Stochastic Distribution Control for the Permanent ... - IEEE Xplore

Stochastic Distribution Control for the Permanent Magnet Synchronous Generator based Variable Speed Wind Energy Conversion Systems Jianhua Zhang*, Ye Tian, Mifeng Ren, Jinfang Zhang, Guolian Hou School of Control and Computer Engineering, North China Electric Power University, Beijing 102206 E-mail: [email protected] Abstract: In this paper, the nonlinear dynamic model of the wind energy conversion system (WECS) is established. Due to the non-Gaussian randomness in the WECS introduced by the wind speed, stochastic distribution controller is designed for both maximum power point tracking (MPPT) and attenuation of disturbance. Compared with conventional PI controller, the effectiveness of the proposed approach is verified by some experimental results. Key Words: Wind Energy Conversion System, Probability Density Function, Entropy, Stochastic Control

1

INTRODUCTION

In recent years, more attention has been paid to wind energy conversion systems (WECSs). There are two types of WECSs classified by speed-control approach: fixed-speed and variable-speed WECSs. Variable speed WECS equipped with power-electronic converters is becoming very promising technologies for large wind farms. Compared with fixed-speed wind turbines, variable speed wind turbines have many advantages in terms of wide operation at maximum power point, high efficiency and power quality. In order to make WECSs more profitable and reliable, the control system plays a vital role in modern WECSs. The control system for a variable-speed WECS needs to meet the following requirements: (1) Electric power output must meet strict power quality standards (power factor, harmonics, flicker, etc.); (2) When a wind turbine operates in full load zone, the utilized wind energy must be maintained at a specific level by manipulating the aerodynamic power and the rotational speed; (3) When a wind turbine operates in partial load zone, the captured wind energy should be maximized; (4) The variance of load should be alleviated to extend lifetime of the mechanical parts. The control of WECSs has been widely studied in the existed literatures using various control techniques. e.g., PI control [1-2]; cascade control [3-4]; gain scheduling control [5-6]; Intelligent control [7-11]; adaptive control [11-13]; predictive control [14-16]; sliding-mode control [17, 10]; Fuzzy control [18,14]; robust control [19]; and LQG control [20-22]. It should be noted that the existed control strategies for WECSs might have achieved better performances. However, these strategies were not cast into stochastic frameworks except the LQG approaches in [20-22]. The This work is supported by National Basic Research Program of China under Grant CB710706, China National Science Foundation under Grant 60974029

c 978-1-4673-5534-6/13/$31.00 2013 IEEE

assumed condition on the stochastic characteristics of WECSs might be a little bit ideal and strict in [20-22]. It is well known that the speed of wind is a random variable which is not necessarily Gaussian, moreover, the WECSs are generally not linear systems. Consequently, it is necessary to make further investigation on control techniques for WECSs with the help of advanced stochastic control theories. Following the recent developments on stochastic distribution theory [23-29], WECSs will be cast into a stochastic framework in this paper. Within this framework a tracking control strategy is further investigated for WECSs characterized by nonlinearity and randomness. The approach presented in this paper is not only an extension of stochastic distribution control, but also a novel method to analyze and design WECSs. The rest of paper is organized as follows: Section 2 presents the system description. Section 3 establishes a dynamic model of a PMSG based variable speed WECS. Section 4 presents a novel stochastic distribution control strategy for the investigated WECS. Section 5 verifies the efficiency and feasibility of the proposed control algorithm using an illustrative example. The last section concludes the paper.

2

SYSTEM DESCRIPTION

The configuration of the PMSG based WECS is shown in Fig. 1. The system is composed of the wind turbine, the drive train, the generator unit including the permanent magnet synchronous generator (PMSG) and the AC-DC-AC converter connected to the electric grid. In this section, the dynamic model of wind energy conversion system will be established. In general, variable-speed wind energy conversion systems (WECSs) operate in two primary regimes: partial load regime, and full load regime. When power production is below the rated power, the turbine operates at variable rotor speed to capture the maximum amount of energy available in the wind. Generator torque provides the control input to vary the rotor speed, and the blade pitch angle is held

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constant. When power production is at rated power, the primary objective is maintaining constant, rated power output. This is generally achieved by regulating the blade pitch angle. In both operation regimes, the turbine response to transient loads must be minimized [30].

d ΩG η = Twt (t ) − TG (t ) (4) dt i where TG (t ) is the generator torque, η the transmission Jh

efficiency, J h the total inertia of the turbine referred to the high-speed shaft, ΩG = i × Ω the generator rotational speed

Drive

and i the drive train ratio.

Train AC DC

DC AC

2.3 Permanent-magnet Synchronous Generator

PMSG

Electric Wind

Grid

Turbine

Fig 1. Atypical PMSG-based variable-speed WECS

3

DYNAMIC MODELING

In this section, the dynamic model of each components in a PMSG-based variable-speed WECS will be built.

2.1 Wind Turbine According to the Betz’s theory, the aerodynamic power extracted from the wind is given as follows [31]

Pwt (t ) = 0.5πρ v 3 (t ) Rt 2 C p (λ (t ), β (t ))

(1)

where ρ is the air density, v(t ) the wind speed, Rt the blade length of wind turbine, β (t ) the blade pitch angle, Ω(t ) the rotational speed of wind turbine rotor and λ (t ) the tip speed ratio defined as λ (t ) = Ω(t ) × Rt / v(t ) . The power coefficient C p (λ (t ), β (t )) is a measure of how efficiently the wind turbine converters the kinetic energy of the wind into mechanical energy of the wind turbine.

Assuming sinusoidal distribution of stator winding, electric and magnetic symmetry, negligible iron losses and unsaturated magnetic circuit, the generator model in the steady-state coordinates can be obtained. Conversion between (a,b,c) and (d,q) coordinates can be realized via the Park transform [32]. Then, after neglecting the homo-polar voltage, by virtue of symmetry, the (d,q) PMSG model becomes < ° ud = − Rid (t ) − Ld id + Lq iq (t )ωS (t ) (5) ® < °uq = − Riq (t ) − Lq iq − ( Ldid (t ) − φm )ωS (t ) ¯ where R is the stator resistance, ωS = p × ωG the stator frequency and p the number of pole pairs. φm is the flux that is constant due to permanent magnets, ud and uq are d and q stator voltage respectively. Ld and Lq are d and q inductance respectively. If the permanent magnets are mounted on the rotor surface, then Ld = Lq , and the electromagnetic torque becomes

TG (t ) = pφm iq (t )

(6)

The aerodynamic torque can be described by Twt =

Pwt = 0.5πρ v 2 (t ) Rt 3 CT (λ (t ), β (t )) Ω

Rs

where CT (λ (t ), β (t )) is the torque coefficient.

Wind

In the partial load regime, the blade pitch angle β is fixed to the optimal value β opt . As such, the torque coefficient CT (λ (t ), β (t )) can be described by a polynomial function of the tip speed ratio λ [32] CT (λ ) = α 0 + α1 λ + α 2 λ 2

Ls

(2)

(3)

where the parameters α i , i = 0,1, 2 , are usually determined by fitting the look-up table representing an experimental torque characteristic in a least squares sense.

2.2 Drive Train The drive train can be viewed as a system by means of which the mechanical energy of the wind turbine is transmitted to the electrical generator. The generator experiences a reduced torque and an increased rotational speed due to the speed multiplier function of the driven train. Neglecting the viscous friction, the motion equation is formulated by

Fig 2. The equivalent PMSG-based WECS

The focus here is on the rotational speed control loop under the assumption that other control loops are operating well. The power electronics dynamic is significantly more rapid than the PMSG based WECS dynamic, and therefore the dynamic of under the assumption can be neglected. The system shown in Fig. 2 represents the equivalent WECS [33]. The parameters of the equivalent load include a constant inductance Ls and a variable resistance Rs which is the manipulated variable. As such, the PMSG model can be formulated by ⋅ ° ( Ld + Ls ) id (t ) = −( R + Rs )id (t ) + p( Lq − Ls )iq (t )ΩG t ) ° ⋅ ®( L + L ) i (t ) = −( R + R )i (t ) − p( L + L )i (t )Ω (t ) q s q s q d s d G ° °+ ( ) p i t φ m q ¯

2013 25th Chinese Control and Decision Conference (CCDC)

(7)

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2.4 PMSG based WECS Combining (1)~(7), the state space model of PMSG based WECS can be obtained as follows < ° x(t ) = f ( x(t )) + g ( x(t )) ⋅ u (t ) (8) ® ° ¯ y (t ) = h( x(t )) where ° ° ° ° ° ° f ( x) = [ f f2 1 ° ° ° ° ° ® ° g ( x ) = [ g1 ° ° ° ° ° ° ° ° ° ° ¯

4

x = ª¬id

iq

Ω º¼

f3 ]

ª º 1 ª − Rx1 + p( Lq − Ls ) x2 x3 º¼ « » Ld + Ls ¬ « » « 1 » =« [ − Rx2 − p( Ld + Ls ) x1 x3 + pΦ m x3 ]» « Lq + Ls » « » 1 2 2 « (d v + d 2 vx3 + d3 x3 − pΦ m x2 ) » «¬ i ⋅ J h 1 »¼

g2

ª 1 T g3 ] = « − x1 ¬« Ld + Ls

T

Fig 3. Proposed stochastic distribution control strategy

T

−

1 x2 Lq + Ls

º 0» ¼»

T

u = Rs h( x ) = x3 = Ω d1 = 0.5ηπρ Rt 3α 0 / i d 2 = 0.5ηπρ Rt 4α1 / i 2 d3 = 0.5ηπρ Rt 5α 2 / i 3

STOCHASTIC DISTRIBUTION CONTROL

When the wind speed is between the cut-in and the rated speed, the wind turbine is operating in partial load regime, the generator control is the only active control. Generator control aims at maximizing the energy captured from the wind and/or at limiting the rotational speed at rated probably by continuously accelerating or decelerating the wind turbine rotor speed in order that the optimum tip speed ratio is tracked. This paper focuses on developing a stochastic distribution control (SDC) algorithm for WECSs operating in partial load regime. The main control objectives can be outlined as maximum power acquisition with smooth mechanical load and electrical output. Control of WECS is challenging because of complex process characterized by nonlinearity and stochastic disturbance induced by wind. Fluctuations of wind turbine power lead to harmful effects on WECS. Frequent variations in the aerodynamic torque and the generator torque can reduce the life time of the mechanical elements of the system. Electric power fluctuations supplied to the grid can cause power quality reduction. In order to prevent fluctuations of wind turbine power, the dispersion of the wind turbine rotor speed should be reduced by a closed-loop control system. The proposed stochastic distribution control strategy is shown as Fig. 3. e = Ω ref − Ω is the tracking error of the closed-loop control system, Ω ref the set-point of the wind turbine rotor speed Ω and γ e the probability density function of tracking error. The optimal control input u is solved by minimizing a proper performance index.

4.1 Performance Index Since the stochastic disturbances coming from wind speed is not Gaussian, it is necessary to consider the high order statistics of tracking error. Entropy is a more general measure of uncertainty for arbitrary random variables. Small entropy of tracking error in a closed-loop control system means that the shape of probability density function of the tracking error is narrow and sharp, corresponding to a small dispersion of the wind turbine rotor speed ΩG . In addition, the mean value of the tracking error and the control energy should also be minimized simultaneously. In our work, quadratic Renyi’s entropy of tracking error b

H R = - log ³ γ e2 ( x)dx

used

to

characterize

the

uncertainty of tracking error due to its computational efficiency. According to the definition of the information b

potential V = ³ γ e2 ( x)dx in [34], the quadratic Renyi’s a

entropy is a monotonic function of the quadratic information potential, hence, minimizing the quadratic Renyi’s entropy is equivalent to minimizing the negative information potential. The discrete performance index at instant k becomes 1 J k = − R1Vk (ek ) + R2 Ek (ek ) + uk2 (9) 2 where Vk and Ek are the information potential and the mean value of the tracking error e respectively. R1 , R2 and R3 are weights that correspond to information potential, the mean value and the control input respectively. The large control input is not expected in practical engineering due to constraint of energy, the amplitude of control input can be limited using R3 . R3 can be usually set to zero or a small value. R1 and R2 can be experimentally tuned such that the control performance is satisfactory.

4.2 Recursive Distribution Control The PDF of the tracking error can be estimated from the samples {e1 , e2 , ..., eN } with kernel function estimation.

1 N (10) ¦ κ (ζ − ei , σ 2 ) N i =1 where κ denotes the Gaussian kernel function. The PDF of the tracking error can be updated at instant k + 1 by adding a new sample ek +1 [35].

γˆe (ζ ) =

^

γe

k +1

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is

a

(ζ ) = (1 − λ ) ⋅ γ e (ζ ) + λ ⋅ κ (ζ − ek +1 , σ 2 ) ^

k


(11)

Consequently, the information potential of tracking error Vk can be obtained recursively by sliding windowing technique.

λ

k

¦ κ ( ei − ek +1 , 2σ 2 ) (12) L i = k − L +1 where L is the window length. At each instant, a sliding window of size L is constructed using the current and the past L − 1 tracking error samples. The aim of the controller design is to find out an optimal control input uk so that the performance index of the closed-loop system can be minimized. The following incremental control law is used at each instant uk = uk −1 + Δuk (13) Denote Sk = − R1Vk + R2 Ek , (13) and the Taylor expansion of Sk are substituted into (9), it turns out to be 1 2 J k ≈ S0 + S1 Δuk + S 2 ( Δuk ) 2 1 2 2 (14) + R3 uk + 2uk −1 Δuk + ( Δuk ) 2 ∂S ∂ 2 Sk where S0 = S k u = u , S1 = k , S2 = . k k −1 ∂uk u = u ∂uk2 u = u Vk +1 ( e ) ≅ (1 − λ ) Vk ( e ) +

(

)

k

k −1

k

GPI ( s ) = 0.035+

comparison. The length of the sliding window used to estimate the tracking error PDF and information potential is set to L = 35 and the forgetting factor λ = 0.2 . The weights of the performance index corresponding to the mean value, information potential and the control input are set to R1 = 0.25 , R2 = 0.74 and R3 = 0.01 respectively. The kernel size parameter is set to σ = 0.6 . For a given wind speed v , in order to maintain λ at its optimal value λopt , the rotor speed must be adjusted to track the reference Ω ref = λopt ⋅ v / Rt . The WECS is in the steady operation at wind speed around 7 m/s initially. At 50s, gust wind acts on the wind energy conversion system, from 7m/s to 7.5m/s.The demand of the wind rotor speed Ω ref increases accordingly . 21.2 21

k −1

* k

Initialize PDF estimation of tracking error in (10).

2)

Estimate the mean value and information potential of the tracking error respectively using sliding windowing technique.

e k: m ean = 0.0824, std.dev. = 1.0091,MSE= 1.0251

yk(rad/s)

20 19.8

e k: m ean = 0.0443, std. dev. = 0.9759,MSE= 0.9543

19.6 0

50

100

150

200

t(s)

Fig 4. Response of the rotor speed.

0.12

0.1

0.08 0

50

100

150

200

150

200

t(s)

Solve the optimal control input using (15).

1.15

Calculate the future control input u k = u k −1 + Δu and impose the input on the WECS and collect the wind turbine rotor speed sequence to recursively update the mean value and information potential of the tracking error. Then repeat the procedure from Step 2 to Step 4 for the next instant, k = k+1.

EXPERIMENTAL RESULTS

The performance of the proposed control strategy is tested and verified in this section. The designed controller is tested with simulation parameters given in [33]. The random component of wind speed obeys the uniform distribution U (−0.05, 0.05) . In addition, some results with the conventional PI controller whose transfer function is

1.1 1.05

k

* k

5

20.4 20.2

J (e)

3)

set point PID SDC

20.6

k

1)

20.8

v (e)

Thus, the optimal control input Δu can be obtained by ∂J k = 0 and the optimal control input Δuk* is solving ∂Δuk solved. S + R3 uk −1 Δuk* = − 1 (15) R3 + S 2 The following inequality can ensure the sufficient condition for optimization R3 + S 2 > 0 (16) In general, the optimal control can be computed and the procedure of implementing stochastic distribution controller is summarized as follows:

0.15 are also described for the sake of s

1 0.95 0.9 0

50

100 t(s)

Fig 5. Information potential and entropy of tracking error

Fig. 4 demonstrates that the fluctuation of the wind turbine rotor speed is smaller using proposed stochastic distribution control algorithm than PI controller. The dot line is the response of the wind turbine rotor speed using the conventional PI controller, in which the mean value and standard deviation are 0.0824 and 1.0091 respectively. On the other hand, the solid line in Fig. 4 indicates the response of the generator rotational speed using the proposed controller, and the mean value and standard deviation are


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0.0443 and 0.9759 respectively. The information potential and the performance index are displayed in Fig. 5.

0.25 0.2

γe

0.15 0.1 0.05 200 -10

150 -5

0

100 5

50

10

15

20

t(s)

Error range(rad/s)

Fig 6. Evolution of PDF of tracking error

8 7.8 7.6 7.4

(rad)

7.2 7 6.8 6.6 6.4 6.2 6

0

50

100

150

200

t(s)

Fig 7. Tip speed ratio

0.48 0.478 0.476 0.474

Cp

0.472 0.47 0.468 0.466 0.464 0.462 0.46

0

50

100

150

200

t(s)

Fig 8. Power coefficient

Fig. 6 shows the range of tracking error and the PDF of tracking error. It is clear that from Fig. 6 that the shape of PDF becomes narrower along with the time. Fig. 7 and Fig.8 demonstrate the evolution of the tip speed ratio and the power coefficient respectively. The maximum value of the power coefficient is 0.476 and the power coefficient ranges from 0.471 to 0.476. The tip speed ratio ranges between 6.55 rad and 7.12 rad. It proves that the proposed SDC strategy can ensure maximum power point tracking under rated wind speed.

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6

CONCLUSIONS

The contributions of this paper are twofold. A nonlinear PMSG-based WECS model for the partial load region is presented in this paper. On the other hand, a novel stochastic distribution control strategy is proposed for variable-speed WECSs. The proposed control law is data-driven in nature which can be extended to other renewable energy conversion systems. The experimental results demonstrate that the proposed stochastic distribution controller can ensure MPPT, provide smooth rotor speed regulation and avoid unnecessary power fluctuations in the presence of random disturbance from wind speed.

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