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Non‐linear autoregressive moving average with exogenous input model‐based adaptive control of a wind energy conversion system

2016; Institution of Engineering and Technology; Volume: 2016; Issue: 7 Linguagem: Inglês

10.1049/joe.2016.0081

2051-3305

Bidyadhar Subudhi, Pedda Suresh Ogeti,

Multilevel Inverters and Converters

The Journal of EngineeringVolume 2016, Issue 7 p. 218-226 ArticleOpen Access Non-linear autoregressive moving average with exogenous input model-based adaptive control of a wind energy conversion system Bidyadhar Subudhi, Corresponding Author Bidyadhar Subudhi bidyadhar@nitrkl.ac.in Department of Electrical Engineering, Centre for Renewable Energy Systems, National Institute of Technology Rourkela, Rourkela, 769008 IndiaSearch for more papers by this authorPedda Suresh Ogeti, Pedda Suresh Ogeti Department of Electrical Engineering, Centre for Renewable Energy Systems, National Institute of Technology Rourkela, Rourkela, 769008 IndiaSearch for more papers by this author Bidyadhar Subudhi, Corresponding Author Bidyadhar Subudhi bidyadhar@nitrkl.ac.in Department of Electrical Engineering, Centre for Renewable Energy Systems, National Institute of Technology Rourkela, Rourkela, 769008 IndiaSearch for more papers by this authorPedda Suresh Ogeti, Pedda Suresh Ogeti Department of Electrical Engineering, Centre for Renewable Energy Systems, National Institute of Technology Rourkela, Rourkela, 769008 IndiaSearch for more papers by this author First published: 01 July 2016 https://doi.org/10.1049/joe.2016.0081Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Wind energy conversion system (WECS) is a stochastic system, since wind speed varies intermittently. Therefore a non-linear autoregressive moving average with exogenous input (NARMAX) model is developed to represent the dynamics of WECS which is used for real-time implementation. In a doubly-fed induction generator (DFIG) WECS, speed and power are the outputs for regulation which are achieved by controlling the torque and pitch angle, repetitively. NARMAX model identifies the structure and significant terms of speed and power of DFIG WECS and its parameters are estimated employing an on-line adaptive recursive least squares algorithm. For optimisation of torque and pitch angle, performance index (PI) is defined in non-linear adaptive model predictive controller (NAMPC) to achieve the control objective, i.e. torque and pitch angle. The weights in PI are updated until the optimised values in control inputs (torque and pitch control) are achieved. Boundedness of the WECS is defined by considering the constraints on the outputs and control inputs. Extensive simulations are carried out with NARMAX structure with NAMPC on DFIG WECS using MATLAB/SIMULINK and the performance is compared with conventional proportional–integral controller and model predictive control. From the obtained results, it is observed that the NARMAX model with NAMPC has minimum deviations from the operating point in power, speed, torque and pitch angle compared to other controllers. 1 Introduction Velocity of wind continuously varies with time, therefore adaptive control is necessary for regulating power and speed of doubly-fed induction generator (DFIG) which are the main objectives. Since speed and power contain a large set of terms, selection of significant terms is important for controlling the torque and pitch angle and reducing the complexity of the wind energy conversion system (WECS). As the load demand increases, large WECS with variable speed variable pitch controlled wind turbine generator systems has been paiying attention to a large extent. Increase in size of wind turbines inevitably causes fluctuations in both aerodynamic power and drive train dynamics. In the past two decades, different non-adaptive controllers such as sliding mode control, linear parameter time varying control, H ∞ control, model predictive control (MPC) for torque and pitch control have been proposed. In [1-3], sliding mode control has been proposed by estimating the aerodynamic torque of DFIG WECS for extracting and regulating maximum power by reducing mechanical stress on the drive train dynamics. In [4], coordination control for disturbances in electrical and mechanical parts of wind turbine has been proposed by using FAST (fatigue, aerodynamics, structures and turbulence code). In [5], robust pitch controller with inverse system controller and robust compensator has been proposed to tolerate large disturbances and fatigue of the pitch actuators. In [6], linear parameter varying (LPV) controller has been proposed by considering quadratic torque speed law in partial load region and LPV pitch controller in full load region. In [7, 8], H ∞ control design based on linear matrix inequality (LMI) has been proposed for both torque and pitch control through minimisation of H ∞ norm. In [9], self-tuning regulator for pitch control in wind turbine has been designed by imposing the limits on pitch angle and identification is done with fuzzy reasoning for unpredictable wind changes. In [10], second-order cone programming has been proposed for power and speed regulation simultaneously by controlling pitch angle and generator torque in the whole operating region (both partial and full load regions) as depicted in Fig. 1. Fig. 1Open in figure viewerPowerPoint Ideal power curve (mechanical power Pm against wind velocity vw) for DFIG WECS 2 Problem statement Two drawbacks are associated with the above controllers. First drawback is non-adaptive in nature which leads to deviations in the regulator power and speed regulation. Further, the second drawback is that WECS is a non-linear system with a large number of non-linear terms. Selection of significant terms and estimation of parameters from this large set of generator speed terms and output power terms is important. In this paper, the above two problems are solved by selecting a non-linear autoregressive moving average with exogenous input (NARMAX) model as identification method with adaptive recursive least squares (RLS) technique for updating the parameters on on-line to avoid deviations which are not being minimised by using non adaptive non-linear controllers. Second problem is rectified by using Gram Schmidt recursive orthogonal decomposition method and error reduction ratio (ERR) method for selecting the significant terms for a large set of non-linear terms. The objective is to regulate the generator speed and output power of DFIG WECS adaptively by designing torque and pitch angle controllers with minimum deviations. The objective has been achieved by using the NARMAX model. Just as parameter estimation, the NARMAX model identifies both the structure and the parameters of an unknown non-linear system. After system identification, selecting the significant model terms using ERR plays a vital role. Thereafter estimating the system parameters using on-line adaptive RLS and finally optimisation is done by selecting the proportional–integral (PI) to achieve the control objective. Section 3 briefs the physical model of DFIG WECS, Section 4 derives the state model based on physical model equations, Section 5 proposes the new control strategy structure (NARMAX structure) on WECS along with polynomial representation, orthogonal decomposition and parameter estimation using RLS, Section 6 covers the optimisation technique for the proposed NARMAX structure and finally in Section 7, results are analysed and concluded in Section 8. 3 Physical model of DFIG-based WECS Fig. 1 shows the two regions in which WECS has to be operated. Region 1 is the partial load region (non-linear region) between cut-in speed v cutin and rated speed v rated. This region is the non-linear region, where maximum power extraction can be done by controlling the generator speed (ω G). Region 2 is the full load region falling between rated speed v rated and cut-out speed v cutout. In this region, output power (P G) regulation and generator (turbine) speed regulation is obtained by controlling the generator torque (Γ G) and pitch angle (β) of wind turbine. In this paper, regulation of generator power and speed of DFIG wind turbine has been developed with adaptive RLS for WECS. Since speed and power depend on controlling the torque and pitch angle, NARMAX structure is used for selecting and determining the significant terms of generator power and speed from a large set of terms reducing the complexity in controlling the torque and pitch angle. WECS comprises mainly of aerodynamic subsystem, pitch actuator subsystem, drive train subsystem, and DFIG with power electronic converters connected to grid as shown in Fig. 2. Wind turbine converts the kinetic energy of wind into mechanical energy which is fed to the DFIG for converting into electrical energy. Drive train dynamics plays a vital role for torque and pitch control of WECS shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Schematic diagram of a WECS 4 State-space model of DFIGWECS The discrete-time state-space model [11, 12] of DFIG WECS is represented as (1) (2) (3) (4) where x (k) is the state vector, u (k) is the control input vector, y (k) is the output vector obtained from the states and inputs through matrix C, w is the disturbance input vector, A is the system matrix which affect the state dynamics, B is the gain distribution matrix, and B v is the disturbance input matrix. The state vectors in x (k) are Δω T, Δω G, ΔΓ D, Δβ, Δv w. These state vectors are calculated from the dynamics of drive train system [1-3] given by (5) where ΓT is the aerodynamic torque, ΓG is the generator torque, ΓD is the drive train torsional torque, ω G is the electrical angular speed, K S is the shaft compliance index, B S is the damping coefficient, θ T, θ G are the angular positions of the shaft at the turbine rotor and generator side, respectively. Aerodynamic torque of wind turbine is given by (6) where P m is the mechanical power obtained from the wind turbine, β is the turbine blade pitch angle, Λ = πR 2, C P is the power coefficient, ω T is the turbine rotational speed, ρ is the air density in gram/m3, Λ is the cross-sectional area of the turbine, v w is the wind velocity and R is the radius of turbine shaft, λ, β are the tip speed ratio and pitch angle, respectively. Since wind speed v w is a function of order three, the expression ΓT [7] has non-linearity, linearising (6) and rewriting as (7) (8) Equation (7) is expanded by using Taylor series at the operating point (OP) as (9) Applying partial derivative on (6), we get (10) where Δω T = ωT − ωT,OP, Δv w = vw − vw,OP, Δβ = β − β OP and Δ is the small signal value (deviation) around (OP). From (5), the state vectors are derived as given in the following equations (11) (12) (13) Pitch system is highly non-linear hydraulic actuator subjected to pitch rate constraints for β and given by (14) where is the derivative of pitch angle, β cmd is the pitch control signal, and T β is the hydraulic lag. Stochastic wind speed in [11] is represented as (15) where m v (k) = d (k), d (k) is the disturbance, v t (k) is the rapidly varying turbulence component, m v (k) is the Gaussian generator white noise, and T v is the time constant. From (11)–(15), the following matrices are obtained as 5 NARMAX structure identification for DFIG WECS 5.1 Structure representation The discrete time representation of (1) using (2)–(4) in non-linear form [13-16] can be expressed as (16) where y i (k) is the autoregressive (AR) variable or system output, u i (k) is an exogenous (X) variable or system input, ξ i (k) is the moving average (MA) variable or white noise. N y, N u, and N ξ represent the order (or maximum delay) in output, input, and moving average variables, respectively. F i [·] represents the multi-input–multi-output non-linear map which contains both process parameters and noise parameters of the WECS. is the i th process model which do not contain noise terms and is the i th process model which contain noise terms. ZOH is zero order hold. . The structure of proposed NARMAX model for DFIG WECS is shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Multivariable self-tuning regulator for DFIG WECS 5.2 Extended polynomial NARMAX model of DFIG WECS NARMAX model can be applied to DFIG WECS as (17) where ω Gi (k), P Gi (k) are the outputs with and as maximum delays. ΓGi (k) and β i (k) are the control inputs with , N β as maximum delays, and ξ i (k) is the driving force of the wind which is considered as white noise from the noise generator of order N ξ. The linear difference equation of the ARMAX model obtained from (17) is represented by (18) (19) Since the above process in WECS is linear in parameters, consider a linear regression model for (19), can be written as (20) where y (k) is the output or dependent variable to be regressed (speed and power of the DFIG to be controlled) is the unknown parameter model to be estimated is the regression matrix and ξ = ξ i (k − 1), ξ i (k − 2), …, ξ i (k − N ξ) is the modelling error. Equation (20) can be rewritten as (21) where 5.3 Orthogonal least squares QR decomposition of the regression matrix The advantage of orthogonal transformation is that it preserves the Euclidian norm of a vector. Orthogonal transformation is numerically stable when applied on inaccurate vector or matrix which results that error will not increase. QR decomposition is applied on linear regression matrix φ having the length N proposed by Gram Schmidt. Let φ T φ is symmetric positive definite, then it can be decomposed as (22) (23) where is an n × n upper triangular matrix and is an N × n orthogonal matrix with orthogonal columns which satisfies where D has positive diagonal entries given by D = diag{d 1 d 2 …d n } with d i = 〈q i, q i 〉 where 〈·〉 denotes the inner product, i.e. (24) Equation (20) can now be written as (25) where 5.4 Normalising the columns of Q Classical Gram Schmidt (CGS) normalises the columns of Q, by taking one column at a time and orthogonalises φ. The process is repeated by augmenting the resultant matrix with N − n further orthonormal columns of Q which covers the full set of orthonormal vectors for N dimensional Euclidian space, a decomposition equivalent to (20) can be obtained as (26) where R 1 is an n × n upper triangular matrix with positive diagonal elements and is an N × N orthogonal matrix, i.e. The estimated value of g is given by (27) where g = [g 1, g 2, …g n]T 5.5 Structure determination (sub set selection) From (25), sum of squares of output is (28) For model structure selection, ERR is defined as proportion of the output variance in terms of q i as (29) ERR can also be defined as (30) Since regression matrix φ contains a large number of terms, selection of subsets containing significant terms plays a vital role. ERR is employed for subset selection through forward regression manner. In ERR, at each step, a term is selected and verify for largest [err]i when compared to remaining terms. Selection of terms is terminated when (31) where ρ (0 < ρ < 1) is the specified tolerance, which leads to subset model of n s (n s < n) terms. Procedure for subset selection in linear regression model φ : Initially user specifies value of ρ and full model set of n terms. At the n s stage (i) Assume each term as n s term in the selected model, compute ERR for each n = n s + 1, and corresponding orthogonalisation is being done. (ii) Term which yields the largest value of ERR is selected. If condition is satisfied, go to (iii), otherwise set n = n s + 1 and go to (i). (iii) Final subset model contains n s terms, then parameter estimate is determined from where R s is an n s × n s upper triangular matrix. (iv) The parameter estimate satisfies g = R Θ and can be estimated by backward substitution (32) 5.6 Parameter estimation (outer loop) The parameter matrix is estimated not only by a backward substitution, but also can be estimated on-line by using RLS algorithm [17] shown in Fig. 4. Within the time limit, the parameters are updated recursively based on a sampling period for converging to the true system parameters. Matrix computations of RLS algorithm are summarised as shown in Table 1. Fig. 4Open in figure viewerPowerPoint Parameter extraction using on-line recursive structure identification Table 1. RLS algorithm i. Parameter estimate is calculated as (33) ii. Model prediction error is achieved based on new output data y (k + 1) and old estimated parameter and regression vector ϕ (k) as (34) iii. Error covariance matrix (35) where is the error signal for predicting outputs y (k)[ω G (k) and P G (k)] based on the parameter estimate iv. Solve the Kalman gain M (k) (blending factor) for the next sample to minimise the mean square error in terms of covariance is given by (36) 6 Optimisation of torque and pitch angle using NMPC technique Inner loop (torque and pitch controller) is derived as shown in Fig. 5. The optimisation parameters are given to NAMPC to obtain the control inputs given in u (k). Performance index (PI) is defined for (18) to achieve the control objective as in [18] (37) Fig. 5Open in figure viewerPowerPoint NAMPC structure with RLS NARMAX identification technique Subjected to constraints of (38) where , are the predefined prediction horizons and N β is the predefined control horizon. z 1 and z 2 are the weighting matrices (observable matrices) of outputs ω G and P G and v 1 and v 2 are weighting matrices (controllable matrices) of control inputs ΓG and β and w is the unit diagonal matrix. The observable matrix Z = [z 1 z 2]T defines the identical input output equivalent subsystem of the original system for all initial states as (39) Controllability matrixV = [v 1 v 2] is constructed from the columns of linearised state space DFIG WECS as given in (40). The control law is defined after parameter estimation of . For deriving the control law, the PI has to be minimised by taking partial derivative of J with respect to control inputs ΓG and β as ∂J /∂ΓG and ∂J /∂β setting it to zero, one obtain the control law as follows: (40) Control input torque command ΓGref (k) and reference pitch angle β cmd (k) is generated from control law as (41) The parameter matrices a i, b j in for i, j = 1, 2, …, n are estimated on-line with RLS algorithm and then the control law is proposed. For saving identification time B 1 is taken as unit matrix. 7 Results and discussion Performance of DFIG WECS using NARMAX structure with recursive NAMPC has been verified with extensive simulations carried using MATLAB/SIMULINK. The tool boxes used for simulation analysis are LMI, MPC, control system, power system and system identification tool box. Simulation parameters used in simulation analysis are given in Table 2. Table 2. Simulation parameters for DFIG WECS [10, 12] Wind turbine and rotor Number of blades 3 Cut in speed v cutin 3 m/s Cut out speed v cutout 25 m/s Rated speed v rated 12 m/s Air density ρ 1.25 kg/m3 Optimum tip speed ratio λ 8 Power coefficient C p 0.49 Rated rotor speed ω T 22 rpm Maximum rotor speed 23 rpm Blade radius 40 m Drive train Gear ratio 250:3 Turbine inertia J T 90 × 106 kgm2 Low speed shaft torsional stiffness K s 160 × 106 Nm/rad Low speed shaft torsional stiffness B s 10 × 106 Nm/rad DFIG Rated power P g 2 MW Maximum generator speed 1500 rpm Generator inertia J G 60 kgm2 Generator torque ΓT 13.4 kNm Pitch actuator Time constant 0.1 s Minimum/maximum pitch angle β min /β max 0°/90° Minimum/maximum pitch rate −10°/10° per second Number of poles 4 Supply frequency 50 Hz Since wind is not a constant parameter, statistical analysis has been pursued for simulation studies by randomly generating the data for wind speed using Gaussian noise generator. Fig. 6 a shows the instantaneous wind speed and the mean wind speed varying in-between 8 and 8.5 m/s. From (37), for minimising the cost function, the weights (z 1, z 2, v 1, v 2) have been relatively adjusted with respect to other weights. For example, for minimising the fluctuations in wind speed, the weight z 1 has to be increased with respect to z 2 and vice versa. Similarly, if the pitch angle has to be kept in a tolerable limit, the control weight v 2 has to be increased with respect to v 1 and vice versa. The weight update has been explained in Figs. 6 b –e. The results obtained are in close approximation with [12]. Fig. 6Open in figure viewerPowerPoint Instantaneous wind speed and the mean wind speed a Response of control and output parameters for variations in instantaneous wind speed b Response of generator speed ω G for different values of weight z 1 c Response of output power P G for different values of weight z 1 d Response of control input ΓG for different values of control weight z 1 e Response of control input β for different values of control weight z 1 Fig. 6 (continued)Open in figure viewerPowerPoint Instantaneous wind speed and the mean wind speed a Response of control and output parameters for variations in instantaneous wind speed b Response of generator speed ω G for different values of weight z 1 c Response of output power P G for different values of weight z 1 d Response of control input ΓG for different values of control weight z 1 e Response of control input β for different values of control weight z 1 Fig. 6 responses of parameters (ω G, P G, ΓG, β) for Gaussian noise disturbance in wind speed from 8 to 8.5 m/s. For validation of the proposed NARMAX method, a comparison has been done with three different controllers (i.e. MPC without NARMAX structure, PI and NAMPC with NARMAX structure). The results have been analysed for variations in both outputs (ω G, P G) and control inputs (ΓG, β). From Fig. 7 a, the deviations in generator speed is large in PI controller, but minimised to some extent using an MPC controller and the deviations are drastically reduced approximating to 1 pu, by using NAMPC with NARMAX. The torsional fluctuations in the drive train have also been reduced avoiding the damage to the wind turbine. In Fig. 7 b, output power has been levelled around 1 pu by appropriate sub set selection using NARMAX as explained in Section 4. In Fig. 7 c, by updating the control weights recursively using RLS and the optimal parameters for are extracted to obtain the optimised control law as explained in Section 5 for torque command generation. Fig. 7 c, shows the minimum torque pulsations in NAMPC when compared to PI and MPC controllers. In Fig. 7 d, by considering the constraints in (38), the parameter matrix b j in for i, j = 1, 2, …, n are estimated on-line with an RLS algorithm for avoiding the damage to the turbine blades. Since the on-line estimation has been used in NARMAX structure, pitching of the blades has been updated according to the instantaneous values instead of mean value which results in poor stability for PI and MPC controllers. Fig. 7Open in figure viewerPowerPoint Performance comparison of PI, MPC and NAMPC with NARMAX structure identification a Generator speed against time b Generator power against time c Generator torque against time d Pitch angle against time 8 Conclusions The performance of the NARMAX structure on DFIG variable speed variable pitch WECS is verified by comparing with the conventional PI and MPC techniques in both partial and full load regions. Deviations are minimised to a large extent in outputs (output power and generator speed regulation) and control inputs (torque and pitch angle) in DFIG WECS. Simulation results confirmed that proposed NARMAX adaptive NAMPC technique given better results around the OP by avoiding the shortcomings when compared to other controllers. In above rated speed region, generator power loss has been reduced with the proposed method. Subset selection in the NARMAX structure has reduced the computational time drastically instead of considering all the terms in the structure. Adaptive RLS has estimated the parameter coefficients and are updated on on-line according to the instantaneous changes in outputs and operated at the OP for running the WECS system very efficiently when compared to the non-adaptive techniques. 9 References 1Beltran B., Ahmed-Ali T., Benbouzid M.E.H.: ‘Sliding mode power control of variable-speed wind energy conversion systems’, IEEE Trans. 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