio of the rotor blade tip speed and the wind speed. The turbine torque is given by T t ¼ Pt =xt . Linearizing T t around the turbine operating speed xt0 yields
dDxg DT e þ ks Dh þ Dtg ðDxt Dxg Þ ¼ 2Hg dt
ð14Þ
dDh ¼ xb ðDxt Dxg Þ dt
ð15Þ
DT t
dDxt DT t ks Dh Dtg ðDxt Dxg Þ ¼ 2Ht dt
ð16Þ
Replacing (19) and (21) into (14) and (16) results in the linearized drive train dynamic as depicted in Fig. 6. Block diagram of Fig. 6 represents the small signal model of the drive train dynamics.
where xt and xg are the turbine and generator speeds in (pu), h is the shaft twist angle in (rad), Hg and Ht are the inertia constants of the turbine and generator in (sec), respectively, ks is the shaft stiffness coefficient in (pu/elec. rad), Dtg is the damping coefficient of the shaft in (pu), Te and Tt are the generator electrical torque and the turbine mechanical torque, respectively, in (pu). Also, the expression ks h þ Dtg ðxt xg Þ in (19) and (20) is known as shaft torsional torque and is depicted by T sh . Considering (14)–(16), the open loop transfer function from the electromagnetic torque, DT e , to the generator speed, xg , is given by
Dx g 1 2Hg s2 þ Dtg s þ ks xb ¼ : 2Hg Ht 2ðHg þ Ht Þs DT e s2 þ Dtg s þ ks xb
ð17Þ
Hg þHt
The open loop transfer function of (17) contains oscillating pole and zero with undamped natural frequency given by (18). The oscillating pole with frequency of x01 (in (18)) is known as torsional mode. Since the damping coefficient Dtg is relatively small, the torsional mode may results in low frequency oscillations if no damping control is provided by the controllers.
x01 x02
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ks xb ðHg þ Ht Þ ¼ 2Hg Ht sffiffiffiffiffiffiffiffiffiffiffi ks xb ¼ 2Ht
ð18Þ
P t0
x2t0
Dxt þ Pwind0
@C P ðk; bÞ dk : Dx t @k dxt
ð21Þ
4. Stability analysis of the drive train dynamics Considering (14)–(16), (19) and (21), the linearized drive train dynamic may be written as
Dx ¼ A Dx þ B Du Dy ¼ C Dx þ D Du where
Dx
is
the
ð22Þ vector
of
state
variables,
i.e.
Dx ¼
½Dxg ; Dh; Dxt ; DT e T , Du is the input variable, i.e. Du ¼ ½DV w and Dy is the output variable, i.e. Dy ¼ ½Dxg . Also, A is the 4 4 state matrix, B is the 4 1 input vector, C is the 1 4 output vector, and D is a scalar value. The matrix A and vectors B, C and D are given in C. In this section small signal stability analysis of the drive train dynamics is presented. The studies are carried out on a DFIG test system with parameters of Appendix A. The system under study is a 4-pole, 710 KW, 690 v, 50 Hz DFIG system consisting of 660 KW wound rotor induction generator with rated generator speed of 1620 rpm. In the studies the bandwidth of the stator active power control loop, aP , is selected equal to 31.4 rad/sec, i.e. 5 Hz.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 6. Linearized block diagram of the drive train dynamics.
The drive train of the turbine-generator has the following parameters: Hg ¼ 0:55 sec, Ht ¼ 3:5 sec, ks ¼ 0:5 pu/elec. rad, Dtg ¼ 1:2 pu, and xb ¼ 314 rad=sec. Hence, according to (18), the undamped natural frequency of the torsional modes is obtained equal to x01 ¼ 12:85 rad=sec (f 01 ¼ 2:045 Hz) with damping ratio of n ¼ 0:049. The element a41 of the state matrix A in (22) is dependent on the sensitivity of the power-speed curve function to the generator speed, i.e.
df ðxg Þ , d xg
and consequently f ðxg Þ plays an important role
on the drive train dynamics. Fig. 18, given in Appendix B, shows the power-speed curve of the DFIG test system with parameters of Appendix A. According to Fig. 18, the power-speed curve is a piecewise function with four operating regions. In the following, we examine the dynamics of the drive train for different operating regions.
4.1. Dynamic analysis at operating region 1 Simulation results show that for wind speeds of 5 and 6 m/s, the DFIG system operates at region 1 of the power-speed curve. Table 1 shows the operating points, system modes, and dominant state variables for wind speed of 6 m/s. According to Table 1, the modes K1;2 ¼ 5:75 j14 are relatively damped oscillatory modes corresponding to the turbine speed Dxt and shaft twist angle Dh. The mode K3 ¼ 20:1 is a damped simple mode corresponding to the state variables DT e and Dxg . This mode is highly dependent on the bandwidth of active power control loop, aP . The mode K4 ¼ 1:45 is a non-oscillatory mode corresponding to the turbine speed Dxt . At region 1, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , dxg
According to Table 2, the modes K1;2 ¼ 1 j120:9 are poorly damped oscillatory modes corresponding to the generator speed Dxg and shaft twist angle Dh. The corresponding damping ratio and natural frequency are n ¼ 0:077 and xn ¼ 12:9 rad/sec (2.05 Hz). Indeed, these modes correspond to open loop torsional oscillating modes with low damping ratio. This is because at region 2, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , dxg
in pu is approximately equal to 0.65 providing a low
damping action on the generator speed. The mode K3 ¼ 30:9 is a damped simple mode corresponding to the state variable DT e . This mode is approximately equal to the bandwidth of active power control loop, aP . The mode K4 ¼ 0:1221 is a simple mode corresponding to the turbine speed Dxt . 4.3. Dynamic analysis at operating region 3 For wind speeds of 9–12 m/sec, the DFIG operates at region 3 of the power speed curve. Table 3 shows the operating points, system modes, and dominant state variables for the wind speed of 12 m/sec. According to Table 3, the modes K1;2 ¼ 13:75 j260:1 are relatively damped oscillatory modes corresponding to the generator speed Dxg and DT e . Also, the modes K3;4 ¼ 2:78 j30:72 are relatively damped oscillatory modes corresponding to the Dxt and Dh. At region 3, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , dxg
in pu is equal to 29 introducing damping
action by making a component of electrical torque proportional to the generator speed. Hence, the mechanical dynamics are well damped and the open loop torsional oscillatory modes are not appeared on the dynamic mechanical response.
in pu is equal to 10 introducing damping
action by making a component of electrical torque proportional to the generator speed.
4.2. Dynamic analysis at operating region 2 For wind speeds of 7 and 8 m/sec, the DFIG operates at region 2 of the power speed curve. Table 2 shows the operating points, system modes, and dominant state variables for the wind speed of 7 m/sec.
4.4. Dynamic analysis at operating region 4 For wind speeds higher than 13 m/sec, the DFIG operates at region 4 of the power speed curve. Table 4 shows the operating points, system modes, and dominant state variables for the wind speed of 14 m/sec. According to Table 4, the modes K1;2 ¼ 0:79 j120:83 are poorly damped oscillatory modes corresponding to the generator speed Dxg and shaft twist angle Dh. The corresponding damping ratio and natural frequency are n ¼ 0:061 and xn ¼ 12:85 rad/sec (2.04 Hz), respectively. Indeed,
Table 1 Operating conditions, system modes, and dominant state variables for the wind speed of 6 m/sec. Operating Conditions System Modes Dominant State variables
C P ¼ 0:3871, P t ¼ 88:5 KW, P o ¼ 79:5 KW, xg ¼ 1236 rpm K3 ¼ 20:1 K1;2 ¼ 5:75 j14 Dxt Dh DT e
Dxg
K4 ¼ 1:45 Dxt
–
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Table 2 Operating conditions, system modes, and dominant state variables for the wind speed of 7 m/sec. C P ¼ 0:425 P t ¼ 155 KW P o ¼ 144:7 KW xg ¼ 1301 rpm K3 ¼ 30:9 K1;2 ¼ 1 j12:9 Dxg Dh DT e
Operating Conditions System Modes Dominant State variables
K4 ¼ 0:1221 Dxt
–
–
Table 3 Operating conditions, system modes, and dominant state variables for the wind speed of 12 m/sec. C P ¼ 0:3525 P t ¼ 647:3 KW P o ¼ 627:3 KW xg ¼ 1585:7 rpm K1;2 ¼ 13:75 j26:1 Dxg DT e
Operating Conditions System Modes Dominant State variables
K3;4 ¼ 2:78 j3:72 Dxt
Dh
Table 4 Operating conditions, system modes, and dominant state variables for the wind speed of 14 m/sec. C P ¼ 0:25 P t ¼ 730 KW P o ¼ 710 KW xg ¼ 1626 rpm K3 ¼ 31 K1;2 ¼ 0:79 j12:83 Dxg Dh DT e
Operating Conditions System Modes Dominant State variables
–
K4 ¼ 0:125 Dxt
–
Fig. 7. Structure of the DFIG active power control considering auxiliary damping control known as wind turbine stabilizer.
Fig. 8. Linearized drive train dynamics with auxiliary damping control known as wind turbine stabilizer.
these modes correspond to open loop torsional oscillating modes with low damping ratio. This is because at region 4, sensitivity of the power-speed curve function to the generator speed,
df ðxg Þ , d xg
is
equal to zero without damping action on the generator speed. The mode K3 ¼ 31:4 is a damped simple mode corresponding to the state variable DT e . This mode is approximately equal to the bandwidth of active power control loop, aP . The mode K4 ¼ 0:125 is a simple mode corresponding to the turbine speed Dxt . 5. Wind turbine stabilizer (WTS) According to the explanations given in Section 4, at operating regions 2 and 4, the generator control does not provide damping action on the mechanical dynamics and consequently open loop torsional oscillation modes with low damping ratio appear as the
closed loop system modes. These modes play an important role on the dynamic response when wind turbine is subjected to a disturbance. In the following, an auxiliary active damping control known as wind turbine stabilizer (WTS), is proposed improving the damping Table 5 system modes and dominant state variables with the proposed WTS for the wind speed of 14 m/sec. System Modes
Dominant State Variables
K1
302
K2;3 K4 K5
4 j10:4 0.12
K6
33
State variable corresponding to the phase compensation term Dh Dxg Dxt – 5.8 State variable corresponding to the high-pass filter dynamics DT e –
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
17
Fig. 9. Time response of the wind turbine for the wind speed of 8 m/sec with and without the wind turbine stabilizer (WTS), (a) generator speed, (b) generator output power, (c) electromagnetic torque, (d) shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 10. Time response of the wind turbine for the wind speed of 12 m/sec with and without the wind turbine stabilizer (WTS), (a) generator speed, (b) generator output power, (c) electromagnetic torque, (d) shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
19
Fig. 11. Time response of the wind turbine for the wind speed of 14 m/sec with and without the wind turbine stabilizer (WTS), (a) generator speed, (b) generator output power, (c) electromagnetic torque, (d) shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
ratio of the drive train oscillatory modes at operating regions 2 and 4. Thus the effect of the WTS is to increase the damping torque component suppressing the torsional oscillations in the drive train dynamics. To provide positive damping, the WTS must provide a component of electrical torque proportional to the rotor speed deviations. Fig. 7 shows the structure of the DFIG active power control considering auxiliary damping control known as wind turbine stabilizer. The main control signal in Fig. 7 is extracted based on the power-speed curve for the generator power control at normal operations. Also, the auxiliary control signal is used for damping improvement of torsional modes when torsional oscillations are excited and appeared on the generator speed. Fig. 8 shows the linearized drive train dynamics of the DFIG system by taking the effect of auxiliary damping control, i.e. wind turbine stabilizer, into account. Fig. 8 is an extension of the block diagram of Fig. 6 including the WTS effect represented by GWTS ðsÞ. The transfer function GWTS ðsÞ in Fig. 8 represents the WTS. Considering Fig. 8, the electrical torque can be given by
DT e ffi
df ðxg Þ aP aP Dxg þ GWTS ðsÞ Dx g s þ aP dxg xg0 s þ aP
ð23Þ
where the second term in (23) is the electrical torque due to WTS and is given by
DT eWTS ffi GWTS ðsÞ
aP Dxg s þ aP
ð24Þ
For producing the damping torque by the WTS, it is required that DT eWTS be proportional to Dxg . Therefore, we select GWTS ðsÞ in the following form given by (25).
GWTS ðsÞ ¼ K WTS
1 þ sT 1 sT H 1 þ sT 2 1 þ sT H
ð25Þ
According to (25), GWTS ðsÞ contains two terms: a gain-phase compensator, and a high-pass filter block. The gian K WTS determines the amount of damping provided by the WTS. The phase compensation term,
1þsT 1 , 1þsT 2
produces the appropriate phase-lead to
compensate for the phase-lag of
aP
sþaP
.
The frequency of interest for reaction of WTS is the torsional oscillation frequency which is approximately 2.05 Hz (12.9 rad/sec) for the test system. Hence, the phase compensation term should provide compensation at this frequency. The high-pass filter with time constant of T H allows only the torsional oscillations in the generator speed to pass, and blocks the steady state changes in the speed to pass. Hence, the WTS responds only when low frequency torsional oscillations appear on the generator speed. The first term of electrical torque in (23) is highly dependent on the power-speed curve and operating conditions. As stated before, for the operating conditions correspond df ðx Þ ing to the region 4 of the power-speed curve, dxgg is zero and generator control does not participate to improve the damping of torsional oscillations. Hence, for the worst case conditions, the first term of (23) is zero, and DT e is equal to DT eWTS .
Fig. 12. Wind turbine time responses against voltage dip, under two different values of the grid short circuit power, at wind speed of 13 m/sec when the WTS is inactive, (a) Generator speed, (b) Shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
For the DFIG test system, the closed loop bandwidth of active power has been designed as ap ¼ 2p 5 rad=sec. Hence, at the torsional frequency of s ¼ j12:9, we have
DT eWTS ffi GWTS ðsÞjs¼j12:9
31:4 Dx g j12:9 þ 31:4
¼ ðGWTS ðj12:9ÞÞ ð0:925\ 220 ÞDxg
ð26Þ
According to (26), if Dxg through the phase compensation term, 1þsT 1 , 1þsT 2
is advanced by 220 at the frequency of 12.9 rad/sec, the
DT eWTS will be in phase with Dxg and thus will improve the damping of torsional oscillations. The amount of damping provided by the GWTS ðsÞ depends on the gain value K WTS . Hence, with the exact compensation of the phase lag of sþaaP P , the damping torque due to
WTS at x ¼ 12:9 rad=sec is DT eWTS ffi K WTS 0:925 Dxg . According to the above explanations, we select the parameters of GWTS ðsÞ for the DFIG test system as follows: K WTS ¼ 10, T 1 ¼ 0:035, T 2 ¼ 0:0032, and T H ¼ 0:177. Table 5 shows the system modes, and dominant state variables with the proposed WTS for the wind speed of 14 m/sec. Considering Table 5, the mode K1 ¼ 302 is a well damped mode corresponding to phase compensation term of WTS. The modes K2;3 ¼ 4 j10:4 are oscillatory modes corresponding to the generator speed Dxg and shaft twist angle Dh. The corresponding damping ratio and natural frequency of these modes are n ¼ 0:35 and xn ¼ 11:14 rad/sec (2.04 Hz). The modes K4 ¼ 0:12 and
21
K6 ¼ 33 are the simple modes associated with the state variables Dxt and DT e , respectively. Also, the mode K5 ¼ 5:8 is the mode corresponding to the high-pass filter of WTS. Comparing Tables 4 and 5 shows that by using the WTS, the poorly damped torsional modes (0:79 j12:83 in Table 4) with n ¼ 0:061 are changed to damped oscillatory modes (4 j100:4 in Table 5) with n ¼ 0:35. In other words, the WTS has increased the damping ratio of the torsional modes by 0.289 and reduced the modes natural frequency by 2.4 rad/sec. Hence, as will be seen in Section 6, the WTS improves the wind turbine dynamic response regarding the torsional oscillations when the wind turbine is subjected to a disturbance. 6. Time domain simulations In this section the effect of proposed wind turbine stabilizer (WTS) on the torsional oscillations of the DFIG based wind turbine is examined by time domain simulations. The system under study, shown in Fig. 1, consists of 710 KW, 690 V, 50 Hz DFIG based WT connected to a 20 kV distribution system. It exports power to the grid through a 10 km-20 kV cable and transformer T1. Transformer T1 is rated 900 KVA with the equivalent series reactance of 5%. The DFIG based WT parameters used in this simulation are given in Appendix A. In the simulations, two cases are considered: the case without the wind turbine stabilizer (WTS), and the case with the proposed WTS.
Fig. 13. Wind turbine time responses against voltage dip, under two different values of the grid short circuit power, at wind speed of 13 m/sec when the WTS is active, (a) Generator speed, (b) Shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 14. DFIG wind turbine connected to the IEEE-39 bus test system [24].
Fig. 15. Wind turbine responses for the step change of the wind speed from 9 to 14 m/sec at t = 4 sec, where DFIG connected at bus 14 of the IEEE 39 bus test system, (a) Generator speed, (b) Shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
Fig. 9 depicts the time responses of the generator speed, output active power, electromagnetic torque, and shaft torsional torque for the wind speed of 8 m/sec. The DFIG average output active power and generator speed at V w ¼ 8 m=sec are Po ¼ 210 kW and xg ¼ 1470 rpm (0.982 pu), respectively, and the DFIG operates at the region 2 of the power-speed curve. Considering Fig. 9, in the case without the WTS, damping of torsional oscillations is lower than that in the case with the proposed WTS. This is because for V w ¼ 8 m=sec, and in the case without the WTS, generator active power control does not provide sufficient damping action on the generator speed. Fig. 10 shows the time responses of the wind turbine for the wind speed of 12 m/sec. The DFIG output active power and generator speed at V w ¼ 12 m=sec before the voltage dip are P o ¼ 627:3 kW (0.88 pu) and xg ¼ 1587:7 rpm (1.058 pu), respectively, and the DFIG operates at the region 3 of the power-speed curve. According to Fig. 10, the torsional oscillations at the both cases of with and without WTS, are well damped. This is because for V w ¼ 12 m=sec the DFIG operates at the region 3 of the power-speed curve, and thus sensitivity of the power-speed curve function to the generator speed is relatively high. Fig. 11 depicts the time responses of the generator speed, output active power, electromagnetic torque, and shaft torsional torque for the wind speed of 14 m/sec. The DFIG output active power and generator speed at V w ¼ 14 m=sec are P o ¼ 710 kW (1 pu) and xg ¼ 1626 rpm (1.084 pu), respectively, and the DFIG operates at the region 4 of the power-speed curve. According to Fig. 11, the frequency of the oscillations appeared after clearing the voltage dip is approximately 2 Hz corresponding to the
23
undamped natural frequency of the torsional modes. This is in agreement with the frequency of torsional modes computed at Section 4. Considering Fig. 11, in the case without the WTS, damping of torsional oscillations is lower than that in the case with the proposed WTS. This is because for V w ¼ 14 m=sec, sensitivity of the power-speed curve function to the generator speed is equal to zero, and thus in the case without the WTS, generator active power control does not provide any damping action on the generator speed. On the other hand, in the case with the proposed WTS, positive damping torque component is provided suppressing the torsional oscillations in the drive train dynamics. According to Fig. 11(d), in the case without the WTS, the shaft torsional torque contains large torsional oscillations with frequency of 2 Hz that may result in fatigue and stress on the wind turbine shaft. Figs. 12 and 13 examine the dynamic performance of the wind turbine under two different values of the grid short circuit power, i.e. Ssc = 4 MVA and 10 MVA. Fig. 12 corresponds to the case without the wind turbine stabilizer, and Fig. 13 corresponds to the case with the wind turbine stabilizer. Figs. 12 and 13 depict the time responses of the generator speed and shaft torsional torque at the wind speed of 13 m/sec when the wind turbine is subjected to a voltage dip. In Fig. 12, the WTS is inactive and after the voltage dip, weakly damped oscillations appear on the wind turbine response. However, according to Fig. 12, in the case with Ssc = 4 MVA, the effect of the grid short circuit power on the wind turbine torsional oscillations is more severe than that in the case with Ssc = 10 MVA. Considering Fig. 12(a) and (b), in the case with Ssc = 4 MVA, the
Fig. 16. Wind turbine responses against 95% voltage dip at t = 10 sec, where DFIG connected at bus 14 of the IEEE 39 bus test system, (a) Generator speed, (b) Shaft torsional torque.
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M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
response of the generator speed is unstable and relatively large oscillations appear on the shaft torsional torque. In Fig. 13, the WTS is active and after the voltage dip, well damped oscillations appear on the wind turbine response. Unlike Fig. 12, in Fig. 13 due to WTS, at both of the Ssc = 4 MVA and Ssc = 10 MVA, the response of the generator speed is stable and well damped oscillations appear on the shaft torsional torque.
6.1. Time domain simulations using IEEE 39-bus New England test system Fig. 14 depicts the IEEE 39-bus New England test system [24], in which the DFIG wind turbine system is connected at bus 14. Figs. 15–17 depict the times responses of the DFIG wind turbine connected at bus 14 of the IEEE 39-bus test system. Fig. 15 compare time responses of the generator speed and shaft torsional torque for the cases with and without the WTS, under the step change of the wind speed from 9 to 14 m/sec at t = 4 sec. Before the wind step change, V w ¼ 9 m/sec, xg ¼ 1:043 pu, and wind turbine operates in region 3 of the power speed curve. After the wind step change, the wind turbine operates in region 4 of the power speed curve. According to Fig. 15(a) and (b), in the case without the wind turbine stabilizer and after the wind step change, torsional oscillations appear on the responses of the generator speed and shaft torsional. This is because the rotor side converter control cannot provide adequate action for damping of torsional oscillations at region 4. On the other hand, in the case with the wind turbine stabilizer, the added auxiliary control (WTS) can
provide positive damping action, and thus the torsional oscillations are well damped. Fig. 16 depicts the time responses of the generator speed and shaft torsional torque under 95% voltage dip at the stator terminal, at the wind speed of 13 m/sec, and for the cases with and without the WTS. Also, Fig. 17 depicts the times responses of the DFIG wind turbine connected at bus 11 of the IEEE 39-bus test system. It shows the time responses of the generator speed and shaft torsional torque under 80% voltage dip, at the wind speed of 13 m/sec, and for the cases with and without the WTS. According to Figs. 16 and 17, in the cased without the WTS, weakly damped oscillations appear on the wind turbine responses after the voltage dip. However, in the cased with the WTS, the torsional oscillations after the voltage dip are well damped.
Fig. 18. Power-speed curve for the DFIG system under study.
Fig. 17. Wind turbine responses against 80% voltage dip at t = 10 sec, where DFIG connected at bus 11 of the IEEE 39 bus test system, (a) Generator speed, (b) Shaft torsional torque.
M. Rahimi / Electrical Power and Energy Systems 95 (2018) 11–25
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7. Conclusion
References
This paper first analytically examines the dynamics of the drive train for different operating regions of the power-speed curve when the wind turbine is controlled in power control mode. It is shown that at operating regions 1 and 3 of the power-speed curve, torsional oscillations are well damped and the DFIG control improves the damping of torsional modes. On the other hand, at operating regions 2 and 4 of the power-speed curve, the generator control does not provide damping action on the poorly damped torsional oscillatory modes. Then an auxiliary active damping control known as wind turbine stabilizer (WTS), is proposed improving the damping ratio of the drive train oscillatory modes at operating regions 2 and 4. The effect of the WTS is to produce a damping torque component suppressing the torsional oscillations in the drive train dynamics. As the future research, it may be possible to propose a comprehensive auxiliary control system for damping of torsional oscillations, subsynchronous oscillations, and power system electromechanical oscillations.
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Appendix A Parameters of the 710 KW, 690 V, 50 Hz, DFIG-WT:
V base ¼ 690 V Sbase ¼ 710 KVA f base ¼ 50 Hz
xb ¼ 2pf b ¼ 314 rad=sec
Hg ¼ 0:55 sec Ht ¼ 3:5 sec Dtg ¼ 1:2 pu ks ¼ 0:5 pu=elec rad xs ¼ 1 pu Appendix B Power-speed curve for the DFIG system under study is shown in Fig. 16. In Fig. 16, the generator speed and stator power are in pu, where the base values of the generator speed and power are 1500 rpm and 710 KVA, respectively. Appendix C Matrix A and vectors B, C and D related to linearized drive train dynamics:
h B¼ 0 0
1 2Ht
3P t0
1
xt0 V w0
wind0 @C P ðk;bÞ Px @k t0
0
k0 V w0
0
iT
C ¼ ½ 1 0 0 0 D ¼ ½0 2
Dtg 2H
ks 2H
Dtg 2H
1 2H
3
g g g g 6 7 6 xb 0 xb 0 7 1 P t0 @C P ðk;bÞ dk 6 7 A¼6 2 þ P wind0 Dtg 7 and N ¼ ks N 0 6 7 x 2H @k d xt0 t t 2Ht 0 4 2Ht 5 df ðxg Þ :aP 0 0 aP dxg