Int J Syst Assur Eng Manag (July-Sept 2014) 5(3):219–229 DOI 10.1007/s13198-013-0150-0
ORIGINAL ARTICLE
Optimal tuning of PI controller using PSO optimization for indirect power control for DFIG based wind turbine with MPPT Youcef Bekakra • Djilani Ben Attous
Received: 24 November 2012 / Revised: 1 March 2013 / Published online: 20 March 2013 The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2013
Abstract In this paper, an artificial intelligence method particle swarm optimization (PSO) algorithm is presented for determining the optimal PI controller parameters for the indirect control active and reactive power of doubly fed induction generator (DFIG) to ensure a maximum power point tracking of a wind energy conversion system. A digital simulation is used in conjunction with the PSO algorithm to determine the optimum parameters of the PI controller. Integral time absolute error, integral absolute error and integral square error performance indices are considered to satisfy the required criteria in output active and reactive power of a DFIG. From the simulation results it is observed that the PI controller designed with PSO yields better results when compared to the traditional method in terms of performance index. Keywords Doubly fed induction generator Wind turbine Field oriented control PI (proportional integral) Particle swarm optimization (PSO) Maximum power point tracking (MPPT)
1 Introduction Wind energy has attracted great attention due to several advantages such as low pollution (Ackermann 2005).
Y. Bekakra D. B. Attous (&) Department of Electrical Engineering, Faculty of Sciences and Technology, University of El Oued, P.O. Box 789, El Oued, Algeria e-mail:
[email protected] Y. Bekakra e-mail:
[email protected]
Among the different alternatives to obtain variable speed wind turbines, the system based on a doubly-fed induction generator (DFIG) has become the most popular (Rahimi and Parniani 2010). The electricity production by wind power is the most predominant source of renewable energy in Europe (Hammons 2008). By 2020 it is expected that wind power generation will supply around 12 % of the total electricity (Lobos et al. 2009). The worldwide concern about the environmental pollution and the possible energy shortage has led to increasing interest in technologies for generation of renewable electrical energy. Among various renewable energy sources, wind generation has been the leading source in the power industry (Qiao et al. 2006). Recently, the DFIG is becoming the main configuration of wind power generation because of its unique advantages. Vector control technology is used to control the generator, and the rotor of DFIG is connected to an AC excitation of which the frequency, phase, and magnitude can be adjusted. Therefore, constant operating frequency can be achieved at variable wind speeds (Guo-qing et al. 2010). In the last decade, various modern control techniques such as adaptive control, variable structure control and intelligent control have been intensively studied for controlling the nonlinear components in power systems. However, these control techniques have few real applications probably due to their complicated structures or the lack of confidence in their stability. Therefore, the conventional PI controllers, because of their simple structures, are still the most commonly used control techniques in power systems, as can be seen in the control of the wind turbines equipped with DFIGs. Unfortunately, tuning the PI controllers is tedious and it might be difficult to tune the PI gains properly due to the nonlinearity and the high complexity of the system (Qiao et al. 2006).
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Modern heuristic algorithms are considered as effective tools for nonlinear optimization problems. The algorithms do not require that the objective function has to be differentiable and continuous. A particle swarm optimization (PSO) is one of the modern heuristic algorithms and can be applied to nonlinear and noncontinuous optimization problems with continuous variables. It has been developed through simulation of simplified social models (Naka et al. 2003). Particle swarm optimization is a population based optimization algorithm which is first introduced by Kennedy and Eberhart (1995). It can be obtained high quality solutions within shorter calculation time and stable convergence characteristics by PSO than other stochastic methods such as genetic algorithm (GA) (Gozde and Cengiz Taplamacioglu 2011). The method is based on the simulation of animal social behaviors such as fish schooling, bird flocking, and swarm theory. Since it is population based and self-adaptive, it has gained an increasing popularity as an efficient alternative to the GAs in solving optimization problems. Moreover, it is shown to be effective in optimizing difficult multidimensional discontinuous problems in a variety of fields. Similar to other populationbased optimization method such as the GA, the PSO algorithm starts with random initialization of a population of individuals in the search space. Each particle in the search space is adjusted by its own flying experience and the other particles flying experience to find the global best solution at each generation. Compared with the GA, PSO has memorial ability to let the knowledge of good solutions be retained by all particles, whereas the previous knowledge need not be considered after each evolution in GA. Furthermore, low computation cost, simplicity of implementation, and quick convergence ability make PSO popular in many applications. In addition, to further enhance the particles learning ability and make it powerful in reasoning, some research has been developed to improve the PSO recently an inertia weight is adopted in PSO to balance the local and global search ability, while the worst experience component is included in PSO to give additional exploration capability (Lin et al. 2009). Through studying the characteristics of wind turbine, the paper proposed the maximum power point tracking (MPPT) control method. Firstly, according to the DFIG character, the paper adopts the vector transformation control method of stator oriented magnetic field to realize the decoupling control of the active and reactive power using PI controllers which is tuned by PSO. In this paper, we investigate the performance of PSO for optimizing the PI controller gains for indirect control active and reactive power of the DFIG with MPPT control algorithm and compared with a PI controller which is tuned manually.
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2 Model of turbine Normally a wind turbine creates mechanical torque on a rotating shaft, while an electrical generator on the same rotating shaft is controlled to produce an opposing electromagnetic torque. The power equation for the wind turbine is as follows (Rahimi and Parniani 2010; Ghedamsi and Aouzellag 2010): 1 Pv ¼ qSw m3 2
ð1Þ
Where q is air density; Sw is wind turbine blades swept the area in the wind; m is wind speed. The output mechanical power of a wind turbine is: 1 Pm ¼ Cp Pv ¼ Cp qSw m3 2
ð2Þ
Where Cp represents the power coefficient. Cp can be described as (Abdin and Xu 2000): Cp ðb; kÞ ¼ ð0:5 0:0167 ðb 2ÞÞ p ðk þ 0:1Þ 0:00184ðk 3Þ sin 18:5 0:3ðb 2Þ ðb 2Þ ð3Þ It is a function of the tip speed ratio k and the blade pitch angle b in a pitch-controlled wind turbine. k is defined as the ratio of the tip speed of the turbine blades to wind speed: k¼
R Xt m
ð4Þ
Where R is blade radius, Xt is the angular speed of the turbine. A figure showing the relation between Cp, b and k is shown in Fig. 1. The maximum value of Cp(Cp max ¼ 0:5:) is achieved for b = 2 degree and for kopt = 9.2.
3 Maximum power point tracking (MPPT) Maximum power variation with rotation speed X of DFIG is predefined for each wind turbine. So for MPPT, the control system should follow the tracking characteristic curve (TCC) of the wind turbine (Abdin and Xu 2000). Each wind turbine has TCC similar to the one shown in Fig. 2. The actual wind turbine, X is measured and the corresponding mechanical power of the TCC is used as the reference power for the power control loop (Eltamaly et al. 2010). The turbine speed should be changed with wind speed so that the optimum tip speed ratio kopt is maintained. This strategy called MPPT, (as shown in Fig. 2).
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0.5
From Fig. 2 we can see there is one specific angular frequency at which the output power of a wind turbine is maximum occurs at the point where Cp is maximized. Connected all the maximum power point of each power curve, the optimal power curve (MPPT curve) is obtained.
X: 9.2 Y: 0.5
0.45
B = 2°
0.4 0.35 B = 3°
Cp
0.3 0.25 0.2
B = 4°
0.15
4 DFIG model
0.1 0.05 0
0
2
4
6
8
10
12
14
16
18
Lamda
Fig. 1 Aerodynamic power coefficient variation Cp against tip speed ratio k and pitch angle b
4000
Turbine mechanicale power (W)
3500 MPPT
7 m/s
Stator and rotor fluxes: 8 /sd ¼ Ls isd þ M ird > > > > < /sq ¼ Ls isq þ M irq > > /rd ¼ Lr ird þ M isd > > : / ¼L i þ Mi
3000 2500 2000
The general electrical state model of the induction machine obtained using Park transformation is given by the following equations (Bekakra and Ben attous 2011; Machmoum and Poitiers 2009; Senthil Kumar and Gokulakrishnan 2011): Stator and rotor voltages: 8 Vsd ¼ Rs isd þ d /sd xs /sq > > < V ¼ R i þ dtd / þ x / sq s sq s sd dt sq ð7Þ Vrd ¼ Rr ird þ dtd /rd ðxs xÞ /rq > > : Vrq ¼ Rr irq þ dtd /rq þ ðxs xÞ /rd
6 m/s
1500
rq
r
1000
0
The electromagnetic torque is given as: 4 m/s
Ce ¼ pMðird isq irq isd Þ 0
100
200
300
400
500
600
700
800
Fig. 2 Turbine powers various speed characteristics for different wind speeds, with indication of the maximum power tracking curve
In order to make full use of wind energy, in low wind speed b should be equal to 2 degrees. Figure 2 illustrates the wind turbine power curve when b is equal to 2 degrees. To extract the maximum power generated, we must fix the advance report kopt is the maximum power coefficient Cp max , the measurement of wind speed is difficult, an estimate of its value can be obtained (Bekakra and Ben attous 2011): R Xt ¼ kopt
ð5Þ
The aerodynamic power reference value must be set to the following value: ref
1 ¼ Cp 2
max
q Sw m3est
ð6Þ
ð9Þ
and its associated motion equation is:
Turbine rotational speed (rpm)
Paer
sq
5 m/s
500
mest
rq
ð8Þ
Ce Cr ¼ J
dX dt
ð10Þ
The state variable vector is then: T X ¼ isd isq ird irq The state model can then be written as: :
X ¼AXþBU
ð11Þ
Where: A B
Must be an n-by-n matrix, where n is the number of states Must be an n-by-m matrix, where m is the number of inputs
with: T : d d d d isd isq ird irq X¼ dt dt dt dt T U ¼ Vsd Vsq Vrd Vrq
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2
a1
ax þ xs
a3
a5 x
6 6 ax xs a1 a5 x a3 6 A¼6 6 a4 a6 x a2 xr þ xs 4 x a6 x a4 a2 r xs 3 2 b1 0 b3 0 7 6 6 0 b1 0 b3 7 7 6 B¼6 7 7 6 b3 0 b 0 2 5 4 0 b3 0 b2
3
The inverter connected to the rotor of the DFIG must provide the necessary complement frequency in order to maintain constant the stator frequency despite the variation of the mechanical speed. The system studied in the present paper is constituted of a DFIG directly connected through the stator windings to the network, and supplied through the rotor by a static frequency converter as presented in Fig. 3.
7 7 7 7; 7 5
6 Designing of PI controller using PSO
where: 1r Rs Rr Rr M ; a1 ¼ ; a2 ¼ ; a3 ¼ ; r rLs Lr rLs rLr Rs M M M a4 ¼ ; a5 ¼ ; a6 ¼ rLs Lr rLs rLr
a¼
b1 ¼
1 1 M M2 ; b2 ¼ ; b3 ¼ ; r¼1 rLs rLr rLs Lr Ls Lr
5 Field oriented control of DFIG In this section, the doubly fed induction machine (DFIM) model can be described by the following state equations in the synchronous reference frame whose axis d is aligned with the stator flux vector /s, (/sd = /s and /sq = 0). The control of the DFIG must allow a control independent of the active and reactive powers by the rotor voltages generated by an inverter. By neglecting resistances of the stator phases the stator voltage will be expressed by (Machmoum and Poitiers 2009): Vsd ¼ 0 and Vsq ¼ Vs
xs /s
ð12Þ
We lead to a decoupled power control; where, the transversal component irq of the rotor current controls the active power. The reactive power is imposed by the direct component ird. Ps ¼ Vs Qs ¼
M irq Ls
M Vs2 Vs ird Ls xs Ls
ð13Þ ð14Þ
The arrangement of the equations gives the expressions of the voltages according to the rotor currents: 8 dird > > gxs rLr irq < Vrd ¼ Rr ird þ rLr dt ð15Þ dirq M > > : Vrq ¼ Rr irq þ rLr þ g Vs þ gxs rLr ird Ls dt With: Tr ¼
Lr Ls xs x ; Ts ¼ ; g ¼ xs Rr Rs
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The PSO as an optimization tool provides a populationbased search procedure in which individuals called particles change their position (state) with time. In a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience (This value is called Pbest), and according to the experience of a neighboring particle (This value is called Gbest), made use of the best position encountered by itself and its neighbor (Lalitha et al. 2010) (as shown in Fig. 4). This modification can be represented by the concept of velocity. The velocity of each agent can be modified by the following equation: vkþ1 ¼ w vk þ c1 rand ðPbest xk Þ þ c2 rand ðGbest xk Þ
ð16Þ
Using the above equation, a certain velocity, which gradually gets close to Pbest and Gbest can be calculated. The current position (searching point in the solution space) can be modified by the following equation: xkþ1 ¼ xk þ vkþ1 ; k
k ¼ 1; 2; . . .; n
ð17Þ k?1
is modified Where x is current searching point, x k k?1 searching point, v is current velocity, v is modified velocity. Pbest is the best solution observed by current particle and Gbest is the best solution of all particles, w is an inertia weight, c1 and c2 are two positive constants, rand is a random generated number with a range of [0,1]. The following inertia weight is used (Lalitha et al. 2010): wmax wmin wðkÞ ¼ wmax k ð18Þ kmax where kmax, k is the maximum number of iterations and the current number of iterations, respectively. Where, wmin and wmax are the minimum and maximum weights respectively. Appropriate value ranges for c1 and c2 are 1–2, but 2 is the most appropriate in many cases. Appropriate values for wmin and wmax are 0.4 and 0.9 (Eberhart and Shi 2000) respectively.
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Fig. 3 DFIG variable speed wind energy conversion MPPT control
Y
Start
Xk+1 Vk+1
Generate initial populations
Vk
Gbestk
VGbest Xk VPbest
Run the DFIG power control model
Pbestk X
Calculate the fitness function
Fig. 4 Concept of a searching point by PSO
The PSO based approach to find the global maximum value of objective function as shown in Fig. 5. The PI controller is a good controller in the field of machine control, but the problem is the mathematical model of the plant must be known. In order to solve problems in the overall system, several methods have been introduced to tuning PI controller. Our proposed method uses the PSO to optimize the active and reactive power PI controller parameters, the PSO is utilized off line to determine the controller parameters (Kp and Ki) (based on quadrature rotor current error irq linked to active power Ps and direct rotor current ird linked to reactive power Qs) of the DFIG as shown in Fig. 6. The performance of the DFIG varies according to PI controller gains and is judged by the value of integral time absolute error (ITAE). The performance index sum (ITAE) is chosen as objective function. The purpose of stochastic algorithms is to minimize the objective function. All particles of the population are decoded for Kp and Ki.
Calculate the Pbest of each particle and Gbest of population
Update the velocity, position, Gbest , Pbest of particles
Calculate parameters [Ki, Kp] of PI controller
No
Maximum iteration number reached ? Yes Print optimal parameters [Ki, Kp]
Stop
Fig. 5 The flowchart of the PSO–PI control system
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ITAE criterion is widely adopted to evaluate the dynamic performance of the control system (Allaoua et al. 2009). The index ITAE is expressed in Eq. (19), as follows: Z1 ITAE ¼ t jeðtÞj dt ð19Þ 0
In this paper a time domain criterion is used for evaluating the PI controller. The performance criteria used for comparison between using a PI controller which is tuned by PSO technique and manually. PI controller includes integration absolute error (IAE) and integrated of squared error (ISE). The IAE and ISE performance criterion formulas are as follows (Allaoua et al. 2009): Z1 IAE ¼ ð20Þ jeðtÞj dt 0
ISE ¼
Z1
e2 ðtÞ dt
ð21Þ
7 Simulation results The DFIG used in this work is a 4 kW, whose nominal parameters are indicated in Appendix. The parameters of PSO algorithm are shown in Table 1. The selecting of the parameters of PSO was as follows: •
• •
After several experiments, we obtained the following ‘‘The augmentation of swarm size leads to complicate the calculation and a prolongation in computing time’’. In our case swarm size = 15 is a good selection. In our case, the number of maximum iteration = 20 is satisfying for obtaining good results as shown in Fig. 7. In many scientific papers c1 = c2 = 2, wmax = 0.9 and wmin = 0.4 [for example in Ref. (Lalitha et al. 2010; Eberhart and Shi 2000)].
The velocity, current position and fitness value of each optimal particle during the simulation are presented in Table 2, after this table the best fitness value is 3.943e ? 005 appeared in iteration number 9, and the optimal gains are
0
Ps Ps *
Ps *
– Ls MV s
+
-
i rq
PSO Algorithm
i rq
*
+
K p , Ki
PI
Udc V rq FOC + Park-1
Qs = 0 + *
V s φs Ls
– Ls MV s
i rd * +
PI
i rd
V rd i rd i rq
Ps
Turbine Win
PWM
θr
Gear
DFIG
Estimation θr
Grid
MPPT
Ωr
5000 8 m/s 4500 4000 Mechanical power (W)
Ps
*
3500 7 m/s 3000 2500 2000
6 m/s
1500 1000 6 m/s 500 0
Fig. 6 Indirect control of DFIG with PSO
123
0
10
20 30 40 Turbine rotational speed (rpm)
50
60
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225 1980
Table 1 Parameters of PSO algorithm 15
Number of maximum iteration
20
c1 = c2
2
wmax
0.9
wmin
0.4
1970 1960
Ki, Kp gain
Swarm size
Ki Kp
1950 1940 1930 1920 1910 1900
5
3.948
x 10
2
4
6
8
10
12
14
16
18
20
Iteration
X: 1 Y: 3.948e+005
3.9475
Fig. 8 The variation of optimal Ki et Kp gain during simulation
3.947
Fitness function
0
3.9465 3.946
current position ¼ randðdim; nÞ ðupbnd lwbndÞ þ lwbnd velocity ¼ randðdim; nÞ
3.9455 X: 5 Y: 3.945e+005
3.945 3.9445 3.944
X: 9 Y: 3.943e+005
X: 20 Y: 3.943e+005
3.9435 3.943
0
2
4
6
8
10
12
14
16
18
20
Iteration
Fig. 7 The fitness function variation during simulation
Ki = 1914.8 and Kp = 1967.4, which are shown clearly in Figs. 7 and 8. Where the initial parameters of PSO are:
Table 2 The velocity, current position and fitness value of each optimal particle Iteration No.
Optimal parameters
Fitness value
1
Velocity(1,1) = 0.1904
3.948e ? 005
Kp = Current_position(1,1) = 1975.5 Velocity(2,1) = 0.0257
where rand Random numbers, dim = 2 Dimension of swarm (Ki and Kp), n = 15 Size of the swarm, upbnd: 1980 The Upper bound for the initial of the swarm, lwbnd: 1900 The Lower bound for the initial of the swarm. The fitness function variations and the variation of optimal Ki et Kp gain during the simulation are presented in Figs. 7 and 8 respectively. In order to evaluate the MPPT control strategy, we proposed a step change in wind speed is simulated in Fig. 9, the wind speed starts at 5 m/s, at 3 s, the wind speed suddenly changing at 6 m/s, as 6 s, the wind speed is 7 m/s. The Fig. 10 presents the turbine speed. Figure 11 presents the power coefficient. Figure 12 presents the stator active power without and with PSO resulting of the MPPT. Figure 13 shows the stator reactive power without and with PSO versus time. Figure 14 stator current without and with PSO and these zoom. Figure 15 presents a comparison of PI controllers without and with PSO for stator current at startup. Figures 16 and 17 present spectrum of phase stator current harmonics without and with PSO respectively.
Ki = Current_position(2,1) = 1975.8 Velocity(1,1) = 22.1064
3.945e ? 005
Current_position(1,1) = 1994.4 [ upbnd
8
Kp = upbnd = 1980
7
Velocity(2,1) = 26.5410
6
Ki = Current_position(2,1) = 1963.2 9
Velocity(1,1) = -42.0899
3.943e ? 005
Kp = Current_position(1,1) = 1967.4 Velocity(2,1) = 6.3232 Ki = Current_position(2,1) = 1914.8 20
Velocity(1,1) = -42.0899 Kp = Current_position(1,1) = 1967.4
3.943e ? 005
Wind speed (m/s)
5
5 4 3 2 1 0
0
1
2
Velocity(2,1) = 6.3232 Ki = Current_position(2,1) = 1914.8
3
4
5
6
7
8
9
Time (s)
Fig. 9 Wind speed profiles
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Turbine speed (rad/s)
120 100 80 60 40 20 0
0
1
2
3
4
5
6
7
8
9
8
9
Time (s)
Fig. 10 Speed turbine according the MPPT with PSO
0.5
Cp
0.4 0.3 0.2 0.1 0
0
1
2
3
4
5
6
7
Time (s)
Fig. 11 Power coefficient Cp variation with PSO
We observe after Figs. 9 and 10 when wind speed v is 5 m/s the optimal turbine speed Xt of DFIG is 87.51 rad/s, when v is 6 m/s, Xt is 100 rad/s and when v is 7 m/s, Xt is 112 rad/s, after each adjustment, the stable turbine speed
totally with the theoretical value. During this adjusting process, realize the maximum wind energy tracking control. As can be seen from the figures, the stator active power is controlled according to the MPPT strategy, and the reactive power is maintained to zero, to guarantee a unity power factor at the stator side. The power coefficient Cp is kept around its optimum (Cp = 0.5), ‘‘as shown in Fig. 11’’. A comparison is done here with the results obtained from conventional PI controllers, which also aims at active and reactive power ripple minimization. The results of the comparison are that the active and reactive powers ripple is reduced considerably with the help of PSO as shown in Figs. 12 and 13. In addition, the stator active in the startup is reduced in case of ‘‘with PSO’’, as shown in Fig. 12b, compared with a case of ‘‘without PSO’’, as shown in Fig. 12a. Table 3 lists the performances of stator active power of the two controllers (the gains of conventional PI controller are calculated from the pole compensation method), from these values obtained it is clearly visible that the error magnitude obtained in different criteria for conventional method is big as compared to the proposed tuning method based on PSO algorithm, which is shown clearly in Fig. 12. Similarly, Stator current with PSO tuned PI controller is smooth as compared with that of conventional PI controller, or the over-current in the stator circuit is reduced in start up when we are using the ‘‘PSO’’ as shown in Fig. 15. From Figs. 16 and 17, it is clear that the stator current in conventional PI has a high THD (with THD = 17.67 %) as compared to the stator current in case when we are used PSO technique (with THD = 15.70 %).
Without PSO
With PSO
8000
8000
reference measured
6000
Stator active power (W)
Stator active power (W)
reference measured
4000
2000
0
-2000
-4000
0
1
2
3
4
5
6
7
8
9
6000
4000
2000
0
-2000
-4000
0
1
2
4
5
Time (s)
a
b
Fig. 12 Stator active power injected in the grid according the MPPT: a without and b with PSO
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3
Time (s)
6
7
8
9
Int J Syst Assur Eng Manag (July-Sept 2014) 5(3):219–229 Without PSO
4000 3000 2000 1000 0 -1000 -2000 -3000 -4000
0
1
2
3
4
5
With PSO
4000 reference measured
Stator reactive power (VAR)
Stator reactive power (VAR)
227
6
7
8
2000 1000 0 -1000 -2000 -3000 -4000
9
reference measured
3000
0
1
2
3
4
Time (s)
5
6
7
8
9
7
8
9
Time (s)
a
b
Fig. 13 Stator reactive power: a without and b with PSO
Without PSO
20
10
-10
4
4.02
4.04
4.06
5 0 -5
0
10
-10
4
4.02
4.04
4.06
3
4
5 0 -5 -10
-10 -15
10
15
0
Stator currents (A)
Stator currents (A)
15
With PSO
20
10
0
1
2
3
4
5
6
7
8
-15
9
0
1
2
Time (s)
5
6
Time (s)
a
b
Fig. 14 Stator current: a without and b with PSO 20
15
Without PSO
Without PSO , THD = 17.67%
With PSO 5
0
-5
-10
-15
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (s)
Magnitude (% of Fundamental)
Stator current (A)
10
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
140
160
180
200
Order of Harmonic (harmonic number)
Fig. 15 Comparison of PI controllers without and with PSO for stator current at startup
Fig. 16 Spectrum of phase stator current harmonics without PSO
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Wind turbine parameters are: R(blade radius) = 3 m, G (Gearbox) = 5.4. Air density: q = 1.22 kg/m3.
Magnitude (% of Fundamental)
With PSO , THD = 15.70% 1
0.8
Appendix B: Nomenclature
0.6
Controller generator response
PI tuned manually Ki = 710.64; Kp = 2,000
IAE
2,933
2,741
v q R Pm Cp Sw k Xt Ce Cr J b Vsd,q Vrd,q isd,q ird,q
ITAE
1.047e ? 004
1.035e ? 004
/sd,q
ISE
3.139e ? 006
2.248e ? 006
/rd,q
0.4
0.2
0
0
20
40
60
80
100
120
140
160
180
200
Order of Harmonic (harmonic number)
Fig. 17 Spectrum of phase stator current harmonics with PSO
Table 3 Comparison between PI tuned manually and PI tuned by PSO PI tuned by PSO Ki = 1914.8; Kp = 1967.4
8 Conclusion In this work, we have presented a complete wind energy conversion system made with a DFIG. This system is constituted of a DFIG with a stator connected directly to the grid while the rotor is connected through inverter PWM. The aim of controlling the rotor side converter is to extract a maximum power from the wind by using a field oriented control and an optimal speed reference which is estimated from the wind speed. The PSO algorithm is then used to find the optimal gains of the PI controllers for the active and reactive power in order to minimize the ripple in the stator current. The performance index for various error criteria for the proposed controller using PSO algorithm is proved to be less than the controller tuned manually. It is clear from the results that there is a reduction of ripple in active and reactive powers as well as stator current when the proposed PSO method is used. Appendix Appendix A: System parameters Rated values: 4 kW, 220/380 V, 15/8.6 A. Rated parameters: Rs = 1.2 X, Rr = 1.8 X, Ls = 0.1554 H, Lr = 0. 1568 H, M = 0.15 H, p = 2.
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Rs , Rr Ls, Ls Ls r p Ts, Tr xs, x g
Wind speed Air density Blade radius Mechanical power of wind speed Power coefficient Swept area Tip speed ratio Angular speed of the turbine Electromagnetic torque Load torque Moment of inertia Bitch angle Stator d-q frame voltage Rotor d-q frame voltage Stator d-q frame current Rotor d-q frame current Stator d-q frame flux Rotor d-q frame flux Stator and rotor resistance Stator and rotor inductance Mutual inductance Leakage factor Number of pole pairs Statoric and rotoric time-constant Stator and rotor d-q reference axes speed Slip coefficient
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