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Recurrent neural network based adaptive integral sliding mode power maximization control for wind power systems
Renewable Energy 145 (2020) 1149e1157
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Recurrent neural network based adaptive integral sliding mode power maximization control for wind power systems Xiuxing Yin a, b, Zhansi Jiang a, *, Li Pan c a
School of Mechanical and Electrical Engineering, Guilin University of Electronic Technology, 541004 Guilin, China School of Engineering, University of Warwick, Coventry CV4 7AL, UK c Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education & Zhejiang Province, Zhejiang, Hangzhou 310014, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 May 2018 Received in revised form 26 December 2018 Accepted 27 December 2018 Available online 4 January 2019
An adaptive integral sliding mode controller is proposed to maximize wind power extraction by maintaining the optimum rotation speed of wind turbine. In the proposed controller, an integral sliding mode control law is designed to track the optimum turbine rotation speed based on a recurrent neural network (RNN) that is used to identify the uncertain wind turbine dynamics. An online update algorithm is then derived to update the weights of the RNN in real time and hence to facilitate the maximum power extraction control. The stability of the overall control system is guaranteed in the sense of Lyapunov stability theory. Comparative experimental results demonstrate that the proposed controller outperforms a conventional control method in tracking the optimum turbine rotation speed and extracting the maximum wind power despite system uncertainties and high nonlinearities. © 2018 Published by Elsevier Ltd.
Keywords: Wind power system Maximum wind power extraction Recurrent neural network Sliding mode control
1. Introduction Wind power systems have recently attracted great attention due to the technological advancements and governmental incentives [1]. The increase of wind power capacity and the reduction of maintenance costs have also made wind power systems more competitive in the energy market. Considering that the electrical energy produced by wind power systems is directly related to wind speed conditions and certain control methods, the power conversion efficiency can be improved by using effective control methods. In this sense, the maximum power point tracking (MPPT) control methods are essential for capturing the maximum power from wind [2]. With the use of these MPPT control methods, wind power systems can continuously adjust its rotation speed according to incoming wind speed. In doing so, the tip speed ratio, which is the ratio of the blade tip speed to the wind speed, can be kept at an optimal value to achieve the maximum power conversion efficiency over a wide range of wind speeds. However, the dynamic characteristics of wind power systems are highly nonlinear in essence due to the turbulent nature of wind, parameter uncertainties and exogenous disturbances, such as
* Corresponding author. E-mail addresses: [email protected] (X. Yin), [email protected] (Z. Jiang), [email protected] (L. Pan). https://doi.org/10.1016/j.renene.2018.12.098 0960-1481/© 2018 Published by Elsevier Ltd.
wind rotor aging, dirt or ice on turbine blades. In this context, classical MPPT control methods [3,4] cannot offer sufficient control accuracy and necessary robustness against parametric uncertainties since their control performances are highly sensitive to parametric variations and the stochastic nature of wind. Therefore, nonlinear MPPT control methods and intelligent control approaches are essentially necessary for guaranteeing the desired MPPT control performance in the presence of high nonlinearities and parametric uncertainties. Neural network-based MPPT control methods can be employed to extract the maximum wind power since neural networks are advantageous in universal approximation, relatively fast learning capability, parallel computation and fault tolerance. In Ref. [5], interval-valued neural network was employed for wind speed estimation against potential drift of wind turbine power coefficient. In Ref. [6], an online-training Wilcoxon radial basis function network with hill-climb searching MPPT strategy was proposed for an inverter-based variable-speed permanent magnet synchronous generator (PMSG) driven by a variable-speed wind turbine. The learning rates of the network were updated by using a modified particle swarm optimization (PSO) method. However, that paper was overfilled with the basic descriptions of the wellestablished Wilcoxon radial basis function network and the PSO, whereas no significant improvements have been made. Furthermore, the incorporation of the PSO method would highly deteriorate the dynamic performance of the MPPT control strategy
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since the PSO method is typically time-consuming and may not be implementable in real time. In Ref. [7], the PSO method was also adopted to update the learning rates of the neural networkbased MMPT controller. The rotation speed was obtained by using a model reference adaptive system observer to enable the sensorless speed control. A speed controller was proposed in Ref. [8] to drive the turbine speed for the maximum wind power extraction by combining a general regression neural network-based torque compensator and an optimal controller. However, the MPPT controllers presented in Refs. [7,8] were basically designed based on the traditional tip speed ratio control method which is highly sensitive to changes in the turbine blade surface and has a significant estimation error of the wind speed, particularly in a large wind farm. In Ref. [9], an adaptive neuro-fuzzy inference controller was designed by combining the Sugeno fuzzy model and neural network to track the maximum power points of a PMSG-based wind energy system. However, the convergence proof of the control method was not presented. Fuzzy logic MPPT control algorithms can also be utilized to extract the maximum wind power and counterbalance other nonlinearities and time variances of the power systems since such algorithms offer significant advantages such as fast convergence, robustness against disturbances and model-free controllability [10]. In Ref. [11], a fuzzy logic model based hill-climbing controller was proposed to track the optimum operating points of a wind power system without knowledge of system characteristics. However, the design procedure and stability analysis of the controller were not detailed. In Ref. [12], a modified fuzzy logic controller was designed to continuously regulate the rotation speed of the PMSG-based wind turbine and hence to force the PMSG to work around the maximum power points below the rated wind speed. However, the dynamics of the active and reactive power control loops were not thoroughly investigated. Generally, fuzzy logic control algorithms are designed and implemented based upon empirical observations and always require prior knowledge of wind turbine dynamics. Furthermore, it is difficult to guarantee that such control algorithms would generate acceptable control actions under all operating conditions. A nonlinear back-stepping speed controller was designed in Ref. [13] to maximize wind energy extraction. An interconnected Kalman observer was proposed to estimate the mechanical state variables for the MPPT control. However, that controller needs complex and timeconsuming computations and was not experimentally validated. In this paper, a new adaptive integral sliding mode controller is proposed to maximize the wind power extraction based on the recurrent neural network capable of representing the uncertain or nonlinear wind turbine dynamics. An online backpropagation update algorithm is also derived based on the wind power system parameters to update the weights of the RNN in real time. Unlike the existing sliding mode control methods, an integral sliding mode surface is incorporated into the proposed controller to significantly eliminate steady-state speed tracking error without necessitating additional mechanical sensors and high-order time derivatives of the turbine rotation speed [20,21]. The integral action is incorporated into the control loop to enhance the robustness of the proposed control scheme with respect to model uncertainties, parametric variations and disturbances. The steady error induced by uncertainties can also be highly eliminated by using the integral action in the control loop. The stability of the overall closed loop control system is guaranteed by using an extended Lyapunov function. Comparative experiments are conducted to evaluate the effectiveness and practical feasibility of the proposed controller in comparison with a conventional control method that was designed based on a three-layer radial basis function network under the
same wind speed inputs. 2. System configuration and characteristics 2.1. System design As schematically illustrated in Fig. 1, the kinetic wind power can be captured by a horizontal-axis wind turbine and consequently, converted into electrical power through a multi-pole three-phase PMSG and a drive-train system including a low-speed shaft, a gearbox and a high-speed shaft. The PMSG is directly interfaced to the utility grid via two back-to-back full power converters. The generator side converter can be controlled to extract the maximum wind power, while the grid side converter is regulated to maintain the rated DC-link voltage. The two converters are controlled independently through the decoupled deq vector control approach. Compared with induction generators, the PMSG has significant advantages such as higher reliability and efficiency due to the absence of rotor losses [14]. Furthermore, the decoupled control performance of the PMSG is less sensitive to parametric variations [15]. 2.2. Wind turbine The captured wind power can be calculated as
Pm ¼
1 prR2 v3 CP ðl; bÞ 2
(1)
where Pm is the extracted wind turbine power, v represents the incoming wind speed, R is the turbine radius, r is the air density, b is the turbine pitch angle, l denotes the tip speed ratio, CP denotes the power capture coefficient and is a nonlinear function of both the turbine speed and the blade pitch angle. The tip speed ratio (TSR) can be expressed as
l¼
ur R
(2)
v
where ur is the turbine rotation speed. The aerodynamic torque of the wind turbine can be described as [3].
Ta ¼
1 C ðl; bÞ 2 rpR3 P ,v 2 l
(3)
where Ta is the turbine aerodynamic torque. For the specific wind turbine, the optimum turbine rotation
DC-Link
Generator-side converter
, r
Low-speed shaft Gearbox
ropt
Pg
PMSG
r
v
Highspeed shaft
Wind turbine
d, q
SVP WM
r
opt
Eq. (4)
ud
Eq. (13) iq
uq
The integral sliding mode control law (32)
dˆ
Recurrent neural network ropt
Fig. 1. Wind power system configuration.
The online updating law
r
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
speed uropt corresponds to the maximum power capture coefficient CPmax and the optimal TSR lopt, and thus indicates the peak power points where the maximum available wind power can be extracted. The optimum turbine rotation speed uropt can be deduced from (2) as follows
uropt ¼
lopt ,v
(4)
R
where lopt denotes optimal TSR. Then, the optimum generator power Pgopt can be calculated as
Pgopt ¼ kopt ,h,u3r
(5)
where h denotes the transmission efficiency of the drive-train, kopt denotes a power control gain that can be described as
kopt ¼
C 1 rpR5 pmax 2 l3
(6)
opt
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since the permanent magnets are uniformly mounted on the rotor [11]. Thus,
3 Tg ¼ np jf iq 2
(11)
In general, the d-axis current id can be adequately regulated equal to zero for unity power factor [16]. Therefore, (9) and (10) can be simplified as
uq Rs j i_q ¼ iq ng np f ur þ Ls Ls Ls
(12)
ud ¼ ng np ur iq Ls
(13)
The time derivative of (8) can be obtained by substituting (11) into (8) and considering (12). Thus,
€r ¼ u
3ng n2p j2f 3n j T_ a Tg Rs Bd þ , u_ r þ ur p f uq Jd Jd Ls Jd 2Jd Ls 2Jd Ls
(14)
Substituting Tg ¼ Ta Jd ,u_ r Bd ,ur from (8) into (14) yields 2.3. Drive train dynamics
€r ¼ u
! 3ng n2p j2f Bd Rs 3n j B Rs T_ a Ta Rs þ , dþ u_ r þ , ur p f u q Jd Jd Ls Jd Ls 2Jd Ls Jd Ls 2Jd Ls
The drive-train system can be represented as two lumped mass with torsion damping and stiffness [6,7]. Thus,
Ta Tg ¼ Jd ,u_ r þ Bd ,ur
Under nominal operating conditions, (15) can be written as
(7)
where Tg denotes the electromagnetic torque of the PMSG, Jd and Bd denote the equivalent moment of inertia and torsion damping coefficient of the drive-train system, respectively. Rearranging (7) yields
T Bd ,ur Tg u_ r ¼ a Jd Jd
(8)
2.4. PMSG dynamics The dynamic characteristics of the PMSG can be described in the deq synchronous rotating reference frame [13]. Thus,
€ r ¼ ða þ DaÞ,u_ r þ ðb þ DbÞ,ur þ ðc þ DcÞ,uq þ u
uq Rs j i_q ¼ iq ng np ur id ng np f ur þ Ls Ls Ls Rs u i_d ¼ id þ ng np ur iq þ d Ls Ls
(15)
€ r ¼ a,u_ r þ b,ur þ c,uq þ u
Ta Rs T_ a , þ Jd Ls Jd
where a, b and c are nominal parameters, and can be represented as
8 > > > Bd Rs > > ; þ a ¼ > > > Jd Ls > > > > < 3ng n2p j2f B Rs d, ; b ¼ > 2J L Jd Ls > d s > > > > > > > c ¼ 3np jf > > > 2Jd Ls :
(17)
Considering un-modelled dynamics, unpredicted uncertainties, parametric variations and errors, (16) can be re-formulated as
Ta Rs T_ a , þ ¼ a,u_ r þ b,ur þ c,uq þ d Jd Ls Jd
(9)
(18)
where Da, Db and Dc denote uncertainties introduced by system parameters and un-modelled dynamics, d denotes the lumped uncertainty.
(10)
where id, iq and ud, uq are the d-axis and q-axis currents and voltages of the PMSG respectively, Rs is the stator resistance (ohm), Ls is the stator inductance (H), jf is the permanent magnet flux, np and ng are the number of the pole pairs and the ratio of the gearbox, respectively. The electromagnetic torque of the PMSG can be described in (11)
(16)
d ¼ Da,u_ r þ Db,ur þ Dc,uq þ
Ta Rs T_ a , þ Jd Ls Jd
(19)
The lumped uncertainty term d is mainly caused by the uncertain aerodynamic torque in (3), parameter variations and unmodelled dynamics from (18). In general, the uncertainty d can be estimated offline roughly since the variation ranges of the parameters Ta, ur, and uq are always known a priori. As shown in (3)
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and (19), the uncertainty d is also directly related to the tip speed ratio l which is negatively affected by the uncertainty d since there is an approximately inverse relationship between them as indicated from (3) and (19). The system dynamics can be represented by using (19) and is uncertain in nature. As indicated in (19), the turbine rotation speed ur can be adequately regulated to maintain the optimum power points and hence to maximize the wind power capture by directly controlling the q-axis voltage uq despite the lumped uncertainty d. 3. Neural identification In practice, the lumped uncertainty d can be identified by using a recurrent neural network (RNN) with online-updated weights since the RNN can approximate a wide range of arbitrary nonlinear functions with good accuracy and remarkable learning speed [17]. The RNN is also insensitive to external disturbances and wind torque fluctuations and hence can achieve the nonlinear dynamic mapping due to the internal feedback loop. Furthermore, the RNN performs faster and more efficiently than other commonly used NNs and is well positioned for real time control and identification applications. Therefore, the RNN is used to identify the lumped uncertainty d by selecting appropriate input signals. As illustrated in Fig. 2, a three-layer RNN, which consists of an input layer, a hidden layer, and an output layer, is employed to approximate the dynamics of the lumped uncertainty d. The RNN has two inputs in the input layer, six neurons in the hidden layer and one output neuron in the output layer. Considering (1)e(3), the turbine aerodynamic torque Ta can be represented as
Ta ¼
Pg
(20)
ur ,h
where Pg denotes the generator power which is related to the turbine rotor power Pm through the drive-train efficiency h, i. e. Pg ¼ Phm . As indicated in (20), the aerodynamic torque Ta depends on both the turbine rotation speed ur and the generator power Pg, and hence can be represented as a nonlinear function of Pg and ur. The lumped uncertainty d in (19) can also be derived from Pg and ur since the uncertainty d is directly related to the torque Ta. Therefore, the two inputs of the RNN can be selected as the generator power Pg and the turbine rotation speed ur, while the output of the RNN is
chosen as the estimate of the lumped uncertainty. The two inputs and outputs in the input layer can be described as
8 > > > < x1 ¼ ur ; x2 ¼ Pg ; expðxi Þ expðxi Þ > : Q ¼ f ðxi Þ ¼ > > : i expðxi Þ þ expðxi Þ
(21)
where xi (i ¼ 1, 2) represents the ith input of the input layer, Qi denotes the ith output of the input layer, f (xi) is a symmetrical hyperbolic tangent sigmoid function. The actually measured turbine rotation speed urm and the generator power Pgm commonly contain measurement noises that will deteriorate the final identification accuracy. Thus, the two inputs ur and Pg should be filtered first from their actual measurements through a low-pass filter as follows:
8 > urm > > > < x1 ¼ 1 þ 0:05s > Pgm > > > : x2 ¼ 1 þ 0:05s
(22)
where s denotes the Laplace operator. In the hidden layer, the inputs and outputs can be represented as
8 6 X > > xij ,Qi Ij ¼ uj ,Oj ðk 1Þ þ > > < j¼1
> > > > : Oj ¼ g Ij ¼
exp Ij exp Ij þ exp Ij
(23)
where Ij and Oj (j ¼ 1, 2, ….., 6) denote the j th input and output of the hidden layer, respectively, uj and xij denote the j th recurrent weight and the connecting weight between the input layer and the hidden layer, respectively, k denotes the k th sampling instant, g (Ij) denotes a non-negative tangent sigmoid function. The output layer has a single neuron that estimates the lumped uncertainty d as the weighted sum of all output signals from the hidden layer. Thus,
b d¼
6 X
uj1 ,Oj
(24)
j¼1
where uj1 denotes the jth connecting weight between the hidden layer and the output layer, b d denotes the output of the RNN and is the estimate of the lumped uncertainty d. The output of the RNN can also be represented as
Hidden layer j
Oj
Input layer
x1
Ij
Q1 f (x1)
j1
.... ....
r ij
x2 f (x2) Pg
b d ¼ uT ,F
g (Ij)
dˆ
6
Q2
I6
g (I8)
Fig. 2. Topology of the recurrent neural network.
O6
(25)
where u ¼ [u11, u21, …, u61]T and F ¼ [O1, O2, …., O6]T denote the collected vectors of uj1 and Oj, respectively. Based on the universal approximation theorem [18], there exists the optimal vector uopt that can represent the lumped uncertainty d. Thus,
d ¼ uTopt ,F
(26)
where uopt is the optimal and constant weight vector. The accuracy of the RNN approximation can be largely increased by increasing the number of the connecting weights and the neurons in the hidden layer. Thus, the approximation error can be
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
significantly reduced to arbitrary small over a compact set by choosing appropriate connecting weights and relatively large number of the neurons in the hidden layer [17].
uq ¼
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1 ½ k,zðtÞ ða þ k1 Þ,u_ r ðb þ k2 Þ,ur þ k2 uropt b d c (32)
4. Adaptive integral sliding mode control Since the power extraction performances of the wind power system are highly influenced by the system uncertainties and external wind loading disturbances, it is necessary to design an adaptive robust control law to maximize the wind power extraction by tracking the optimum turbine rotation speed uropt despite system uncertainties and external disturbances. Therefore, an integral sliding mode controller is designed to track the optimum turbine rotation speed uropt and hence to extract the maximum wind power since the sliding mode control is an effective nonlinear robust control approach and provides sufficient robustness against system uncertainties [19]. The proposed controller can also be used to compensate for the external wind disturbances by using online parameter update law and can guarantee the fast online update speed and high convergence speed. In addition, the proposed controller can also attenuate the chattering problems since no switching functions are used.
The sliding mode surface can be selected to model the desired closed-loop speed tracking performance and maintain insensitivity to external disturbances. Thus, the integral sliding mode surface can be defined as
ðt
ur uropt dt
(27)
0
where z(t) denotes the sliding mode surface, k1 and k2 denote the sliding mode coefficients and are positive constants, t denotes a dummy variable for the integration. The time derivative of (24) is
_ ¼u € r þ k1 ,u_ r þ k2 ur uropt zðtÞ
(28)
Setting (28) equal to zero yields
_ ¼ k,zðtÞ þ d b zðtÞ d
€ r þ k1 ,u_ r þ k2 ur uropt ¼ 0 u
_ ¼ k,zðtÞ þ u ~ T ,F zðtÞ
ur k2 ¼ uropt s2 þ k1 s þ k2
1 1 T 1 ~ ,G ,u ~ VðtÞ ¼ z2 ðtÞ þ u 2 2
(35)
where G is a positive definite constant adaptation matrix. Considering (34), the time derivative of (35) can be derived as
(36)
Therefore, the weight update law can be formulated as
u_ ¼ G,F,zðtÞ
(37)
Substituting (37) into (36) yields
_ VðtÞ ¼ k,z2 ðtÞ 0
(38)
_ _ Since VðtÞ 0, VðtÞ is negative semi-definite. Define the func_ tion WðtÞ ¼ k,z2 ðtÞ ¼ VðtÞ and integrate the function W(t) with respect to time
ðt
WðtÞ dt ¼ Vð0Þ VðtÞ
(39)
0
Since V(0) is bounded, and V(t) is non-increasing and bounded, the following result can be obtained:
ðt lim
t/∞
WðtÞ dt < ∞
(40)
0
(30)
As indicated in (30), the sliding mode coefficients k1 and k2 can be adequately chosen to obtain the desired stable transfer function between ur and uropt regardless of system uncertainties and external disturbances. Since this transfer function is stable, the turbine rotation speed ur will be eventually equal to the optimum _ ¼ 0 can be reached. speed uropt when the sliding mode surface zðtÞ Relatively large values of the coefficients k1 and k2 can be selected to guarantee the robustness of the controller and minimize the time required to reach the sliding mode surface from the initial states. Substituting (18) into (28) yields
_ ¼ ða þ k1 Þ,u_ r þ ðb þ k2 Þ,ur k2 uropt þ c,uq þ d zðtÞ
(34)
~ ¼ uopt u denotes the estimation error of the weight where u vector u. Define a Lyapunov function as
(29)
The Laplace transformation of (29) is
(33)
Substituting (25) and (26) into (33) yields
i h _ ~ T , zðtÞ,F G1 ,u _ VðtÞ ¼ k,z2 ðtÞ þ u
4.1. Controller design
zðtÞ ¼ u_ r þ k1 ,ur þ k2
where k is a positive constant control gain. Substituting (32) into (31) yields
(31)
The control law can be designed to force the system state trajectories toward the sliding mode surface and slide along it to a stable equilibrium point. Thus,
By using Barbalat’s lemma [20,21], it can be shown that lim WðtÞ ¼ 0. Thus, z(t)/0 as t/∞. Therefore, the speed tracking t/∞ performance and stability of the controller can be guaranteed. The system dynamics can also be directed towards the sliding mode surface z(t) ¼ 0 and can reach a stable equilibrium point where _ ¼ 0 in finite time. z(t) ¼ 0 and zðtÞ Applying the final-value theorem [22] to (30) yields
lim
ur
t/∞ uropt
¼ lim
s/0 s2
k2 ¼1 þ k1 s þ k2
(41)
Thus, the final steady-state value of the turbine rotation speed
ur will eventually converge to the optimum value uropt despite their time-varying values. The convergence of the weight vector u and the lumped uncertainty d can also be proved based on the following persistent excitation condition [23]. By persistent excitation of F, there exist a strictly positive constant s and a positive time period T such that for any time t > 0,
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tþT ð
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
mT GFðtÞ,FT ðtÞmdt s; ct > 0
(42)
t
where m is an arbitrary unit vector with ||m|| ¼ 1. This convergence can be proved by contradiction. Assuming that ~ k ε. for any ε > 0, there exists a time constant t1, such that ku Consider the following Lyapunov function
Vu~ ðtÞ ¼
i 1h T ~ ðt þ TÞu ~ ðt þ TÞ u ~ T ðtÞu ~ ðtÞ u 2
(43)
The time derivative of (43) is
tþT ð
i dh T ~ ðtÞ dt ~ ðtÞu u dt
¼
i dh T ~ ðtÞGFzðtÞ dt u dt
t tþT ð
¼
_ z dt ~ T GFz u ~ T GF zT FT G2 Fz þ ku
~ GFF u ~ dt u T
_ z dt 1 sε2 ; ct t ~ T GFz u ~ T GF zT FT G2 Fz þ ku 2 2
(45)
t
~ k ε and (42) indicate that the second inThe assumption ku tegral in (44) satisfies
1 ε2
tþT ð
t
~ T GFFT u ~ dt u
vOj vVðtÞ vVðtÞ v b d vOj ¼ ¼ hðtÞ,uj1 , b vxij vO v x vxij j ij vd
(50)
vO
_ is bounded as indicated from (23) and z(t)/0 as t/∞, Since F there will also exist a time constant t2 such that
(49)
where vujj can be calculated by using (21) and (23). Further, the online update law of the connecting weight between the input layer and the hidden layer can be obtained as
x_ ij ¼ T
(48)
vO
(44)
t
tþT ð
vVðtÞ v b d vVðtÞ G, ¼ G, ,Oj vb d vuj1 vb d
vOj vVðtÞ vVðtÞ v b d vOj ¼ ¼ hðtÞ,uj1 , b vuj vO v u vuj j j vd
t tþT ð
vVðtÞ ¼ vuj1
u_ j ¼
t tþT ð
u_ j1 ¼ GhðtÞ,Oj ¼ G,
where vVðtÞ ¼ hðtÞ denotes the system Jacobian. d vb The j th recurrent weight in the hidden layer can be updated as
~ T ðt þ TÞu ~ ðt þ TÞ u ~ T ðtÞu ~ ðtÞ V_ u~ ðtÞ ¼ u ¼
recurrent weights and the connecting weights between the input layer and the hidden layer. All the weights can be updated in the negative direction of the gradient of the Lyapunov function in (35) to minimize this Lyapunov function [25]. By considering (37), the j th connecting weight uj1 between the hidden layer and the output layer can be updated as
tþT ð
t
~T ~ u u dt s; ct t1 GFFT ~k ~k ku ku
where vx j can be calculated by using (21) and (23). ij Therefore, the weights of the RNN at the present sampling instant can be obtained by using (48)e(50) and their values at the previous sampling instants. The control law for uq can also be calculated in real time based on (32) and the updated values of the RNN weights since the sampling frequency can be chosen large enough as compared with the slowly time-varying uncertainty d in practical applications. The generated control voltages uq and ud can then be employed to control the generator side power converter to track the maximum power points and hence to maximize wind power extraction (Fig. 1).
(46)
Combining (44)e(46) yields
1 V_ u~ ðtÞ sε2 ; ct maxðt1 ; t2 Þ 2
(47)
The above inequality (47) clearly contradicts the assumption ~ k ε, hence, there exists the time constant t max(t1, t2), such ku ~ k < ε for any ε > 0, or equivalently, lim u ~ ¼ lim ðuopt uÞ ¼ that ku t/∞ t/∞ 0. Therefore, the weight vector u will asymptotically converge to b the optimum value uopt and the uncertainty d will also asymptotically converge to its real value d as indicated in (25) and (26). The interested readers can also refer to Ref. [23], pp. 367e370] for more details. 4.2. Online update algorithm In practice, an online update algorithm should be derived to facilitate the optimum speed tracking control and hence the maximum power extraction control by updating the weights of the RNN in real time. This online update algorithm can not only be used to update the values of the connecting weights between hidden and output layer, but also can be used to adjust the values of the
Fig. 3. Implementation of the proposed controller in the experimental setup.
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
5. Experiments and discussions 5.1. Experimental setup Comparative experiments have been conducted to verify the feasibility and effectiveness of the proposed controller and the online update algorithm of the RNN weights. As shown in Fig. 3, the experimental setup mainly consists of an AC servo motor, a 4 kW PMSG and an industrial computer. The torque-controlled AC servo motor is employed to represent the dynamic characteristics of a variable speed wind turbine and is directly connected to the PMSG through a mechanical coupling. The proposed integral sliding mode control law, the RNN algorithm and the associated online update algorithm are fully programmed in the industrial computer by using LabVIEW software and its add-on toolkits that provide user friendly control panels and virtual instruments for parameter tuning and online monitoring. Three-phase voltages and currents of the PMSG are measured by using Hall-effect current and voltage transducers and can then be sampled by the multi-channel data acquisition cards installed in the industrial computer to calculate the generator power and generate the optimum turbine rotation speed uropt by using (4). The actual turbine rotation speed ur can be measured by a tachometer mounted on the rotating shaft of the servo motor. The integral sliding mode control law can be designed based on (32) to calculate the q-axis control voltage uq, while the weights of the RNN can be online updated by using (48)e(50). The d-axis control voltage ud can also be generated based on (13) to complete the closed control loop. The calculated voltage commands uq and ud are then sent to a PWM driver and a full-bridge IGBT module with a switching frequency of 10 kHz to implement the control actions. The sampling frequency is chosen as 1 ms to minimize the implementation time and round off error and accelerate the convergence rate of the controller. The controller parameters are chosen to achieve the best transient control performances while considering the stability requirements. The RNN weights are all initialized with random values and will be updated to the optimal values by using the online update algorithm since this tuning algorithm can be guaranteed based on the Lyapunov stability theorem. The experimental results of the proposed control method have
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also been compared with a conventional control method [24] to further evaluate its control efficiency. As shown in Fig. 4, this conventional control method was designed by using a nonlinear three-layer radial basis function network (RBFN) to achieve the MPPT control for a PMSG-based wind power system. The RBFN was adopted to generate the optimum q-axis current for controlling the PMSG and its weights were trained by using supervised backpropagation learning algorithm. The RBFN parameters such as the learning rates were optimized by using a modified particle swarm optimization (MPSO) algorithm in the back-propagation process to improve the learning capability. Since this conventional control method has the same control objective as our work and was also designed based on neural network for real-time implementation, this control method can be chosen as the baseline control method for comparison. The main control parameters are G ¼ diag{0.58, 0.98, 0.88, 0.78, 0.62, 0.87}, k1 ¼128, k2 ¼ 186, k ¼ 286. 5.2. Results and discussions As illustrated in Fig. 5, a 60 s representative data set of actual wind speed v(t) with inherent stochastic nature is used as input for the experiments. The selected wind speed time series varies between 0.5 m/s and 12.5 m/s, and can readily be used to represent below the rated wind speed conditions for MPPT control since the simulated wind turbine has the rated wind speed of 12.5 m/s. As shown in Fig. 6, the optimum turbine rotation speed can be accurately tracked by using the proposed controller, while the speed exhibits significant oscillations when the conventional control method is applied. Therefore, the proposed controller can be employed to better track the optimum speed and hence to maintain the optimum power points despite the uncertainties in the wind power system as compared with the conventional control method. As shown in Fig. 7, th proposed controller varies smoothly, while the q-axis control output voltage of the conventional control method exhibits large fluctuations and has obvious chattering phenomena. Therefore, relatively small control efforts can be generated by the proposed controller to ensure the system stability and accelerate the speed tracking speed as compared with the conventional control method. As illustrated in Fig. 8, the proposed controller can be employed to precisely track the optimum generator power, whereas the generator power deviates significantly from the optimum power points when the conventional control method is used. Thus, the proposed controller can be used to better maintain the optimum generator power and hence to extract the maximum wind energy regardless of external disturbances as compared to the conventional control method. As illustrated in Fig. 9, the TSR can be smoothly maintained
Fig. 4. Implementation of the conventional controller in the experimental setup.
Fig. 5. Wind speed time series.
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X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
Fig. 6. Turbine rotation speed variations.
Fig. 10. Estimate of the lumped uncertainty.
time for implementing the MPPT control. 6. Conclusion
Fig. 7. q-axis voltage variations.
The adaptive integral sliding mode control law and the associated online update algorithm of the RNN weights have been presented to maximize wind power extraction by tracking the optimum turbine rotation speed. The RNN has been employed to estimate the unknown lumped uncertainty in real time for the controller design. The Lyapunov synthesis method has also been used to guarantee the stability of the overall control system despite the existence of uncertainties, and alleviate the chattering phenomena in the control efforts. Comparative experimental results have demonstrated that the proposed controller possesses a remarkable learning capability of the RNN weights and can be used to better maintain the optimum generator power. Acknowledgement
Fig. 8. The generator power variations.
This work is supported in part by the National Natural Science Foundation of China (No. 51565008), Natural Science Foundation of Guangxi (No. 2017JJA160071z). This project is also supported by Zhejiang Natural Science Foundation No. LY18E050024. References
Fig. 9. Variations of the tip speed ratio.
around the optimal value by using the proposed controller, whereas the TSR fluctuates significantly and cannot be kept around the optimal values when the conventional control method is employed. Thus, the proposed controller can be used to better maintain the optimal TSR and hence the peak power points when compared with the conventional control method. As illustrated in Fig. 10, the identified lumped uncertainty (normalized) can accurately track the real value over a wide range of operating conditions based on the online update algorithm. Therefore, the lumped uncertainty can be accurately estimated by using the RNN and can be used to generate the control efforts in real
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Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Recurrent neural network based adaptive integral sliding mode power maximization control for wind power systems Xiuxing Yin a, b, Zhansi Jiang a, *, Li Pan c a
School of Mechanical and Electrical Engineering, Guilin University of Electronic Technology, 541004 Guilin, China School of Engineering, University of Warwick, Coventry CV4 7AL, UK c Key Laboratory of E&M, Zhejiang University of Technology, Ministry of Education & Zhejiang Province, Zhejiang, Hangzhou 310014, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 May 2018 Received in revised form 26 December 2018 Accepted 27 December 2018 Available online 4 January 2019
An adaptive integral sliding mode controller is proposed to maximize wind power extraction by maintaining the optimum rotation speed of wind turbine. In the proposed controller, an integral sliding mode control law is designed to track the optimum turbine rotation speed based on a recurrent neural network (RNN) that is used to identify the uncertain wind turbine dynamics. An online update algorithm is then derived to update the weights of the RNN in real time and hence to facilitate the maximum power extraction control. The stability of the overall control system is guaranteed in the sense of Lyapunov stability theory. Comparative experimental results demonstrate that the proposed controller outperforms a conventional control method in tracking the optimum turbine rotation speed and extracting the maximum wind power despite system uncertainties and high nonlinearities. © 2018 Published by Elsevier Ltd.
Keywords: Wind power system Maximum wind power extraction Recurrent neural network Sliding mode control
1. Introduction Wind power systems have recently attracted great attention due to the technological advancements and governmental incentives [1]. The increase of wind power capacity and the reduction of maintenance costs have also made wind power systems more competitive in the energy market. Considering that the electrical energy produced by wind power systems is directly related to wind speed conditions and certain control methods, the power conversion efficiency can be improved by using effective control methods. In this sense, the maximum power point tracking (MPPT) control methods are essential for capturing the maximum power from wind [2]. With the use of these MPPT control methods, wind power systems can continuously adjust its rotation speed according to incoming wind speed. In doing so, the tip speed ratio, which is the ratio of the blade tip speed to the wind speed, can be kept at an optimal value to achieve the maximum power conversion efficiency over a wide range of wind speeds. However, the dynamic characteristics of wind power systems are highly nonlinear in essence due to the turbulent nature of wind, parameter uncertainties and exogenous disturbances, such as
* Corresponding author. E-mail addresses: [email protected] (X. Yin), [email protected] (Z. Jiang), [email protected] (L. Pan). https://doi.org/10.1016/j.renene.2018.12.098 0960-1481/© 2018 Published by Elsevier Ltd.
wind rotor aging, dirt or ice on turbine blades. In this context, classical MPPT control methods [3,4] cannot offer sufficient control accuracy and necessary robustness against parametric uncertainties since their control performances are highly sensitive to parametric variations and the stochastic nature of wind. Therefore, nonlinear MPPT control methods and intelligent control approaches are essentially necessary for guaranteeing the desired MPPT control performance in the presence of high nonlinearities and parametric uncertainties. Neural network-based MPPT control methods can be employed to extract the maximum wind power since neural networks are advantageous in universal approximation, relatively fast learning capability, parallel computation and fault tolerance. In Ref. [5], interval-valued neural network was employed for wind speed estimation against potential drift of wind turbine power coefficient. In Ref. [6], an online-training Wilcoxon radial basis function network with hill-climb searching MPPT strategy was proposed for an inverter-based variable-speed permanent magnet synchronous generator (PMSG) driven by a variable-speed wind turbine. The learning rates of the network were updated by using a modified particle swarm optimization (PSO) method. However, that paper was overfilled with the basic descriptions of the wellestablished Wilcoxon radial basis function network and the PSO, whereas no significant improvements have been made. Furthermore, the incorporation of the PSO method would highly deteriorate the dynamic performance of the MPPT control strategy
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since the PSO method is typically time-consuming and may not be implementable in real time. In Ref. [7], the PSO method was also adopted to update the learning rates of the neural networkbased MMPT controller. The rotation speed was obtained by using a model reference adaptive system observer to enable the sensorless speed control. A speed controller was proposed in Ref. [8] to drive the turbine speed for the maximum wind power extraction by combining a general regression neural network-based torque compensator and an optimal controller. However, the MPPT controllers presented in Refs. [7,8] were basically designed based on the traditional tip speed ratio control method which is highly sensitive to changes in the turbine blade surface and has a significant estimation error of the wind speed, particularly in a large wind farm. In Ref. [9], an adaptive neuro-fuzzy inference controller was designed by combining the Sugeno fuzzy model and neural network to track the maximum power points of a PMSG-based wind energy system. However, the convergence proof of the control method was not presented. Fuzzy logic MPPT control algorithms can also be utilized to extract the maximum wind power and counterbalance other nonlinearities and time variances of the power systems since such algorithms offer significant advantages such as fast convergence, robustness against disturbances and model-free controllability [10]. In Ref. [11], a fuzzy logic model based hill-climbing controller was proposed to track the optimum operating points of a wind power system without knowledge of system characteristics. However, the design procedure and stability analysis of the controller were not detailed. In Ref. [12], a modified fuzzy logic controller was designed to continuously regulate the rotation speed of the PMSG-based wind turbine and hence to force the PMSG to work around the maximum power points below the rated wind speed. However, the dynamics of the active and reactive power control loops were not thoroughly investigated. Generally, fuzzy logic control algorithms are designed and implemented based upon empirical observations and always require prior knowledge of wind turbine dynamics. Furthermore, it is difficult to guarantee that such control algorithms would generate acceptable control actions under all operating conditions. A nonlinear back-stepping speed controller was designed in Ref. [13] to maximize wind energy extraction. An interconnected Kalman observer was proposed to estimate the mechanical state variables for the MPPT control. However, that controller needs complex and timeconsuming computations and was not experimentally validated. In this paper, a new adaptive integral sliding mode controller is proposed to maximize the wind power extraction based on the recurrent neural network capable of representing the uncertain or nonlinear wind turbine dynamics. An online backpropagation update algorithm is also derived based on the wind power system parameters to update the weights of the RNN in real time. Unlike the existing sliding mode control methods, an integral sliding mode surface is incorporated into the proposed controller to significantly eliminate steady-state speed tracking error without necessitating additional mechanical sensors and high-order time derivatives of the turbine rotation speed [20,21]. The integral action is incorporated into the control loop to enhance the robustness of the proposed control scheme with respect to model uncertainties, parametric variations and disturbances. The steady error induced by uncertainties can also be highly eliminated by using the integral action in the control loop. The stability of the overall closed loop control system is guaranteed by using an extended Lyapunov function. Comparative experiments are conducted to evaluate the effectiveness and practical feasibility of the proposed controller in comparison with a conventional control method that was designed based on a three-layer radial basis function network under the
same wind speed inputs. 2. System configuration and characteristics 2.1. System design As schematically illustrated in Fig. 1, the kinetic wind power can be captured by a horizontal-axis wind turbine and consequently, converted into electrical power through a multi-pole three-phase PMSG and a drive-train system including a low-speed shaft, a gearbox and a high-speed shaft. The PMSG is directly interfaced to the utility grid via two back-to-back full power converters. The generator side converter can be controlled to extract the maximum wind power, while the grid side converter is regulated to maintain the rated DC-link voltage. The two converters are controlled independently through the decoupled deq vector control approach. Compared with induction generators, the PMSG has significant advantages such as higher reliability and efficiency due to the absence of rotor losses [14]. Furthermore, the decoupled control performance of the PMSG is less sensitive to parametric variations [15]. 2.2. Wind turbine The captured wind power can be calculated as
Pm ¼
1 prR2 v3 CP ðl; bÞ 2
(1)
where Pm is the extracted wind turbine power, v represents the incoming wind speed, R is the turbine radius, r is the air density, b is the turbine pitch angle, l denotes the tip speed ratio, CP denotes the power capture coefficient and is a nonlinear function of both the turbine speed and the blade pitch angle. The tip speed ratio (TSR) can be expressed as
l¼
ur R
(2)
v
where ur is the turbine rotation speed. The aerodynamic torque of the wind turbine can be described as [3].
Ta ¼
1 C ðl; bÞ 2 rpR3 P ,v 2 l
(3)
where Ta is the turbine aerodynamic torque. For the specific wind turbine, the optimum turbine rotation
DC-Link
Generator-side converter
, r
Low-speed shaft Gearbox
ropt
Pg
PMSG
r
v
Highspeed shaft
Wind turbine
d, q
SVP WM
r
opt
Eq. (4)
ud
Eq. (13) iq
uq
The integral sliding mode control law (32)
dˆ
Recurrent neural network ropt
Fig. 1. Wind power system configuration.
The online updating law
r
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
speed uropt corresponds to the maximum power capture coefficient CPmax and the optimal TSR lopt, and thus indicates the peak power points where the maximum available wind power can be extracted. The optimum turbine rotation speed uropt can be deduced from (2) as follows
uropt ¼
lopt ,v
(4)
R
where lopt denotes optimal TSR. Then, the optimum generator power Pgopt can be calculated as
Pgopt ¼ kopt ,h,u3r
(5)
where h denotes the transmission efficiency of the drive-train, kopt denotes a power control gain that can be described as
kopt ¼
C 1 rpR5 pmax 2 l3
(6)
opt
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since the permanent magnets are uniformly mounted on the rotor [11]. Thus,
3 Tg ¼ np jf iq 2
(11)
In general, the d-axis current id can be adequately regulated equal to zero for unity power factor [16]. Therefore, (9) and (10) can be simplified as
uq Rs j i_q ¼ iq ng np f ur þ Ls Ls Ls
(12)
ud ¼ ng np ur iq Ls
(13)
The time derivative of (8) can be obtained by substituting (11) into (8) and considering (12). Thus,
€r ¼ u
3ng n2p j2f 3n j T_ a Tg Rs Bd þ , u_ r þ ur p f uq Jd Jd Ls Jd 2Jd Ls 2Jd Ls
(14)
Substituting Tg ¼ Ta Jd ,u_ r Bd ,ur from (8) into (14) yields 2.3. Drive train dynamics
€r ¼ u
! 3ng n2p j2f Bd Rs 3n j B Rs T_ a Ta Rs þ , dþ u_ r þ , ur p f u q Jd Jd Ls Jd Ls 2Jd Ls Jd Ls 2Jd Ls
The drive-train system can be represented as two lumped mass with torsion damping and stiffness [6,7]. Thus,
Ta Tg ¼ Jd ,u_ r þ Bd ,ur
Under nominal operating conditions, (15) can be written as
(7)
where Tg denotes the electromagnetic torque of the PMSG, Jd and Bd denote the equivalent moment of inertia and torsion damping coefficient of the drive-train system, respectively. Rearranging (7) yields
T Bd ,ur Tg u_ r ¼ a Jd Jd
(8)
2.4. PMSG dynamics The dynamic characteristics of the PMSG can be described in the deq synchronous rotating reference frame [13]. Thus,
€ r ¼ ða þ DaÞ,u_ r þ ðb þ DbÞ,ur þ ðc þ DcÞ,uq þ u
uq Rs j i_q ¼ iq ng np ur id ng np f ur þ Ls Ls Ls Rs u i_d ¼ id þ ng np ur iq þ d Ls Ls
(15)
€ r ¼ a,u_ r þ b,ur þ c,uq þ u
Ta Rs T_ a , þ Jd Ls Jd
where a, b and c are nominal parameters, and can be represented as
8 > > > Bd Rs > > ; þ a ¼ > > > Jd Ls > > > > < 3ng n2p j2f B Rs d, ; b ¼ > 2J L Jd Ls > d s > > > > > > > c ¼ 3np jf > > > 2Jd Ls :
(17)
Considering un-modelled dynamics, unpredicted uncertainties, parametric variations and errors, (16) can be re-formulated as
Ta Rs T_ a , þ ¼ a,u_ r þ b,ur þ c,uq þ d Jd Ls Jd
(9)
(18)
where Da, Db and Dc denote uncertainties introduced by system parameters and un-modelled dynamics, d denotes the lumped uncertainty.
(10)
where id, iq and ud, uq are the d-axis and q-axis currents and voltages of the PMSG respectively, Rs is the stator resistance (ohm), Ls is the stator inductance (H), jf is the permanent magnet flux, np and ng are the number of the pole pairs and the ratio of the gearbox, respectively. The electromagnetic torque of the PMSG can be described in (11)
(16)
d ¼ Da,u_ r þ Db,ur þ Dc,uq þ
Ta Rs T_ a , þ Jd Ls Jd
(19)
The lumped uncertainty term d is mainly caused by the uncertain aerodynamic torque in (3), parameter variations and unmodelled dynamics from (18). In general, the uncertainty d can be estimated offline roughly since the variation ranges of the parameters Ta, ur, and uq are always known a priori. As shown in (3)
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and (19), the uncertainty d is also directly related to the tip speed ratio l which is negatively affected by the uncertainty d since there is an approximately inverse relationship between them as indicated from (3) and (19). The system dynamics can be represented by using (19) and is uncertain in nature. As indicated in (19), the turbine rotation speed ur can be adequately regulated to maintain the optimum power points and hence to maximize the wind power capture by directly controlling the q-axis voltage uq despite the lumped uncertainty d. 3. Neural identification In practice, the lumped uncertainty d can be identified by using a recurrent neural network (RNN) with online-updated weights since the RNN can approximate a wide range of arbitrary nonlinear functions with good accuracy and remarkable learning speed [17]. The RNN is also insensitive to external disturbances and wind torque fluctuations and hence can achieve the nonlinear dynamic mapping due to the internal feedback loop. Furthermore, the RNN performs faster and more efficiently than other commonly used NNs and is well positioned for real time control and identification applications. Therefore, the RNN is used to identify the lumped uncertainty d by selecting appropriate input signals. As illustrated in Fig. 2, a three-layer RNN, which consists of an input layer, a hidden layer, and an output layer, is employed to approximate the dynamics of the lumped uncertainty d. The RNN has two inputs in the input layer, six neurons in the hidden layer and one output neuron in the output layer. Considering (1)e(3), the turbine aerodynamic torque Ta can be represented as
Ta ¼
Pg
(20)
ur ,h
where Pg denotes the generator power which is related to the turbine rotor power Pm through the drive-train efficiency h, i. e. Pg ¼ Phm . As indicated in (20), the aerodynamic torque Ta depends on both the turbine rotation speed ur and the generator power Pg, and hence can be represented as a nonlinear function of Pg and ur. The lumped uncertainty d in (19) can also be derived from Pg and ur since the uncertainty d is directly related to the torque Ta. Therefore, the two inputs of the RNN can be selected as the generator power Pg and the turbine rotation speed ur, while the output of the RNN is
chosen as the estimate of the lumped uncertainty. The two inputs and outputs in the input layer can be described as
8 > > > < x1 ¼ ur ; x2 ¼ Pg ; expðxi Þ expðxi Þ > : Q ¼ f ðxi Þ ¼ > > : i expðxi Þ þ expðxi Þ
(21)
where xi (i ¼ 1, 2) represents the ith input of the input layer, Qi denotes the ith output of the input layer, f (xi) is a symmetrical hyperbolic tangent sigmoid function. The actually measured turbine rotation speed urm and the generator power Pgm commonly contain measurement noises that will deteriorate the final identification accuracy. Thus, the two inputs ur and Pg should be filtered first from their actual measurements through a low-pass filter as follows:
8 > urm > > > < x1 ¼ 1 þ 0:05s > Pgm > > > : x2 ¼ 1 þ 0:05s
(22)
where s denotes the Laplace operator. In the hidden layer, the inputs and outputs can be represented as
8 6 X > > xij ,Qi Ij ¼ uj ,Oj ðk 1Þ þ > > < j¼1
> > > > : Oj ¼ g Ij ¼
exp Ij exp Ij þ exp Ij
(23)
where Ij and Oj (j ¼ 1, 2, ….., 6) denote the j th input and output of the hidden layer, respectively, uj and xij denote the j th recurrent weight and the connecting weight between the input layer and the hidden layer, respectively, k denotes the k th sampling instant, g (Ij) denotes a non-negative tangent sigmoid function. The output layer has a single neuron that estimates the lumped uncertainty d as the weighted sum of all output signals from the hidden layer. Thus,
b d¼
6 X
uj1 ,Oj
(24)
j¼1
where uj1 denotes the jth connecting weight between the hidden layer and the output layer, b d denotes the output of the RNN and is the estimate of the lumped uncertainty d. The output of the RNN can also be represented as
Hidden layer j
Oj
Input layer
x1
Ij
Q1 f (x1)
j1
.... ....
r ij
x2 f (x2) Pg
b d ¼ uT ,F
g (Ij)
dˆ
6
Q2
I6
g (I8)
Fig. 2. Topology of the recurrent neural network.
O6
(25)
where u ¼ [u11, u21, …, u61]T and F ¼ [O1, O2, …., O6]T denote the collected vectors of uj1 and Oj, respectively. Based on the universal approximation theorem [18], there exists the optimal vector uopt that can represent the lumped uncertainty d. Thus,
d ¼ uTopt ,F
(26)
where uopt is the optimal and constant weight vector. The accuracy of the RNN approximation can be largely increased by increasing the number of the connecting weights and the neurons in the hidden layer. Thus, the approximation error can be
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
significantly reduced to arbitrary small over a compact set by choosing appropriate connecting weights and relatively large number of the neurons in the hidden layer [17].
uq ¼
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1 ½ k,zðtÞ ða þ k1 Þ,u_ r ðb þ k2 Þ,ur þ k2 uropt b d c (32)
4. Adaptive integral sliding mode control Since the power extraction performances of the wind power system are highly influenced by the system uncertainties and external wind loading disturbances, it is necessary to design an adaptive robust control law to maximize the wind power extraction by tracking the optimum turbine rotation speed uropt despite system uncertainties and external disturbances. Therefore, an integral sliding mode controller is designed to track the optimum turbine rotation speed uropt and hence to extract the maximum wind power since the sliding mode control is an effective nonlinear robust control approach and provides sufficient robustness against system uncertainties [19]. The proposed controller can also be used to compensate for the external wind disturbances by using online parameter update law and can guarantee the fast online update speed and high convergence speed. In addition, the proposed controller can also attenuate the chattering problems since no switching functions are used.
The sliding mode surface can be selected to model the desired closed-loop speed tracking performance and maintain insensitivity to external disturbances. Thus, the integral sliding mode surface can be defined as
ðt
ur uropt dt
(27)
0
where z(t) denotes the sliding mode surface, k1 and k2 denote the sliding mode coefficients and are positive constants, t denotes a dummy variable for the integration. The time derivative of (24) is
_ ¼u € r þ k1 ,u_ r þ k2 ur uropt zðtÞ
(28)
Setting (28) equal to zero yields
_ ¼ k,zðtÞ þ d b zðtÞ d
€ r þ k1 ,u_ r þ k2 ur uropt ¼ 0 u
_ ¼ k,zðtÞ þ u ~ T ,F zðtÞ
ur k2 ¼ uropt s2 þ k1 s þ k2
1 1 T 1 ~ ,G ,u ~ VðtÞ ¼ z2 ðtÞ þ u 2 2
(35)
where G is a positive definite constant adaptation matrix. Considering (34), the time derivative of (35) can be derived as
(36)
Therefore, the weight update law can be formulated as
u_ ¼ G,F,zðtÞ
(37)
Substituting (37) into (36) yields
_ VðtÞ ¼ k,z2 ðtÞ 0
(38)
_ _ Since VðtÞ 0, VðtÞ is negative semi-definite. Define the func_ tion WðtÞ ¼ k,z2 ðtÞ ¼ VðtÞ and integrate the function W(t) with respect to time
ðt
WðtÞ dt ¼ Vð0Þ VðtÞ
(39)
0
Since V(0) is bounded, and V(t) is non-increasing and bounded, the following result can be obtained:
ðt lim
t/∞
WðtÞ dt < ∞
(40)
0
(30)
As indicated in (30), the sliding mode coefficients k1 and k2 can be adequately chosen to obtain the desired stable transfer function between ur and uropt regardless of system uncertainties and external disturbances. Since this transfer function is stable, the turbine rotation speed ur will be eventually equal to the optimum _ ¼ 0 can be reached. speed uropt when the sliding mode surface zðtÞ Relatively large values of the coefficients k1 and k2 can be selected to guarantee the robustness of the controller and minimize the time required to reach the sliding mode surface from the initial states. Substituting (18) into (28) yields
_ ¼ ða þ k1 Þ,u_ r þ ðb þ k2 Þ,ur k2 uropt þ c,uq þ d zðtÞ
(34)
~ ¼ uopt u denotes the estimation error of the weight where u vector u. Define a Lyapunov function as
(29)
The Laplace transformation of (29) is
(33)
Substituting (25) and (26) into (33) yields
i h _ ~ T , zðtÞ,F G1 ,u _ VðtÞ ¼ k,z2 ðtÞ þ u
4.1. Controller design
zðtÞ ¼ u_ r þ k1 ,ur þ k2
where k is a positive constant control gain. Substituting (32) into (31) yields
(31)
The control law can be designed to force the system state trajectories toward the sliding mode surface and slide along it to a stable equilibrium point. Thus,
By using Barbalat’s lemma [20,21], it can be shown that lim WðtÞ ¼ 0. Thus, z(t)/0 as t/∞. Therefore, the speed tracking t/∞ performance and stability of the controller can be guaranteed. The system dynamics can also be directed towards the sliding mode surface z(t) ¼ 0 and can reach a stable equilibrium point where _ ¼ 0 in finite time. z(t) ¼ 0 and zðtÞ Applying the final-value theorem [22] to (30) yields
lim
ur
t/∞ uropt
¼ lim
s/0 s2
k2 ¼1 þ k1 s þ k2
(41)
Thus, the final steady-state value of the turbine rotation speed
ur will eventually converge to the optimum value uropt despite their time-varying values. The convergence of the weight vector u and the lumped uncertainty d can also be proved based on the following persistent excitation condition [23]. By persistent excitation of F, there exist a strictly positive constant s and a positive time period T such that for any time t > 0,
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tþT ð
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
mT GFðtÞ,FT ðtÞmdt s; ct > 0
(42)
t
where m is an arbitrary unit vector with ||m|| ¼ 1. This convergence can be proved by contradiction. Assuming that ~ k ε. for any ε > 0, there exists a time constant t1, such that ku Consider the following Lyapunov function
Vu~ ðtÞ ¼
i 1h T ~ ðt þ TÞu ~ ðt þ TÞ u ~ T ðtÞu ~ ðtÞ u 2
(43)
The time derivative of (43) is
tþT ð
i dh T ~ ðtÞ dt ~ ðtÞu u dt
¼
i dh T ~ ðtÞGFzðtÞ dt u dt
t tþT ð
¼
_ z dt ~ T GFz u ~ T GF zT FT G2 Fz þ ku
~ GFF u ~ dt u T
_ z dt 1 sε2 ; ct t ~ T GFz u ~ T GF zT FT G2 Fz þ ku 2 2
(45)
t
~ k ε and (42) indicate that the second inThe assumption ku tegral in (44) satisfies
1 ε2
tþT ð
t
~ T GFFT u ~ dt u
vOj vVðtÞ vVðtÞ v b d vOj ¼ ¼ hðtÞ,uj1 , b vxij vO v x vxij j ij vd
(50)
vO
_ is bounded as indicated from (23) and z(t)/0 as t/∞, Since F there will also exist a time constant t2 such that
(49)
where vujj can be calculated by using (21) and (23). Further, the online update law of the connecting weight between the input layer and the hidden layer can be obtained as
x_ ij ¼ T
(48)
vO
(44)
t
tþT ð
vVðtÞ v b d vVðtÞ G, ¼ G, ,Oj vb d vuj1 vb d
vOj vVðtÞ vVðtÞ v b d vOj ¼ ¼ hðtÞ,uj1 , b vuj vO v u vuj j j vd
t tþT ð
vVðtÞ ¼ vuj1
u_ j ¼
t tþT ð
u_ j1 ¼ GhðtÞ,Oj ¼ G,
where vVðtÞ ¼ hðtÞ denotes the system Jacobian. d vb The j th recurrent weight in the hidden layer can be updated as
~ T ðt þ TÞu ~ ðt þ TÞ u ~ T ðtÞu ~ ðtÞ V_ u~ ðtÞ ¼ u ¼
recurrent weights and the connecting weights between the input layer and the hidden layer. All the weights can be updated in the negative direction of the gradient of the Lyapunov function in (35) to minimize this Lyapunov function [25]. By considering (37), the j th connecting weight uj1 between the hidden layer and the output layer can be updated as
tþT ð
t
~T ~ u u dt s; ct t1 GFFT ~k ~k ku ku
where vx j can be calculated by using (21) and (23). ij Therefore, the weights of the RNN at the present sampling instant can be obtained by using (48)e(50) and their values at the previous sampling instants. The control law for uq can also be calculated in real time based on (32) and the updated values of the RNN weights since the sampling frequency can be chosen large enough as compared with the slowly time-varying uncertainty d in practical applications. The generated control voltages uq and ud can then be employed to control the generator side power converter to track the maximum power points and hence to maximize wind power extraction (Fig. 1).
(46)
Combining (44)e(46) yields
1 V_ u~ ðtÞ sε2 ; ct maxðt1 ; t2 Þ 2
(47)
The above inequality (47) clearly contradicts the assumption ~ k ε, hence, there exists the time constant t max(t1, t2), such ku ~ k < ε for any ε > 0, or equivalently, lim u ~ ¼ lim ðuopt uÞ ¼ that ku t/∞ t/∞ 0. Therefore, the weight vector u will asymptotically converge to b the optimum value uopt and the uncertainty d will also asymptotically converge to its real value d as indicated in (25) and (26). The interested readers can also refer to Ref. [23], pp. 367e370] for more details. 4.2. Online update algorithm In practice, an online update algorithm should be derived to facilitate the optimum speed tracking control and hence the maximum power extraction control by updating the weights of the RNN in real time. This online update algorithm can not only be used to update the values of the connecting weights between hidden and output layer, but also can be used to adjust the values of the
Fig. 3. Implementation of the proposed controller in the experimental setup.
X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
5. Experiments and discussions 5.1. Experimental setup Comparative experiments have been conducted to verify the feasibility and effectiveness of the proposed controller and the online update algorithm of the RNN weights. As shown in Fig. 3, the experimental setup mainly consists of an AC servo motor, a 4 kW PMSG and an industrial computer. The torque-controlled AC servo motor is employed to represent the dynamic characteristics of a variable speed wind turbine and is directly connected to the PMSG through a mechanical coupling. The proposed integral sliding mode control law, the RNN algorithm and the associated online update algorithm are fully programmed in the industrial computer by using LabVIEW software and its add-on toolkits that provide user friendly control panels and virtual instruments for parameter tuning and online monitoring. Three-phase voltages and currents of the PMSG are measured by using Hall-effect current and voltage transducers and can then be sampled by the multi-channel data acquisition cards installed in the industrial computer to calculate the generator power and generate the optimum turbine rotation speed uropt by using (4). The actual turbine rotation speed ur can be measured by a tachometer mounted on the rotating shaft of the servo motor. The integral sliding mode control law can be designed based on (32) to calculate the q-axis control voltage uq, while the weights of the RNN can be online updated by using (48)e(50). The d-axis control voltage ud can also be generated based on (13) to complete the closed control loop. The calculated voltage commands uq and ud are then sent to a PWM driver and a full-bridge IGBT module with a switching frequency of 10 kHz to implement the control actions. The sampling frequency is chosen as 1 ms to minimize the implementation time and round off error and accelerate the convergence rate of the controller. The controller parameters are chosen to achieve the best transient control performances while considering the stability requirements. The RNN weights are all initialized with random values and will be updated to the optimal values by using the online update algorithm since this tuning algorithm can be guaranteed based on the Lyapunov stability theorem. The experimental results of the proposed control method have
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also been compared with a conventional control method [24] to further evaluate its control efficiency. As shown in Fig. 4, this conventional control method was designed by using a nonlinear three-layer radial basis function network (RBFN) to achieve the MPPT control for a PMSG-based wind power system. The RBFN was adopted to generate the optimum q-axis current for controlling the PMSG and its weights were trained by using supervised backpropagation learning algorithm. The RBFN parameters such as the learning rates were optimized by using a modified particle swarm optimization (MPSO) algorithm in the back-propagation process to improve the learning capability. Since this conventional control method has the same control objective as our work and was also designed based on neural network for real-time implementation, this control method can be chosen as the baseline control method for comparison. The main control parameters are G ¼ diag{0.58, 0.98, 0.88, 0.78, 0.62, 0.87}, k1 ¼128, k2 ¼ 186, k ¼ 286. 5.2. Results and discussions As illustrated in Fig. 5, a 60 s representative data set of actual wind speed v(t) with inherent stochastic nature is used as input for the experiments. The selected wind speed time series varies between 0.5 m/s and 12.5 m/s, and can readily be used to represent below the rated wind speed conditions for MPPT control since the simulated wind turbine has the rated wind speed of 12.5 m/s. As shown in Fig. 6, the optimum turbine rotation speed can be accurately tracked by using the proposed controller, while the speed exhibits significant oscillations when the conventional control method is applied. Therefore, the proposed controller can be employed to better track the optimum speed and hence to maintain the optimum power points despite the uncertainties in the wind power system as compared with the conventional control method. As shown in Fig. 7, th proposed controller varies smoothly, while the q-axis control output voltage of the conventional control method exhibits large fluctuations and has obvious chattering phenomena. Therefore, relatively small control efforts can be generated by the proposed controller to ensure the system stability and accelerate the speed tracking speed as compared with the conventional control method. As illustrated in Fig. 8, the proposed controller can be employed to precisely track the optimum generator power, whereas the generator power deviates significantly from the optimum power points when the conventional control method is used. Thus, the proposed controller can be used to better maintain the optimum generator power and hence to extract the maximum wind energy regardless of external disturbances as compared to the conventional control method. As illustrated in Fig. 9, the TSR can be smoothly maintained
Fig. 4. Implementation of the conventional controller in the experimental setup.
Fig. 5. Wind speed time series.
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X. Yin et al. / Renewable Energy 145 (2020) 1149e1157
Fig. 6. Turbine rotation speed variations.
Fig. 10. Estimate of the lumped uncertainty.
time for implementing the MPPT control. 6. Conclusion
Fig. 7. q-axis voltage variations.
The adaptive integral sliding mode control law and the associated online update algorithm of the RNN weights have been presented to maximize wind power extraction by tracking the optimum turbine rotation speed. The RNN has been employed to estimate the unknown lumped uncertainty in real time for the controller design. The Lyapunov synthesis method has also been used to guarantee the stability of the overall control system despite the existence of uncertainties, and alleviate the chattering phenomena in the control efforts. Comparative experimental results have demonstrated that the proposed controller possesses a remarkable learning capability of the RNN weights and can be used to better maintain the optimum generator power. Acknowledgement
Fig. 8. The generator power variations.
This work is supported in part by the National Natural Science Foundation of China (No. 51565008), Natural Science Foundation of Guangxi (No. 2017JJA160071z). This project is also supported by Zhejiang Natural Science Foundation No. LY18E050024. References
Fig. 9. Variations of the tip speed ratio.
around the optimal value by using the proposed controller, whereas the TSR fluctuates significantly and cannot be kept around the optimal values when the conventional control method is employed. Thus, the proposed controller can be used to better maintain the optimal TSR and hence the peak power points when compared with the conventional control method. As illustrated in Fig. 10, the identified lumped uncertainty (normalized) can accurately track the real value over a wide range of operating conditions based on the online update algorithm. Therefore, the lumped uncertainty can be accurately estimated by using the RNN and can be used to generate the control efforts in real
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