Materials
and processes for energy: communicating current research and technological developments (A. Méndez-Vilas, Ed.)
____________________________________________________________________________________________________
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
2
Universidade Federal do ABC (UFABC), R. Santa Adelia, 166, 09210-170 Santo Andre, SP, Brasil.
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-voltageoriented [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 statorflux-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).
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


Rλ1dq

+ jω1 λ1dq
v1dq = R1 i1dq +
dt



Rλ 2 dq

+ j (ω1 − PPω mec )λ 2 dq
v 2 dq = R2 i2 dq +
dt
The relationship between fluxes and currents is given by Eq. (3) and Eq. (4).



λ1dq = L1i1dq + LM i2 dq



λ2 dq = LM i1dq + L2 i2 dq
(1)
(2)
(3)
(4)
The machine dynamics is given by Eq. (5).
j
(
)
 
dω mec 3
= PP Im i1dq λ*1dq − TM
dt
2
(5)
The generator active and reactive power is given by Eq. (6) and Eq. (7).
(
)
(6)
(
)
(7)
P=
3
v1d i1d + v1q i1q
2
Q=
3
v1q i1d − v1d i1q
2
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
i1d =
λ1
L1
i1q = −
−
LM
i2 d
L1
(8)
LM
i2 q
L1
(9)

where λ1d = λ1 = λ1dq . The active (6) and reactive (7) power can be computed by using (8) and (9).
3 L
P = − v1 M i2 q
2 L1
(10)

3  λ1 LM
v1  −
i2 d 
(11)
2  L1 L1


where v1 = v1q = v1dq . Thus, if the rotor currents are controlled, so the stator active and reactive power control is
Q=
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
Qref = Pref
1 − pf 2
pf
(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
*
) is calculated taking into account
and the developed electric power (Pelec). The optimum electric power reference ( Popt
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
i 2 d ref and i2 qref . The expressions to calculate the rotor current references are done by:
ki 

i2 d = Qref − Q  kp + 
s

(13)
ki 

i2 q = Pref − P  kp + 
s

(14)
ref
ref
(
)
(
)
Again, PI controllers process the error between the rotor current reference ( i2 d ref and
i2 qref ) and the actual rotor
current components ( i2 d and i2 q ) so it generates references of rotor vector components v2 dref and v2 qref , that allow the
active and reactive power to reach the references values. The expressions to calculate the rotor voltage vector are done
by:
(
ref
(
ref
)
(15)
)
(16)
ki 

v2 d = i2 d − i2 d  kp + 
s

ref
ki 

v2 q = i2 q − i2 q  kp + 
s

ref
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, v2 dref and v2 qref 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 vgd and vgq by
controlling the grid current of references i gdref and i gqref . In this case, igd controls the link DC voltage. So the
expression to control the link DC voltage the igdref is calculated by:
(
)
ki 

igd = Vdc − Vdc  kp + 
s

ref
ref
(17)
The grid voltage vgd and vgq are achieved by:
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v gd
ref
(
)
(18)
)
(19)
ki 

= i gd − i gd  kp + 
s

(
ref
ki 

v gq = igq − i2 g  kp + 
s

ref
ref
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).
min f ( x )
s .t g i ( x ) = 0 ,
h j ( x ) ≤ 0,
i = 1,...,m
(20)
j = 1,...,r
x min ≤ x ≤ x max
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 simplebound 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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min f ( x )
s.t g i ( x ) = 0 ,
i = 1,...,m
h j ( x ) + z j = 0,
1
j = 1,...,r
max
x+s = x
(21)
x − s 2 = x min
zj ≥0
s1 ≥ 0
s2 ≥ 0
1
1
2
2
where: (s1)t=(s 1 , ... ,s n ), with s 1l ≥ 0 , and (s2)t=(s 1 , ... ,s n ), with s l2 ≥ 0 , 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:
n
n
1 r
min f ( x ) − δ  ln sk1 − δ  ln sk2 + υ ( h j ( x ) + z j )2
2 j =1
l =1
l =1
s.t g i ( x ) = 0 ,
i = 1,...,m
h j ( x ) + z j = 0,
1
x+s = x
j = 1,...,r
(22)
max
x − s 2 = x min
zj ≥ 0
The problem (22) can be associated with the following augmented Lagrangian [18]:
La( x,ψ,π,μ ) = f ( x ) − δ
n
 ln s
1
k
−δ
l =1
m
n
ψ g ( x ) + π
i
i
l =1
n
 ln s
2
k
l =1
1
1
1
1
1
+ υ
2
max
1
( x1 + s − x
)+
l =1
r
( h ( x ) + z
j
j
)2 +
j =1
n
π
2
1
2
1
min
1
( x1 − s − x
l =1
)+
(23)
r
 μ ( h ( x )+ z
j
j
j
)
j =1
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,
∇ z La( x,ψ,π,μ ) = 0 ,
j
j = 1, , r
(24)
we obtain:
zj = −
μj
− h j ( x ), j = 1, ,r
υ
(25)
As zj ≥ 0, we have:
μj
 μj
− h j ( x ), se −
− hj( x )≥ 0
 −
υ
υ
zj = 
μj

0,
se −
− hj( x ) ≤ 0

υ
j = 1, r
(26)
Substituting (26) into (23) we obtain the PDLBALF:
La( x,ψ,π,μ ) = f ( x ) − δ
n
 ln s
l =1
1
k
−δ
n
m
n
 ln s + ψ g ( x ) + π
2
k
l =1
i
i
l =1
1
1
( x1 + s11 − x1max ) +
l =1
μj

1 2
μ j h j ( x ) + υh j ( x ), if h( x ) ≥ −
υ
2
π 12 ( x1 − s12 − x1min ) + 
μ 2j
μj
l =1
j =1 
−−
, if h( x ) ≤ −
υ
2υ

n

r

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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 :
∇ x La( x,ψ,π,μ ) = 0

∇ s La( x,ψ,π,μ ) = 0

∇ λ La( x,ψ,π,μ ) = 0
∇π La( x,ψ,π,μ ) = 0
(28)
This system of equations can be represented as follows:

μj

r ( μ j +υ h j ( x )) ∇ x h j ( x ), se h j ( x ) ≥ −


υ =0
∇ x f ( x )t +ψ t J ( x ) + ( π 1 )t I + ( π 2 )t I + 
μ
j

j =1 
0 , se h j ( x ) ≤ −


υ

δ
1
− 1 + π l = 0 , l = 1,...,n

sl
(29)

δ

2
− 2 − π l = 0 , l = 1,...,n

sl

g
i ( x ) = 0 , i = 1,..., m


x + s 1 − x max = 0


x − s .2 − x min = 0

where J ( x )T = ( ∇ x g 1 ( x ), ,∇ x g m ( x )) 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:
W .Δd = −∇La
(30)
 ∇ 2xx La
J ( x )t
0
0

δ( S1 )
0
0
 0

2
δ( S )
0
0
where W =  0
 J( x )
0
0
0

I
0
0
 I
 I
I
−
0
0

I 

 1
0 
 1 2

− I  , with 1  ( s1 )
S =
0 


 0
0 

0 
I
I
0
0
0
0

 1

 2 2

 ( s1 )
2

, S =

1 

1 2 
 0
( sn ) 

0




 and
1 

( sn2 )2 
0
μ

μ j + υh j ( x ) ∇ 2x h j ( x ) + υ ( ∇ x h j ( x ))t ∇ x h j ( x ), if h j ( x ) ≥ − j


υ , where H(x) is the
∇ 2xx La = ∇ 2xx f ( x ) +ψ t H ( x ) + 
μ
j
j =1 
0 , if h j ( x ) ≤ −
υ

r

(
)
(
)
Hessian of the constraint g(x); Δd T = Δx , Δs 1 , Δs 2 , Δψ , Δπ 1 , Δπ 2 ; and

μj 



r ( μ j + υ h j ( x )) ∇ x h j ( x ), se h j ( x ) ≥ −

t
t
1
t
2
t
υ
 ∇ f ( x ) +ψ J ( x ) + ( π ) I + ( π ) I +


μj
 x

j =1 
0 , se h j ( x ) ≤ −



υ


δ


− 1 + π l1 , l = 1,...,n
.
sl
∇La = 


δ


− 2 − π l2 , l = 1,...,n
sl




g i ( x ), i = 1,...,m


1
max
x + s −x




x − s 2 − x min


1
Using the search directions obtained from (30), the vectors of the variables, x, s and s2 and of the Lagrange
multipliers, ψ, π1 and π2 are updated, as follows:

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x k +1 = x k + α p Δx k
k +1
= s1 + α p Δ s1
k +1
= s2 +α pΔ s2
s1
s2
k
k
k
k
(31)
ψ k +1 =ψ k + α d Δλk
π1
k +1
= π 1 + α d Δπ 1
π2
k +1
= π 2 + α d Δπ 2
k
k
k
k
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]:
s1
α p = min { σ ( min
Δs 1
Δs 1 <0
α d = min { σ ( min
−π1
Δπ 1 >0
Δπ 1
, min
Δs 2 <0
, min
Δπ 2 <0
s2
) ,1 }
(32)
π2
) ,1 }
Δπ 2
(33)
Δs 2
where σ = 0.9995 is an empirical value which, according to Wright [19], can be derived from the formula 1−
1
9 m
,
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]:
 k
μ kj
k
k +1
k +1
μ j + υ h j ( x ), se h j ( x ) ≥ − k
υ
μ kj +1 = 
k
μ
j

0 , se h j ( x k +1 ) ≤ − k

υ
j = 1, , r
(34)
and the penalty factor v and the barrier factor δ, as follows:
υ k +1 = β υ k , β > 1
δ k +1 =
(35)
δk
, α >0
α
(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.
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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.
Without reactive
power support
With reactive
power support
1.15
1.10
1.05
1.00
0.95
0.90
0.85
1.12
Voltage magnitude (pu)
Voltage magnitude (pu)
1.20
0.80
6 m/s
1.10
7 m/s
1.08
8 m/s
1.06
9/ms
10/ms
1.04
11/ms
1.02
12/ms
1.00
13/ms
14/ms
0.98
1
2
3
4
5
6
7
8
9
10 11 12 13 14
1
Bus
2
3
4
5
6
7
8
9 10 11 12 13 14
Bus
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.
532
©FORMATEX 2013
60.00
12.00
50.00
10.00
40.00
30.00
Bus 1
20.00
Bus 2
10.00
Bus 8
Active Losses (MW)
Reactive Power Injection (Mvar)
Materials
and processes for energy: communicating current research and technological developments (A. Méndez-Vilas, Ed.)
____________________________________________________________________________________________________
8.00
6.00
4.00
2.00
0.00
0.00
-10.00
6
7
8
9
10
11
12
13
6
14
7
8
9
10
11
12
13
14
Speed (m/s)
Speed (m/s)
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
acknowledged.
The support by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) is gratefully
References
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Appendix
A. Wind farm electrical systems parameters
Table A.1
Doubly-fed induction generator characteristic.
R1 (pu)
L1 (pu)
R2 (pu)
L2 (pu)
Mm (pu)
H (s)
Number of Pole
0.01
0.1
0.01
0.08
3
0.5
4
Table A.3
Min. Rotor Speed - variable speed (rpm))
Nom. Rotor Speed – variable speed (rpm)
Rotor diameter (m)
Area covered by rotor (m2)
Nom. Power (MW)
Nom. Wind Speed – variable speed (rpm)
Gear box ratio - variable speed
Inertia constant (s)
Shaft stiffness – fixed speed (pu/ el rad)
Table A.2 DFIG reactive power capability.
Wind
Q
S
P (MW)
FP
(m/s)
(Mvar)
(MVA)
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.
534
Turbine characteristic.
©FORMATEX 2013
9
14
75
4418
2
14
1:100
2.5
0.3
Materials
and processes for energy: communicating current research and technological developments (A. Méndez-Vilas, Ed.)
____________________________________________________________________________________________________
B. Modified IEEE 14 bus system
Table B.1 Data line.
from
to
r (pu)
1
2
1.94
1
5
5.4
2
3
4.7
2
4
5.81
2
5
5.7
3
4
6.7
4
5
1.34
4
7
0.01
4
9
0.01
5
6
0.01
6
11
9.5
6
12
12.29
6
13
6.62
7
8
0.01
7
9
0.01
9
10
3.18
9
14
12.71
10
11
8.2
12
13
22.09
13
14
17.09
Fig. B.1
Table B.2
x (pu)
5.92
22.3
19.8
17.63
17.39
17.1
4.21
20.91
55.62
25.2
19.89
25.58
13.03
17.62
11.00
8.45
27.04
19.21
19.99
34.80
bsh (pu)
5.28
5.28
4.38
3.74
3.40
3.46
1.28
0
0
0
0
0
0
0
0
0
0
0
0
0
bus
1
2
8
Data buses with reactive control.
Q (Mvar)
Qmin (Mvar)
0
-200
12.7
-40
shown in Table A.1
Qmax (Mvar)
200
50
Modified IEEE 14 bus system configuration.
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535