Modeling and dynamic response of double-feed induction generator and back-to-back converters in unbalanced grid voltage conditions

Abstract

Double-feed induction generators are one of the most commonly used generators in wind power plants. Therefore, studying this type of generators in different grid conditions such as unbalanced grid voltage is of great importance. In this article, double-feed induction generators, in synchronous reference frame, and back-to-back converters are reviewed and modeled. The advantage of this model, compared with previous ones, is that it can be used in unbalanced grid voltage conditions. Hence, different parts of the generator in synchronous positive reference frame are studied and governing equations in such conditions are analyzed. Stator output power, rotor side converters, grid side converters, and electromagnetic torque, mentioned in the model, are analyzed, as well. The model is also applied in balanced conditions. Therefore, the model proposed in this article is perfect for analyzing wind turbine-based power plants with double-feed induction generators. The accuracy of the suggested function was confirmed through simulation.

Keywords

Double-feed induction generator unbalance grid voltage grid side converter rotor side converter

Introduction

In recent years, double-feed induction generator (DFIG)-based wind power plants have been increasingly used. These types of generators can transmit about 30% of their nominal power from rotor to grid through connected back-to-back converters (see Figure 1). Accordingly, power losses and the final price of converters have reduced significantly, compared to generators with converter in their stator circuit. Many wind turbines are installed in remote areas in which there are multiple sources of voltage unbalancing, including heavy nonsymmetric loads (single-phase loads), nonsymmetric impedances of transmission lines, and voltage dips. If small voltage oscillations resulting from the mentioned factors in a grid with DFIGs are not avoided, serious consequences may follow in the electrical and mechanical parts of wind power plants, such as severe oscillation in active and reactive powers, torque oscillation in generator’s shaft, high current in rotor, increased direct current (DC) link voltage, harmonic stator current, and turbine speedup (Zhang et al., 2012). These consequences can also affect the generator’s operation and result in increased temperature of windings, increased losses, and considerable life loss of expensive power plant equipment (Zandzadeh and Vahedi, 2014). Therefore, connection of this type of wind power plants with DFIG to grid, with no proper control, to eliminate destructive effects of unbalanced voltage, can result in their disconnection from the grid under such conditions (Petersson, 2005). While, according to the criteria, around 2% voltage unbalancing in the grid is permissible and grid systems are known to have a reasonable function in such conditions (Michalke, 2008). To eliminate the destructive effects mentioned above, an appropriate model is required to analyze DFIGs as well as their converters in unbalanced voltage conditions of grid (Choudhury, 2011; Hansen and Hansen, 2007). Many studies have focused on voltage unbalancing and function of control systems based on different DFIG models in such conditions (Tazil et al., 2010; Zhang et al., 2012). Decomposing the positive and negative sequences of rotor current components as well as current control loop of negative sequence and studying rotor side converter (RSC) model, Ganti et al. (2012) and Mishra et al. (2012) have focused on reduced torque oscillation in such conditions.

In Hu et al. (2009, 2013), resonance controllers in grid side converters (GSCs) and their model were applied to control the positive and negative sequences of current, without decomposing it into positive and negative components, which reduced computations. In Liu et al. (2011) and Lopez et al. (2008), slip mode control strategy in direct power control method without decomposing it into positive and negative components and zero-order DFIG model were applied. In Kiani and Lee (2010), Fan et al. (2010), Hu et al. (2011), and Liserre et al. (2006), a dynamic review of GSC was carried out to eliminate oscillating stator output power. In this research, direct power control-based DFIG with no rotor component measurements to estimate flux are investigated and modeled. Accordingly, first, DFIG and then, back-to-back converters in unbalanced grid voltage conditions are investigated, modeled, and analyzed and power oscillation components in synchronous reference frame in unbalanced voltage conditions are included in equations and analysis of each section. Finally, the model’s function on a DFIG is studied through simulation.

Identifying and calculating power in unbalanced grid voltage conditions

In unbalanced grid voltage co decomposed into three symmetric positive, negative, and zero components. Given that machine terminals usually have a $Y /$ ∆ connection with no ground transformer, zero component can be ignored. Accordingly, calculating the positive and negative sequences to find the oscillation components is adequate. If f is the function (f can be voltage, current, flux, or any desired quantity), then in double-phase αβ we have

\begin{array}{l} f_{α β} (t) = f_{α β +} (t) + f_{α β -} (t) = \\ | f_{α β +} (t) | e^{j (ω_{s} + ϕ_{+})} + | f_{α β -} (t) | e^{- j (ω_{s} + ϕ_{-})} \end{array}

(1)

where + and − are the positive and negative sequences, $ω_{s}$ is the speed of synchronous reference frame, and $ϕ$ is the initial phase. Figure 2 displays 2-phase frame $α β$ in $d q^{+}$ and $d q^{-}$ frames.

According to Figure 2, the following equations are obtained

f_{d q}^{+} = f_{d q}^{-} . e^{- j 2 ω_{s} t}

(2)

f_{d q}^{-} = f_{d q}^{+} . e^{j 2 ω_{s} t}

(3)

According to equations (2) and (3) and Figure 2, the following equation is obtained

f_{d q}^{+} = f_{d q +}^{+} + f_{d q -}^{+} = f_{d q +}^{+} + f_{d q -}^{-} . e^{- j 2 ω_{s} t}

(4)

As is seen, the oscillation term is in negative frame and negative sequence and the oscillation frequency is twice as much as the synchronous frequency. Therefore, expounding equation (4), we have

[\begin{matrix} f_{d}^{+} \\ f_{q}^{+} \end{matrix}] = [\begin{matrix} f_{d +}^{+} \\ f_{q +}^{+} \end{matrix}] + [\begin{matrix} f_{d \sim}^{+} \\ f_{q \sim}^{+} \end{matrix}]

(5)

where

[\begin{matrix} f_{d \sim}^{+} \\ f_{q \sim}^{+} \end{matrix}] = [\begin{matrix} f_{d -}^{-} & f_{q -}^{-} \\ f_{q -}^{-} & - f_{d -}^{-} \end{matrix}] [\begin{matrix} \cos 2 ω_{s} t \\ \sin 2 ω_{s} t \end{matrix}]

(6)

Also, to calculate power in a 2-phase environment, we have

[\begin{matrix} P \\ Q \end{matrix}] = \frac{3}{2} [\begin{matrix} V_{d}^{+} & V_{q}^{+} \\ V_{q}^{+} & - V_{d}^{+} \end{matrix}] [\begin{matrix} I_{d}^{+} \\ I_{q}^{+} \end{matrix}]

(7)

Now, according to equations (5) and (6) and the calculated power from equation (7), we have

[\begin{matrix} P \\ Q \end{matrix}] = [\begin{matrix} V_{d +}^{+} + V_{d \sim}^{+} & V_{q +}^{+} + V_{q \sim}^{+} \\ V_{q +}^{+} + V_{q \sim}^{+} & - V_{d +}^{+} - V_{d \sim}^{+} \end{matrix}] [\begin{matrix} I_{d +}^{+} + I_{d \sim}^{+} \\ I_{q +}^{+} + I_{q \sim}^{+} \end{matrix}]

(8)

To simplify equation (8), it is proved that

\begin{array}{l} V_{d \sim}^{+} I_{d \sim}^{+} + V_{q \sim}^{+} I_{q \sim}^{+} = \\ (V_{d -}^{-} c o s 2 ω_{s} t + V_{q -}^{-} s i n 2 ω_{s} t) (I_{d -}^{-} c o s 2 ω_{s} t + I_{q -}^{-} s i n 2 ω_{s} t) + \\ (V_{q -}^{-} c o s 2 ω_{s} t - V_{d -}^{-} s i n 2 ω_{s} t) (I_{q -}^{-} c o s 2 ω_{s} t - I_{d -}^{-} s i n 2 ω_{s} t) = \\ V_{d -}^{-} I_{d -}^{-} + V_{q -}^{-} I_{q -}^{-} \end{array}

(9)

As a similar proof

V_{q \sim}^{+} I_{d \sim}^{+} - V_{d \sim}^{+} I_{q \sim}^{+} = V_{q -}^{-} I_{d -}^{-} - V_{d -}^{-} I_{q -}^{-}

(10)

Therefore, the equations for active and reactive powers are obtained as follows

[\begin{matrix} P \\ Q \end{matrix}] = [\begin{matrix} P_{D C} \\ Q_{D C} \end{matrix}] + [\begin{matrix} P_{1 \sim} + P_{2 \sim} \\ Q_{1 \sim} + Q_{2 \sim} \end{matrix}]

(11)

where $Q_{D C}$ and $P_{D C}$ are constant terms of active and reactive powers, $P_{1 \sim}$ and $P_{2 \sim}$ are oscillating components of active power in unbalanced grid voltage conditions and $Q_{1 \sim}$ and $Q_{2 \sim}$ are oscillated components of reactive power in such conditions. Equation (11) is explained as

\begin{array}{l} P_{D C} = \frac{3}{2} (V_{d +}^{+} I_{d +}^{+} + V_{q +}^{+} I_{q +}^{+} + V_{d -}^{-} I_{d -}^{-} + V_{q -}^{-} I_{q -}^{-}) \\ P_{1 \sim} = \frac{3}{2} [\begin{matrix} V_{d \sim}^{+} & V_{q \sim}^{+} \end{matrix}] [\begin{matrix} I_{d +}^{+} \\ I_{q +}^{+} \end{matrix}] \\ P_{2 \sim} = \frac{3}{2} [\begin{matrix} V_{d +}^{+} & V_{q +}^{+} \end{matrix}] [\begin{matrix} I_{d \sim}^{+} \\ I_{q \sim}^{+} \end{matrix}] \\ Q_{D C} = \frac{3}{2} (V_{q +}^{+} I_{d +}^{+} - V_{d +}^{+} I_{q +}^{+} + V_{q -}^{-} I_{d -}^{-} + V_{d -}^{-} I_{q -}^{-}) \end{array}

(12)

\begin{array}{l} Q_{1 \sim} = \frac{3}{2} [\begin{matrix} V_{q \sim}^{+} & V_{d \sim}^{+} \end{matrix}] [\begin{matrix} I_{d +}^{+} \\ I_{q +}^{+} \end{matrix}] \\ Q_{2 \sim} = \frac{3}{2} [\begin{matrix} V_{q +}^{+} & V_{d +}^{+} \end{matrix}] [\begin{matrix} I_{d \sim}^{+} \\ I_{q \sim}^{+} \end{matrix}] \end{array}

(13)

where $V_{d q +}^{+}$ , $I_{d q +}^{+}$ , $I_{d q -}^{-}$ , $V_{d q -}^{-}$ are constant values and $I_{d q \sim}^{+}$ and $V_{d q \sim}^{+}$ are oscillating values which oscillate in voltage unbalancing conditions with frequency of $2 ω_{s}$ . The above equations display a power model in nonsymmetric grid voltage conditions and this unbalancing results in oscillation components with twice as much as the synchronous frequency in active, reactive, and current in grid.

Modeling and studying DFIG behavior in unbalanced grid voltage conditions

To model and study the DFIG behavior in unbalanced grid voltage conditions, equations and models are investigated first for double-feed induction machine (DFIM) and then for GSCs and RSCs.

Modeling DFIM in unbalanced grid voltage conditions

Figure 3 displays the DFIM model in synchronous reference frame (Okedu et al., 2012) according to which, basic equations in positive frame can be written as follows

V_{s}^{+} = R_{s} I_{s}^{+} + \frac{d Ψ_{s}}{d t} + j ω_{s} Ψ_{s}^{+}

(14)

V_{r}^{+} = R_{r} I_{r}^{+} + \frac{d Ψ_{r}}{d t} + j (ω_{s} - ω_{r}) Ψ_{s}^{+}

(15)

[\begin{matrix} Ψ_{s}^{+} \\ Ψ_{r}^{+} \end{matrix}] = [\begin{matrix} L_{s} & L_{m} \\ L_{m} & L_{r} \end{matrix}] [\begin{matrix} I_{s}^{+} \\ I_{r}^{+} \end{matrix}]

(16)

With unbalanced grid voltage and following unbalanced stator voltage (according to Figure 1, the DFIG stator is connected directly to the grid), oscillation components in power equations, mentioned in equation (11), are included in machine equations, as follows

[\begin{matrix} P_{s} \\ Q_{s} \end{matrix}] = [\begin{matrix} P_{s D C} \\ Q_{s D C} \end{matrix}] + [\begin{matrix} P_{s 1 \sim} + P_{s 2 \sim} \\ Q_{s 1 \sim} + Q_{s 2 \sim} \end{matrix}]

(17)

$P_{s}$ and $Q_{s}$ , respectively, are active and reactive powers from stator, each of which contains constant and oscillating terms in unbalanced grid voltage condition.

In addition, electromagnetic torque in synchronous reference positive frame (Suh and Lipo, 2006) is

T_{e m} = \frac{3}{2} P (Ψ_{s d}^{+} I_{s q}^{+} - Ψ_{s q}^{+} I_{s d}^{+})

(18)

Given the unbalanced grid voltage, the positive and negative sequence range is proved to be constant (Komatsu and Kawabata, 1995), therefore

\frac{d}{d t} Ψ_{s d q +}^{+} = \frac{d}{d t} Ψ_{s d q -}^{+} = 0

(19)

Ignoring stator resistance and simplifying equation (14), the following equation is obtained

\begin{array}{l} V_{s d q}^{+} = V_{s d q +}^{+} + V_{s d q \sim}^{+} = j ω_{s} Ψ_{s d q}^{+} + \frac{d Ψ_{s d q}^{+}}{d t} = \\ j ω_{s} (Ψ_{s d q +}^{+} + Ψ_{s d q -}^{-} e^{- j 2 ω_{s} t}) + \frac{d}{d t} (Ψ_{s d q +}^{+} + Ψ_{s d q -}^{-} e^{- j 2 ω_{s} t}) = \\ j ω_{s} (Ψ_{s d q +}^{+} - Ψ_{s d q -}^{+}) \end{array}

(20)

Following is the expounded equation (20)

\begin{array}{l} Ψ_{s d +}^{+} = \frac{1}{ω_{s}} V_{s q +}^{+}, Ψ_{s d \sim}^{+} = - \frac{1}{ω_{s}} V_{s q \sim}^{+} \\ Ψ_{s q +}^{+} = - \frac{1}{ω_{s}} V_{s d +}^{+}, Ψ_{s q \sim}^{+} = \frac{1}{ω_{s}} V_{s d \sim}^{+} \end{array}

(21)

Accordingly, electromagnetic torque in such conditions, based on equations (17) and (18), is as follows

T_{e m} = \frac{p}{ω_{s}} (P_{T D C} - P_{s 1 \sim} + P_{s 2 \sim}) = \frac{p_{e}}{ω_{s}}

(22)

P_{T D C} = \frac{3}{2} (V_{s q +}^{+} I_{s q +}^{+} + V_{s d +}^{+} I_{s d +}^{+} - V_{s q -}^{-} I_{s q -}^{-} - V_{s d -}^{-} I_{s d -}^{-})

(23)

Therefore, in unbalanced grid voltage condition, the electromagnetic torque will also contain constant and oscillating terms.

Investigating model and behavior of GSC and RSC

Equations for stator power are similar for converters, as is displayed in Figure 4

P_{g} = P_{g D C} + P_{g 1 \sim} + P_{g 2 \sim} + P_{g D C} + P_{g \sim}

(24)

According to power direction in equations (17) and (24) and Figure 1, rotor power can be calculated using the following equation

\begin{array}{l} P_{r} = P_{e} - P_{s} \\ = ((1 - S) P_{T D C} - P_{S D C}) - (2 - S) P_{s 1 \sim} - S P_{s 2} \\ = P_{r D C} + P_{r \sim} \end{array}

(25)

where S is machine slip calculated by the following equation

S = \frac{ω_{s} - ω_{m}}{ω_{s}}

(26)

where $ω_{m}$ is rotor shaft speed. To calculate power for GSC, according to Figure 4, the following equation is obtained

\begin{array}{l} C \frac{d V_{d c}}{d t} . V_{d c} = P_{D C} = P_{g} - P_{r} \\ = (P_{g D C} - P_{r D C}) + (P_{g \sim} + P_{r \sim}) \end{array}

(27)

Therefore, in grid voltage unbalancing conditions, DC link oscillations are symbolized by $P_{r \sim}$ and $P_{g \sim}$ .

Simulation results

The model proposed here was developed in MATLAB/SIMULINK and its results were studied. Simulations on a 2 MW generator with its parameters are available in Table 1. It is worth noting that, initially, the input of DFIG is speed which is proved to be 1.2 per unit.

Table 1.

Generator parameters used in simulation.

Section	Parameters	Value
DFIG section	Rated power	2 MW
	Stator voltage	690 V
	Stator/rotor turn ratio	0.3
	$R_{s}$	0.0121 pu
	$R_{r}$	0.0108 pu
	$L_{m}$	3.362 pu
	$L_{l s}$	0.102 pu
	$L_{l r}$	0.11 pu
	H	0.5 s
Converter section	Pole pair. No	2
	DC link voltage	1200 V
	DC link capacitor	16 mF
	GSC inductance	0.4 mH

DFIG: double-feed induction generator; DC: direct current; GSC: grid side converter.

Figure 5 displays the grid voltage which, as seen, is unbalanced at t = 0.5, due to large single-phase load added. The unbalancing, occurred in phase b, resulted in a 20% reduction of the phase b voltage range.

Due to the conditions created in grid voltage, Figures 6 and 7 display the rotor and stator currents during and before unbalanced conditions. In the balanced condition, according to Figure 7, before t = 0.5 s, the generator was in its nominal load and the current from stator was equal to its nominal value. This is a proof of the validity of the model proposed for DFIG in unbalanced conditions.

Also, rotor current is transmitted to rotor based on the ratio of generator conversion. The process is displayed in Figure 6. Figure 8 displays active and reactive powers from stator in balanced and unbalanced grid voltage conditions. First, the powers are free from any oscillations; however, after grid voltage is unbalanced, they oscillate at twice the synchronous frequency. Figure 8 shows the validity of the model proposed for GSC to investigate power from the stator side.

Figure 9 shows harmonic monitoring of the direct control method based on the proposed model. Low harmonic pollution in unbalanced conditions is a reason for the efficiency of the proposed model in control method during unbalanced conditions.

Figure 10 displays the DC link voltage generated by GSC before and after grid voltage unbalancing using the proposed model and direct power control method.

Electromagnetic torque, caused by RSC operation before and after unbalancing grid voltage is displayed in Figure 11, which is obtained based on the proposed model using direct power control method. However, as given in Figure 12, the DFIG input with 30% change in wind speed and unbalanced conditions occurred at t = 0.5 s, while Figure 13 displays stator flux in the αβ reference frame. Figure 14 shows the power from stator and stability of control system based on the proposed dynamic model.

Figures 15 and 16 show stator and rotor currents, respectively, based on changes in wind speed, like Figure 12 and the unbalancing occurred at t = 0.5 s. Figure 17 displays DC link voltage with changes in incoming wind speed and constant power received, based on proposed model for GSC, which proves the accuracy of the proposed model for wind power plants with DFIG and its appropriate dynamic response.

Conclusion

In this article, dynamic modeling of DFIG and its converters based on direct power control method in unbalanced grid voltage conditions was considered. As was shown, with grid voltage unbalancing, active and reactive powers from stator, GSC, and DC link voltages are oscillated that can lead to generator’s trip or disconnection from grid by relays in wind turbine-based power plants. Considering frequent unbalancing in grid caused by large loads or single-phase loads, a suitable mechanism is required for optimal utilization of these power plants. In this article, first, equations for oscillations were presented considering grid voltage unbalancing and, next, DFIG and back-to-back converters were studied. The results show that, during unbalanced conditions, oscillation components appeared in power and current from stator and converters and can lead to inefficiency of generators when infusing power into the grid. Therefore, an appropriate model is required for these power plants in order to investigate their operation in unbalanced grid voltage conditions. Besides, due to ever-changing wind speed, dynamic function of the proposed model is very important. In this article, a dynamic model was proposed for DFIG and its converters in synchronous reference frame and different components formed in such conditions were considered in different sections. The validity of the model was analyzed through simulation, as well.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Choudhury

(2011) Performance analysis of doubly-fed induction generator in wind energy conversion system. Master’s Thesis, Electrical Engineering (Power Control & Drives), National Institute of Technology Rourkela, Rourkela, June.

Fan

Yuvarajan

Kavasseri

(2010) Harmonic analysis of a DFIG for a wind energy conversion system. IEEE Transactions on Energy Conversion 25(1): 181–190.

Ganti

Singh

Aggarwal

, et al. (2012) DFIG-based wind power conversion with grid power leveling for reduced gusts. IEEE Transactions on Sustainable Energy 3(1): 12–20.

Hansen

(2007) Market penetration of different wind turbine concepts over the years. In: Proceedings of the 2007 European wind energy conference and exhibition, Milan, 7–10 May, vol. 10, pp. 81–97. Brussels: European Wind Energy Association (EWEA).

, et al. (2009) Improved control of DFIG systems during network unbalance using PI–R current regulators. IEEE Transactions on Industrial Electronics 56(2): 439–451.

Nian

, et al. (2011) Dynamic modeling and improved control of DFIG under distorted grid voltage conditions. IEEE Transactions on Energy Conversion 26(1): 163–175.

(2013) Coordinated control of DFIG’s RSC and GSC under generalized unbalanced and distorted grid voltage conditions. IEEE Transactions on Industrial Electronics 60(7): 2808–2819.

Kiani

Lee

(2010) Effects of voltage unbalance and system harmonics on the performance of doubly fed induction wind generators. IEEE Transactions on Industry Applications 46(2): 562–568.

Komatsu

Kawabata

(1995) A control method of active power filter where system voltage contains negative-phase-sequence component or zero-phase-sequence component. In: Proceedings of the 1995 international conference on power electronics and drive systems (PEDS’95), Singapore, 21–24 February, pp. 583–586. New York: IEEE.

10.

Liserre

Teodorescu

Blaabjerg

(2006) Multiple harmonics control for three-phase grid converter systems with the use of PI-RES current controller in a rotating frame. IEEE Transactions on Power Electronics 21(3): 836–841.

11.

Liu

Blaabjerg

Chen

, et al. (2011) Stator current harmonic control with resonant controller for doubly fed induction generator. IEEE Transactions on Power Electronics 27(7): 3207–3220.

12.

Lopez

Gubia

Sanchis

, et al. (2008) Wind turbines based on doubly fed induction generator under asymmetrical voltage dips. IEEE Transactions on Energy Conversion 23(1): 321–330.

13.

Michalke

(2008) Variable speed wind turbines—modelling, control, and impact on power systems. PhD Thesis, Department of Renewable Energies, Darmstadt University of Technology, Darmstadt.

14.

Mishra

(2012) Coordinated tuning of DFIG-based wind turbines and batteries using bacteria foraging technique for maintaining constant grid power output. IEEE Systems Journal 6(1): 16–26.

15.

Okedu

Muyeen

Takahashi

, et al. (2012) Wind farms fault ride through using DFIG with new protection scheme. IEEE Transactions on Sustainable Energy 3(2): 242–254.

16.

Petersson

(2005) Analysis, modeling and control of doubly-fed induction generators for wind turbines. PhD Thesis, Division of Electric Power Engineering, Department of Energy and Environment, Chalmers University of Technology, Göteborg.

17.

Suh

Lipo

(2006) Modeling and analysis of instantaneous active and reactive power for PWM AC/DC converter under generalized unbalanced network. IEEE Transactions on Power Delivery 21(3): 1530–1540.

18.

Tazil

Kumar

Bansal

, et al. (2010) Three-phase doubly fed induction generators: An overview. IET Electric Power Applications 4(2): 75–89.

19.

Zandzadeh

Vahedi

(2014) Modeling and improvement of direct power control of DFIG under unbalanced grid voltage condition. International Journal of Electrical Power & Energy Systems 59: 58–68.

20.

Zhang

Sun

Lin

, et al. (2012) Coordinated frequency regulation by doubly fed induction generator-based wind power plants. IET Renewable Power Generation 6(1): 38–47.