analysis of reactive power capability for doubly-fed...

Analysis of Reactive Power Capability for Doubly-Fed Induction Generator of Wind Energy Systems Using an Optimal Reactive Power Flow

E. A. Belati1, A. J. Sguarezi Filho1, M. B. C. Salles2 1 Universidade Federal do ABC (UFABC), R. Santa Adelia, 166, 09210-170 Santo Andre, SP, Brasil. 2 Universidade de São Paulo (USP), Av. Professor Luciano Gualberto, trav.3 – nº158, São Paulo, SP – Brasil.

This approach aims optimal operation of power system with reactive power control in wind turbines. In this work was used a reactive power control to Doubly-Fed Induction Generator (DFIG) and analyzed the benefits to the power system provided by optimal reactive power injecting control performed via Optimal Reactive Power Flow (ORPF). The approach is divided in two parts. In the first, the DFIG reactive power control is achieved using stator field orientation control and PI controllers. In the second part, the ORPF based in the Primal-Dual Logarithmic Barrier Augmented Lagrangian Function (PDLBALF) is used to optimize reactive power dispatch aiming to minimize active power losses system. Case studies on the modified IEEE 14 bus system are carried out to verify reactive power control influence in DFIG for power systems using the ORPF shown.

Keywords: reactive power control; optimal reactive power flow; wind energy; doubly-fed induction generator.

1. Introdution

The use of renewable sources for electric power generator has experienced great changes in the last few years. The renewable energy systems and specially wind energy have attracted interest due to the increasing concern about CO2 emissions. The wind energy systems using a doubly-fed induction generator (DFIG) have some advantages due to variable speed operation, as well as the four quadrant active and reactive power capabilities compared with fixed speed induction squirrel cage and with the conventional synchronous generators [1, 2]. The power control of DFIG in wind turbine is traditionally based on either stator-flux-oriented [1] or stator-voltage-oriented [2] vector control. In this case, the scheme decouples the rotor current into active and reactive power components to be achieved by the rotor current control. The proportional plus integral (PI) controllers and the stator-flux-oriented applied to the DFIG power control have been presented in [3]. The voltage profile and the active power losses in the power system can be optimized with reactive power sources, what motivates studies related to the reactive power control for DFIG [4, 5]. The reactive power capability of the DFIG is limited due the maximum current of the power converter [6-8]. In most of cases this value is 1.1 pu and the maximum power factor of the generator is 0.9. These works do not consider optimal reactive power that the power system needs. Thus, the optimal reactive power injection has to be evaluated in function of the reactive power that can be generated by the wind farm for each wind speed. Using DIFG reactive power control the optimal reactive power injection can be evaluated respecting the DFIG constraints for each wind speed, specified load and generation and constraints of the power system. This optimal can be evaluated via optimal reactive power flow (ORPF). ORPF is a non convex static nonlinear programming problem; it is one of the most powerful tools to analyses static systems of electrical energy. The ORPF used has the objective of minimizing a function and, at the same time, of satisfying a set of physical and operational constraints in power systems, e.g. reactive power injection constraint. As a solution, it provides the optimal operation point for the electrical network for a given load and generation configuration of the system satisfying all system constraints. It was proposed by Carpentier in the early 1960s based on the economic dispatch problem [9]. Since then, many papers have been written in an attempt to solve the problem [10-14] This work considers the reactive power injection capacity of a wind farm using DFIG to optimize the active power losses in a power system. In this way is proposed to use an ORPF for a system with DFIG. The DFIG power control is achieved by using stator flux orientation and PI controllers. The ORPF used in this paper is based in the Primal-Dual Logarithmic Barrier Augmented Lagrangian Function (PDLBALF) approach [15]. Thus, the contribution of this work is the analyzes of the benefits of the reactive power injection, by an wind farm with reactive power control, to the power system provided by optimal reactive power injecting control performed via ORPF.

2. Machine Model and Rotor Current Vector Control

2.1 Doubly Fed Machine Model - DFIM

The DFIM model in synchronous reference frame is given in[16] and shows by Eq. (1) and Eq. (2).

Materials and processes for energy: communicating current research and technological developments (A. Méndez-Vilas, Ed.)____________________________________________________________________________________________________

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dq11dq1

dq11dq1 jdt

RiRv λω

λ

++= (1)

( ) dq2mec1dq2

dq22dq2 PPjdt

RiRv λωω

λ

−++= (2)

The relationship between fluxes and currents is given by Eq. (3) and Eq. (4).

dq2Mdq11dq1 iLiL

+=λ (3)

dq22dq1Mdq2 iLiL

+=λ (4)

The machine dynamics is given by Eq. (5).

( ) M*

dq1dq1mec TiImPP

2

3

dt

dj −= λω

(5)

The generator active and reactive power is given by Eq. (6) and Eq. (7).

( )q1q1d1d1 iviv2

3P += (6)

( )q1d1d1q1 iviv2

3Q −= (7)

The subscripts 1 and 2 represent the stator and rotor parameters respectively, ω1 is the synchronous speed, ωmec is the machine speed, R1 and R2 are the stator and rotor windings per phase electrical resistance, L1, L2 and LM are the proper

and mutual inductances of the stator and rotor windings, v

is the voltage vector, i

is the current vector, λ

is the flux vector, PP is the machine number of pair of poles, J is the load and rotor inertia moment and TM is the mechanical torque. The DFIG power control is achieved by rotor current control and hence independent stator active P and reactive Q power control. In this case, P and Q are computed by each individual rotor current. By using stator flux oriented, that decouples dq axis (3) becomes

d21

M

1

1d1 i

L

L

Li −= λ

(8)

q21

Mq1 i

L

Li −= (9)

where dq11d1 λλλ

== . The active (6) and reactive (7) power can be computed by using (8) and (9).

q21

M1 i

L

Lv

2

3P −= (10)

−= d2

1

M

1

11 i

L

L

Lv

2

3Q

λ (11)

where dq1q11 vvv

== . Thus, if the rotor currents are controlled, so the stator active and reactive power control is

achieved. For the power factor control, the reactive power is computed by using the active power reference and the desired power factor (pf) as given by

pf

pf1PQ

2

refref

−= (12)

The reactive power capability of the DFIG is limited due the maximum current of the converter [6, 7]. In most of cases it is 1.1 pu and the maximum power factor is 0.9.


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2.2 Power Control

The DFIG power control is achieved using a power electronic converter as shown in Figure 1 and using the theory presented in section 2.1. This converter consists in a back-to-back converter connecting the grid and the rotor windings, the stator circuits are connected directly to the grid. a) Rotor Side Converter: In normal operation, the rotor-side converter (RSC) regulates the reactive power injection

and the developed electric power (Pelec). The optimum electric power reference ( *optP ) is calculated taking into account

the optimal rotor speed for the incoming wind considering the Cp curves presented in [17]. To aim this control the PIs controllers are used. The PI controllers process the error between the active and reactive power references and the computed active and reactive power, so it generates references of rotor current components in direct and quadrature axis

refd2i and refq2i . The expressions to calculate the rotor current references are done by:

( )

+−=

s

kikpQQi refd2 ref

(13)

( )

+−=

s

kikpPPi refq2 ref

(14)

Again, PI controllers process the error between the rotor current reference (refd2i and

refq2i ) and the actual rotor

current components ( d2i and q2i ) so it generates references of rotor vector components refd2v and

refq2v , that allow the

active and reactive power to reach the references values. The expressions to calculate the rotor voltage vector are done by:

( )

+−=

s

kikpiiv d2d2d2 refref

(15)

( )

+−=

s

kikpiiv q2q2q2 refref

(16)

where kp is the proportional gain and the ki is the integral gain of the PI controller. After a dq0-to-abc transformation using the stator flux and rotor positions,

refd2v and refq2v are sent to the PWM (Pulse-width Modulation) signal

generator. The block diagram of this strategy [17], is shown in Figure 2.

Fig. 1 Configuration of DFIG connected directly on grid. Fig. 2 RSC control diagram for DFIG in normal operation. b) Grid Side Converter: In normal operation, the grid side converter (GSC) controls the DC link voltage of the back-to-back converter [17]. This strategy uses PI controllers which generate the grid voltage gdv and gqv by

controlling the grid current of references refgdi and

refgqi . In this case, gdi controls the link DC voltage. So the

expression to control the link DC voltage the refgdi is calculated by:

( )

+−=

s

kikpVVi dcdcgd refref

(17)

The grid voltage gdv and gqv are achieved by:


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( )

+−=

s

kikpiiv gdgdgd refref

(18)

( )

+−=

s

kikpiiv g2gqgq refref

(19)

The controller also uses a PLL (Phase Locked Loop) that provides the angle (φ ) to the abc-to-dq0 (and dq0-to-abc)

transformations and it allow to synchronize the three-phase voltages at the converter output with the zero crossings of the fundamental component of the phase-A of the terminal voltage. The GSC control strategy is shown in Figure 3.

Fig. 3 GSC control diagram for DFIG in normal operation.

3. Optimal Reactive Power Flow

The ORPF is a variation of Optimal Power Flow (OPF). The OPF is a nonconvex large-scale constrained nonlinear problem, and is the recommended means of finding the best set of operating conditions in an electrical power system, by simultaneously making optimal adjustments to all the control variables.

3.1 The Problem

The optimal reactive power flow problem can be mathematically described by Eq. (20).

maxmin

j

i

r,...,1j,0)(h

m,...,1i,0)(gt.s

)(fmin

xxx

xx

x

≤≤

=≤==

(20)

where x ϵ Rn is the control and state variable vector representing voltage magnitudes, voltage angles and Load Tap Changer (LTC) taps. The objective function f(x) represents the active power losses in the transmission system. The equality constraints g(x)=0 represent the power flow equations for scheduled load and generation. The inequality constraints h(x)≤0 represent the functional constraints of the power flow, e.g. limits of active and reactive power flows in the transmission lines and transformers and limits of reactive power injections for reactive control buses. This is a typical nonlinear and no convex problem. The ORPF used employs the formulation presented in Baptista et al. [15]. In the formulation of the problem described here, we handle the equality constraints by Newton’s method, the simple-bound voltage and tap (LTC) constraints by the logarithmic barrier method and the remaining simple-bound and inequality constraints by the augmented Lagrangian method.

3.2 Primal-Dual Logarithmic Barrier and Augmented Lagrangian Function

The definition presented in this section is a combination of ideas from two methods, barrier and augmented Lagrangian, applied to problem (20). This problem may be described by Eq. (21).


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0

0

0z

r,...,1j,0z)(h

m,...,1i,0)(gt.s

)(fmin

j

min

max

jj

i

≥

≥

≥=−

=+

==+==

2

1

2

1

ss

xsxxsx

xx

x

(21)

where: (s1)t=(s 1

1, ... ,s 1

n ), with s 01l ≥ , and (s2)t=(s 1

2, ... ,s 2

n ), with s 02l ≥ , l= 1,.....,n. The variables zj, j=1,...,r, as well

as the components of vectors s1 and s2, are slack variables. s1 and s2 are auxiliary vectors. Problem (2) can be modified to incorporate the strictly positive variables and the constraint hj(x) + zj = 0 into the objective function, the former via the logarithmic barrier function and the latter via the penalty function. After modification, the problem becomes:

0z

r,...,1j,0z)(h

m,...,1i,0)(gt.s

)z)(h(2

1slnsln)(fmin

j

min

max

jj

i

r

1j

2jj

n

1l

2k

n

1l

1k

≥=−

=+

==+==

++−− ===

xsxxsx

xx

xx

2

1

υδδ

(22)

The problem (22) can be associated with the following augmented Lagrangian [18]:

=== =

===

++−−+−++

+++−−=

r

1jjjj

n

1l

min1

211

21

m

1l

n

1l

max1

111

11ii

r

1j

2jj

n

1l

2k

n

1l

1k

)z)(h()xsx()xsx()(g

)z)(h(2

1slnsln)(f),,,(La

xx

xμπψx

μππψ

υδδ x(23)

where: ψ, π1, π2, and μ are vectors of Lagrange multipliers, υ > 0 is the penalty factor and δ the barrier factor. Minimizing (23) with respect to zj, j=1,...,n, and applying the necessary conditions for optimality,

r,,1j,0),,,(Lajz ==∇ μπψx (24)

we obtain:

r,,1j),(hz jj

j =−−= xυμ

(25)

As zj ≥ 0, we have:

=≤−−

≥−−−−= r,1j

0)(hse,0

0)(hse),(hz

jj

jj

jj

j

x

xx

υμυμ

υμ

(26)

Substituting (26) into (23) we obtain the PDLBALF:

==

= ===

−≤−−

−≥++−−

+−+++−−=

r

1j j2j

j2jjjn

1l

min1

211

21

m

1l

n

1l

max1

111

11ii

n

1l

2k

n

1l

1k

)(hif,2

)(hif),(h2

1)(h

)xsx(

)xsx()(gslnsln)(f),,,(La

υμ

υμ

υμ

υμπ

πψδδ

x

xxx

xxμπψx

(27)


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3.3 Solution of the PDLBALF

Applying conditions of optimality to the problem (27) associated with problem (20), we obtain the nonlinear system of equations (28), the solution of which consists of the search directions Δx, Δs1, Δs2, Δλ, Δπ1 and Δπ2 :

=∇=∇=∇=∇

0),,,(La

0),,,(La

0),,,(La

0),,,(La

s

x

μπψxμπψxμπψxμπψx

π

λ (28)

This system of equations can be represented as follows:

=−−=−+

==

==−−

==+−

=

−≤

−≥∇+++++∇

=

0

0

m,...,1i,0)(g

n,...,1l,0s

n,...,1l,0s

0)(hse,0

)(hse),(h))(h()(I)()(J)(f

min

maxi

2l2

l

1l1

l

r

1j jj

jjjxjj

t2t1ttx

xsxxsx

x

x

xxxIxx

.2

1

πδ

πδυμ

υμ

υμππψ

(29)

where ))(g,),(g()( mx1xT xxxJ ∇∇= and I the identity matrix.

The solutions to the nonlinear equations (29) are found by Newton’s method, which results in a matrix system that, simplified, can be written:

aLd.W −∇=Δ (30)

where

−

−

∇

=

0000

0000

00000)(

00)(00

000)(0

)(00La t2xx

IIII

xJIS

ISIIxJ

W2

1

δδ

, with

=

21n

211

)s(

10

0)s(

1

1S ,

=

22n

221

)s(

10

0)s(

1

2S and

( )

=

−≤

−≥∇∇+∇+++∇=∇

r

1j jj

jjjx

tjxj

2xjj

t2xxa

2xx

)(hif,0

)(hif),(h))(h()(h)(h)()(fL

υμ

υμ

υυμψ

x

xxxxxxHx , where H(x) is the

Hessian of the constraint g(x); ( )2121T ,,,s,s,xd πΔπΔψΔΔΔΔΔ = ; and

−−−+=

=−−

=+−

−≤

−≥∇+++++∇

=∇

=

min2

max1

xsxxsx

x

x

xxxIIxJ

m,...,1i),(g

n,...,1l,s

n,...,1l,s

)(hse,0

)(hse),(h))(h()()()()x(f

L

i

2l2

l

1l1

l

r

1j jj

jjjxjj

t2t1ttx

a

πδ

πδυμ

υμ

υμππψ

.

Using the search directions obtained from (30), the vectors of the variables, x, s1 and s2 and of the Lagrange multipliers, ψ, π1 and π2 are updated, as follows:


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k2d

k21k2

k1d

k11k1

kd

k1k

2p

22

1p

11

kp

k1k

kk1k

kk1k

sss

sss

xxx

πΔαππ

πΔαππ

λΔαψψ

Δα

Δα

Δα

+=

+=

+=

+=

+=

+=

+

+

+

+

+

+

(31)

where αp and αd are the updating rates used for the primal and dual variables, respectively. These rates are chosen so as to keep every component of the auxiliary vectors s1 and s2 strictly positive and elements of the dual vectors ψ, π1 and π2 with unchanged signs. A recommended strategy for the calculation of the maximum rate is that used by Granville [10]:

(32)

(33)

where σ = 0.9995 is an empirical value which, according to Wright [19], can be derived from the formula m9

11− ,

where m is the number of constraints in the problem. The updating process continues until convergence is achieved, within an error specified for Newton’s method. When the new point that has been reached is found to satisfy the KKT conditions, the process ends, otherwise it goes on to update the Lagrange multipliers μ, using the rule given by [20]:

r,,1j

)x(hse,0

)x(hse),x(h

k

kj1k

j

k

kj1k

j1k

jkk

j1k

j =

−≤

−≥+=

+

++

+

υμ

υμ

υμμ (34)

and the penalty factor v and the barrier factor δ, as follows:

1,k1k >=+ βυβυ (35)

0,k

1k >=+ ααδδ (36)

where β and α are called correction parameters and in the present work were chosen empirically. The updated linear system is solved again and the steps continue until the KKT conditions are satisfied.

3.3.1 Simplified algorithm

Initialization Step • Given problem (22), construct the PDLBALF (27); • Let k = 0; • Choose initial values for the problem variables: x0, ψ0, (π1)0, (π2)0, (s1)0, (s2)0, μ0, υ0, δ0; • Go to Main Steps.

Main Steps 1. Obtain system (28), solve it and go to step 2; 2. Update the vectors x, ψ, s1, s2, π1 and π2, using (31) and go to step 3; 3. If the stopping criterion for Newton’s method is satisfied, go to step 4, otherwise, return to step 1; 4. If problem variables satisfy KKT, END, otherwise, go to step 5; 5. Update the Lagrange multipliers, penalty factors and barrier factors, using (34), (35) and (36), set k = k + 1 and return to step 1.

Steps 1 to 3 represent the internal iteration, i.e. Newton’s method, and step 5 the external iteration.

}1,)s

smin,

s

smin({min

2

2

0s1

1

0sp 21 ΔΔ

σαΔΔ <<

=

}1,)min,min({min2

2

01

1

0d 21 πΔ

ππΔπσα

πΔπΔ <>

−=


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4. Simulation Results

The reactive power control strategy was simulated using MATLAB/SimPowerSystems package. The DFIG parameters are shown in Appendix (Tables A.1 – A.3). Figure B.1 shows the schematic of the implemented system and DFIG power control are presented in Figures 1 – 3. The inverter was modeled as controlled voltage source. It was simulated the wind energy system making the FP = 0.95 and several wind operation as shown in Table A.2 due to the fact in this work the maximum power factor is 0.95. The reactive power Q for each wind speed (more than 9m/s) can be adjusted from FP=1 till FP=0.95. The simulations results of the reactive power control made in steady state are presented in Table A.2. These results will be used by the ORPF algorithms. The analysis of a power system using the ORPF is presented in section 3 with DFIGs forming a wind farm. The ORPF was implemented in FORTRAN using double-precision arithmetic. The computational work was performed on an Intel Core i5 CPU 2.5 GHz microprocessor. The studies were performed on the modified IEEE 14 bus systems. The systems data are shown in the Tables B.1 - B.2). For each test the solution was obtained with a precision of 10-5 pu for the power balance equations (internal iteration). The upper and lower voltage limits considered in the studies were 0.9 and 1.1 pu. The tests were accomplished with the following initial conditions: Vk=1.0 and θk=0.0. The Lagrange multipliers associated with equality and inequality, ψ = 0 and μ = π1 = π2 = 0 respectively. The barrier factor δ = 0.001, its correction parameter α = 10 and the slack variables s1 = s2 = 0.1. The penalty factor associated with the inequality constraint, υ = 1, with a correction parameter β = 1.2.

4.1 Validation of ORPF

The performance of the ORPF can be seen in the following comparative test in which system losses and voltage magnitude to modified IEEE 14 bus systems were compared with the Power Flow (PF) solution by Newton’s Method [21]. The Figure 4 shows two voltage magnitude curves for the system using the algorithm ORPF and PF for wind speed of 14 m/s according to Table A.2. The active power losses obtained by PF totaled 7.241 and by ORPF 5.472 MW having a gain of 24.43%. The ORPF optimized the reactive power injection making the system more efficient. The liquid reactive power injection obtained via ORPF and PF was 86.599 and 146.637 Mvar respectively.

4.2 System performance considering the DFIG

Considering the wind farm connected at bus 8 was performed simulations for the wind conditions of 6 m/s to 14 m/s considering the data generator according to the Table A.1 for all wind speed. Figure 5 shows in a clear way that reactive power injection contributes to improving voltage profile. From 9 m/s the generator provides reactive power. From this speed the voltage profile improves resulting in an active power losses reduction.

Fig. 4 Voltage magnitude to the modified IEEE 14 bus systems using the algorithms ORPF and PF.

Fig. 5 Magnitude voltage to the modified IEEE 14 bus systems in relation to wind speed.

The optimal reactive power injection in the system is illustrated in Figure 6. The wind farm (bus 8) presents a great contribution in the reactive power injection from 9 m/s. The bus 2 generated maximum reactive power, 50 Mvar, until 9 m/s. From this speed, the bus 8 start reactive power injection causing a decrease in the reactive power generation of the bus 2. The slack bus contributed with the reactive power balance. Figure 7 illustrate the active power losses in relation to wind speed range. It shows that from speed of 13 m/s the active losses began to increase. This is due to location of the wind farm and amount of active power generated. In this situation we need to evaluate the cost of MWh to each generation. Considering that the wind generation has a low cost per MWh, this situation for system operation may be viable, even causing an increase in active power losses.

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Volta

ge m

agni

tude

(pu)

Bus

Without reactive power support With reactive power support

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Volta

ge m

agni

tude

(pu)

Bus

6 m/s

7 m/s

8 m/s

9/ms

10/ms

11/ms

12/ms

13/ms

14/ms


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Fig. 6 Reactive power injection in the modified IEEE 14 bus systems in relation to wind speed.

Fig. 7 Active power losses to the modified IEEE 14 bus systems in relation to wind speed.

5. Conclusions

This paper presents an approach to optimal operation of power system with reactive power control in wind turbines. In this work is used a reactive power control for DFIG-based wind turbine using stator field orientation for high control performance. The steady state simulations results are used in ORPF algorithms. An ORPF based in the Modified Barrier Lagrangian Function approach to optimize reactive power dispatch aiming to minimize active power losses system was utilized. The ORPF was able to optimize the reactive power dispatch of the system considering the operational constraints. In the tests performed with the modified IEEE 14 bus system was observed a better voltage profile and power loss which shows the importance of injecting reactive power provided from wind generators. Therefore it is evident the benefits of using wind generators with reactive power control for optimize the system.

Acknowledgements The support by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) is gratefully acknowledged.

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-10.00

0.00

10.00

20.00

30.00

40.00

50.00

60.00

6 7 8 9 10 11 12 13 14

Reac

tive

Pow

er In

ject

ion

(Mva

r)

Speed (m/s)

Bus 1

Bus 2

Bus 8

0.00

2.00

4.00

6.00

8.00

10.00

12.00

6 7 8 9 10 11 12 13 14

Activ

e Lo

sses

(MW

)

Speed (m/s)


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[16] Leonhard W. Control of Electrical Drives: Springer-Verlag Berlin Heidelberg New York; 1985. [17] Kundur P. Power System Stability and Control: McGraw-Hill; 1994. [18] Santos A, Jr., Da Costa GRM. Optimal-power-flow solution by Newton's method applied to an augmented Lagrangian function.

Generation, Transmission and Distribution, IEE Proceedings- 1995, 142:33-36. [19] Wright M. Why a Pure Primal Newton Barrier Step May be Infeasible. SIAM Journal on Optimization 1995, 5:1-12. [20] Hestenes M. Multiplier and gradient methods. Journal of Optimization Theory and Applications 1969, 4:303-320. [21] Tinney WF, Hart CE. Power Flow Solution by Newton's Method. Power Apparatus and Systems, IEEE Transactions on 1967,

PAS-86:1449-1460.

Appendix

A. Wind farm electrical systems parameters

Table A.1 Doubly-fed induction generator characteristic. Table A.3 Turbine characteristic.

R1 (pu) 0.01 L1 (pu) 0.1 R2 (pu) 0.01 L2 (pu) 0.08 Mm (pu) 3 H (s) 0.5 Number of Pole 4

Table A.2 DFIG reactive power capability.

Wind (m/s)

P (MW) Q (Mvar)

S (MVA)

FP

6 16.3 0 16.3 1 7 23.75 0 23.75 1 8 33.85 0 33.85 1 9 64.36 21.33 67.85 0.95 10 80 26.34 84.22 0.95 11 98.55 32.5 103.77 0.95 12 124.24 40.93 130.80 0.95 13 157.32 51.76 165.61 0.95 14 164.64 54.11 173.30 0.95 The reactive power Q for each wind speed (more than 9m/s) can be adjusted from FP=1 so Q = 0 Mvar till FP=0.95 lead.

Min. Rotor Speed - variable speed (rpm)) 9 Nom. Rotor Speed – variable speed (rpm) 14 Rotor diameter (m) 75 Area covered by rotor (m2) 4418 Nom. Power (MW) 2 Nom. Wind Speed – variable speed (rpm) 14 Gear box ratio - variable speed 1:100 Inertia constant (s) 2.5 Shaft stiffness – fixed speed (pu/ el rad) 0.3


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B. Modified IEEE 14 bus system

Table B.1 Data line. from to r (pu) x (pu) bsh (pu)

1 2 1.94 5.92 5.28

1 5 5.4 22.3 5.28

2 3 4.7 19.8 4.38

2 4 5.81 17.63 3.74

2 5 5.7 17.39 3.40

3 4 6.7 17.1 3.46

4 5 1.34 4.21 1.28

4 7 0.01 20.91 0

4 9 0.01 55.62 0

5 6 0.01 25.2 0

6 11 9.5 19.89 0

6 12 12.29 25.58 0

6 13 6.62 13.03 0

7 8 0.01 17.62 0

7 9 0.01 11.00 0

9 10 3.18 8.45 0

9 14 12.71 27.04 0

10 11 8.2 19.21 0

12 13 22.09 19.99 0

13 14 17.09 34.80 0

Table B.2 Data buses with reactive control.

bus Q (Mvar) Qmin (Mvar) Qmax (Mvar)

1 0 -200 200

2 12.7 -40 50

8 shown in Table A.1

Fig. B.1 Modified IEEE 14 bus system configuration.


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analysis of reactive power capability for doubly-fed...

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