Modeling of Grid-Connected DFIG-Based Wind Turbines ... - IEEE Xplore

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 6, NO. 4, OCTOBER 2015

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Modeling of Grid-Connected DFIG-Based Wind Turbines for DC-Link Voltage Stability Analysis Jiabing Hu, Senior Member, IEEE, Yunhui Huang, Student Member, IEEE, Dong Wang, Hao Yuan, and Xiaoming Yuan, Senior Member, IEEE

Abstract—The electromagnetic stability issues of the gridconnected doubly fed induction generator (DFIG) system are usually overlooked. This paper presents a reduced order smallsignal model that can be used to analyze the stability of DFIG’s dc-link voltage control system, especially under weak ac grid conditions. This model neglects DFIG flux and fast current control dynamics. However, the effects of operating points, grid strengths and control loops’ interactions on system dynamic performance are taken into account. An eigenvalue comparison shows the proposed model holds dominant oscillation mode featured by the detailed model and is suitable for stability analysis of dc-link voltage control system of DFIG. Influence coefficients reflecting control loops’ interactions are also presented. Application studies of the proposed model show it is suitable for illustrating the effect of grid strength on dynamic performance of the DFIG’s dc-link voltage control system. Meanwhile, phase-locked loop (PLL) and rotor-side converter (RSC) active power control (APC)/reactive power control’s (RPC) effect on system stability are also explored. Index Terms—DC-link voltage control, doubly fed induction generator (DFIG), modeling, phase-locked loop (PLL), smallsignal stability, weak ac grids.

N OMENCLATURE Lls , Llr Ls , Lr Lm Rs , Rr Ps , Q s Pr , Q r Pc , Q c Pg , Q g ω1 ωr

Stator and rotor leakage inductances. Stator and rotor inductances. Magnetizing inductance. Stator and rotor resistances. Stator active and reactive powers. Rotor active and reactive powers. Active and reactive powers delivered to grid by grid-side converter (GSC). Total active and reactive powers sent to grid. Synchronous angular frequency. Rotor angular frequency.

Manuscript received October 01, 2014; revised January 12, 2015 and March 07, 2015; accepted April 03, 2015. Date of publication June 11, 2015; date of current version September 16, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 51277196, in part by the National Natural Science of China for Excellent Young Scholars under Grant 51322704, in part by the Program for MOE New Century Excellent Talents in University under Grant NCET-12-0221, and in part by the National Basic Research Program of China (973 Program) under Grant 2012CB215100. Paper no. TSTE-00535-2014. The authors are with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, and School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; [email protected]; dongwanghust@ gmail.com; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSTE.2015.2432062

θr θp , θpll θ, θp ωp C Lc Lg , R g U s , θs U g , θg PIj = kpj + kij /s

Superscript p Subscripts dq s, r c, g

Rotor angle. PLL output angle in abc frame, PLL output angle in synchronous dq frame. Vector’s phase in synchronous dq frame, in PLL reference frame. Angular frequency of PLL reference frame. DC-link capacitance. Filter inductance of GSC. Transmission line inductance and reactance. Terminal voltage magnitude and angle. Grid voltage magnitude and angle. Transfer function of a generic PI controller (j = 1, 2, . . . , 8).

Signals measured in PLL reference frame. Direct- and quadrature-axis components in rotating dq reference frame. Stator and rotor components. GSC and grid components. I. I NTRODUCTION

D

OUBLY fed induction generator (DFIG)-based wind turbine (WT) has become most installed at present due to its high efficiency, capability of providing reactive power support, relatively small rating of power converters, etc. [1]. In a commercial DFIG WT, there exist different types of energy storage elements, such as rotor mass of generator and dc-link capacitor of voltage source converter (VSC) for DFIG excitation. Corresponding hierarchical control is adopted to keep the states of these storage elements in a certain level, namely proper rotor speed and fixed dc-link voltage, respectively. However, there exist various dynamics and even stability problems in these multiple storage elements or masses due to their complicated control structure in multitime scale. Previous studies are usually focused on the effect of DFIG behaviors on power system stability in electromechanical time scale since the drive chain and pitch control bandwidths of DFIG WT are close to the rotor swing mode of synchronous generator [2]–[4]. DC-link voltage control system is seldom taken into account under this consideration due to

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its faster response. However, when dc-link voltage dynamics is concentrated specifically in a DFIG WT, stability issues in the corresponding time scale may result. In addition, in highly penetrated wind farms, dc-link voltage dynamic interactions among multiple WTs may exist. To the best knowledge of the authors, these phenomena have not been studied extensively. Furthermore, due to the distributions of wind resources, large-scaled wind farms are located in remote areas far away from the load centers. As a result, the strength, typically described by the value of short-circuit ratio (SCR), of an ac grid that WTs are attached to becomes weak. Loss stability of VSC in a full-capacity WT and HVDC has been studied, respectively, under weak ac grid conditions [5]–[9]. As indicated in [5] and [6], challenging wind power projects can be confronted by SCR < 2 and even operating near physical stability limit are of practical relevance. Loss of stability of VSC proved to be a practical phenomenon and needs to be expected in case of wind power projects with an SCR below a critical level. Taking converter control effect into account, [7] and [8] study the effect of phase-locked loop (PLL) and reactive power control (RPC) on the stability of dc-link voltage, respectively, in grid-connected VSC system. In [9], the impact of SCR and PLL parameters on the behavior of a VSC-HVDC is investigated, which suggests the SCR boundary between “weak” and “strong” ac grids for a VSC-HVDC system be in the range 1.3–1.6. As a partialcapacity VSC interfaced WT, DFIG’s dc-link voltage control system stability, which is determined by both GSC and rotorside converter (RSC) controls, has not been reported and will be investigated in this paper. In order to study the dynamic stability characteristic of dclink voltage control system in a grid-connected DFIG system, it is necessary to present a model which can reflect system behavior and internal mechanism. Different models exist for grid-connected DFIG WTs [10]–[13]. In [10], a detailed modeling of DFIG WT is given and electromechanical modes are studied by eigenvalue analysis, while DFIG’s control dynamics are not involved. In [11]–[13], modeling work performed by the WECC Wind Generation Modeling Group and the IEEE Working Group is summarized. These models neglect flux dynamics and dc-link voltage control for the purpose of power system stability studies on the interactions between DFIG and SG in electromechanical time scale, whereas when it comes to the electromagnetic dynamics, in particular related to dc-link voltage control time scale, defined as a time scale mainly concerning converter outer control loop dynamics, the assumptions for simplifying should be reconsidered and verified in this paper. To establish a proper model for dc-link voltage stability analysis, some crucial factors should be considered. First, different operating points and grid strengths influence DFIG dc-link voltage dynamic performance [5], [14]. Second, interactions among control loops in DFIG control system can affect dclink voltage stability in weak grid. Terminal voltage is sensitive to DFIG output power variations in weak grid. The dynamics of DFIG’s converter control affect the output power dynamics. In turn, terminal voltage dynamics affect DFIG’s converter control through influencing the dynamics of PLL, which provides phase reference for converter control. Therefore, DFIG’s


control loops are coupled in weak ac system, making dc-link voltage control system dynamics more complicated. This paper proposes a model for stability analysis of dc-link voltage control system for grid-connected DFIG system. This paper is organized as follows. Section II gives a brief introduction of the mathematical model and control system of DFIG. In Section III, a reduced order small-signal model is proposed. Influence coefficients which reflect control loops interactions are presented. The feasibility of the proposed model is verified by comparing the eigenvalues with those of detailed model in different situations. In Section IV, effect of grid strength, PLL, and RSC active power control (APC)/RPC on system stability is analyzed with the proposed model. Conclusion is drawn in Section V. II. DFIG S YSTEM B EHAVIOR BY T YPICAL V ECTOR C ONTROL For a DFIG-based wind power generation system, the main task of RSC is to control DFIG’s stator output active and reactive powers while the GSC controls the common dc-link voltage. System behaviors via typical vector control based on either stator flux orientation (SFO) [15], [16] or stator voltage orientation (SVO) [1], [17] have been thoroughly investigated in the last decades. Thus, only a brief description is given here by an example of SVO-based vector control due to its superior stability to the SFO [18]. Dynamics of PLL is highlighted to present the system’s synchronizing characteristics. A. DFIG System Model According to the DFIG system configuration shown in Fig. 1(a), the complex vector equivalent circuits of the generator and its GSC and RSC in the synchronous dq reference frame in which the d-axis is orientated to the stator voltage vector and rotates at an angular speed of ω1 are shown in Fig. 2(a) and (b), respectively. From Fig. 2, the voltage, flux, and power of the entire DFIG system including its GSC and RSC in the dq reference frame can be summarized as s Us = −Rs Is + dΨ dt + jω1 Ψ s (1) dΨ r Ur = −Rr Ir + dt + j(ω1 − ωr )Ψ r Ψ s = −Ls Is − Lm Ir (2) Ψ r = −Lm Is − Lr Ir Pr = 1.5 (urd ird + urq irq ) Pc = 1.5(usd icd + usq icq ) dUdc . Pr − Pc = CUdc dt

(3) (4) (5)

The equations for the interface between DFIG system and grid can be written as Ig = Is + I c Us = Ug + jω1 Lg Ig + Ig Rg Pg = 1.5 (usd igd + usq igq ) .

(6) (7) (8)

HU et al.: MODELING OF GRID-CONNECTED DFIG-BASED WTs

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Fig. 3. Scheme of a typical PLL.

Fig. 1. Scheme of a grid-connected DFIG WT with SVO-based vector control for both GSC and RSC: (a) system topology, (b) control strategy.

thus Qc = 0. The equations for RSC and GSC control can be obtained as ki1 iprdref = (Pgref − Pg ) kp1 + (9) s ki2 (10) iprqref = (Usref − Us ) kp2 + s ki3 (11) ipcdref = (Udcref − Udc ) kp3 + s ki5 (12) uprd = (iprdref − iprd ) kp5 + s ki6 kp6 + (13) uprq = iprqref − iprq s ki7 (14) upcd = (ipcdref − ipcd ) kp7 + s ki8 . (15) upcq = −ipcq kp8 + s C. PLL Dynamics

Fig. 2. Complex vector equivalent circuits in synchronous dq reference frame: (a) DFIG and (b) GSC and RSC.

B. SVO-Based Vector Control Scheme Fig. 1(b) depicts the SVO-based vector control scheme of DFIG’s RSC and GSC [1]. Both RSC and GSC active and reactive control loop includes two cascaded control loops. Typical proportional-integral (PI) controllers are applied in RSC and GSC control loops. The superscript p means the dq components of variables are in PLL-dq reference frame. The RSC control includes two cascaded control loops, one for APC and the other for RPC. APC is aimed at controlling active power sent to the grid from both stator and GSC tracking the reference given from maximum power tracking curve for given rotor speeds [19]. Terminal voltage control, as one scheme of RPC, is adopted to maintain the voltage of WT terminal bus within given values [20]. GSC controller controls active and reactive power flow between the GSC and the grid. APC mainly keeps dc-link voltage constant by controlling active power delivered to grid, the dc-link voltage reference Udcref is given as a constant value. The reactive power is set to zero to reduce GSC rating,

As shown in Fig. 1, PLL is widely employed to synchronize the DFIG WTs with grid. [9] and [21] showed that PLL gains influence VSC’s performance greatly, particularly at low SCRs. Similarly, its dynamics play an important role in grid-connected DFIG system. Fig. 3 presents a typical scheme of PLL [22]. Since the dynamics of PLL contribute to the dynamics of reference current generation, it should be slower than current control dynamics. Fig. 4 depicts the spatial relationship between the synchronous dq reference frame and the PLL-dq reference frame. In synchronous dq reference frame, the steady-state terminal voltage phase θs0 is defined to be zero. θ0 is the steady-state phase difference between the terminal voltage and the grid voltage. The PLL-dq reference frame coincides with the synchronous dq frame in steady state. F represents vectors such as voltage, current, and flux, respectively. The relationship of variables in synchronous dq frame and measured values in PLL reference frame is given below fd = fdp cos θpll − fqp sin θpll (16) fq = fdp sin θpll + fqp cos θpll where F can be the vector of Is , Ic , Ig , Us . The absolute phase angle θ of F equals its phase θp in PLL-dq reference frame plus PLL reference frame phase angle θpll , where θpll = θp − ω1 t θ = θp + θpll .

(17) (18)

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Fig. 5. Equivalent circuit model for a DFIG system connected to grid.

Fig. 4. Spatial relationships between synchronous dq reference frame and PLL-dq reference frame.

III. P ROPOSED DFIG M ODEL A full-order model to describe the behavior of a gridconnected DFIG can be obtained based on the equations presented in Section II. However, it is difficult to conduct the mechanism analysis and find the determinant effect on system stability of dc-link voltage control system with this high-order model. This section proposes a reduced-order small-signal model, which is more convincing for system stability analysis. Compared to dc-link voltage dynamics, the fast current control and flux dynamics are omitted in the model. Influence coefficients that reflect the interactions among control loops of dc-link voltage, RSC’s active/reactive power, and PLL are presented. Eigenvalue comparison with detailed model verifies the feasibility of the presented model to conduct small-signal stability analysis. Since this paper focuses on the time-scale issue of dclink voltage dynamics, which is much faster than the system electromechanical dynamics, both rotor speed and mechanical power input into the generator are assumed to be constant throughout this paper. A. Proposed Mathematical Model With vector control, the inner current control time constant is about 20 ms [11], [15], the associated current dynamics are very fast compared to dc-link voltage control time scale we are interested in. Therefore, we make two assumptions. The first one is that the current in PLL reference frame can track its reference instantaneously. This assumption is applied to both GSC and RSC currents, namely p Irp = Irref p Icp = Icref .

(19) (20)

The second assumption is neglecting stator and rotor flux dynamics, thus dΨ s /dt = 0, dΨ r /dt = 0. As a result, (1) can be simplified as an algebraic equation Us = −Rs Is + jω1 Ψ s (21) Ur = −Rr Ir + j (ω1 − ωr ) Ψ r . We define the stator internal voltage similar to synchronous machine [23] Es = −jω1 Lm Ir .

(22)

Fig. 6. Proposed model of a grid-connected DFIG system. A: (22); B: (2), (3), (21), and (23); C: (4); D: (16); and E: (7) and (8).

Then stator side can be expressed as a controlled voltage source behind an inductance series with resistance. Stator voltage can be rewritten as Us = Es − jω1 Ls Is − Rs Is .

(23)

The GSC can be equivalent to a controlled current source connected to grid. Neglecting all the resistance, the whole system can be modeled as an electric circuit in Fig. 5. The circuit model indicates system’s intrinsic characteristic. Terminal voltage is more sensitive to stator internal voltage and grid current variation for larger grid impedance. Thus, grid impedance can reflect grid strength. Fig. 6 represents a mathematical model reflecting the relationship between control loops and physical circuit. We consider RSC APC, RSC terminal voltage control, GSC dc-link voltage control, and PLL. Blocks A, B, C, and E represent algebraic equations describing physical circuit. Block D and PLL represent the conversion of variables between PLL-dq reference frame and synchronous dq reference frame, which interfaces the DFIG control system with grid. The overall control system acts as a link with the physical parts and determines system dynamic performance for given operating condition. The RSC mainly controls stator internal voltage Es and the GSC mainly controls grid converter current Ic . However, they are based on PLL reference frame, and as depicted by (18), their absolute phases equal phases in PLL reference frame plus PLL output angle θpll . From the equivalent circuit model in Fig. 5, stator internal voltage Es , grid converter current Ic , and grid voltage Ug in synchronous dq reference frame codetermine terminal voltage vector Us , which is the input of the PLL, thus making the overall system strongly coupled. From the above analysis, PLL plays an important role in interfacing DFIG control system with grid.


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Fig. 7. Proposed linearized small-signal model of a grid-connected DFIG system.

B. Linearized Small-Signal Model and Influence Coefficients This mathematical model of the overall system considers physical structure, related control, and grid strength. It is a nonlinear six-order system. A small-signal model can be obtained by linearizing all the related equations at an operating point. The state-space equations of the small-signal model can be obtained as Δx˙ = AΔx + BΔu

(24)

Δy = CΔx + DΔu.

(25)

The state vector is Δx = [Δiprd Δiprq ΔUdc Δipcd Δωpll Δθpll ]T . Fig. 7 shows the block diagram of the small-signal model. This model can be used to study the effects of operating point, grid strength, and control loop interactions on the system stability in dc-link voltage control time scale. G1 −G13 are the influence coefficients, which are determined by DFIG’s parameters, operating point, and grid strength. Their mathematical expressions are given in Appendix A. They can reflect system’s intrinsic characteristic. G1 −G4 are defined as self-influence coefficients, which reflect control loops characteristics on their own under different conditions. G5 −G13 are defined as mutual influence coefficients, which reflect the degree of control loops interactions. It is obvious that these control loops are coupled with each other. Taking PLL as an example, the feedback branch through G4 is called self-influence branch, the other branch through G12 or G13 is called mutual influence branch. The dynamic characteristics of a loop are codetermined by its self-influence branch and mutual influence branch. Fig. 8 gives several influence coefficients varying with different operating points and grid strengths in the case of constant terminal voltage. Influence coefficients vary under different conditions. It means the open loop gains of self-influence branch or mutual influence branch change, which might make the dynamics of different control loops vary.

Fig. 8. Influence coefficients. (a) G3 ; (b) G4 ; (c) G6 ; and (d) G13 varying with different operating points and grid strengths.

C. Eigenvalue Comparison for Verification In this section, we compare the eigenvalues of detailed model and presented model in different situations. The system and controllers’ parameters are given in the Appendix B. Table I compares the eigenvalues between detailed model and proposed model at the operating condition of Pgref = 1 p.u., Us = 1.0 p.u., and SCR = 3. Related participating factors and response mode are also given. First, it is shown that the presented model can hold the main pairs of eigenvalues of the detailed one, namely λ1 −λ6 . The dominant oscillation mode λ1,2 is associated with PLL. λ3,4 is associated with GSC dc-link voltage control and the nonoscillation mode λ5,6 is associated with RSC outer control. Second, λ7 −λ14 of the detailed model obviously decays much faster than λ1,2 , so it is reasonable to neglect the dynamics of flux and current control. Tables II and III show the concerned two oscillatory eigenvalues of detailed and proposed model under different operating points and grid strengths, respectively. The proposed modeldominant eigenvalues are quite close to the detail model for a wide range of operating points and grid strengths. It means the proposed model holds similar stable region to the detailed model and is suitable for small-signal stability analysis in the concerned time scale. IV. S YSTEM S TABILITY A NALYSIS W ITH THE P ROPOSED M ODEL This section will give case study of the proposed model application. Effect of grid strengths and controller gains, including PLL, RSC’s APC, and RPC, on system dynamic behavior in dc-link voltage control time scale is studied by eigenvalue analysis and influence coefficients analysis with the proposed model. The results are verified with time domain simulation with detailed model.

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TABLE I E IGENVALUES C OMPARISON OF D ETAILED M ODEL AND P ROPOSED M ODELS FOR SCR = 3

Fig. 9. Eigenvalue locus for varying grid strength. TABLE IV S YSTEM M ODES (SCR = 4 AND 1.5)

TABLE II C ONCERNED O SCILLATORY E IGENVALUES OF D ETAILED M ODEL AND P ROPOSED M ODELS IN D IFFERENT O PERATING P OINTS FOR SCR = 1.5

TABLE III C ONCERNED O SCILLATORY E IGENVALUES OF D ETAILED AND P ROPOSED M ODELS FOR VARIATIONS OF SCR (Pgref = 1 p.u., ωr = 1.1ω1 )

A. Grid Strength Effect It is known that grid strength significantly impacts the system dynamics. The proposed small-signal model is used to

TABLE V PARTICIPATION FACTORS C ORRESPONDING TO THE O SCILLATORY M ODES

investigate the effect of grid strength on system dynamics. The operating point is set as: Pgref = 1.0 p.u., Us = 1.0 p.u., and SCR varying from 5 to 1.2. The eigenvalue locus of the proposed system is depicted in Fig. 9. Eigenvalues in SCR = 4 and 1.5 are given in Table IV and participating factor corresponding to oscillatory modes is given in Table V. In Table IV, √ damping ratio of an oscillation mode is defined as ζ = −σ/ σ 2 + ω 2 , oscillation frequency f = ω/2π [23] (where λ = σ ± jω is the eigenvalues of the mode). The system has two oscillation modes λ1,2 and λ3,4 . As grid strength decreases, the eigenvalues of mode λ3,4 almost stay intact; participation factor analysis shows that this mode is totally determined by dc-link voltage control and almost unaffected by grid strength and other control loops. The other mode λ1,2 is the dominant oscillation mode of system. This pair of eigenvalues moves to the right half-plane visibly with the decrease of grid strength. Damping of this mode gradually decreases. This mode is mainly related to PLL for SCR = 4, while RSC’s APC and RPC participate in a large proportion of this mode for SCR = 1.5. This means that as SCR decreases, the interactions between RSC control and PLL


Fig. 10. DC-link voltage responses with detailed model in different grid strengths.

become stronger. Fig. 10 shows dc-link voltage responses of a corresponding detailed model for a step change of power reference at t = 15 s in different grid strengths. The damping of mode λ1,2 is about 0.7 for SCR > 2, so there is little overshoot of dc-link voltage after the initial fast current dynamics decayed. However, the damping of mode λ1,2 decreases dramatically for SCR < 2 and oscillation of dc-link voltage becomes severe, which is coherent with the eigenvalue locus analysis in Fig. 9. It can be concluded from Tables IV and V that control loop interactions become more noticeable as SCR decreases. The following will investigate the effects of the gains of PLL and RSC control on system behavior in weak ac grid.

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Fig. 11. Eigenvalue locus of the proposed model as PLL bandwidth varying from 2 to 24 Hz.

Fig. 12. DC-link voltage responses with detailed model for different PLL bandwidths.

B. PLL Effect This section gives an analysis of PLL effects on dc-link voltage control system in very weak ac grid with varying PLL bandwidths. Fig. 11 shows the eigenvalue locus of the presented small-signal model. The operating condition is chosen as: Pgref = 1.0 p.u., Us = 1.0 p.u., and SCR = 1.2. Fig. 11 shows that the dominant eigenvalues first move toward the right-half plane and then turn back as the bandwidth of PLL varies from 2 to 24 Hz with keeping equivalent damping ratio. It indicates that PLL’s bandwidth has pronounced influence on system stability. System damping first reduces then increases. Fig. 12 shows the simulation results obtained from the detailed model. The dc-link voltage responses for a small power reference step change at t = 15 s are given for different PLL’s bandwidths. System is unstable for PLL bandwidth of 8 Hz, but becomes stable for 4 or 16 Hz, which coincide with the trend of eigenvalue locus of the proposed model.

C. RSC’s APC and RPC Effect As indicated in Table V, RSC’s APC and RPC also participate in the dominant oscillation mode in weak grid. This section will evaluate the effect of RSC’s APC and RPC on system dynamic performance by changing its gains but keeping other controller gains constant. The operating condition is set the same as that in Section IV-B.

Fig. 13. Eigenvalue locus of the proposed model as RSC APC bandwidth varying from 0.5 to 12 Hz.

Eigenvalue locus of the dominant oscillation mode with APC bandwidth changing from 0.5 to 12 Hz with the proposed model is plotted in Fig. 13. This oscillatory eigenvalues move toward the right-half plane as APC bandwidth increases. System becomes unstable when APC bandwidth exceeds 6 Hz. Simulation result in Fig. 14 with detailed model shows good match with the eigenvalue locus. Therefore, it indicates that RSC APC has obvious effect on the dominant mode and its gain is suggested to be reduced for weak grid.

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Fig. 14. DC-link voltage responses with detailed model for different RSC APC bandwidths.


Fig. 16. DC-link voltage responses with detailed model for different RSC RPC bandwidths.

Fig. 17. Diagrams for illustrating dynamics of (a) DC-link voltage control related mode and (b) PLL-related mode.

Fig. 15. Eigenvalue locus of the proposed model as RSC RPC bandwidth varying from 0.2 to 16 Hz.

RSC RPC’s effect is studied similarly. Eigenvalue locus of dominant oscillation mode for RSC RPC bandwidth varies from 0.2 to 16 Hz as given in Fig. 15. This pair of eigenvalues first moves toward the right-half plane and then goes back. Damping of the dominant oscillation mode first decreases and then increases with RPC gain increasing. Time domain simulation results, as shown in Fig. 16 with detailed model, are coherent with the trend of eigenvalue locus. D. Further Explanation for Grid Strength and Controller’s Gain Effect In this section, influence coefficients are used to give a deeper explanation of the effect of grid strength and controller’s gain on system dynamic behavior in Sections IV-A, IV-B, and IV-C. Fig. 17 gives a block diagram of dc-link voltage control and PLL, respectively, for clear illustration of the concerned two oscillation modes. As seen in Fig. 17(a), the branch with G3 represents own characteristics of dc-link voltage control loop in different conditions, and F1 (s) stands for the transfer function introduced by other control loops, including RSC APC/RPC and PLL. In terms of grid strength effect, for the dc-link voltage control related mode, bode diagram in Fig. 18(a) shows that the gain of

Fig. 18. (a) Bode plot of F1 (s) with different grid strengths. (b) Bode plot of G4 and F2 (s) with different grid strengths.

G3 in Fig. 8 is over ten times larger than that of F1 (s) at the oscillation frequency of this mode (about 10 Hz calculated in Table IV). This means that the self-influence branch with G3 dominates this mode. Moreover, Fig. 8 shows that G3 is almost unaffected by operating point and grid strength. As a result, it can be concluded that the dynamics of dc-link voltage controlrelated mode change little with grid strength varying, which is coherent with the eigenvalue locus in Fig. 9. The dominant oscillation mode is mainly related to PLL. Likewise, PLL model is presented in Fig. 17(b) for analyzing this mode. The dc-link voltage control loop is omitted here for simplified analysis because the state variables ΔUdc and Δipcd hardly participate in the PLL-related mode seen from Table V. As shown in Fig. 18(b), the branch with G4 is no longer a unit feedback loop considering the effect of grid strength. In relatively strong grid, for SCR = 4, G4 is much larger than the gain of F2 (s), so this mode is mainly related with PLL and


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Fig. 19. (a) PLL defining synchronizing component ΔθS and damping component ΔθD . (b) Phasor diagram expression.

there is no apparent interaction between PLL and converter outer control loops. However, G4 decreases as grid strength becomes weaker. G4 decreases dramatically for SCR < 2; this can explain the reason why the damping of this oscillation mode deteriorates as SCR decreases in Fig. 9. Conversely, the gain of F2 (s) increases with decrease in SCR and gradually becomes comparable with G4 . More specifically, as shown in Fig. 8, the absolute value of G6 increases with reduction in SCR. This means that the coupling between RSC’s APC and RPC becomes stronger. Similarly, the increase in G13 as SCR decreases indicates that terminal voltage phase becomes more sensitive to RSC APC. Therefore, the RSC APC and RPCs participate more in this mode as SCR decreases, as seen in Table V. As a result, the effects of control loop interactions on this mode get more dominant, as grid becomes weaker. To get a better understanding of controllers’ gain impact, the synchronizing and damping torque analysis method [24], first introduced based on a physical understanding of the torqueangle loop of synchronous generator and widely used for power system stability analysis, is applied to PLL diagram in Fig. 17(b). Considering the self-influence branch of PLL, we define the damping component ΔθD in phase with speed and synchronizing component ΔθS in phase with angle shown in Fig. 19(a), and modal oscillation frequency is considered to be mainly determined by this branch. The synthetic feedback angle vector can be expressed as Δθ1 = ΔθS + ΔθD .

(26)

Impact of other control loops on PLL is evaluated by the additional synthetic vector Δθ2 of mutual influence branch. As shown in Fig. 19(b), monotonic instability would occur if the synchronizing components provided by the two branches are negative in quadrant II or III, and negative damping components in III or IV would cause oscillatory instability. The open-loop transfer function in Fig. 17(b) can be written as F (s) = E(s)G4 + E(s)F2 (s)

Δθ1

1

ki4

(27)

Δθ2

where E(s) = s kp4 + s . Then, the effect of controllers’ gains can be evaluated by their impact on the synthetic vector. For different PLL bandwidths, bode plots of E(s) and F2 (s) are given in Fig. 20. As PLL bandwidth increases, modal oscillatory frequency increases. Phase

Fig. 20. Phasor diagram for illustrating PLL bandwidth effect. (a) Bode plot of E(s). (b) Bode plot of F2 (s). (c) Phasor diagram of Δθ1 , Δθ2 .

of Δθ1 provided by self-influence branch keeps unchanged as shown in Fig. 20(a), but corresponding Δθ2 of mutual influence branch differs because the gain and phase of F2 (s) differ at different oscillation frequencies. Although the gain of F2 (s) decreases with an increase in oscillatory frequency, the phase shift makes the negative damping component of Δθ2 first increase and then decrease, as shown in Fig. 20(c). Actually, in terms of 8 Hz of PLL bandwidth, the negative damping component provided by mutual influence branch exceeds the positive damping component of self-influence branch, thus leading system oscillatory instability shown in Fig. 12. Likewise, effects of RSC’s APC and RPC are studied and depicted in Figs. 21 and 22, respectively. Modal oscillation frequency varies little with RSC control gain change (around 4 Hz shown in Figs. 13 and 15). With an increase in RSC APC bandwidth, the gain of F2 (s) increases, whereas its phase hardly changes as shown in Fig. 21(a); thus, the gain of Δθ2 becomes larger. Negative damping component provided by mutual influence branch gradually increases, as shown in Fig. 21(b), making system unstable with APC bandwidth equal to 6 Hz as seen in Fig. 14. For RSC RPC, gain of F2 (s) changes little with increasing RPC bandwidth, whereas its phase varies by a wide margin. Δθ2 first moves close to negative axis of Δω, and then goes away. Negative damping component provided by mutual influence branch first increases and then reduces. As a result, system damping first reduces and then increases with increasing RSC RPC bandwidth, as shown Fig. 16. Synchronizing and damping components are presented to analyze the interactions of control loops quantitatively, which gives a clear explanation of the eigenvalue locus results in Sections IV-B and IV-C. It declares that the interactions of control loops can introduce negative impact on system dynamic performance in weak grid, and the gains of PLL, RSC’s APC, and RPC have noticeable effect on this.

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weak grid. All the analyses above with the proposed model are verified by time domain simulation with detailed model. In addition, for the purpose of studying a wind farm’s dynamic behavior in dc-link voltage control time scale, how to extend the developed model to a multi-DFIG system remains a problem, which needs further research and will be reported in the near future.

A PPENDIX Fig. 21. Phasor diagram for illustrating RSC APC bandwidth effect. (a) Bode plot of F2 (s). (b) Phasor diagram of Δθ1 , Δθ2 .

A. Influence Coefficients 3Xm (igq Xg + Us ) −Xg Xm G2 = 2 (Xg + Xs ) Xg + Xs (Us − igq Xs ) Xg 3 srXm Xg irq = Us + G4 = 1 − 2 (Xg + Xs ) (Xg + Xs ) Us 3Xs (igq Xg + Us ) 3Xm igd Xg = G6 = − 2 (Xg + Xs ) 2 (Xg + Xs ) 2 3 igd Xg Xs + (igq Xs − Us ) (igq Xg + Us ) =− 2 (Xg + Xs ) −igd Xg Xs = Xg + Xs 2 3 Xm = ird Rr + srirq Xr − + urd 2 Xg + Xs 2 Xm 3 = − Xr + urq irq Rr + srird 2 Xs Xg Xm − (srXm ird + icd Xs ) Xs (Xg + Xs ) 3 (srXm ird + icd Xs ) igd Xg + srXm irq (igq Xg + Us ) =− 2 (Xg + Xs ) Xs Xg Xm Xg = G13 = . (Xg + Xs ) Us (Xg + Xs ) Us

G1 = G3 G5 G7 G8 Fig. 22. Phasor diagram for illustrating RSC RPC bandwidth effect. (a) Bode plot of F2 (s). (b) Phasor diagram of Δθ1 , Δθ2 .

G9 G10

V. C ONCLUSION A reduced-order small-signal model of grid-connected DFIG system for stability analysis of dc-link voltage control system is proposed. The model was established to investigate the effect of operating points, grid strengths, and the interactions control loops on the dc-link voltage control system dynamic performance. The main assumption of the model is that the fast current control and flux dynamics are neglected. Eigenvalue comparison indicates that the proposed model can retain the concerned oscillation mode of the detailed model. Therefore, it is suitable for conducting small-signal stability analysis in dc-link voltage control time scale. With the proposed model, effects of grid strength and controller gains, including PLL, RSC’s APC, and RPC, are first investigated by eigenvalue locus analysis. Two oscillation modes are concerned in the interest frequency scale. It shows that the mode related to dc-link voltage control is hardly affected by grid strength, but the damping of PLL-related mode reduces with the decrease in grid strength and gradually becomes the dominant oscillation mode of system. Meanwhile, the interactions of control loops become more noticeable, and the gains of PLL, RSC’s APC, and RPC have significant effect on this mode and system dynamic stability in weak grid. Furthermore, the presented influence coefficients effectively explain the varying trend of the two concerned modes as grid strength changes. Further, effects of the interactions of control loops on system stability are investigated quantitatively by the presented synchronizing and damping component analysis. This analysis develops insight into the effects of the interactions of control loops on system dynamic behavior and can provide guidance for parameters tuning to enhance system stability in

G11 G12

where sr =

ωr −ω1 ω1 .

B. 2-MW DFIG Parameters Sbase = 2 MW

Ubase = 690 V (phase to phase rms value)

ωbase = 2πfbase Lls = 0.171 p.u. Udcref = 1200 V

fbase = 50 Hz Rs = 0.022 p.u. Rr = 0.009 p.u. Llr = 0.156 p.u. Lm = 3.9 p.u. Lc = 0.475 p.u. C = 0.02F Ug = 1 p.u. Rg = 0.1ω1 Lg

Controllers parameters values (p.u.) RSC APC kp1 = 0.4 ki1 = 40 RSC terminal voltage control kp2 = 0.25 ki2 = 25 DC-link voltage control kp3 = 1.5 ki3 = 100 PLL kp4 = 60 ki4 = 1400 RSC current control kp5 = kp6 = 0.6 ki5 = ki6 = 80 GSC current control kp7 = kp8 = 8 ki7 = ki8 = 200 R EFERENCES [1] S. Muller, M. Deicke, and R. W. De Doncker, “Doubly fed induction generator systems for wind turbines,” IEEE Ind. Appl. Mag., vol. 8, no. 3, pp. 26–33, May/Jun. 2002. [2] J. G. Slootweg and W. L. Kling, “The impact of large scale wind power generation on power system oscillations,” Electr. Power Syst. Res., vol. 67, no. 1, pp. 9–20, Oct. 2003.


[3] G. Tsourakis, B. M. Nomikos, and C. D. Vournas, “Contribution of doubly fed wind generators to oscillation damping,” IEEE Trans. Energy Convers., vol. 24, no. 3, pp. 783–791, Sep. 2009. [4] J. M. Rodriguez et al., “Incidence on power system dynamics of high penetration of fixed speed and doubly fed wind energy systems: Study of the Spanish case,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1089–1095, Nov. 2002. [5] V. Diedrichs, A. Beekmann, K. Busker, S. Nikolai, and S. Adloff, “Control of wind power plants utilizing voltage source converter in high impedance grids,” in Proc. IEEE Power Energy Soc. (PES) Gen. Meeting, San Diego, CA, USA, Jul. 2012, pp. 1–9. [6] V. Diedrichs, A. Beekmann, and S. Adloff, “Loss of (angle) stability of wind power plants- the underestimated phenomenon in case of very low short circuit ratio,” in Proc. 10th Int. Workshop Large-Scale Integr. Wind Power Power Syst. Transmiss. Netw. Offshore Wind Power Plants, Aarhus, Denmark, Oct. 25–26, 2011, pp. 393–340. [7] P. Zhou, X. Yuan, and J. Hu, “Stability of DC-link voltage as affected by phase locked loop in VSC when attached to weak grid,” in Proc. IEEE Power Energy Soc. (PES) Gen. Meeting, Washington, DC, USA, Jul. 2014, pp. 1–5. [8] Y. Huang, X. Yuan, and J. Hu, “Effect of reactive power control on stability of DC-Link voltage control in VSC connected to weak grid,” in Proc. IEEE Power Energy Soc. (PES) Gen. Meeting, Washington, DC, USA, Jul. 2014, pp. 1–5. [9] J. Z. Zhou, H. Ding, S. Fan, and Y. Zhang, “Impact of short-circuit ratio and phase-locked-loop parameters on the small-signal behavior of a VSCHVDC converter,” IEEE Trans. Power Del., vol. 29, no. 5, pp. 2287– 2296, Oct. 2014. [10] F. Mei and B. C. Pal, “Modeling and small signal analysis of a grid connected doubly fed induction generator,” in Proc. IEEE Power Energy Soc. (PES) Gen. Meeting, San Francisco, CA, USA, 2005, pp. 358–367. [11] A. Ellis, Y. Kazachkov, E. Muljadi, and P. Pourbeik, “Description and technical specifications for generic WTG models—A status report,” in Proc. Power Syst. Conf. Expo. (PSCE), Mar. 2011, pp. 1–8. [12] P. Pourbeik, A. Ellis, J. Sanchez-Gasca, and Y. Kazachkov, “Generic stability models for type 3 & 4 wind turbine generators for WECC,” in Proc. IEEE Power Energy Soc. (PES) Gen. Meeting, Jul. 2013, pp. 1–5. [13] Western Electricity Coordinating Council. (Aug. 2010). WECC Wind Power Plant Dynamic Modeling Guide, Prepared by WECC Renewable Energy Modeling Task Force [Online]. Available: http://www.wecc.biz/ committees/StandingCommittees/PCC/TSS/MVWG/REMTF/Shared %20Documents/WECC%20Wind%20Power%20Plant%20Dynamic%20 Modeling%20Guidelines%20-%20August%202010%20draft.pdf [14] T. Ackermann, Wind Power in Power Systems. Hoboken, NJ, USA: Wiley, 2005. [15] R. Pena, J. C. Clare, and G. M. Asher, “Doubly fed induction generator using back-to-back PWM converters and its application to variable speed wind-energy generation,” Proc. Inst. Electr. Eng. Electr Power Appl., vol. 143, no. 3, pp. 231–241, May 1996. [16] A. Tapia, G. Tapia, J. Ostolaza, and J. Saenz, “Modeling and control of a wind turbine driven doubly fed induction generator,” IEEE Trans. Energy Convers., vol. 18, no. 2, pp. 194–204, Jun. 2003. [17] H. Akagi and H. Sato, “Control and performance of a doubly-fed induction machine intended for a flywheel energy storage system,” IEEE Trans. Power Electron., vol. 17, no. 1, pp. 109–116, Jan. 2002. [18] A. Petersson, L. Harnefors, and T. Thiringer, “Comparison between stator-flux and grid-flux-oriented rotor current control of doubly-fed induction generators,” in Proc. 35th IEEE Power Electron. Spec. Conf. (PESC), Jun. 2004, vol. 1, pp. 482–486. [19] H. A. Pulgar-Painemal and P. W. Sauer, “Power system modal analysis considering doubly-fed induction generators,” in Proc. Bulk Power Syst. Dyn. Control (iREP) Symp., Aug. 2010, pp. 1–7. [20] K. Clark, N. W. Miller, and J. J. Sanchez-Gasca, Modeling of GE Wind Turbine-Generators for Grid Studies, Version 4.5, General Electric International, Inc., New York, NY, USA, Apr. 2010. [21] T. Midtsund, J. A. Suul, and T. Undeland, “Evaluation of current controller performance and stability for voltage source converters connected to a weak grid,” in Proc. 2nd IEEE Int. Symp. Power Electron. Distrib. Gen. Syst., 2010, pp. 382–388. [22] A. Gole, V. K. Sood, and L. Mootoosamy, “Validation and analysis of a grid control system using D-Q-Z transformation for static compensator system,” in Proc. Can. Conf. Elect. Comput. Eng., Montreal, QC, Canada, Sep. 1989, pp. 745–748. [23] P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994. [24] F. P. De Mello and C. Concordia, “Concepts of synchronous machine stability as affected by excitation control,” IEEE Trans. PAS, vol. 88, no. 4, pp. 316–329, Apr. 1969.

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Jiabing Hu (S’05–M’10–SM’12) received the B.Eng. and Ph.D. degrees from the College of Electrical Engineering, Zhejiang University, Hangzhou, China, in 2004 and 2009, respectively. From 2007 to 2008, he was funded by the Chinese Scholarship Council (CSC) as a Visiting Scholar with the Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, U.K. From April 2010 to August 2011, he was a Postdoctoral Research Associate with Sheffield Siemens Wind Power (S2WP) Research Center and the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, U.K. Since September 2011, he has been a Professor with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, and School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. He is the author/coauthor of more than 70 peer-reviewed technical papers and one monograph “Control and Operation of Grid-Connected Doubly-Fed Induction Generators,” and holds more than 20 issued/pending patents. His research interests include gridintegration of large-scale renewables, modular multilevel converter (MMC) for HVDC applications, and transient analysis and control of semiconducting power systems. Dr. Hu serves as Domestic Member of the Editorial Board for Frontiers of Information Technology and Electronic Engineering (FITEE), formerly known as Journal of Zhejiang University-SCIENCEC. He was the recipient of the 2014 TOP TEN Excellent Young Staff Award from Huazhong University of Science and Technology, and is currently supported by the National Natural Science of China for Excellent Young Scholars and the Program for New Century Excellent Talents in University from Chinese Ministry of Education. Yunhui Huang (S’12) was born in Wuhan, Hubei, China, in 1986. He received the B.S. degree in electrical engineering and automation from Wuhan University of Technology, Wuhan, China, in 2009. He is currently pursuing the Ph.D. degree at the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, and School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. His research interests include modeling and control of grid-integrated power converters, in particular on stability analysis and control of grid-connected renewable generations in electromagnetic timescale.

Dong Wang was born in Huanggang City, Hubei Province, China, in 1988. He received the B.Eng. degree in electrical engineering and automation from the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China, in 2012. He is currently pursuing the M.Eng. degree in electrical engineering at the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, and School of Electrical and Electronic Engineering, Huazhong University of Science and Technology. His research interests include modeling and control of wind turbines, in particular on small signal stability analysis and control of grid-integrated DFIG-based wind turbines in electromagnetic timescale.

Hao Yuan was born in Xiantao City, Hubei Province, China, in 1990. He received the B.S. degree from the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China, in 2012. He is currently pursuing the Ph.D. degree in electrical engineering at the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, and School of Electrical and Electronic Engineering, Huazhong University of Science and Technology. His research interests include control and stability analysis of grid-integration of large-scale renewables and VSC-HVDC, in particular on the transient behavior analysis of dc-link voltage of multiwind turbines in weak grid.

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Xiaoming Yuan (S’97–M’99–SM’01) received the B.Eng. degree from Shandong University, Jinan, China, in 1986, the M.Eng. degree from Zhejiang University, Hangzhou, China, in 1993, and the Ph.D. degree from Federal University of Santa Catarina, Florianópolis, Brazil, in 1998, all in electrical engineering. He was with Qilu Petrochemical Corporation, Zibo, China, from 1986 to 1990. From 1998 to 2001, he was a Project Engineer with the Swiss Federal Institute of Technology, Zurich, Switzerland. From 2001 to 2008, he was with the GE Global Research Center, Shanghai, China, as the Manager of the Low Power Electronics Laboratory. From 2008 to 2010, he was with the GE Global Research Center, Niskayuna, NY, USA, as an Electrical Chief Engineer. In 2010, he joined Huazhong University of Science and Technology, Wuhan, China. His research interests include stability and control of power system with multimachines multiconverters, control and gridintegration of renewable energy generations, and control of high-voltage dc transmission systems. Dr. Yuan was the recipient of the First Prize Paper Award from the Industrial Power Converter Committee of the IEEE Industry Applications Society in 1999. He is a Distinguished Expert of the National Thousand Talents Program of China, and the Chief Scientist of the National Basic Research Program of China (973 program).
