This article was published online on [17 September 2012]. Errors were subsequently identified in the affiliation of the first and second author. This notice is included in the online and print versions to indicate that both have been corrected [27 June 2013].

Research Article

# Difference equation matrix model (DEMM) for the control of wind turbines^{†}

Article first published online: 17 SEP 2012

DOI: 10.1002/we.1556

Copyright © 2012 John Wiley & Sons, Ltd.

Additional Information

#### How to Cite

Avello, A. J., Al-Hadithi, B. M., Garcia, M. I. G. and Rubio, J. M. L. (2014), Difference equation matrix model (DEMM) for the control of wind turbines. Wind Energ., 17: 57–74. doi: 10.1002/we.1556

^{†}

#### Publication History

- Issue published online: 11 DEC 2013
- Article first published online: 17 SEP 2012
- Manuscript Accepted: 8 AUG 2012
- Manuscript Revised: 8 JUN 2012
- Manuscript Received: 28 MAR 2012

#### Funded by

- Spanish Ministry of Innovation and Science. Grant Number: ARABOT project DPI2010-21247-C02-01

- Abstract
- Article
- References
- Cited By

### Keywords:

- predictive control;
- wind turbine

### ABSTRACT

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

In this paper, a new algorithm, difference equation matrix model (DEMM), in the framework of model predictive control (MPC) is introduced. Instead of the standard dynamic matrix control (DMC), which is based upon step response method that has been used in most research works, we propose a new approach based upon a DEMM for model prediction. It has shown that DEMM has proven to be less computational and thus faster than the original DMC for real time applications. Thus, the drawbacks of DMC for online identification or adaptive design could be avoided. The control of wind turbines is carried out in order to decrease the cost of wind energy by increasing the efficiency, and thus the energy capture, or by reducing structural loading and increasing the lifetimes of the components and turbine structures. Modeling of wind turbine has been carried out. Effect of noise and disturbance on the system has been also studied. The results obtained show that the proposed DEMM minimizes the effect of the disturbance and produces an accurate and smooth control. Significant improvements in the regulation of rotor speed at high wind speeds are obtained from the proposed DEMM, where control set points are obtained ahead of the disturbance, saving the turbine of the negative effects of them and thus increasing its lifetime. Copyright © 2012 John Wiley & Sons, Ltd.

### INTRODUCTION

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

Model predictive control (MPC) was developed to meet control challenges in the industry. With the passage of time, it has become the one of the most effective advanced control technique for a wide range of industries. The advantages of MPC are most evident when it is used as a multivariable controller integrated with an optimizer. Dynamic matrix control (DMC) was the first MPC algorithm developed by Shell Oil Company in the 1970s. The advantages of these methods have already been proven, and these methods have been found to work satisfactorily for long durations of time. DMC is available in all industrial process control systems and on a number of control simulation platforms. DMC is particularly advantageous for multiple-input–multiple-output (MIMO) systems.

The DMC has been used successfully in industry for the last decade. It can deal with constraints and unusual dynamic behavior directly. It also shows a good control performance for the servo problem. Motivated by industrial applications, we investigate the so-called DMC strategy for the control of wind turbines.

In Moon *et al*.,[1] an adaptive control algorithm that combines the recursive least squares system identification algorithm and the generalized predictive control (GPC) design algorithm is presented. It is defined as recursive generalized predictive control (RGPC.) Two new parameters are defined to describe the effects of the prediction and control horizons and those parameters provide the effective ranges of the horizons. The RGPC algorithm adjusts the control penalty on the basis of the stability of a closed-loop system model. A time-varying algorithm for the control penalty allows designing an aggressive controller. The multi-sampling rate algorithm is added between the system identification and the control design in order to design a higher order controller. The main difference between the work presented in Moon *et al*.[1] and our proposal is that our algorithm can be considered as a mixture of DMC and the one proposed in Moon *et al*.[1] We use the incremental control action and correction of model [*y*_{k measured} − *y*_{k calculated}] such as DMC, and we apply the difference equation and matrix inversion as the work developed in Moon *et al*.[1] The advantages obtained from the proposed algorithm are twofold. First, in comparison with the DMC, our algorithm has the advantage of simplicity and computational reduction, which makes it compatible for real time applications. Second, comparing the suggested algorithm with the one presented in Moon *et al*.,[1] our algorithm is characterized by having zero steady-state position error, good and robust performance in front of disturbances and uncertainties of the model.

Clarke and Mohtadi[2] presented a derivation of prediction equations using an observer polynomial and a demonstration that the selection of particular horizons (the “costing horizons” and the “control horizon”) leads to well-understood basic techniques such as dead-beat, pole-placement and linear quadratic regulator (LQR).

A robust algorithm that is suitable for challenging adaptive control applications is described in Clarke *et al*..[3] The method is simple to derive and to implement in a computer; indeed for short control horizons GPC can be mounted in a microcomputer. A simulation study shows that GPC is superior to currently accepted adaptive controllers when used on a plant that has large dynamic variations.

The relationship between GPC and LQR designs is investigated in Clarke *et al*.[4] to show the computational reduction of the new approach. The roles of the output and control horizons are explored for processes with non-minimum phase, unstable and variable dead time models. The robustness of the GPC approach to model overparameterization and underparameterization and to fast sampling rates is demonstrated by further simulations. This paper introduces similar polynomials to GPC for specifying a desired closed-loop model and for tailoring the controlled responses to load disturbances, and the derivation of GPC is expanded to include the more general controlled auto-regressive integrated moving average model.

Wind turbines are large and flexible structures operating in uncertain environments and lend themselves nicely to advanced control solutions. Advanced controllers can help achieve the overall goal of decreasing the cost of wind energy by increasing the efficiency, and thus the energy capture, or by reducing structural loading and increasing the lifetimes of the components and turbine structures.

This paper is the result of collaboration between Gamesa Corporación Tecnológica, SA and the Universidad Politécnica de Madrid. With more than 15 years of experience, Gamesa is a global technology leader in the market for design, manufacture, installation and maintenance of wind turbines, with more than 23,000 MW installed in 30 countries and about 15,000 MW in maintenance. Figure 1 shows Gamesa Wind Park

Of the three levels of control that exist in wind turbine system, the wind turbine control is on the middle level, which includes generator torque control, pitch control and yaw control. The quality of the pitch control algorithm has a decisive influence on the cost of energy produced by wind turbines. Much research on this control and torque control has been carried out.[5-8] However, there is still a need to introduce more “intelligence” in the operation of wind turbines.

Currently, most control algorithms depend on the measurements made from the structure of the turbine and gear train in the feedback control. Often these measurements are unreliable or show a delay in responding to perturbations. This forces the control to react to complex atmospheric disturbances after its effects have been felt by the turbine. Thus, there is a lag between the time at disturbance arrives and the time when the control actuator begins to mitigate the resulting loads.[9]

Significant improvements in the regulation of rotor speed at high wind speeds can be obtained from the predictive control, where control set points are obtained ahead of the disturbance, saving the turbine of the negative effects of them and thus increasing its lifetime, whereas greater savings is achieved in technology investment.[10]

In Kadali and Huang,[5] a new subspace identification-based method for the estimation of the dynamic matrix of the deterministic input(s) directly from the closed-loop data is proposed. A dynamic matrix is a lower triangular matrix containing the step response coefficients of the deterministic input used in the MPC schemes such as the dynamic matrix controller. Subspace matrices (defined in subspace state-space identification methods) corresponding to the deterministic input and the stochastic input contain the impulse response coefficients of the deterministic and stochastic models, respectively. The noise model is simultaneously obtained from the closed-loop data in the impulse response form. The method is extendable to the case of measured disturbances.

All of the results presented in this paper are applicable to the multivariate systems. Guidelines for the practical implementation of the algorithm are also presented in this paper. The proposed method is illustrated through matlab simulations and an application on a pilot-scale plant.

An extension of a DMC to handle different operating regimes and to reject parameter disturbances is presented in Aufderheide and Bequette.[7] This is carried out with the use of two new multiple MPC schemes: one based on actual step response tests and the other on a minimal knowledge-based first order plus dead time models. Both approaches do not require fundamental modeling. As a benchmark comparison, the two controllers are compared with a nonlinear model predictive controller (NLMPC) using an extended Kalman filter with no initial model/plant mismatch. The application example is the isothermal Van de Vusse reaction, which exhibits challenging input multiplicity. Simulations include disturbances in the feed concentration, kinetic parameters and additive input and output noise. The two controllers have comparable performance to NLMPC and, in the case of multiple disturbances, can outperform NLMPC.

Recently, a fuzzy internal model control (FIMC) scheme with the novel crisp consequent fuzzy relational model (ccFRM) was presented by Postlethwaite and Edgar (2000). The design of the FIMC is based on linear internal model control (IMC). The FIMC was shown to be easy to formulate and possessed all the benefits of linear IMC with the added capability of handling nonlinear systems. However, when the FIMC is applied to MIMO systems with different dead times, serious dynamic errors are observed in the controller response. Process interactions and dead times can be intrinsically handled with MPC schemes such as DMC. In Gormandy and Postlethwaite,[6] the procedure for the local linearization of ccFRM is presented. Transfer functions are then easily extracted and this facilitates the incorporation of the ccFRM into a DMC scheme. The proposed FMPC is compared with the FIMC by using a liquid level system and a binary distillation column.

In De Almeida *et al*.,[8] genetic algorithms (GA) with the elitism strategy is used to optimize the tuning parameters of the dynamic matrix controller for single-input–single-output (SISO) and MIMO processes with constraints. A comparison is made between the computational method proposed here and the tuning guidelines described in the literature, showing advantages of the GA method.

Moon and Lee[9] present the application of DMC to a drum-type boiler–turbine system. Two types of step response models (SRMs) for DMC are investigated in designing the DMC; one is developed with the linearization of the nonlinear plant model and the other is developed with the process step response data. Then, the control performances of the DMC based on both types of models are evaluated. Because of severe nonlinearity of drum water-level dynamics, it is observed that the simulation with the SRM based on the test data shows satisfactory results, whereas the linearized model is not suitable for controller design of the drum-type boiler–turbine system.

In Moon and Lee,[10] DMC is made adaptive for a boiler–turbine system by using online interpolated SRM with fuzzy inference. A plant is described as an SRM in a conventional DMC. However, a nonlinear boiler–turbine system cannot be effectively modeled as a single SRM. In this paper, the plant is modeled with nine SRMs at various operating points, and then they are interpolated with fuzzy inference rules. The interpolation is performed at every sampling step online in order to find the best SRM for an arbitrary operating point. Therefore, the proposed adaptive DMC can achieve a bumpless control for nonlinear systems. Simulation results show satisfactory result with a wide range operation of a boiler–turbine system.

In Nippert,[11] a series of online instructional modules that allow students to compare the performance of DMC controllers with conventional control schemes using proportional-integral-derivative controller (PID) control are detailed. Examples of the behavior of DMC and comparisons to PID are presented.

In Besenyei and Simon,[12] it is shown that if the process is governed by a one-dimensional stable dynamical system, then the method drives the output of the sampled system into the desired set point as time goes to infinity, that is the system is asymptotically output controllable with DMC. For two-dimensional systems, sufficient condition on the asymptotic output controllability is given.

It has been shown that relatively, DMC cannot reject disturbances systematically. In Sung and Lee,[13] a modified DMC method to control the regulatory process more efficiently is presented. The proposed DMC method makes the control output by subtracting the estimated disturbance from the control output of the original DMC. Here, the disturbance is estimated by a new disturbance estimator. It shows better control performances than the original DMC.

In Kim *et al*.,[14] an application of DMC for controlling steam temperatures in a large-scale once-through boiler–turbine system is developed. A spray and a damper, as two controllers, are chosen to control the steam temperatures. The SRM for the DMC is generated for the two major output variables, superheater and reheater temperatures, by performing step input tests. Online optimization is performed for the DMC using the SRM. Proposed controller is implemented in a large-scale power plant simulator, and the simulation results show satisfactory performance of the proposed DMC technique.

An application of an online self-organizing fuzzy logic controller to a boiler–turbine system of fossil power plant is presented in Moon and Lee.[15] The control rules and the membership functions of the proposed fuzzy logic controller are generated automatically without using a plant model. A boiler–turbine system is described as a MIMO nonlinear system in this paper. Then, three single-loop fuzzy logic controllers are designed independently. Simulation shows robust results for various kinds of electric load demand changes and parameter variations of boiler–turbine system.

### DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

The difference equation of the plant is described using linear combinations of past output and input measurements as states. For a SISO system,

- (1)

where *y _{k}* is the output value at sample

*k*,

*u*the input control signal and

_{k}*a*,

_{i}*i*= 1,2,…,

*n*and

*b*,

_{j}*j*= 1,2,…,

*n*are the model coefficients.

Incremental inputs ∆*u _{k} = u_{k}-u*

_{k−1}will be introduced in the discrete model given in equation (1) by subtracting from it the same equation for

*k*− 1:

- (2)

obtaining

- (3)

which can be rewritten as

- (4)

We may write equation (3) in matrix formulation, taking *h* as the prediction horizon, as

- (5)

Expression that in a compact form will look like

- (6)

where *f* stands for future outputs–inputs and *p* for past outputs–inputs. Since *A*_{1}is invertible, it is possible to obtain *y _{f}* and the predicted model of the plant will be

- (7)

where

Considering the trajectory tracking problem with the reference *r*(*t + k*) the optimization function handled by the controller is

- (8)

The matrix *Q* represents the error in the followed trajectory whereas *R* weights the effort in the control signals. Note that this index has a unique minimum value, the Jacobian matrix should be positive definite, therefore it is sufficient to take *Q* and *R* as positive definite matrices. When no constraints are imposed on equation (7), we obtain the analytic solution to the optimization:

- (9)

In general, it is convenient to take *h* ≥ *n* at least to assess the direct control action *Δu*_{k} on the response of the system. Also, if the system has a pure time delay of *d* units,

The matrix will have its last *d* rows as zeros, so that the last *d* elements of vector Δ*u*_{f} become zeros. It is necessary therefore that *h* > *d* so that it can obtain a nonzero solution.

There is a great similarity with the LQR, only that we are minimizing the output error and not the state error. Moreover, as we are calculating the incremental value of the input, the controller will have an integral action and, like any regulator with integral action, without steady-state error.

We can express the control action as

- (10)

where

being *y*_{l} the free response of system, i.e. when Δ*u*_{f} = 0,

- (11)

This prediction can be corrected with the possible variations of the model by adding a vector of *h* equal terms *y*_{k measured} − *y*_{k calculated}

- (12)

and then

In detail, the algorithm will be as follows:

- Calculate matrices
- Select the weight matrices
*Q*and*R* - Calculate
*M*and selection of the first row- (13)

*m*_{1}=*M*(1, :) - Measure
*y*_{k measured} - Calculate
*y*_{k calculated}using equation (1) - Calculate
- Define
*r* - Calculate
- Calculate

The points 1–3 can be performed offline, whereas points 4–9 must be run at each sampling instant. If we extend equation (7) to MIMO systems with *i* outputs and *p* inputs

- (14)

The general form of the DEMM algorithm for multivariable systems becomes

- (15)

which has the same structure of the monovariable case.

### DEMM VERSUS DMC

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

The DMC is based on step response. The model parameters are the output samples for a step input up to the settling time. As the sampling period is set for this time, there are almost 50–80 samples which will have parameters approximately. DEMM, on the other hand, is based on the discrete transfer function. A system that can be modeled as a third order will have six parameters, three for the output and another three for the input. In the following, we will compare both models from various points of view.

#### Dynamic matrix control

In order to make a soft introduction to the proposed DEMM algorithm and for the sake of comparison, the main features of the DMC are herewith reviewed. The process model employed is the step response of the plant:

where *g _{i}* is the sampled output values for the step input. The output can be calculated as follows:

- (16)

where Δ*u*(*k*) = *u*(*k*) − *u*(*k* − 1). The predicted value of the output will be

- (17)

where the first term contains the future control actions to be calculated and the second contains past values of the control actions and is therefore known.

The vector of future outputs is

It can then be calculated as follows:If we suppose that *N* is the number of samples taken up to the settling time, we will confirm that the left part of the previous matrix will be constant and can be expressed as follows:

- (18)

and its model becomes

- (19)

where

and *y*_{B} is a vector of constants with the base value of the vector of outputs.

#### Identification

The data in both models are a set of input and output samples. The error of estimation of the parameters depends in one hand on the number of parameters to be estimated and on the other hand on the size of the sample; therefore, to obtain the same confidence level, the sample size should be proportional to the number of parameters to be estimated. If we take a minimum of 10 samples per parameter, the DMC would need about 600 samples, whereas the DEMM would need around 60.

In both cases, the algorithm used is based on least squares estimation, being necessary to form a matrix Φ of dimension equal to the number of samples × number of parameters. So in DMC, this matrix would be of 600 × 60 whereas DEMM would be of 60 × 6. To calculate the estimated parameters, it is necessary to calculate the inverse of Φ^{t}Φ matrix of 60 × 60 size in the first case and of 6 × 6 size in the second one.

#### Model

Assuming a prediction horizon *h*, in both models we obtain the following vector of future outputs:

In the case of DMC, this vector will be a function of

and its model becomes

where *y*_{B} is a vector of constants with the base value of the vector of outputs.

In the case of DEMM, the vector is a function of

and its model is

- (20)

Comparing both models, the DMC needs the *y*_{B} vector, whereas the DEMM does not need it taking into account the previous outputs and that it is an incremental one. For a prediction horizon of 20 and the previous values of the number of model parameters, the DMC matrices would be

i.e. 1580 coefficientsand for DEMM

i.e. 640 coefficients

#### Algorithm

Both algorithms have a first part that can be run offline, on the basis of the inverse of *B*_{f} which is equivalent. However, in both algorithms, the online calculation should be performed from the free response of the system without variations in the input. The DMC will basically be

- (21)

whereas the DEMM is

#### Multivariable case

For multivariable case, the difference increases, because in the case of the DMC, each input–output relation should be modeled by step response of each output for each input. That is a system with four inputs needs, in the case of DMC, in the order of 240 parameters for each output, whereas modeling by DEMM using a fourth order system needs 20 output parameters for each output. This means that DMC grows linearly with the number of entries, whereas the DEMM increases less proportional. For the set-up of a DMC to control a distillation column, it is required in the order of a month, which is personnel costs and loss in production.

We can conclude that the DMC through programmable logic controller (PLC) can be hardly applied to a system with a minimum of complexity and cannot fulfill completely some adjusted sampling periods.

The comparison carried out in this section between DMC and DEMM algorithms show that the proposed DEMM reduces the computational cost of DMC, which is based upon step response method and thus reduces the time consumption in real time application. The dynamic behavior of both algorithms is the same, i.e. accurate, stable and smooth. Both the dynamic behavior and stability depend on the proper choice of the matrices *Q* and *R*.

### DEMM CONTROL OF THE PITCH OF WIND TURBINE

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

In the case under study, the system model is given in difference equation form as follows:

- (22)

where *y*_{k} is the output (the angular velocity of the turbine) at sample*k*, *u*_{k} the input control signal (the pitch reference) and *v*_{k} the input disturbance (the velocity of the wind).

Following the previous design procedure, we obtain

- (23)

Equation (23) depends on *v*_{f}, future disturbance values

Thus, we would estimate and the known past values *v*_{p}.

- (24)

If the noise is correlated, its prediction can be made from its past values. This prediction is based primarily on the Wold theorem: “For any second-order stationary process v (k) that does not have deterministic part, there is a stable linear system H(z) such that, fed with white noise N (0, σ), gives as output a process whose moments are the process v (k).[25]” If, as in the case of wind, the process is not of zero mean, this theorem will fulfill it with an incremental process above the average

Using an auto-regressive (AR) model, noise sequence *z* transform can be expressed as

- (25)

where *w*(*k*) is white noise *N*(0, *σ*_{w}). It can be assumed also that the denominator is in the form of

- (26)

and then

Since *w*(*k*) is white noise, the optimal prediction is the mean

so the optimal prediction of incremental noise Δ*v* will be

- (27)

The parameters of this equation can be calculated from a time series of values of the wind in the area, using one of the classical methods, such as the least squares method, the forward–backward approach, etc.

The real values of the variable, will be

- (28)

Then

- (29)

To obtain the vector , we will calculate

- (30)

In order to calculate the parameters,

Different methods, such forward–backward approach, least squares method, Yule–Walker method, etc., can be used from a representative and long enough sample.

In the particular case of the of wind prediction control, a calculation can be realized on the basis of samples taken in a year. However, this approach would not distinguish between smooth winds and turbulent ones. A better approach is to take into account that the wind is maintained with the same characteristics for long periods, so that the calculation of parameters can be made adaptively using the last recorded data and therefore can be considered valid for all types of wind, soft and turbulent.

Our proposal is using recursive least square method with forgetting factor *λ* ≈ 0.95

sequence of matrices *P*_{k} are used with dimension (*m* + 1,*m* + 1). Initially, we suppose

*P*_{0} = *cI* with a comparatively high value of *c*.

Thus, the parameters can be calculated adaptively as follows:

The algorithm can malfunction *P*_{k} *∞* when the parameters are excessively adjusted, therefore they must be updated only when the prediction error is greater than a certain bound

### SIMULATION MODEL

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

For variable speed wind turbines, the pitch control loop is active in region 3 of operation, at high wind speeds. The main objective of the control loop is to change the pitch of the blade pitch angle in order to control energy production and the rotor speed, while reducing the mechanical loads as much as possible. The methodology of control depends on the complete configuration of the turbine. The most popular method is the pitch-to-feathering.[16] In this case, at the operating range of pitch angle, there is a linear relationship between the pitch angle and aerodynamic forces. If the pitch angle increases in this range, the aerodynamic forces are reduced. Normally, the control loop is designed to operate in the linear range, preventing problems from the stall values, at high angles of attack, where increased uncertainty comes and disappears linearly.[17]

The DEMM control will be applied to regulate generator speed in the region of high wind speeds. The goal is to keep the generator speed at rated value, through the inputs of pitch control obtained with the DEMM.

#### Simulation structure and tools

In this paper, three-dimensional turbulent wind has been simulated by TurbSim. Fatigue aerodynamics structures and turbulence (FAST) program has been used to simulate the wind turbine mechanical parts. Both programs have been developed at the National Renewable Energy Laboratory, Golden, CO. The controllers have been designed by Simulink and programming in matlab (Figure 2).

#### TurbSim to simulate wind

TurbSim is a stochastic, full field and turbulent-wind simulator. It numerically simulates time series of three-dimensional wind velocity vectors at points in a vertical rectangular grid. TurbSim output can then be used as input into AeroDyn-based codes such as FAST, YawDyn or MSC.ADAMS.[18] Its purpose is to provide the wind turbine designer with the ability to drive design code simulations of advanced turbine designs with simulated inflow turbulent environments that incorporate many of the important fluid dynamic features known to adversely affect turbine aeroelastic response and loading.

#### FAST for simulation of mechanical parts of wind turbine

The FAST code is a comprehensive aeroelastic simulator capable of predicting both the extreme and fatigue loads of two-bladed and three-bladed horizontal-axis wind turbines. It is proven that the structural model of FAST is of higher fidelity than other codes.[19] During time-marching analysis, the FAST makes it possible to control the turbine and model-specific conditions in many ways. Five basic methods of control are available: pitching the blades, controlling the generator torque, applying the High Speed Shaft (HSS) brake deploying the tip brakes and yawing the nacelle.

It is possible to incorporate complex control methods, by writing specific routines, compiling them and linking them with the rest of the program.

An interface has also been developed between the FAST and the Simulink with matlab, enabling users to implement advanced turbine controls in Simulink convenient block diagram form. The FAST subroutines have been linked with a matlab standard gateway subroutine in order to use the FAST equations of motion in an *S*-function that can be incorporated in a Simulink model.

#### Simulink

Simulink is a software package for modeling, simulating and analyzing dynamic systems. It supports linear and nonlinear systems, modeled in continuous time, sampled time or a hybrid of the two. The pitch controller is written as a matlab function, has been applied the DEMM algorithm and is incorporated into the Simulink model blocks to simulate the system.

#### DEMM algorithm

The first point of the control process is to identify the system.

- (31)

with

*y*_{k}generator speed at time

*k**u*_{k}pitch angle at time

*k**v*_{k}disturbance, wind speed at time

*k*

It is aimed to calculate the set of parameters *θ* that best describe the dynamics of the system

These parameters are tested to identify different procedures. The first parameter tested was from a linearized model around an operating point. The model thus obtained did not show a good performance in implementing the DEMM. It was found that as demonstrated in other studies[21] that identification by linearizing models does not reflect the real dynamics of the turbine for different operating points and the application of planned gain[18-20] did not show good result for the control operation of the DEMM. The closed-loop identification overcomes this problem. It has been shown experimentally[22, 24] that using appropriate algorithms, the models obtained by identification in closed loop are better than those obtained by identification in open loop in terms of control performance, as it guarantees the security and integrity of the turbine in all conditions wind.

Steps wind signals are used as inputs from the beginning to the end of region 3, from 14 to 21 m s^{−1}. To obtain the parameters, we used an algorithm based on least squares method,[23] and a second order model was chosen. The matrices were calculated with these parameters and with a prediction interval *h* = 4. Both the identification and the obtaining of matrices were made offline. Whereas the following steps of the DEMM algorithm were made online by programming a matlab function that was incorporated to Simulink blocks for simulation.

### SIMULATION RESULTS

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

First, we will simulate the wind turbine system under non-turbulent wind. Figure 3 shows the wind speed response to a reference speed input of 1800 rpm. The non-turbulent wind varies from 18 to 19 m s^{−1} at 40 s.

Second, we will repeat the previous simulations under turbulent wind. The wind speed response under turbulent wind input of 18 m s^{−1}, with a reference speed input of 1800 rpm, is shown in Figure 6.

In Figure 7, it is shown that DEMM controller is significantly a robust and accurate controller, which as can be observed follows closely to the reference velocity input of 1800 rpm. The mean square error is 0.47, which is approximately 0.026%.

The DEMM action to turbulent wind is shown in Figure 8.

Third, the robustness of the proposed DEMM is examined when the system is subjected to set point changes for the turbulent wind case. In Figure 9, it is shown that DEMM controller acts perfectly when the system is subjected to disturbances. The system follows closely a velocity step reference input from 1700 to 1800 rpm at 40 s. with a turbulent wind of 18 m s^{−1}.

Figure 10 shows response to a velocity ramp reference input from 1700 to 1800 rpm, from 40 to 50 s. with a turbulent wind of 18 m s^{−1}.

The effect of the horizon prediction on the system response is analyzed. Figures 11-13 show responses to a velocity reference input from 1700 to 1800 rpm at 40 s. with a non-turbulent wind of 18 m s^{−1}. The horizon prediction is 1, 5 and 10, respectively.

To examine the robustness of the system when it is subjected to turbulent wind with varying speed as shown in Figure 14, the prediction error in four steps is shown in Figure 15 with a mean square error of 0.0205. The transient response remains robust and stable as shown in Figure 16 with a mean square error of 0. 0230.

### CONCLUSIONS

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

The application of a new predictive algorithm to control the pitch of a wind turbine has been presented. This algorithm obtains a smooth and fast control. It is based on a new algorithm, DEMM, in the framework of MPC. It has been shown that DEMM is more appropriate for extreme loads, when the voltage drops or at very high wind speeds. It could be an alternative to expert control used to wind gusts where rapid control response is vital to mitigate the effects of these high loads. It has shown that DEMM has proven to be less computational and thus faster than the original DMC for real time applications. Thus, the drawbacks of DMC for online identification or adaptive design could be avoided. The control of wind turbines has been handled in order to decrease the cost of wind energy by increasing the efficiency, and thus the energy capture, or by reducing structural loading and increasing the lifetimes of the components and turbine structures. Effect of noise and disturbance on the system has been also studied. The results obtained show that the proposed DEMM minimizes the effect of the disturbance and producing an accurate and smooth control. Significant improvements in the regulation of rotor speed at high wind speeds are obtained from the proposed DEMM, where control set points are obtained ahead of the disturbance, saving the turbine of the negative effects of them and thus increasing its lifetime. DEMM control will be applied to regulate generator speed in the region of high wind speeds. The goal is to keep the generator speed at rated value, through the inputs of pitch control obtained with the DEMM.

### ACKNOWLEDGEMENT

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

This work is funded by Spanish Ministry of Innovation and Science (ARABOT project DPI2010-21247-C02-01).

### REFERENCES

- Top of page
- ABSTRACT
- INTRODUCTION
- DIFFERENCE EQUATION MATRIX MODEL (DEMM) ALGORITHM
- DEMM VERSUS DMC
- DEMM CONTROL OF THE PITCH OF WIND TURBINE
- SIMULATION MODEL
- SIMULATION RESULTS
- CONCLUSIONS
- ACKNOWLEDGEMENT
- REFERENCES

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