Electromechanical Dynamics of Controlled Variable-Speed Wind ...

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Electromechanical Dynamics of Controlled Variable-Speed Wind Turbines Sudipta Ghosh, Student Member, IEEE, and Nilanjan Senroy, Member, IEEE

Abstract—Variable-speed wind turbines are increasingly penetrating into the electrical grid, replacing the conventional synchronous-generator-based power plants and thus decreasing the available inertial response for primary frequency stability. This paper offers a deeper understanding of variable-speed wind turbine generators (WTGs) in the context of maximum power point tracking and obtaining primary frequency response. Linearized models have been obtained between the wind velocity and the system frequency versus the power output. System complexity has been studied from the point of view of modal analysis of a two-mass drive train model of a WTG, as well as Hankel singular values. Finally, individual WTG models have been combined to form wind farms, whose complexity has again been found to depend on the nature of modeling of the WTG drive trains. Index Terms—Deloading, linearization, maximum power point tracking (MPPT), model order reduction (MOR), pitch control, primary frequency control, transfer function, wind power generation, wind turbine generator (WTG).

I. I NTRODUCTION

T

HE role of wind in determining the frequency characteristics of electric power networks is coming under scrutiny, with wind turbine generators (WTGs) being increasingly employed for bulk power generation in grid-connected mode and in standalone microgrids. The generators employed in WTGs involve technologies significantly different from the traditional synchronous alternators used in thermal power, nuclear power, and hydropower plants. As a result, wind power frequently appears “inertialess” to the network [1]–[3]. Furthermore, the “fuel” for WTGs is wind, which is renewable, yet unpredictable by nature. In the context of the inherent uncertainty associated with wind as a resource, the ability of the WTG to service loads reliably and economically is a challenge for power engineers. A key step in confronting this challenge is to model the response of WTGs to variations in wind velocity and network frequency. The need is for dynamic models relevant at pertinent timescales ranging from tens of seconds to minutes. The focus of this paper is on the following: 1) the response of the WTG to network frequency disturbances; and 2) the effect

Manuscript received September 1, 2012; revised March 10, 2013; accepted August 18, 2013. This work was supported in part by the Department of Science and Technology, Government of India under Grant RP02322. S. Ghosh is with the Department of Electrical Engineering, Indian School of Mines, Dhanbad 826004, India (e-mail: [email protected]). N. Senroy is with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: [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/JSYST.2013.2280037

of wind velocity fluctuations on the WTG active power output. The answer to these questions depends on the type of WTG in operation. Fixed-speed WTGs, employing squirrel cage induction generators, are directly connected to the electrical network whose frequency affects the slip of the machine. Therefore, fixed-speed WTGs react to grid frequency disturbances in a manner to mitigate the disturbance. On the other hand, in doubly fed induction generator (DFIG)-based variable-speed WTGs, the power electronic converters in the generator–grid interface effectively delink the generator speed (and hence power output) from the system frequency. The reaction of such a WTG to a system frequency disturbance is negligible [4]. Variable-speed WTGs are more controllable. For instance, at low wind speeds, maximum power point tracking (MPPT) employed to maximize the wind power harvested result in greater variations in WTG active power output due to wind speed changes. Such fluctuations in the WTG speed and power are tempered to some extent by the inertia of the high- and low-speed turbine shafts. Another control strategy for variablespeed WTGs is to modulate its active power output in response to network frequency excursions [5]–[8]. The aim is to emulate a synchronous machine behavior by providing temporary support to the network during disturbances. Whatever control strategy is used, a fundamental understanding of its efficacy is possible only if appropriate dynamic models are developed for their testing and development. In this paper, simplified linearized models of controlled variable-speed WTGs are presented that relate its active power output to the wind speed and network frequency deviations. From the linearized models that focus on the electromechanical dynamics, a deeper understanding of both MPPT and primary frequency response strategies is obtained. The target application of these models is load frequency control studies, where the role of reactive power is limited and the focus is more on the relationship between active power injection and system frequency. With focus on DFIG-based WTGs, special emphasis is laid on the two mass behavior characteristics of the drive train. A state-space modeling approach is adopted to identify the modal stability properties and the particular states that highlight input–output relationships. Using Hankel singular values (HSVs), the reducibility of the WTG dynamic models is linked to the relative inertia of the turbine (low-speed) and generator (high-speed) shafts. Finally, it is shown that the dynamic reducibility of wind farms (modeled from individual WTGs) also shows similar dependence. The rest of this paper is organized as follows. Section II presents a preliminary model of a variable-speed WTG, which focuses on the electromechanical dynamics of a single-mass

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uses Cp −λ curves specific to the turbine aerodynamics [9] to calculate the mechanical torque. This torque interacts with the electrical torque of the generator to set the rotor speed. For ease of calculations, the turbine inertia is initially lumped with the generator inertia. The generator sets its power output according to an MPPT lookup table tuned to the specific turbine. The following subsections describe the MPPT control and pitchangle-based deloading strategies in further detail. A. MPPT Fig. 1. Reduced-order model of a single-mass variable-speed WTG. H represents the combined turbine and generator shaft inertia; and Kb and Kf are the gains for the pitch and electrical power control loops, respectively.

At any given wind velocity, vwind (in meters per second) below rated wind speed, the power harvested by the WTG is controlled by adjusting the turbine speed. The ratio of the turbine blade tip speed and wind speed is called the tip speed ratio, i.e., λ. The mechanical power, in watts, developed by the turbine is Pturb =

1 3 ρ · A · Cp (λ, β) · vwind 2

(1)

where ρ is the density of air, and A is the area swept by the turbine. The coefficient of performance of the wind turbine, i.e., Cp (λ, β), is determined from the Cp −λ curves at different pitch angles, i.e., β [9]. Maximum power extraction occurs at an optimum tip speed ratio, i.e., λnom , when Cp (λ, β) is maximum. The actual optimum values of Cp (λ, β) and λnom depend on the mechanical design of the turbine and are used as base values during normalization. Accordingly, the normalized tip speed ratio is Fig. 2. MPPT operation combined with pitch-angle-based deloading. All quantities, including Cp , are in per unit.

turbine. Both control strategies, i.e., MPPT and primary frequency response, are discussed using this model. Section III expands the model to include a two-mass turbine representation and derives transfer functions between the wind velocity/ network frequency variations and active power output. Critical aspects of the transfer function, as well as the linearized model of the two-mass WTG for load frequency studies, are also discussed in Section III. Section IV discusses the reducibility of the derived models by exploring its Hankel singular values, which are then applied for model order reduction (MOR) of wind farms in Section V. Section VI is the concluding section. II. WTG DYNAMIC M ODEL Here, a reduced-order WTG model is considered, which demonstrates both MPPT control and responds to grid frequency changes. Fig. 1 presents the schematic of a variablespeed WTG, operating under a combination of MPPT strategy and pitch-control-based deloading. The turbine block has three inputs—rotor speed, pitch angle change, and wind speed; the pitch angle reference is fixed at βREF . βREF is found by slowly increasing the pitch until the deloading of the WTG is equal to a desired value, for example, 90%. For this paper, βREF was taken to be 1◦ , which gave a deloading of 0.9029 per unit (p.u.). The wind turbine block, shown in Fig. 2, essentially

λpu =

λ λnom

=

ωturb[pu] . vw[pu]

(2)

Here, vw refers to the per unit wind speed on a base of 12 m/s. The angular speed of the turbine, i.e., ωturb [pu], is equal to the speed of the generator, i.e., ω[pu]. The coefficient of performance can be normalized to its optimum value at a minimum pitch Cp , and the mechanical torque from the turbine, i.e., Tm , is (in per unit) Tm =

3 kp C p v w . ωturb

(3)

The scaling factor kp (= ρACpMAX vwBASE /2PBASE = 0.73) indicates the maximum turbine output power at base wind speed [10]. Operation at MPPT requires setting the turbine speed ωturb such that the tip speed ratio λpu is at its nominal value of 1 p.u. With a blade pitch angle of 0◦ , the coefficient of performance is Cp_pu = 1 pu. At this operating condition, the output torque of the wind turbine in per unit is Tm =

3 kp v w . ωturb

(4)

The generator (see Fig. 1) controls the turbine speed by setting its electrical power output as per a lookup table. The table models the MPPT characteristic passing through the maxima of the turbine characteristics for different wind velocities at zero

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pitch [9] (see Fig. 2). The equation of the generator torque for MPPT characteristics is [11] T e = kp ω 2 .

(5)

At any wind speed, the steady-state operating power is determined by the intersection of the MPPT line and the turbine characteristics for that particular wind speed. B. Pitch-Angle-Based Deloading A WTG operating under MPPT control extracts the maximum power from the wind if its pitch angle β is set to zero. However, if the pitch is set to a positive value, i.e., βREF > 0, it reduces the coefficient of performance Cp and thereby the power output [12], [13]. Thus, with the generator under MPPT control and the turbine pitch controlled to create a power margin, the WTG is said to be deloaded. Fig. 2 shows the two operating conditions—MPPT and deloaded operating conditions— at two different wind speeds. The solid curves represent the turbine characteristics obtained at zero pitch, and the dashed curves represent the deloaded turbine characteristics for a pitch β = βREF = 1◦ . Two wind speeds, i.e., va and vb , are considered. The MPPT characteristic curve is shown as a dotted line. With the pitch angle set to zero, the MPPT controller establishes the operating point at point A(B) for wind speed vA (vB ). Since vw = vA for both points A and A and CpA = 1 at point A, therefore, at point A , CpA < 1 is the deloading factor. Similarly, at wind speed vB , the deloading factor is CpB . When operating at A (or B ), the mechanical turbine power is equal to the generator electrical power. Using (3) and (5) yields λ3 = Cp (λ, β) or λ = f (β).

(6)

The preceding equation is valid in the per unit domain only. The direct conclusion of (6) is that, under MPPT control, the steadystate tip speed ratio depends only on the steady-state pitch angle βREF . Furthermore, at any operating point, Cp is a function of the tip speed ratio and the pitch angle only; therefore, the extent of the deloading of a WTG is independent of wind speed and depends only on βREF . As part of the primary frequency response control strategy, the network frequency deviation is used to modulate both the pitch angle from βREF and the electrical power Pe . The operating point accordingly moves between A (B ) and A(B). This control strategy is better understood using linearized WTG models, as discussed in the next section. III. WTG L INEARIZED M ODEL The response of variable-speed WTGs to network frequency excursions may be controlled by modulating the electrical power and/or the mechanical power. The advantage of modulating the electrical torque is the availability of a power reserve immediately after the disturbance is detected (see Fig. 3). The response decays unless the mechanical torque is also controlled. If both torques are modulated (as shown in Fig. 1), a primary

Fig. 3. Response of a controlled variable-speed WTG to a 0.01-p.u. rise in network frequency.

frequency response is obtained (see Fig. 3). If there is a small change in system frequency fΔ , Te is controlled by adding a value of −Kf FΔ /ω to Te , whereas Tm is controlled by adding βΔ = Kb · fΔ to the initial pitch angle βREF . At the MPPT point, βREF = 0; accordingly, the mechanical torque cannot be controlled, i.e., kb = 0. Let CpREF and λREF correspond to the operating values of Cp and λ at βREF . As shown in Section II-B, these values remain constant in steady state, irrespective of the wind speed under MPPT control. Let ω0 be the current rotor speed and v0 the current wind speed. Linearizing (3) and (5) for small changes in v and ω yields kp kCp v 2 kp v 3 CpREF TmΔ = − ωΔ ω02 ω0 3kp CpREF v02 vΔ kp kCp λREF v 2 + − vΔ ω0 ω0 kp kβ kb v 3 + fΔ (7) ω0 Kf f Δ TeΔ = 2kp ω0 ωΔ − (8) ω0 ∂Cp ∂Cp CpΔ = λΔ + βΔ . (9) ∂λ λ=λREF ∂β λ=λREF β=βREF

β=βREF

Both ∂Cp /∂λ and ∂Cp /∂β are calculated at (λREF , βREF ) and, hence, are constants, for example, KCp and Kβ , respectively. The values of KCp and Kβ may be calculated from the slope of the Cp −λ curve for β = βref and the Cp −β curve for λ = λREF , respectively. Alternatively, the variation in Tm (and thus in Cp ) for a small change in λ (keeping other parameters constant) and then for β (keeping other parameters constant) can yield the values of KCp and Kβ , respectively. It is observed that (λREF + λΔ ) =

(ω0 + ωΔ ) (v0 + vΔ )

(10)

which yields λΔ /λREF = ωΔ /ω0 − vΔ /v0 .

(11)



Fig. 4. Bodè magnitude plot for transfer functions obtained for different wind speeds and rotor inertia. (a) Wind speed: 10 m/s, H = 2.5 s. (b) Wind speed: 8 m/s, H = 2.5 s. (c) Wind speed: 10 m/s, H = 10 s. (d) Wind speed: 8 m/s, H = 10 s.

Substituting the value of λΔ in (9) yields ωΔ vΔ − CpΔ = λREF KCp + Kβ Kb f Δ . ω0 v0

(12)

A. Single-Mass Representation The linearized swing equation may be written as [14] sωΔ =

TmΔ − TeΔ . 2H

(13)

Using (7), (8), and (12) in (13) and solving for ωΔ in terms of vΔ and fΔ , the change in electrical power output may be written as in (14), shown at the bottom of the page. Individual transfer functions between the wind speed or the network frequency deviation and the WTG power output can be obtained from (14). Further discussions on these transfer functions follow in the next section. Inspection of (14) gives further insight into the effect of wind velocity and network frequency variations on the power output of a controlled WTG. The transfer function for straightforward MPPT control without any deloading and without any electrical and mechanical power modulator controller may be obtained by setting βREF , Kf , and Kb to zero in (14). Further approximation can be done by considering CpΔ to be zero for small vΔ . The transfer function thus obtained is

Fig. 5. Simplified model of a variable-speed two-mass model WTG. Kb and Kf represent the droop constants for the pitch and electrical power outputs (Pe ), respectively.

at 12 m/s, a 1% change in wind speed results in a 2.19% change in WTG output power. However, at 10 m/s, a 1% change in wind speed results in a 1.52% change in WTG output power. This finding may be verified from the Bodè magnitude plot of (15) at different wind speeds and for WTGs of varying inertia shown in Fig. 4. The WTG behaves like a first-order low-pass filter, with a cutoff frequency determined by the inertia constant of the WTG and the ambient rotor speed. B. Two-Mass Representation In Section III-A, a lumped-mass model of the WTG was considered, similar to steam turbo generators. More accurate results may be obtained by increasing the number of masses used to represent the physical characteristics of an actual wind turbine. A simplified two-mass representation [15] considers the wind turbine, low-speed shaft, and gearbox as a single-mass system and the high-speed shaft and the generator rotor as the other single-mass system (see Fig. 5). Neglecting shaft damping, the following equations describe the two-mass model [16]–[18]:

PeΔ =

2Hω02 s kp (2ω03 +v03 )

vΔ .

(15)

+1

The relationship between the dc gain and the ambient wind speed (v0 ) in (15) is quadratic. This indicates that, for a given wind velocity variation, the resultant variation in output power is much more pronounced at higher wind speeds. For instance,

3kp ω03 v02 (3CpREF −KCp λREF ) (2ω03 +v03 CpREF −KCp v02 ω0 )

PeΔ = (

2Hω02 /kp 2ω03 +v03 CpREF −KCp v02 ω0

)

s+1

vΔ −

(16)

Te dω Ks θtw = − dt 2Hg ω 2Hg

(17)

dθtw = (ωt − ω) dt

9kp ω03 v02

(2ω03 +v03 )

Pt Ks θtw dωt = − dt 2Ht ωt 2Ht

(18)

where ωt and ω[pu] are the turbine and generator speeds, respectively; θtw [pu] is the shaft twist angle; and Ht and Hg [s] are the turbine and generator inertia, respectively. Pt [pu] is the turbine power, Ks is the shaft stiffness (in per unit torque per radian), and Te [pu] is the generator torque, respectively. For small deviations in the wind speed and the grid frequency, the

2HKf ω02 /kp s (2ω03 +v03 CpREF −KCp v02 ω0 )

(

+ Kf −

3ω03 (Kf +kp Kβ Kb v03 )

(2ω03 +v03 CpREF −KCp v02 ω0 )

2Hω02 /kp 2ω03 +v03 CpREF −KCp v02 ω0

)

s+1

fΔ

(14)


5

Fig. 7. Response of a controlled variable-speed WTG to a lower wind velocity of 0.5 m/s (from 12 to 11.5 m/s). Linear and nonlinear models are compared. TABLE I M ODAL A NALYSIS OF A WTG W ITH D ISSIMILAR T URBINE AND S HAFT I NERTIA (Ht = 4.5 s, Hg = 0.5 s, Ks = 2.32 p.u. per electric radian) Fig. 6. Comparison of linear model (13) with an actual DFIG-based wind turbine simulated in DIgSILENT. The perturbations are (a) 0.001-, (b) 0.01-, and (c) 0.1-p.u. rises in frequency for 0.1 s.

generator and turbine speeds, shaft twist angle, and generator torque deviate from their initial values as ω = ω0 + ωΔ , ωt = ωt0 + ωtΔ , θtw = θtw0 + θtwΔ , and Te = Te0 + TeΔ . The linearized WTG model of (16)–(18) is obtained in state-space form using (7) and (8) as ⎤ ⎤ ⎡ 1 kp KCp v02 ⎤ ⎡ kp v03 Cp0 −ks ⎡ 0 − ωtΔ ˙ ωtΔ 2 2Ht ωt0 2Ht ωt0 ⎢ ⎥ ⎣ ω˙Δ ⎦=⎣ −kp ω0 k s ⎦ ⎣ ωΔ ⎦ 0 Hg 2Hg ˙ θtwΔ θtwΔ 1 −1 0 ⎡ ⎤ kp v02 (3Cp0 −KCp λREF ) kp Kβ v03 Kb

2Ht ωt0 2Ht ωt0 ⎥ v ⎢ Δ K f +⎣ (19) ⎦ 0 fΔ 2Hg ω0 0 0 PeΔ = 3kP ω02 ωΔ − Kf fΔ .

(20)

C. Validation of the Linearized Model The linear model of (19) and (20) was validated against an actual DFIG-based WTG simulated in DigSILENT Power Factory [19]. Fig. 6 compares the responses of a WTG to a temporary swell in network frequency for 0.1 s. The values of wind speed were set at 10 m/s, β = 10 , Kf = 15, and Kb = 600, i.e., turbine torque was controlled at deloading condition. The linearized model approximates the nonlinear model dynamics well for small perturbations. The gains of the controllers of the q-axis current loop of the rotor-side converter play an active role in determining the response of the generator. In this particular example, rudimentary tuning of the controller gains was carried out. A further comparison of the step response (see Fig. 7) of the two models to a wind speed change reveals that the steady-state error manifests itself significantly after about 5 s, as is expected for a linearized model.

IV. E FFECT OF D RIVE -T RAIN PARAMETERS ON S YSTEM C OMPLEXITY A complex dynamic system is difficult to reduce. System complexity is indicated by the number of states required to model it. Thus, if some states do not participate in the system dynamics, they may be ignored to arrive at a reduced-order model. This section highlights the importance of the WTG drive-train parameters in determining the complexity and reducibility of the model. The matrices of (19) are useful to understand the drive-train dynamics. Linearization at a wind speed of 10 m/s, i.e., v0 = 0.833, ωt0 = ω0 = 0.8054, kp = 0.73, β = 10 , Kf = 15, Cp0 = 0.9029, λREF = 0.96648, KCp = 0.0771, Kβ = −0.1421, and Kb = 600, yields the eigenvalues shown in Table I. The shaft stiffness was taken as Ks = 2.32 p.u./rad. The inertia of the turbine and generator shafts were taken as highly dissimilar, i.e., Ht ≈ 10 Hg [8], [16], as is usual in WTGs, where the lowspeed turbine is of high inertia. With similar shaft inertia of both turbine and generator and high shaft stiffness, the drive train behaves like a single-mass system. Table II shows the eigenvalues obtained in this case. This is the usual case with conventional synchronous generators. It may be observed that, for all the modes, the WTG with unequal turbine and generator masses shares dissimilar



TABLE II M ODAL A NALYSIS OF A WTG W ITH S IMILAR T URBINE AND S HAFT I NERTIA (Ht = 2.5 s, Hg = 2.5 s, Ks = 50 p.u. per electric radian)

Fig. 9. Incremental power output of the two-mass model WTG (of similar and dissimilar inertia) and the lumped-mass model WTG due to an increase in frequency of 0.001 p.u. TABLE III C LASSIFICATION OF THE T EST W IND FARMS , BASED ON THE I NERTIA OF THE 100 C ONSTITUENT WTG S

Fig. 8. Response of turbine and generator speed deviations due to a 0.001-p.u. step rise in frequency. The upper (or lower) figure is for a drive train consisting of unequal (or equal) turbine and generator shaft inertia.

eigenvectors and participation factors for the two masses. The alternate scenario in which the two masses have equal inertia shows equal participation in all the modes. The torsional mode, in which the torsional angle θtw participates, is also of higher frequency when both masses are equal. Fig. 8 shows the response of turbine and rotor speed for a step rise in frequency of 0.001 p.u., for both scenarios. In fact, a WTG with equal turbine and generator masses behaves similarly to a single-mass drive train (lumped H = 5 s), as shown in Fig. 9. The eigenvector in Table II shows that the equal two masses act in antiphase close to about 180◦ of equal magnitude in the oscillating mode. The model of the WTG is considered for MOR based on balanced truncation to highlight the electromechanical dynamics further [20], [21]. It is found that the dynamic model of the WTG drive train is more reducible if the turbine and generator masses are unequal, compared with when the two masses are equal. For the scenario of dissimilar turbine and generator inertia, it can be concluded from the distribution of the HSVs that only two states actively contribute to the system dynamics. The system is thus reducible from three to two. Conversely, in a WTG with similar turbine and generator inertia, it may not be prudent to reduce the order, as all three states are equally important in determining the system dynamics. The scope for order

reduction assumes greater significance, when wind farms consisting of hundreds of WTGs are modeled from (19) and (20). V. W IND FARM L INEARIZED M ODEL A dynamic model of a wind farm consisting of n WTGs may be obtained from (19) and (20), as a multiple-input–singleoutput incremental model, where each individual WTG is linearized around its steady-state operating point. The model becomes a single-input–single-output system if all the WTGs experience identical wind speed variations or frequency variation. Adopting the two-mass drive train to model individual WTGs, two wind farms were considered for the study (see Table III), each consisting of 100 variable-speed WTGs. Wind Farm I consists of WTGs of randomly distributed dissimilar inertia, whose turbine and generator masses were unequal. Wind Farm II consists of WTGs with drive trains comprising of randomly distributed equally sized turbine and generators and a high shaft spring constant. The HSVs were obtained for both Wind Farms I and II, as shown in Fig. 10. For Wind Farm II, only the first five HSVs are of significant magnitude, indicating that the farm may be reduced to a fifth-order dynamic model. For Wind Farm I, three HSVs are of significance, indicating the potential to reduce the order to three. Upon reducing the wind farm, the step response was used to gauge the efficacy of the MOR. Fig. 11 shows the response of the output of Wind Farms I


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VI. C ONCLUSION

Fig. 10. Largest HSVs of wind farms modeled from WTGs having similar or dissimilar turbine and generator inertia.

Simplified linearized models for controlled variable-speed WTG have been developed in this paper, which simulate the electromechanical dynamics in response to changes in wind speed and network frequency deviations. The models are valid at low wind speeds, when the only purpose to modulate the pitch of the turbine from its preset value is to enable the deloaded WTG to provide network frequency support. The model has been linearized at a particular wind velocity to obtain state-space representation changes in wind velocity and network frequency, to the electrical power output variations. The effect of drive-train parameters—inertia of the turbine and generator on the model complexity—has been studied. It is shown that, if the generator and turbine masses vary significantly, the wind farm model exhibits very uncomplicated dynamics, which may be represented by a third-order model. Furthermore, a wind farm consisting of WTGs whose generator and turbine inertia are of comparable mass would acquire greater degrees of freedom, and consequently, the power output shows greater oscillations due to input variations, which may be represented by a fifth-order model. Fundamental conclusions have been drawn with regard to the behavior of the WTG under varying wind conditions and about the controllers suitable for primary frequency support by WTGs. The results for frequency variations of linear and nonlinear models agree well for the cases discussed in this paper, thereby providing validation of the linearized model. The applications of the models developed in this paper are in the area of design of WTG controllers and system-wide frequency control strategies. ACKNOWLEDGMENT The authors would like to thank Mr. Abhinav Kr. Singh for the contribution toward the development of some of the concepts in this paper. R EFERENCES

Fig. 11. Comparison of response of original and reduced models of Winds Farm I and II to change in wind speed. (a) Simultaneous increase in wind velocity by 0.5 m/s at all WTGs. (b) Gradual drop in wind velocity of up to 0.5 m/s.

and II for an increase in wind speed of 0.5 m/s. The transfer function of Wind Farm I was calculated as −0.96s2 + 2.26s + 153.2 vΔ s3 + 1.71s2 + 3.02s + 0.7 −1500s3 + 1088s2 − 4246s − 2050 fΔ . + s3 + 1.71s2 + 3.02s + 0.7

PeΔ_WF [pu] =

(21)

In a wind farm, local weather phenomena may introduce random variations in the wind speed seen by a particular WTG. Modeling the wind farm as a multiple-input system can incorporate the effect of multiple wind speeds encountered across the wind farm. Fig. 11(b) shows the total output power deviations of the wind farms due to different wind speeds seen by WTGs in the farm. The first WTG sees no change in wind speed, whereas the hundredth WTG sees a drop of 0.5 m/s. Adjacent WTGs see wind speed deviations that differ by 0.005 m/s. Inspection of the figures reveals that the reduced model approximates the original one satisfactorily.

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Sudipta Ghosh (S’12) received the Bachelor’s degree from National Institute of Technology (NIT) Jamshedpur (formerly Regional Institute of Technology Jamshedpur), Jamshedpur, India, the M.Tech degree from NIT Durgapur, and the Ph.D. degree from Indian Institute of Technology, Delhi, India. He was with Bajaj Electricals Ltd. and Havells India Ltd. He is currently an Assistant Professor with the Department of Electrical Engineering, Indian School of Mines, Dhanbad, India. His research interests include the areas of power system small-signal stability, power system model order reduction, and dynamic analysis of wind power system problems.

Nilanjan Senroy (M’06) received the Ph.D. degree from Arizona State University, Tempe, AZ, USA. He was a Postdoctoral Fellow with the Center for Advanced Power Systems, Florida State University, Tallahassee, FL, USA. He is currently an Associate Professor with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India. His research interests include power system stability and control and renewable energy.