Adaptive PID Control of Wind Energy Conversion Systems Using RASP1 Mother Wavelet Basis Function Networks

— In this paper a PID control strategy using neural network adaptive RASP1 wavelet for WECS’s control is proposed. It is based on single layer feedforward neural networks with hidden nodes of adaptive RASP1 wavelet functions controller and an infinite impulse response (IIR) recurrent structure. The IIR is combined by cascading to the network to provide double local structure resulting in improving speed of learning. This particular neuro PID controller assumes a certain model structure to approximately identify the system dynamics of the unknown plant (WECS’s) and generate the control signal. The results are applied to a typical turbine/generator pair, showing the feasibility of the proposed solution.

control and an RBF network-based adaptive controller has proposed in [6].
This paper proposes a adaptive PID controller using neural network frame RASP1 wavelets for WECS's control.It consists of a single layer feedforward neural network with hidden nodes of adaptive wavelet functions controller and an infinite impulse response (IIR) recurrent structure.The IIR is combined by cascading to the network to provide double local structure resulting in improving speed of learning.

II. WIND ENERGY CONVERSION SYSTEMS
In this paper the most common type of wind turbine, that is, the horizontal-axis type, is considered.
The output mechanical power available from a wind turbine is [1].It can be pointed that max P C , the maximum value for P C , is a constant for a given turbine [6].That value, when replaced in (1), gives the maximum output power for a given wind speed.This corresponds to an optimal relationship opt λ between ω and ω V .The torque developed by the wind turbine is: Fig. 1 shows the torque/speed curves of a typical wind turbine, with ω V as a parameter.Note that maximum generated power ( max P C ) points do not coincide with maximum developed torque points.
Optimal performance is achieved when the turbine operates at the Here the double output induction generator (DOIG) is considered.In this generator, slip power is injected to the AC line using a combination of rectifier and inverter known as a static Kramer drive.Changes on the firing angle ( α ) of the inverter can control the operation point of the generator, in order to develop a resistant torque that places the turbine in its optimum (maximum generation) point.The torque developed by the generator/Kramer drive combination is [6]: The dominant dynamics of the whole system (turbine plus generator) are those related to the total moment of inertia.Thus ignoring torsion in the shaft, generator's electric dynamics, and other higher order effects, the approximate system's dynamic model is: where J is the total moment of inertia.Regarding (2) and (3), this is highly nonlinear model.Moreover, it is well known that certain generator parameters are strongly dependent on factors such as temperature and aging.Thus a nonlinear adaptive control strategy seems very attractive.Its objective is to place the turbine in its maximum generation point ( max Cp ), i.e.The general form of (4) is ) , , where h a nonlinear function is accounting for the turbine and generator characteristics.The system is usually designed so that the maximum turbine torque corresponds to 0.5 to 0.7 of the peak generator torque.In that region a simple linearization of the generator expression can be made.The resulting expression after linearization of the generator characteristics for the whole system is then

A. Structure and Algorithms
Before beginning tracking operation using a neuro network based PID controller, the unknown nonlinear WECS must be identified according to a certain model.In this particular identification process, the model consists of a neural network topology with the wavelet transform embedded in the hidden units.In cascades with the network is a local infinite impulse response (IIR) block structure as shown in Fig. 2. The algorithm of neural network adaptive wavelets is similar to those in [8] where any desired signal )  Assume that the network output function satisfies the admissibility condition and that the network sufficiently distributes K sets of the mother wavelet basis functions, evenly portioning the interest region.The approximated signal of the network ) ( ˆt y can be modeled by [9]: (6) where To minimize E we may use the method of steepest descent and each coefficient vector andd of the network is updated in accordance with the rule (10) Where the subscripted f μ values are fixed learning rate parameter and

B. System Model and PID Controller Design
Consider a general SISO dynamical system similar to (4) is represented by the state equations That in discrete domain rewritten following as: C g f ∈ .The only accessible data are the input u and output y.It has been [9] that if the linear system around the equilibrium state is observable, an input-output representation exists which has the form .In view of this, a neural network model ϕ ˆ can be trained to approximate ϕ over the domain interest.The considerations are based on the neural network controller design of the control system.The following alternative model of an unknown plant that can simplify the computation of the control input is described by the equation: If the nonlinearity terms (.) φ and (.) Γ are known exactly, the required control ) (k u for tracking a desired output ) 1 ( + k r can be computed at every time instant using the formula However, if (.)   φ and (.) Γ are unknown, the idea is to use the neural network adaptive wavelets model to approximate the system dynamics i.e., Comparing the model of Eq. ( 17) with the one of Eq. ( 6) we can conclude that After the nonlinearities (.) where P, I, and D are proportional, integral, and differential gains, u(k) is a plant input at kT, where T is a sampling interval, and (21) P, I, D parameters are considered as part of the function of E and can be optimized and updated according to the cost function E of Eq. ( 9), Where term, Γ ˆ comes from Eq. ( 19), and μ is the fixed learning rate of each adaptive PID parameter.Fig. 3 depicts the block diagram of the resulting network topology based on the PID controller for self-tuning controls WECS's.The optimum shaft rotational speed opt ω is obtained, for each wind speed w V , and used as a reference for the closed loop.Note that wind speed acts also as a perturbation on the turbine's model.Actually, the turbine is linked with the generator's shaft using a gearbox, which imposes an additional transform relation in the model.Dynamics of this gearbox are considered unknown.

A. Approximation of WECS
In this section, it will be shown through simulations RASP1 mother wavelet basis functions perform their learning ability.Using the data from the WECS extracted from [10], the wavenet network with different size and RASP1 mother wavelets is employed to approximate the WECS data.IIR block structure with feed forward coefficients M=3 and feedback coefficients N=3 is also implemented.Moreover, wavelets are local basis functions that provide less interfering than global ones, leading to a noncomplex dependency in the NN parameters.This section will confirm this idea by providing several observations derived from the results of the MATLAB simulations.Assuming the training data are stationary and sufficiently rich, good performance can usually be achieved with a small learning rate.Thus, all learning rate parameters for weights, dilations, translations, IIR feed forward coefficients, and feedback coefficients are fixed at 0.005, 0.025, 0.0250, 0.01, and 0.01, respectively.All initial weights k w and dilations k a are set to 0 and 10, respectively.Note that if the dilation parameters are set too wide, they can cause several overlapping partitions and thus cannot be rallied.Setting The learning epoch will terminate when the desired normalized error of 0.0324 is reached.I illustrates the approximation of WECS performance.The results show that wavenet employing 36 RASP1 wavelets can reach the error goal of 0.032 in the fastest time among all mother wavelets experimented at 60 iterations.However, again when we oversize the number of wavelets to K = 60 and 75, the normalized error starts to oscillate steadily resulting in prolong time consumption to reach the fine error target.From Table I, the best number of wavelets to employ in the wavenet network for approximation of the WECS model turns out to be 36.Similar to these results have obtained for unknown voice model [8].

B. Control
After the identification model is completed, the tracking operation takes command of the neuro process PID control to track the desired set point opt ω .The co-input ) (t v is set to 0.95.In Fig. 6 the results of the WECS control using the proposed self-tuning neuro wavenet controller with 36 RASP1 compare with the results of the WECS control using the combined RBF/supervisor control [6].In this figure, a step sequence of step-shaped wind gusts is applied to the system.The resulting evolution of the closed loop converges rapidly to the desired optimal rotational speed with simple first-order dynamics.Fig. 7 depicts the turbines developed torque versus rotational speed, for the same input sequence.Superimposed (in dotted line) the turbines characteristic curves are displayed.It can be seen that the torque trajectories of the controlled system converge to points belonging to the maximum torque curve.In Fig. 6 is showed that proposed method is better than RBF method [6] regarding simulation accuracy.The proposed controller can be efficiently implemented on digital signal processors.V. CONCLUSION This paper discussed the application of wavenet networks in the implementation of adaptive controllers for WECS's.The approach used, based on a single layer feed forward neural networks with hidden nodes of adaptive RASP1 wavelet functions PID controller and an infinite impulse response (IIR) recurrent structure , allowed fast convergence to a simple linear dynamic behavior, even in the presence of parameter changes and model uncertainties.The resulting controller showed to be simple enough to be synthesized using signal processors.
the "power coefficient," and is given as a nonlinear function of the parameter γ are constructive parameters for a given turbine.

Fig. 1
Fig. 1 Torque/speed curves (solid) of a typical wind turbine.The curve of max P C

,
in despite of wind gusts and generator's parameter changes.
dilation, a, and translation, b, from a mother wavelet :

Fig. 2 IIR
Fig. 2 IIR Adaptive wavelet network structure: (a) neural network architecture (b) IIR model and the output at the kth instant of time.

ka
too narrow may result in longer convergence.Initial translation parameters k b are spaced equally apart throughout the training data to provide non-overlapping partitions throughout the neighboring intervals.Finally, the initial IIR coefficients c and d should be set so that system has poles inside the unit circle, thus both are set to 0.1.

TABLE I NUMBER
OF ITERATIONS VS.NUMBER OF RASP1 WAVELETS EMPLOYED