www.ietdl.org Published in IET Renewable Power Generation Received on 7th September 2009 Revised on 17th July 2010 doi: 10.1049/iet-rpg.2009.0137
ISSN 1752-1416
Multiple model multiple-input multiple-output predictive control for variable speed variable pitch wind energy conversion systems M. Soliman O.P. Malik D.T. Westwick Department of Electrical and Computer Engineering, University of Calgary, Canada E-mail:
[email protected]
Abstract: A multivariable control strategy based on model predictive control techniques for the control of variable-speed variable pitch wind energy conversion systems (WECSs) in the above-rated wind speed zone is proposed. Pitch angle and generator torque are controlled simultaneously to provide optimal regulation of the generated power and the generator speed while minimising torsional torque fluctuations in the drive train and pitch actuator activity. This has the effect of improving the power quality of the electrical power generated by the WECS and increasing the life time of the mechanical parts of the system. Furthermore, safe and acceptable operation of the system is guaranteed by incorporating most of the constraints on the physical variables of the WECS in the controller design. In order to cope with the non-linearity in the WECS and the continuous variation in the operating point, a multiple model predictive controller is suggested to provide near optimal performance within the whole operating region.
1
Introduction
Control systems play a very important role in wind energy conversion systems (WECSs). A well-designed control system for a WECS enables more efficient energy generation, better power quality and the alleviation of aerodynamic and mechanical loads resulting in increased life of the installation. Consequently, such a control system will have a direct impact on the cost of energy produced by the system [1, 2]. During the past 20 years, a variety of WECS configurations have been used [1 – 3]. The wind turbine can be equipped with a blade pitch control mechanism, or the stall effect can be utilised to limit the power at high wind speeds (above rated wind speed). WECSs can operate at variable speed using power electronic converters or they can have only one fixed speed related to the grid frequency. Both induction and synchronous generators have been used in WECSs. In general, most WECSs of today are of the variable-speed variable pitch type where the generator torque and the pitch angle of the turbine blades can be controlled independently. Despite the relative complexity of their control systems, variable-speed variable pitch WECSs allow better performance to be achieved [4, 5]. The variable speed operation allows more efficient energy extraction from the system, less torque pulsations on the drive train and better smoothing of the electrical power injected to the grid. In addition, allowing variable pitch operation yields better power regulation and lower dynamic loads [6]. In general, a variable-speed variable pitch WECS has two operating regions with different control objectives as shown 124 & The Institution of Engineering and Technology 2011
in Fig. 1 [1, 4]. The partial load regime (Region I) is defined at all wind speeds falling between the cut in wind speed vci and the rated wind speed vr (wind speed at which the system-rated power is achieved). In this region, the control system is required to adjust the generator speed such that maximum energy is extracted. Region II is the full load regime which is defined at all wind speeds falling between vr and the cut out speed vco . In this region, the control system is required to regulate both the output power and the generator (turbine) speed to their rated values in the presence of input power fluctuations resulting from wind speed variations. Speed and power regulation is achieved by controlling the generator torque and the pitch angle of the wind turbine’s blades. These days there exists an increasing interest in the control of variable-speed variable pitch WECS operating in the full load regime [6 – 8] and this paper is focused on that aspect. Design of the control system for variable-speed variable pitch WECSs operating in the full load regime is not a straightforward task. One of the main reasons for this is that the controlled system becomes a multiple-input multipleoutput (MIMO) system with strongly coupled variables when operating in the full load regime. Furthermore, the system non-linearity, the stochastic variations of the input power and the presence of physical constraints on the variables of the controlled system, such as limits on the pitch angle, pitch angle rate, render the control design task more difficult. The multivariable nature of the problem is ignored in a majority of the work reported in the literature dealing with controller design in the full load regime. The common IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124 –136 doi: 10.1049/iet-rpg.2009.0137
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Fig. 1 Ideal power curve for a WECS
approach is to use two separate controllers to regulate the generator speed and power independently, forming two separate control loops [4, 7 – 11]. Classical proportional integral (PI) controllers are used within the power and speed control loops in [8– 10]. The controllers are designed for a single operating point and hence, performance degradation can result when the system is not working at this operating point. To obtain good performance within the whole operating region, a gain scheduling technique is used in [4] to adjust the PI controller parameters. Robust control theory is used in [11] to design a robust pitch control system that can tolerate turbine parameter uncertainties and non-parametric perturbations. A PI controller in the power control loop and a learning adaptive controller in the form of a self-tuning regulator are used in [7]. It was discussed in [1, 5, 6] that much superior performance is obtainable when the multivariable nature of the problem is recognised and a
proper MIMO controller is designed. To the knowledge of the authors, the only work that describes the design of a multivariable controller that can cope with the wide range of variation in the operating point can be found in [1, 6], where H1 optimal control theory is applied. To cope with the system non-linearity, local controllers are designed at different operating points, and gain scheduling techniques are used for interpolating between these controllers. In this approach, the physical constraints in the system do not enter explicitly into the design and the tuning of the controller achieving the required compromise between power and speed regulation, while respecting the constraints of the system, can be difficult. In this paper, a new control strategy based on model predictive control (MPC) techniques is proposed for controlling variable-speed variable pitch WECSs in the full load regime. The main advantage of the proposed strategy is that it is a multivariable control method that effectively uses the full capability of the controlled system to obtain the desired regulation of both generator speed and power while keeping the system variables within safe operating limits.
2
Modelling of WECS
A model of the entire WECS can be structured as several interconnected subsystems as shown in Fig. 2 [1, 10]. The aerodynamic subsystem describes the transformation of kinetic energy stored in the wind into mechanical power via the wind turbine rotor. The drive train subsystem represents the mechanical parts that transfer the aerodynamic torque on the blades to the generator shaft. The pitch actuator
Fig. 2 A typical variable – speed variable pitch WECS a Physical diagram of the WECS b WECS model IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124– 136 doi: 10.1049/iet-rpg.2009.0137
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www.ietdl.org subsystem models the pitch control system that controls the pitch angle of the wind turbine’s blades. Finally, the electrical subsystem describes the electric generator, the power electronic converters and the generator control system. In general, the generator control system is based on a fieldoriented vector control strategy where the machine variables are expressed in a synchronously rotating reference frame. Vector control stems from decoupled flux-current and torque-current control in AC drives. Detailed descriptions of vector control of variable-speed variable pitch WECSs can be found in [12, 13]. In Fig. 2, the input signals coming from the turbine control system are the generator torque set point Tg∗ and the desired pitch angle bd . The measured outputs are assumed to be the generator speed, vg , and the generator power, Pg . The wind speed v is the disturbance signal affecting the WECS. Details of individual blocks are given in Appendix 1 [1– 3, 7, 14]. Linearised WECS model: The overall WECS model described in (31) – (42) is non-linear. The main non-linearity is because of the turbine torque expression in (35). Linearising the turbine torque equation in (35) yields )dvt + Lv ( )dv + Lb ( )db dTt = Lv ( vt , v, b vt , v, b vt , v, b (1) ∂Tt ∂Tt vt , v, b) = ; L ( v , v, b) = Lv ( ∂vt (vt ,v,b ) v t ∂v (vt ,v,b ) ) = ∂Tt Lb ( vt , v, b ∂b (vt ,v,b )
(2)
The symbol d is used for representing the deviation of a variable from its operating point value, whereas the over denotes the value of the variable at the operating bar in † point. In the case of WECS, the operating point is completely defined by v [1]. Now, the linearised state-space representation of the WECS, (31) – (42), can be written as (3) – (5) where def x = [dvt dvg dTtw dTg db]T [ R5 is the state vector, u = [dTg∗ dbd ]T [ R2 def
is
the
control
input
and
def
y = [dvg dPg ]T [ R2 is the measured output. Bv dv(t) x(t) = Ax(t) + Bu u(t) +
(3)
y(t) = Cx(t) ⎡
Lv Jt
⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ iB ⎢ A = ⎢ ks i + s Lv ⎢ Jt ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0
0
−ks
i Jt 1 Jg
−
0
i2 Bs Bs − + Jt Jg
0
0
0
0
0 −
1 Jg
0 −
1 tg
0
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⎤ Lb Jt ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ iBs ⎥ L ⎥ Jt b ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 1 ⎦ − t (4)
⎡
⎤ ⎡ L ⎤ 0 0 v ⎢ 0 0⎥ ⎢ ⎥ J ⎢ ⎥ ⎢ t ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ 0 0⎥ ⎢ iB ⎥ ⎢ ⎥ ⎢ ⎥ s = , B Bu = ⎢ 1 ⎥ v L ⎢ ⎥ ⎢ 0⎥ ⎢ Jt v ⎥ ⎢ tg ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 ⎦ ⎣ 1⎦ 0 0 t
0 1 0 0 0 C= g 0 0 T g 0 v
and
(5)
It can be seen from the linearised model that it is a MIMO system. Furthermore, it is apparent that the dynamics of the system vary when the average wind speed varies.
3
Control problem description
The main control objectives in the full load regime are to regulate both the generator power Pg and the generator speed vg at their rated values Pg,rat and vg,rat , respectively. These objectives can be achieved by manipulating the desired pitch angle bd and/or the generator torque set point Tg∗ . This can be inferred from (6), where the aerodynamic power Pt extracted from the wind is determined by the power coefficient, CP (l, b). This coefficient can be interpreted as a variable gain controlled by l and b. Thus, in the full load regime, to regulate the power at its rated value, the power coefficient should be reduced by increasing b, decreasing l, or changing both variables. Consequently, manipulating the pitch angle results in deviations in the power extracted by the wind turbine and, indirectly, induces deviations in the turbine speed via the drive train dynamics. Similarly, the generator torque can affect the turbine speed through the drive train dynamics and can be used for controlling the power extracted by the wind turbine by controlling l. 1 Pt = CP (l, b) rpR2 v3 2
(6)
The use of pitch control to regulate the power is limited by the constraints on the amplitude and speed of the pitch servos. In general, good power smoothing and minimisation of transient loads require large pitch activity that might cause fatigue damage to some mechanical parts of the system. On the other hand, the control of the generator torque is fast, but when it is applied to regulate the power, stability problems may arise [6]. If a multivariable controller that can harmonise the use of both pitch and torque control is used, the transient response of the system can be enhanced and the pitch activity can be reduced. There are many challenges for designing an effective control system for the WECS. The system variables must be regulated in the presence of severe fluctuations in the input turbine power Pt caused by erratic variations in the wind speed. Fluctuations in Pt can lead to harmful effects on the system [7, 8]. Large variations in the drive train torsional torque Ttw can occur, thus reducing the life time of the mechanical parts of the system. Input power fluctuations can result in electric power fluctuations supplied to the grid. This, in turn, can cause voltage flicker problems and a reduction in the power quality. Another challenge is the presence of non-linearity in the system dynamics and the continuous variation of the operating point depending on IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124 –136 doi: 10.1049/iet-rpg.2009.0137
www.ietdl.org the average wind speed vm (t). The control system must cope with these variations to ensure good performance over the whole range of operation of the WECS. It is clear that there are many design aspects that must be considered in the design of an effective control system for WECSs in the full load regime. The most important ones are outlined below. † Minimising transient mechanical loads on the drive train and the control activity of the pitch actuator system. † Smoothing the electrical power supplied to the system. † Ensuring good performance of the closed-loop system over the whole range of operation of the WECS. † Keeping system variables within acceptable and safe limits. There are maximum limits on the generated power and the system speed that are dictated by safety and operational issues. Furthermore, there are some physical actuator limits such as the ones on the pitch actuator rate and the pitch angle given in (32) – (33). Ignoring such constraints during the controller design phase can lead to performance degradation of the system [15]. Despite the multivariable nature of the control problem, the majority of papers in the literature dealing with controller design in the full load regime use a decentralised approach as shown in Fig. 3a [4, 7 – 10]. In this approach, two separate controllers are designed to regulate the generator speed and power independently. Commonly, the power is regulated using the pitch actuator system and the speed is regulated using the generator control. This approach has many disadvantages. First, designing these two controllers is a difficult task owing to the presence of interaction between these two control loops. Second, using this control configuration, large torsional torque variations and electric power fluctuations occur. This poor performance is owing to the slow response of the pitch actuator and consequently of the power control loop in comparison with the generator response and the speed control loop. As a consequence, the generator speed is kept almost constant. Owing to this (almost) fixed speed operation, the aerodynamic power fluctuations cause large variations in the generated electric power and the drive train torsional torque. Furthermore, large control activity in the pitch actuator can occur [5, 6]. These drawbacks can be substantially attenuated if variable-speed operation is allowed during the transients [1, 5]. As a result, a part of the energy captured in excess is transiently stored as kinetic energy, thereby reducing the energy supplied to the grid [10]. This can be achieved by designing a single multivariable MIMO control system that
controls Pg and vg simultaneously by manipulating Tg∗ and bd as shown in Fig. 3b [6].
4
Proposed control strategy
The proposed control strategy based on multiple model predictive control (MMPC) is described below. 4.1
Review of MPC
MPC is the only one among all the advanced control techniques (more advanced than PID controllers) which has been tremendously successful in industrial applications in recent decades [15 – 18]. The main reasons for this success are [15]: † MPC is based on optimal control techniques. In fact, under some conditions, the MPC controller coincides with the famous Linear Quadratic Gaussian (LQG) controller; † MPC based on state-space models is very suitable for MIMO control problems. † MPC algorithms can directly take into account constraints on the system variables. The idea behind MPC can be explained as follows. A dynamic model for the system to be controlled and any physical constraints on the system variables need to be known first. Then, a performance index that reflects the system dynamic behaviour must be selected. At each sampling interval, the future outputs of the system are predicted within a predefined prediction horizon, Np . These predictions are expressed in terms of the control signal within a predefined control horizon, Nc . The set of future control signals is calculated by solving a standard optimisation problem involving the system constraints and the preselected performance index. The first control signal in the optimal sequence is applied to the system, and the entire calculation is repeated at subsequent control intervals. The main drawback of using MPC controllers is the requirement to solve a quadratic programming (QP) problem on line. This has restricted the use of MPC to applications with slow dynamics. However, owing to the advances in the computational power of computers and in the optimisation algorithms, MPC can be used now for systems with fast dynamics [15]. As an example, it was shown in [17] that an MPC algorithm involving 12 states, three controls and a horizon of 30 time steps (which entails solving a QP with 450 variables and 1284 constraints) can be solved in about 5 ms.
Fig. 3 Control schemes used in full load regime a Decentralised control b Multivariable control IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124– 136 doi: 10.1049/iet-rpg.2009.0137
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www.ietdl.org 4.2 Single model predictive control (SMPC) for variable-speed variable pitch WECS The proposed control strategy based on MMPC is shown Fig. 4. For simplicity, the suggested control strategy is first explained in this subsection based on a single model while the MMPC will be explained in Section 4.3. It is assumed in this section that only one model will be used to represent the dynamics of the system in the whole operating range of the WECS in the full load regime (i.e. the number of models M is one and the model selection block is removed). This model can be chosen to be the model (3) linearised at wind speed v taken in the middle of the wind speed range in the full load regime (i.e. v = (vr + vco )/2). For any MPC algorithm, the main components that must be chosen by the designer are the prediction model, the optimisation problem and the state estimator (observer) [15, 19, 20]. Details of these selections for the case of variablespeed variable pitch WECS are given below. Prediction model: The MPC algorithm used in this paper is based on a discrete linear time invariant state-space model of the form (7) that represents the plant dynamics. x(k + 1) = Ax(k) + Bu u(k) + Bd d(k) y(k) = Cx(k) + Dd d(k)
(7)
def
y(k) = [dvg (k) dPg (k)]
(8)
T
The fictitious unmeasured disturbance d(k) [ Rnd is used to represent the effect of actual unmeasured disturbances on the plant and it is modelled as the output of the linear time invariant system (9). d (k) + Bn d (k) xd (k + 1) = Ax d (k) + Dn d (k) d(k) = Cx
=B = I2 =C A
and
=0 D (10)
This selection of the disturbance model ensures zero steady state errors in the regulated generator power and speed in the presence of step disturbances in the wind speed and modelling errors [20]. The matrices A, Bu and C of the discrete time model (7) with sampling period Ts can be obtained by discretising the linearised continuous plant model in (3) as shown in (11). A=e
ATs
, Bu =
Ts
eAt dt Bu
and
C= C
(11)
0
x(k + 1) xd (k + 1)
=
A Bu
x(k)
xd (k) x(k)
0 I2
y(k) = C 0
+
Bu 0
u(k) +
0 I2
nd (k)
xd (k) (12)
The augmented state space model (12) can be written compactly in (13) – (14).
def
x(k) = [dvt (k) dvg (k) dTtw (k) dTg (k) db(k)]T def
Bd = Bu , Dd = 0,
Combining (7), (9) and (10), the augmented prediction model used in the MPC formulation is given as
For the case of the WECS described in Section 2, the state vector x(k) [ R5 , the control input vector u(k) [ R2 and the measured output vector y(k) [ R2 at the sampling instant k are defined in (8).
u(k) = [dTg∗ (k) dbd (k)]T
It is assumed that nd (k) is random Gaussian noise having zero mean and unit covariance. In this paper, it will be assumed that the unmeasured disturbances d(k) [ R2 are entering at the input of the system and they are integrated white noise (random walks). This can be achieved by using (10).
(9)
x(k + 1) = A x(k) + Bu(k) + Bn nd (k) y(k) = C x(k) x A Bu Bu , x= , A= , B= 0 I2 0 xd 0 and C = [C 0] Bn = I2
(13)
(14)
Remark 1: It is inherently assumed in the above formulation that the continuous time model matrices in (4) – (5) are known
Fig. 4 Proposed control strategy using MMPC 128 & The Institution of Engineering and Technology 2011
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www.ietdl.org at the operating point of interest. The drive train model parameters Jt , Jg , ks and Bs can be identified from ) and experimental data [3]. Furthermore, Lv ( vt , v, b Lb ( vt , v, b) can be determined numerically at certain operating point from CP , which is usually given in tabular form [1, 6]. However, if this information is not known, the discrete state-space model in (7) can be identified directly from input output data using subspace identification methods or prediction error methods [21]. Optimisation problem: Assuming the knowledge of the current state of the plant x(k) and the disturbance model xd (k) or their estimates, the MPC controller solves the optimisation problem given by (15) and (16). (see (15)) Subject to : Dbmin ≤ Dbd (k + j) ≤ Dbmax , bmin ≤ bd (k + j) ≤ bmax ,
j = 1, 2, . . . , Nc j = 1, 2, . . . , Nc
Pg (k + j) ≤ Pg, max ,
j = 1, 2, . . . , Np
vg (k + j) ≤ vg, max ,
j = 1, 2, . . . , Np
(16)
In the full load regime, the set point values for v∗g (k + j) and Pg∗ (k + j) are set to vg,rat and Pg,rat , respectively. The desired pitch angle control move Dbd (k) is defined as bd (k) − bd (k − 1). A similar definition is used for DTg∗ (k). The limits on the desired pitch angle control move, Dbmax and Dbmin , can be related to the actual pitch angle rate limits b˙ max and b˙ min in (33) by (17). Dbmax = b˙ max Ts ,
Dbmin = b˙ min Ts
(17)
Using the model in (12), the optimisation problem (15) and (16) can be recast as a standard QP [15]. The weights q1 , q2 , r1 and r2 are non-negative scalars that represent the relative importance of the generator speed regulation, the generator power regulation, the generator torque command activity and the desired pitch angle activity, respectively. Increasing the value of one of these weights relative to the others adds more emphasise on minimising the corresponding quantity with respect to the other variables. For example, if the value of the weight q1 is increased with respect to the value of q2 , more emphasis is added on reducing the power deviations from its rated value. Consequently, it should be expected that the power fluctuation around the rated value will be reduced while allowing more fluctuations in the generator speed. Therefore better smoothing of the generated electric power is achieved with smaller overshoots in the drive train torque, Ttw . Furthermore, if it is required to achieve less activity in the pitch actuator, increasing the weight r2 with respect to the other weights achieves this objective. Finally, the physical constraints of the system given in (16) are explicitly incorporated in the controller formulation within the optimisation problem solved at each sampling instant. It can be seen that all design aspects stated in Section 3 can be achieved using MPC techniques. The only aspect that is not dealt with is how to ensure good performance of the j=Np min
∀DTg∗ (k+j),Dbd (k+j) j=0,1,...,Nc −1
closed-loop system over the whole range of operation of the WECS. This aspect will be considered in the next subsection. Remark 2: In the absence of constraints (16) and for sufficiently large values of Np and Nc , it can be shown that the MPC control law coincides with a state feedback linear quadratic regulator [15]. If, in addition, a Kalman state observer is used as a state estimator, the MPC-observer control law is equivalent to an LQG controller. State estimation: The optimisation problem in (15) – (16) requires knowledge of the current state of the plant x(k) and the disturbance model xd (k) before it can be solved. In general, not all these states are measurable and an estimator (observer) should be used for reconstructing them. In this paper, the estimates are computed using the linear state observer (18) – (19).
x(k|k) x(k|k − 1) = + K(y(k) − C x(k|k − 1)) xd (k|k) xd (k|k − 1) (18) A Bu Bu x(k + 1|k) x(k|k) u(k) (19) = + 0 I2 0 xd (k + 1|k) xd (k|k) The estimates of the plant and disturbance states, given the data up to time k, are denoted by x(k|k) and xd (k|k), respectively. The gain K is designed using Kalman filtering techniques [19, 21] based on the model (12) with added white noise at the output. 4.3
MMPC for variable speed variable pitch WECS
4.3.1 Multiple model predictive control: The use of linear MPC with a non-linear plant, such as the one considered in Section 2, in which the operating point is continuously changing can lead to degradation in the closed-loop performance [15]. There has been extensive research to extend the applicability of MPC to non-linear systems [22, 23]. One of the most straightforward approaches is to use MMPC [24 – 26]. In the MMPC approach, the whole operating region of interest is divided into M operating sub-regions with M linearised models that adequately represent the local system dynamics within each sub-region. A linear MPC controller based on each model is designed using the same guidelines discussed in Section 4.2. Finally, a criterion by which the control system switches one controller to another as the change in operating condition is defined. At any sampling instant, there is only one MPC controller that is active, and its corresponding QP is solved. However, to reduce bumps when switching between different MPC controllers, all inactive controllers automatically receive the current control signal and measured output signals so that their internal state estimates are kept up to date, thus reducing the transients in the state estimates when switching between MCP controllers. In the case of variable-speed variable pitch WECS, the operating point is completely dependent on the value
q1 (Pg∗ (k + j) − Pg (k + j))2 + q2 (v∗g (k + j) − vg (k + j))2 j=Nc −1 + j=1 r1 DTg∗2 (k + j) + r2 Db2d (k + j)
j=1
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Fig. 5 Response to a step change in wind speed from 20 to 21 m/s using four different SMPC controllers a b c d
Generator speed Generator power Torsional torque Pitch angle
of the operating wind speed v. In this paper, the full load regime is partitioned into M operating sub-regions defined in (20).
25 m/s for 600 s average, 28 m/s for 30 s average and 30 m/s for 3 s average. In (20), vco denotes the maximum of these wind speeds.
full load regime: vr ≤ v ≤ vco ⎧ sub-region 1(R1 ): vr ≤ v , v1 ⎪ ⎪ ⎪ ⎪ ⎨ sub-region 2(R2 ): v1 ≤ v , v2 ⇔ .. ⎪ ⎪ . ⎪ ⎪ ⎩ sub-region M (RM ): vM −1 ≤ v , vco (20)
4.3.2 Partitioning the full load regime: The most important building block of an MPC controller is the prediction model. The performance of an MPC controller is highly dependent on the ability of the model to predict the future response of the actual system. Multiple local linear models can be used for representing the non-linear dynamics of the WECS. However, special care must be taken when partitioning the whole operating region. In general, increasing the number of partitions and, consequently, reducing the range of each sub-region will enhance the linear approximation and the prediction accuracy. This comes at the cost of increasing the controller complexity and the computational burden. Many approaches can be used for partitioning the full load regime. One way is to use the non-linear torque expression in (35) – (37) and compare it with the linear approximation in (1) and partition the full load regime such that the accuracy of the linear approximation is bounded by a user-defined value. This approach assumes the knowledge of expressions (35) – (37) which may not be known in practice. The other approach, which is used in this paper, assumes that a large grid of linear models in (7), linearised at different wind speeds {v1 , v2 , . . . , vN } covering the full load regime, is known. These models can be calculated using (2) – (5) and (11) if expressions (35) – (37) are known or they can be estimated
where vr , v1 , v2 · · · , vM −1 , vco . The switching between different MPC controllers is based on (20). The current operating wind speed v can be determined online by filtering the wind speed measured by the anemometer located at the nacelle of the wind turbine. In Subsection 4.3.2, an algorithm is proposed to partition the whole operating region of the WECS in the full load regime into M sub-regions. Remark 3: In most commercial wind turbines, there is no single value for the cut-out wind speed at which the WECS is stopped. In practice, the wind turbine is stopped when the WECS is operating at certain average wind speed over a specific period of time. For example, one of the commercial wind turbines has cut-out wind speeds of 130 & The Institution of Engineering and Technology 2011
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www.ietdl.org directly using identification techniques. The basic idea is to partition the full load regime such that the distance between different prediction models in the same sub-region is bounded by a user-defined value, 1. Assume that a model (7), with parameters Ai , Bu,i and C i , linearised at vi is used by the MMPC controller. Using (13) recursively, the prediction vector yi (k + 1), based on this model, can be related to the current state and the future control inputs by (21) – (23).
Fig. 6 Wind speed profile used in the simulation
yi (k + 1) = Gi (k) x(k) + Hi u (21) ⎡ ⎤ ⎡ ⎤ yi (k + 1) u(k) ⎢ y (k + 2) ⎥ ⎢ u(k + 1) ⎥ ⎢ i ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ def def ⎢ ⎥ u(k + 2) ⎥ yi (k + 1) = ⎢ ⎢ yi (k + 3) ⎥, u(k) = ⎢ ⎢ ⎥ ⎢ ⎥ .. .. ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ . . ⎣ ⎦ yi (k + Np ) u(k + Nc − 1) (22) ⎡
C i Ai
⎤
⎢ 2 ⎥ ⎢ Ai ⎥ ⎢ C i ⎥ 3 ⎥ def ⎢ ⎢ Gi = ⎢ C i Ai ⎥ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ Np C i Ai ⎡ C i 0 0 Bi ⎢ C i Bi 0 ⎢ C i Ai Bi ⎢ def ⎢ 2 C i Ai Bi C i Bi Hi = ⎢ C i A i B i ⎢ .. .. .. ⎢ ⎣ . . . Np −1 Np −2 Np −3 Bi C i Ai Bi C i Ai Bi C i Ai
···
0
···
0
···
0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Np −Nc Ai ··· C i Bi (23)
The prediction equation (21) is written compactly in (24), where Ji and w(k) are defined in (25). yi (k + 1) = Ji w(k) def
Ji = [Gi Hi ],
def
T T w(k) = [ x (k) uNc (k)]T
(24) (25)
If the WECS is operating at different wind speeds, vj , while the MPC controller is using a prediction model linearised at vi , the prediction mismatch using those two models and its norm can be written in (26) and (27), respectively, where def √ x2 = xT x and s (X ) is the maximum singular value of X. yi (k + 1) − yj (k + 1) = (Ji − Jj )w(k)
(26)
yi (k + 1) − yj (k + 1)2 = (Ji − Jj )w(k)2 ≤ Ji − Jj 2 w(k)2 = s (Ji − Jj )w(k)2 (27) Fig. 7 System response using PI and SMPC controllers (results are shown for only 60 s) a b c d
Generator speed Generator power Torsional torque Pitch angle
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It can be seen from (27) that if the distance between two different models Jj and Ji is small, it should be expected that the difference between their predictions is small. The algorithm used for partitioning the full load regime is given below. 131
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Fig. 9 Wind speed profile
Fig. 8 Histograms a For generator speed b For generator power c For torsional torque
Table 1 Controller
Different partitioning of the full load regime Number of partitions, M
Sub-regions
SMPC MMPC1
1 3
MMPC2
7
R1 = [12.5, 27.5] R1 = [12.5, 17.5], R2 = [17.5, 22.5], R3 = [22.5, 27.5] R1 = [12.5, 13.6], R2 = [13.6, 16.0], R3 = [16.0, 18.0], R4 = [18.0, 20.1], R5 = [20.1, 22.3], R6 = [22.3, 24.6], R7 = [24.6, 27.5]
Algorithm Input: Desired accuracy 1, parameters (Ai , Bu,i and C i , i = 1, 2, . . . , N) of model (7) linearised at vi where v1 , v2 . . . , vN and the grid of wind speed {v1 , v2 , . . . , vN } covers the full load regime. Output: Sub-regions Rk where k = 1, 2, . . . , M Initialisation: k ¼ 1, j ¼ 1, calculate J1 from A1 , Bu,1 and C 1 using (14), (23) and (25) 132 & The Institution of Engineering and Technology 2011
Fig. 10 Performance comparison between SMPC and MMPC in Sub-regions I and II a Generator speed b Generator power c Pitch angle
1. for i = 1, 2, . . . , N , do 2. calculate Ji from Ai , Bu,i and C i using (14), (23) and (25); 3. if s (Ji − Jj ) . 1/2, then Rk = [vj , vi−1 [; j i − 1; k k+1 4. endfor Using the above algorithm and applying the triangle inequality for matrices, it can be inferred that for any vs , vt [ Rk and s, t [ {1, 2, . . . , N }, inequality (28) is satisfied. Therefore any model linearised at vs [ Rk and s [ {1, 2, . . . , N } can be used by the MPC controller as a IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124 –136 doi: 10.1049/iet-rpg.2009.0137
www.ietdl.org designed controllers are tested on the non-linear WECS model described by (31) – (42) with data given in Appendix 2. It will be assumed that vr is 12.5 m/s and vco is 27.5 m/s. 5.1 Designing an SMPC for variable-speed variable pitch WECS An SMPC is designed in this subsection based on a linearised model of the system at an operating wind speed v of 20 m/s (in the midrange of the full load regime). The other SMPC parameters are selected as Ts = 50 ms, Np = 20, Nc = 10, r1 = 0, r2 = 1, q2 = 100 (29) In order to study the effect of the weights in (15), four different SMPCs based on the same model and parameters of (29) but with different values of q1 taken as 2, 4, 7 and 10 are compared. These controllers are implemented using the Model Predictive Control Toolboxw [19]. Simulations are shown in Fig. 5 where vg , Pg and Ttw have been normalised with their rated values. As expected, when increasing q1 , Pg is better regulated and the overshoots in Ttw are effectively reduced. However, this comes at the cost of more fluctuation in vg and more pitch activity. In the following studies, the SMPC controller with q1 = 10 will be used. Remark 6: The computational time of the SMPC controller to calculate the new control input is recorded during simulations which are performed on a 1.66 GHZ dual core PC. It was found that, each MPC step is carried out in less than 25 ms (about 50% of the sampling period). Fig. 11 Performance comparison between SMPC and MMPC in Sub-region III a Generator speed b Generator power c Pitch angle
candidate model representing sub-region Rk . Js − Jt 2 = Js − Jj + Jj − Jt 2 ≤ Js − Jj 2 + Jj − Jt 2 ≤ 1
(28)
Remark 4: Selecting large values of 1 in the proposed algorithm means that it is accepted to have large prediction mismatch between different linearised models within the same sub-region. In this case, it should be expected that the algorithm will produce a small number of partitions with large ranges. Reducing the value of 1 increases the number of partitions calculated by the proposed algorithm and enhances the linear approximation within each sub-region. Remark 5: To determine the minimum number of partitions used by the MMPC to obtain satisfactory performance for the WECS, one can use the proposed algorithm iteratively, by starting with a large value of 1 and incrementally reducing it until the designed MMPC gives the desired performance over the full load regime.
5
Simulation results
The performance of the proposed control strategy is assessed in this section. In all simulations provided in this section, the IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124– 136 doi: 10.1049/iet-rpg.2009.0137
5.2 Single multivariable MPC controller against two decentralised PI controllers The objective of this subsection is to compare the performance of the SMPC designed in the previous subsection with two decentralised PI controllers used for regulating the generator power and speed independently as shown in Fig. 3a. Both PI controllers are implemented in the parallel form with an anti-windup strategy. The tuning of these PI controllers is not straightforward. The main challenge with a multivariable system is the presence of interactions between system variables. This prevents the design of each control loop being performed independently. For the studies reported in this paper, an exhaustive search was made to tune the two PI controllers to give approximately the same settling time as obtained when using the SMPC controller designed in Section 5.1. The transfer functions of the power controller Gc,power (s) and the speed controller Gc,speed (s) are Gc,power (s) = −9.7 −
29 67 , Gc,speed (s) = −39.71 − (30) s s
In this simulation, a stochastic wind speed signal with average wind speed v of 20 m/s is used. The wind speed model used is based on [1, 27]. The stochastic wind speed signal is shown in Fig. 6, whereas the system response zoomed in the period of 1 min is shown in Fig. 7. It is clear from Fig. 7 that by using the multivariable SMPC controller, the power is effectively smoothed and the fluctuations in the drive train torsional torque are 133
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Fig. 12 Response of the WECS system with the MMPC controller when the system is excited with a realistic wind speed signal a b c d
Wind speed Generator speed Generator power Pitch angle and pitch angle rate
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www.ietdl.org significantly reduced in comparison with the two decentralised PI controllers. These advantages are achieved while the pitch activities of both controllers are almost the same. The histograms of vg , Pg and Ttw shown in Fig. 8 confirm the superiority of performance of the multivariable SMPC controller. 5.3
SMPC controller against MMPC controller
Performance of the SMPC controller designed in Section 5.1 is compared with two different MMPC controllers given in Table 1. The first MMPC uses three linearised models based on partitioning the full load regime into three equal sub-regions (M ¼ 3). The second MMPC uses seven linearised models based on partitioning the full load regime into seven sub-regions (M ¼ 7) using the algorithm described in Section 4.3.2, with 1 ¼ 0.5 and a bank of models obtained by linearising the WECS at wind speeds {vci + iDv|i = 0, 1, . . . , (vco − vci )/Dv; Dv = 0.1}. To test the performance of these controllers over the full load regime, a staircase wind speed signal is used as shown in Fig. 9. To enhance visualisation of the results, the simulation results of the first 40 s are shown in Fig. 10 and the rest are shown in Fig. 11. It can be seen from Figs. 10 and 11 that the performance of the SMPC deteriorates significantly when the system is operating at wind speeds far from the one at which the SMPC is designed. By comparing the results, in the same figures, of the two MMPC controllers described in Table 1, it is clear that good performance is maintained by both controllers over the full load regime. A very slight enhancement in performance in terms of damping can be observed when using the MMPC with higher number of models (M ¼ 7). However, the controller complexity and the computational requirement of the MMPC with M ¼ 7 is higher than the MMPC with M ¼ 3. Finally, the performance of MMPC1 in Table 1 is assessed when the WECS is excited by a realistic non-stationary wind speed signal, for which vm (t) in (43) is varying with time as described in [1, 2]. The WECS is simulated for 1 h. Full simulation results are shown in the left of Fig. 12, whereas, the system response zoomed for a period of 30 s, where the MMPC switches between two different models, is shown in the right of Fig. 12. It can be seen from the right of Fig. 12, that the controller switching occurs without bumps and that the performance of the WECS is maintained over a wide range of wind speeds.
6
Conclusions
A multivariable control strategy based on MPC techniques is proposed to control variable speed variable pitch WECSs in the full load regime. The MPC controller can be designed to provide the desired trade-off between power smoothing and speed regulation while reducing the drive train torsional torque fluctuations and pitch actuator activity. Furthermore, the MPC controller provides good performance of the WECS while keeping the system variables within safe operating limits. Performance of the multivariable MPC controller is compared with the traditional use of two PI controllers for independent control of the generator power and speed. Simulation results show that the MPC controller provides much better smoothing for the generated power and much less torsional torque variations in the drive train. This has its effect on improving the power quality of the electrical power generated by the WECS and increasing the IET Renew. Power Gener., 2011, Vol. 5, Iss. 2, pp. 124– 136 doi: 10.1049/iet-rpg.2009.0137
life time of the mechanical parts of the system. To cope with the system non-linearity and the continuous variation of the operating point of the WECS, an MPC based on multiple models of the WECS is suggested. It is shown that using this approach, good performance of the closed-loop system is achieved over the whole operating region in the full load regime. Extension of this work to the whole operating region of the WECS (both partial and full load regimes) is currently in progress.
7
Acknowledgment
This paper is based on work supported by the Alberta Ingenuity Fund (File # 200800408).
8
References
1 Bianchi, F.D., Battista, H., Mantz, R.J.: ‘Wind turbine control systems: principles, modelling and gain scheduling design’ (Springer-Verlag, London, 2007, 1st edn.) 2 Munteanu, I., Bratcu, A.I., Cutululis, N.A.: ‘Optimal control of wind energy systems’ (Springer-Verlag, 2008, 1st edn.) 3 Akhmatov, V.: ‘Induction generators for wind power’ (Multi-Science Publishing Co. Ltd., 2007, 1st edn.) 4 Hansen, A.D., Sorensen, P., lov, F., Blaabjerg, F.: ‘Control of variable speed wind turbines with doubly-fed induction generators’, Wind Eng., 2004, 28, pp. 411 –443 5 Salle, D.L., Reardon, D., Leithead, W.E., Grimble, M.J.: ‘Review of wind turbine control’, Int. J. Control, 1990, 52, pp. 1295– 1310 6 Bianchi, F.D., Mantz, R.J., Christiansen, C.F.: ‘Power regulation in pitch-controlled variable-speed WECS above rated wind speed’, Renew. Energy, 2004, 29, pp. 1911– 1922 7 Muhando, E.B., Senjyu, T., Yona, A., Kinjo, H., Funabashi, T.: ‘Disturbance rejection by dual pitch control and self-tuning regulator for wind turbine generator parametric uncertainty compensation’, IET Control Theory Appl., 2007, 1, pp. 1431–1440 8 Bossanyi, E.A.: ‘The design of closed loop controllers for wind turbines’, Wind Energy, 2000, 3, pp. 149– 163 9 Wu, F., Zhang, X.-P., Godfrey, K., Ju, P.: ‘Small signal stability analysis and optimal control of a wind turbine with doubly fed induction generator’, IET Gener. Transm. Distrib., 2007, 1, pp. 751 –760 10 Muljadi, E., Butterfield, C.P.: ‘Pitch-controlled variable-speed wind turbine generation’, IEEE Trans. Ind. Appl., 2001, 37, pp. 240– 246 11 Geng, H., Yang, G.: ‘Robust pitch controller for output power levelling of variable-speed variable-pitch wind turbine generator systems’, IETRenew. Power Gener., 2009, 3, pp. 168– 79 12 Pena, R., Clare, J.C., Asher, G.M.: ‘Doubly fed induction generator using back-to-back PWM converters and its application to variablespeed wind-energy generation’, IEE Proc. Electr. Power Appl., 1996, 143, pp. 231–41 13 Conroy, J.F., Watson, R.: ‘Low-voltage ride-through of a full converter wind turbine with permanent magnet generator’, IET Renew. Power Gener., 2007, 1, pp. 182– 189 14 Pal, B.C., Mei, F.: ‘Modelling adequacy of the doubly fed induction generator for small-signal stability studies in power systems’, IET Renew. Power Gener., 2008, 2, pp. 181– 190 15 Maciejowski, J.: ‘Predictive control with constraints’ (Prentice Hall, 2000, 1st edn.) 16 Qin, S.J., Badgwell, T.A.: ‘A survey of industrial model predictive control technology’, Control Eng. Pract., 2003, 11, pp. 733–764 17 Wang, Y., Boyd, S.: ‘Fast model predictive control using online optimization’. Proc. 17th Int. Federation of Automatic Control Conf., Seoul, Korea, pp. 6974–6980 18 Bemporad, A.: ‘Model predictive control design: new trends and tools’. Proc. 45th IEEE Conf. on Decision and Control, Piscataway, USA, 13– 15 December 2006, pp. 6678– 6683 19 Bemporad, A., Morari, M., Ricker, N.L.: ‘Model predictive control toolbox 3 – user’s guide’ (The Mathworks, Inc., 2008), http://www. mathworks.com/access/helpdesk/help/toolbox/mpc/) 20 Muske, K.R., Badgwell, T.A.: ‘Disturbance modeling for offset-free linear model predictive control’, J. Process Control, 2002, 12, pp. 617–632 21 Verhaegen, M., Verdult, V.: ‘Filtering and system identification: a least squares approach’ (Cambridge University Press, 2007, 1st edn.) 22 Gattu, G., Zafiriou, E.: ‘Nonlinear quadratic dynamic matrix control with state estimation’, Ind. Eng. Chem. Res., 1992, 31, pp. 1096– 1104 135
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www.ietdl.org 23 Henson, M.A.: ‘Nonlinear model predictive control: current status and future directions’, Comput. Chem. Eng., 1998, 23, pp. 187–202 24 Dougherty, D., Cooper, D.: ‘A practical multiple model adaptive strategy for multivariable model predictive control’, Control Eng. Pract., 2003, 11, pp. 649– 664 25 Kuure-kinsey, M., Bequette, B.W.: ‘Multiple model predictive control: a state estimation based approach’. 2007 American Control Conf., New York City, USA, 9 –13 July 2007, pp. 3739–3744 26 Konakom, K., Kittisupakorn, P., Mujtaba, I.M.: ‘Batch control improvement by model predictive control based on multiple reducedmodels’, Chem. Eng. J., 2008, 145, pp. 129– 134 27 Petru, T., Thiringer, T.: ‘Modeling of wind turbines for power system studies’, IEEE Trans. Power Syst., 2002, 17, pp. 1132– 1139
shaft as [2] dvt i 1 = − Ttw + Tt dt Jt Jt
(38)
dvg 1 1 = Ttw − Tg (39) dt Jg Jg dTtw i2 Bs Bs iB B = ks ivt − ks vg − + Ttw + s Tt + s Tg dt Jt Jg Jt Jg (40)
9
Appendix 1: detailed WECS model
Pitch actuator system: The pitch actuator is modelled as a first-order dynamic system with saturation in the amplitude and derivative of the pitch angle b as [1, 3, 7] 1 1 b˙ = − b + bd t t
(31)
bmin ≤ b ≤ bmax
(32)
bmin ≤ b˙ ≤ b˙ max
(33)
Here, t is the time constant of the pitch system and †max (†min ) is the maximum (minimum) limit of †. Aerodynamic system: The output of the model is the harvested mechanical power of the wind turbine Pt or the turbine torque Tt (at the low-speed shaft). The model is given as 1 Pt = CP (l, b) rpR2 v3 2 P C (l, b) 1 rpR3 v2 Tt = t = P vt l 2
(35)
116 CP (l, b) = 0.5176 − 0.4b − 5 e−21/li + 0.0068l li (36) (37)
Drive train model: For control purposes, it is sufficient to represent the drive train by two masses with a flexible
(41)
Here, Jt and Jg are the inertia of the turbine and the generator, respectively; utw is the shaft twist angle; i is the gear ratio; ks , Bs are the shaft stiffness and damping coefficients, respectively. Generator model: In general, the electrical transients and the generator control system are much faster than the mechanical transients and the turbine controller [4, 7]. For that reason, simple models are used for representing the generator control system and the generator dynamics for the purpose of designing the wind turbine controller [4 – 8]. In this paper, a first-order system with a time constant tg and a unity DC gain is used for representing the generator control system and the generator model blocks shown in Fig. 2.
(34)
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def
Ttw = ks utw + Bs (ivt − vg )
Tg = −
Here, R is the blade length of the wind turbine; CP (l, b) is the def power coefficient, l = vt R/v is the tip speed ratio and vt is the speed of the low-speed shaft. In this paper the nonlinear functions given by (36) – (37), as suggested in [7], are used to describe CP (l, b).
1 1 0.035 = − 3 li l + 0.08b b + 1
The drive train torsional torque experienced by the flexible shaft Ttw is defined as
1 1 T + T∗ tg g t g g
(42)
Wind speed model: Wind speed v(t) is modelled in the literature as a non-stationary random process (43), yielded by superposing two components; a low-frequency component vm (t) (describing long-term, low-frequency variations) and a turbulence component vt (t) (corresponding to high-frequency variations) [1, 2]. v(t) = vm (t) + vt (t)
10
(43)
Appendix 2: WECS data
Wind turbine Pt,rat = 2MW , vt,rat = 3.0408 rad/s, R = 33.29 m Pitch actuator t ¼ 0.1 s, bmin = 08, bmax = 458, bmin = −108/s, b˙ max = 108/s Drive train i ¼ 74.38, Jt = Jt0 = 1.86E + 06 kg m2 , Jg = 56.29 kg m2 , ks = ks0 = 31.8E + 04 Nm/rad, Bs = Bs0 = 212.2 Nm.s/rad Generator tg = 20 ms, number of poles ¼ 4, supply frequency ¼ 60 Hz
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