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Estimator based adaptive fuzzy logic control technique for a wind turbine–generator system
Energy Conversion and Management 44 (2003) 135–153 www.elsevier.com/locate/enconman
Estimator based adaptive fuzzy logic control technique for a wind turbine–generator system Andrea Dadone, Lorenzo Dambrosio * DIMeG, Sezione di Macchine ed Energetica, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy Received 28 June 2001; accepted 26 November 2001
Abstract The control of a wind power plant, operating as an isolated power source, is analyzed. The plant consists of a wind turbine and a three phase synchronous electric generator, connected by means of a gear box. The mathematical models of the wind turbine and of the electrical generator are indicated. The use of an estimator based adaptive fuzzy logic control technique to govern the system is proposed. The results of a control test case are shown in order to demonstrate the reliability of the proposed control technique. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Wind turbine; Electric generator; Adaptive control; Fuzzy logic
1. Introduction Wind power plants are generally used to convert wind energy into electrical energy. These plants consist of a wind turbine and an electrical generator connected by means of a gear box. The wind turbine converts wind kinetic energy into mechanical energy, which is transformed into electrical energy by the electrical generator. These plants are controlled by an appropriate control system. The flow field about a wind turbine is unsteady and three dimensional. Mathematical models for such complex flows are available in the literature. They may be divided into momentum, vortex and finite difference models. The momentum models rely on actuator surfaces to approximate wind turbine effects and subdivide the flow domain into a finite number of stream tubes [1,2]. Vortex models simulate the wind turbine blades by means of bounded distributed vortices *
Corresponding author. Tel.: +39-80-5963406; fax: +39-80-5963411. E-mail address: [email protected] (L. Dambrosio).
0196-8904/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 1 9 9 - 6
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[3]. Finally, finite difference models solve the fluid dynamic governing equations by means of finite volume or finite difference methods [4]. The wind turbine drives a synchronous three phase electric generator. Since the stator and rotor windings of the electric generator together with the external load are balanced, a per-phase analysis can be adopted [5]. The wind power plant is controlled by an appropriate control system, which must yield an asymptotic disturbance reduction in order to minimize the differences between the actual and the reference values of the output voltage and frequency. Open loop [6] and closed loop [7,8] classical control techniques can be used to control wind power plants. In the present paper, we introduce the estimator based adaptive fuzzy logic (EAFL) control, which represents an innovative type of adaptive fuzzy logic control system [9,10]. The EAFL controller continuously optimizes the internal parameters of the fuzzy logic rule base, in order to adapt these parameters to the system dynamic behavior. Indeed, starting from a first glance fuzzy logic rule base, the EAFL minimizes an appropriate cost function in order to adapt the fuzzy controller parameters to the controlled wind system. This control technique employs a gradient descent algorithm to minimize the cost function, which is based on the control error estimated one step ahead. Such errors are estimated by using the least square algorithm (LSA) in recursive form [11–13]. The EAFL requires the observability of the controlled plant, while it does not require any a priori knowledge of its deterministic model. Nonetheless, the mathematical models of the wind turbine and of the synchronous electrical generator are given in the present paper because the proposed EAFL control technique is here applied to a numerically simulated wind system instead of being used to control a real wind plant. In particular, a rather simple momentum model is adopted for the wind turbine, while a complete dynamic model of the rotor and stator windings of the electrical generator is employed. Taking into account that the power plant is assumed to be an isolated power source, the disturbances here considered are the wind speed variability as well as the external load changes. Obviously, the EAFL control has to counterbalance such disturbances in order to control the system outputs represented by the voltage and the rotational speed of the electric generator. The required control is obtained by acting on two appropriate control variables. The first one is the transmission ratio of the gear box. The second one is the voltage of the generator rotor windings of the synchronous generator configuration. In the next sections, the mathematical models of the horizontal wind turbine as well as of the electrical generator are first outlined. Then, the fuzzy logic system and the estimator based adaptive fuzzy logic control technique are described. Finally, some results pertaining to the application of the EAFL control algorithm to the numerically modelled wind system are presented. For comparison, the results of a PI controller applied to the wind system that undergoes the same disturbance pattern have been reported.
2. Mathematical model of the horizontal axis wind turbine The mathematical model of the horizontal axis wind turbine here considered is based on stream tube discretization, which relies on momentum conservation in both the axial and tangential
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directions. The aerodynamic characteristics of the wind turbine blades are represented by the lift (CL ) and drag (CD ) coefficients, which depend on the inflow angle. According to [1], the thrust and torque acting on a blade element dr, at a distance r from the center of the rotor, are given by cos / CD 2 2 1þ tan / dr; dT ¼ qprrð1 aÞ Vw CL 2 CL sin / ð1Þ sin / CD 1 2 2 0 2 1 dQ ¼ qpr rð1 þ a Þ ðXrÞ CL dr; cos2 / CL tan / where r ¼ Ab =A is the solidity ratio, while A and Ab are the rotor area and the total blade area, respectively. Moreover, q, a, a0 and / represent the air density, the axial and tangential induced velocity coefficients and the inflow angle, respectively. Finally, X is the rotor angular speed and Vw is the undisturbed wind speed. Following Eqs. (1), the wind turbine torque, QT , and power, P , are given by Z Rmax dQ; ð2Þ QT ¼ Rmin
P ¼ QT X;
ð3Þ
where Rmin and Rmax represent the internal and external rotor radii, respectively. The wind turbine power coefficient, CP , is given by: CP ¼
P : ð1=2ÞqAVw3
ð4Þ
In the present paper, we consider a three blade horizontal axis wind turbine. Its external Rmax and internal Rmin rotor radii are equal to 5 m and 0.5 m, respectively. The considered standard conditions are: Vw ¼ 10 m/s; P ¼ 22 kW; tip speed ratio k ¼ 7, with k ¼ XRmax =Vw . The maximum power and maximum airfoil efficiency criteria are adopted to design the outer section of the rotor blade, while the constant chord criterion is used for its inner section. An NACA 0012 airfoil profile is selected for the rotor blade, and the analytical expressions for the lift and drag coefficients are taken from [14]. The corresponding power coefficient CP versus tip speed ratio k is shown in Fig. 1. 3. Mathematical model of the three phase synchronous generator Based on the usual hypothesis of balanced external load as well as balanced stator and rotor windings, a fixed axis reference system dq can be used again to determine the governing equations of both the stator and the rotor windings of a synchronous electrical generator [5]: dif 3 did þ Laf ; dt 2 dt dif did Ld þ Lq iq x; vd ¼ ra id þ Laf þ dt dt diq vq ¼ ra iq Lq þ ðLaf if Ld id Þx; dt vf ¼ rf if þ Lff
ð5Þ
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Fig. 1. Power coefficient vs. tip speed ratio.
where ra and rf are the stator and rotor internal resistances, respectively, while Ld and Lq are the synchronous inductances with respect to the d and q axes. Moreover, Laf represents the mutual inductance between the stator and the rotor windings, Lff is the rotor self-inductance, x is the angular frequency, vf and if are the voltage and current of the rotor windings. Finally, the variables vd , vq , id and iq are related to the phase voltages and currents. The external load is represented by an inductance, LL , and a resistance, RL , so that the corresponding governing equations in the dq reference system are v d ¼ LL
did ; dt
v q ¼ LL
diq ; dt
vd ¼ RL ðid iLd Þ;
vq ¼ RL ðiq iLq Þ;
ð6Þ
where iLd and iLq are the inductance external load currents in the d–q system. The dynamic equilibrium of the rotating parts for both the wind turbine and the synchronous generator can be expressed as dxg QT QT 3 ð7Þ ¼ QE ¼ p½ðLaf if þ Ld id Þ Lq iq id ; J 4 dt s s where J , s and t represent the moment of inertia of all the rotating parts (referring to the electrical generator shaft), the transmission ratio of the gear box and the time, respectively. Moreover, QE is the rotor electromagnetic torque, QT the wind turbine torque computed by means of Eq. (2), x the shaft rotational speed and p the number of polar pairs. Eqs. (5)–(7) represent a set of eight equations which can be solved with respect to the eight unknown variables id , iq , if , vd , vq , iLd , iLq and x by means of a Runge–Kutta iterative method. Finally, the phase voltages va , vb and vc can be computed from vd and vq by means of the following relations:
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va ¼ vd cosðpxtÞ vq sinðpxtÞ; vb ¼ vd cosðpxt 2=3pÞ vq sinðpxt 2=3pÞ;
ð8Þ
vc ¼ vd cosðpxt þ 2=3pÞ vq sinðpxt þ 2=3pÞ;
4. Fuzzy logic system The fuzzy logic system (FLS) employs a set of N fuzzy linguistic rules. These rules may be provided by experts or can be extracted from numerical data. In either case, the engineering rules in the FLS are expressed as a collection of IF–THEN statements. Therefore, a fuzzy rule base R containing N fuzzy rules can be expressed as R ¼ ½Rule1 ; Rule2 ; . . . ; Rulei ; . . . ; RuleN ;
ð9Þ
where the ith rule is ~ THEN ½uðkÞ is bi ; Rulei : IF ½zðkÞ is A
ð10Þ
where k refers to the variable values at time t ¼ kDt. Moreover, the vector T
zðkÞ ¼ ½z1 ðkÞ; . . . ; zl ðkÞ
ð11Þ
represents all the l fuzzy inputs to the FLS. On the other hand, uðkÞ and bi represent the fuzzy output of the FLS and its fuzzy set, respectively. In the antecedent of the ith rule, the term ~ ¼ ½A~1 ; . . . ; A~l T A i i
ð12Þ
represents the vector of the fuzzy sets referring to the input fuzzy vector zðkÞ. The membership ~ of the fuzzy sets are Gaussian and assume functions of both the input vector zðkÞ and the vector A the following expressions: lzj ðkÞ ¼ expð1=2½ðzj ðkÞ ^zj Þ=rzj 2 Þ;
lA~ij ðkÞ ¼ expð1=2½ðzj ðkÞ A^ij Þ=rA~ij 2 Þ;
ð13Þ
where ^zj and rzj are the mean value and the variance of the Gaussian membership function of the jth input, zj ðkÞ. Likewise, A^ij and rA~ij are the mean value and the variance of the Gaussian membership function of the jth fuzzy set referring to the ith fuzzy rule, A^ij . The terms ^zj and rzj are known constants, while A^ij and rA~ij represent the unknown parameters of the FLS. As it will be shown, these parameters will be adapted to the controlled wind system by minimizing an appropriate cost function. The output of the fuzzy controller, uðkÞ, assumes the following expression [9]: PN
Q bi lj¼1 lQij ½zj;max ðkÞ ; uðkÞ ¼ PN Ql i¼1 j¼1 lQij ½zj;max ðkÞ i¼1
ð14Þ
where lQij ½zj ðkÞ ¼ lzj ðkÞlA~ij ðkÞ :
ð15Þ
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Fig. 2. Layout of the fuzzy control system.
Moreover, zj;max ðkÞ ¼
^zj r2zj þ A^ij r2A~i
j
r2zj þ r2A~i
ð16Þ
j
is the value of the jth input that maximizes Eq. (15). The maximization of Eq. (15) represents the supremum operation in the sup–star composition of the ith rule [9]. This fuzzy controller appears to be parameterized by hðkÞ ¼ f A^ij ðkÞ; rA~ij ðkÞ; bi ðkÞ; i ¼ 1; 2; . . . ; N ; j ¼ 1; 2; . . . ; lg:
ð17Þ
In the next section, a procedure that allows an online adaptation of the parameters hðkÞ to the controlled wind system will be introduced. The fuzzy logic control system adopted in the present paper is represented in Fig. 2. The fuzzy input vector is defined as zðkÞ ¼ ½yðk 1Þ; rðkÞ; uðk 1ÞT ;
ð18Þ
where yðkÞ is the output of the plant (controlled variable), uðkÞ is the control variable (output of the fuzzy controller) and rðkÞ represents a reference signal for yðkÞ. 5. Estimator based adaptive fuzzy logic In general, an adaptive fuzzy logic (AFL) control starts from an initially assumed set of parameters hð0Þ, whose only requirement is to stabilize the plant. Then, at each time step, the AFL control adapts the set of parameters hðkÞ in order to minimize the cost function: 1 J ðkÞ ¼ e2y ðkÞ; 2 where ey ðkÞ is the control error defined as ey ðkÞ ¼ rðkÞ yðkÞ:
ð19Þ
ð20Þ
The control error ey ðkÞ can be determined only if a deterministic model of the controlled system is available. In the present paper, we suppose that no a priori deterministic model of the controlled system is available. The EAFL control here suggested allows solving this class of problems. Indeed, instead
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of deriving the appropriate change in each internal parameter from the control error ey ðkÞ, the EAFL control refers to an approximate estimation of the control error e^y ðkÞ ¼ rðkÞ y^ðkÞ
ð21Þ
and to the corresponding cost function 1 J^ðkÞ ¼ e^2y ðkÞ: 2
ð22Þ
In Eq. (21), y^ðkÞ represents the estimated value of the output at the time k to be evaluated. The present system can be expressed in an auto regressive moving average (ARMA) as follows [11]: yðkÞ ¼ ak yðk 1Þ þ bk uðk 1Þ;
ð23Þ
where ak and bk represent the time varying coefficients of model (23). If the controlled plant is observable, then Eq. (23) represents its model in state space notation. In such a case, the model coefficients ak and bk are unknown. These coefficients can be estimated online by applying the LSA in recursive form [11–13]. As a consequence, the basic scheme of the fuzzy control system has to be modified as shown in Fig. 3, where the LSA estimator evaluates the coefficients a^k and b^k . Assuming that such coefficients do not change from the time k to the time k þ 1, the estimated model of the controlled system one step ahead, i.e. at time k þ 1, assumes the following expression: y^ðk þ 1Þ ¼ a^k yðkÞ þ b^k uðkÞ;
ð24Þ
which is the output of the LSA parameter estimator (Fig. 3). The signal y^ðk þ 1Þ is compared to the reference signal rðk þ 1Þ, and the difference determines the modification of the fuzzy controller parameters hðkÞ. This is implemented by rewriting the cost function J^ at time k þ 1 as 1 2 ak yðkÞ þ b^k uðkÞg : ð25Þ J^ðk þ 1Þ ¼ frðk þ 1Þ ½^ 2 The minimization of the cost function J^ðk þ 1Þ can be easily accomplished by using the gradient descent algorithm as follows:
Fig. 3. Layout of the estimator based adaptive fuzzy logic control system.
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hðkÞ ¼ hðk 1Þ g
oJ^ðk þ 1Þ ; oh
ð26Þ
where the sensitivity derivatives of J^ðk þ 1Þ with respect to h (refer to Eq. (17)) are given by Ql oJ^ðk þ 1Þ j¼1 wij ðkÞ ^ ; ¼ bk e^y ðk þ 1Þ PN Ql obi i¼1 j¼1 wij ðkÞ Ql PN vij ðkÞ i¼1 cij j¼1 wij ðkÞ½bi uðkÞ oJ^ðk þ 1Þ ^ ; ¼ bk e^y ðk þ 1Þ PN Ql ð27Þ o A^ij i¼1 j¼1 wij ðkÞ Q P rA~ij v2ij ðkÞ Ni¼1 cij lj¼1 wij ðkÞ½bi uðkÞ oJ^ðk þ 1Þ ; ¼ b^k e^y ðk þ 1Þ PN Ql orA~ij i¼1 j¼1 wij ðkÞ where A^ij ^zj vij ðkÞ ¼ 2 ; rA~i þ r2zj
wij ðkÞ ¼ exp½1=2ð A^ij ^zj Þ2 =ðr2A~i þ r2zj Þ: j
ð28Þ
j
The coefficient g is the rate of descent which can be chosen arbitrarily. Moreover, cij is equal to 1 if the ith rule is dependent on the jth input, otherwise it is equal to 0.
6. Results The wind power plant configuration considered to test the control system is composed of a three blade horizontal axis wind turbine connected to a three phase synchronous generator by means of a gear box with variable transmission ratio. In the present application, such a component has been simulated by means of the following second order transfer function: GðsÞ ¼
x2n ; s2 þ 2dxn s þ x2n
ð29Þ
where xn ¼ 10 and d ¼ 4. Such a second order transfer function refers to a gear box actuator. It is characterized by a large rising time Ts ¼ 1:73. In the computed applications, we have considered a controlled system starting from rest. We have also assumed that the power plant reaches a steady state condition after a starting time interval. During the steady state time interval, we have hypothesized that appropriate disturbances perturb the system status. Such disturbances are represented by the instantaneous changes of the wind speed (Vw ) and the resistance load (RL ), shown in Fig. 4. In particular, first, the disturbances act separately: the wind speed perturbation is followed by a load perturbation occurring after a separating time interval. Then, we analyze a more severe situation corresponding to disturbances acting in a rapid sequence: the wind speed perturbation ends exactly at the beginning of the load perturbation. During the starting time interval, the control system has to guarantee the required time sequence of the system status. During the steady state time interval, the control system has to
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Fig. 4. Time history of the disturbance changes.
counteract the disturbance effects in order to preserve the desired steady state conditions. The aimed target is to control the output voltage and the rotational speed of the electric generator. The required control is obtained by acting on two appropriate control variables. The first one is the transmission ratio, s, of the gear box. The second one is the voltage of the generator rotor windings, Vf , for the synchronous generator configuration [5]. Fig. 5 shows the time history of the rotational speed, xg , of the synchronous generator, while Fig. 6 presents the corresponding percentage control error, Dx%. In Fig. 5, dotted and solid lines represent the target and the actual rotational speed, respectively. A quick glance at these figures allows us to state that the EAFL control technique tunes itself in order to maintain a small
Fig. 5. Rotational speed of the synchronous generator.
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Fig. 6. Percentage control error of the synchronous generator rotational speed.
percentage control error. Except for the very beginning of the starting period, where small absolute errors lead to large percentage errors, the only evidence of the disturbance actions is represented by the small spikes in Fig. 6 (Dx% < 2:5%) due to the instantaneous variation of the disturbance variables. Fig. 7 presents the output voltage, V, while Fig. 8 shows the corresponding percentage error, DV %. Again, the actual and the target output voltages are practically coincident for most of the considered time interval (Fig. 7), with the exception of the control errors observed during the starting period, which are due to the low excitation of the synchronous generator. Moreover,
Fig. 7. Voltage of the synchronous generator.
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Fig. 8. Percentage control error of the synchronous generator voltage.
Fig. 8 shows no significant spikes during the steady state time interval, therefore indicating no practical effect of the instantaneous changes of the resistance load and the wind speed. Finally, Figs. 9 and 10 show the time variations of the control variables, i.e. the gear box transmission ratio, s, and the rotor voltage, Vf . Such control variables vary in order to counterbalance the disturbance effects. In particular, Fig. 9 reports not only the actual value of the transmission ratio but also the signal computed by the EAFL controller. Comparing the actual
Fig. 9. Time history of the transmission ratio (control variable).
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Fig. 10. Time history of the rotor voltage (control variable).
value and the signal of the transmission ratio, it can be concluded that the EAFL controller takes into account the time delay of the gear box. Though the previous test case does span the whole map of the wind system (coefficient of power vs. tip speed ratio), a second test case characterized by a different pattern of disturbance variables has also been considered (Fig. 11). Figs. 12 and 13 show the synchronous generator rotational speed, xg , and the corresponding percentage control error, Dx%, respectively. Again, a quick glance at these figures allows us to state that the EAFL control technique retunes itself in order to maintain a small percentage control error: the only evidence of the disturbance actions is repre-
Fig. 11. Time history of the disturbance changes.
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Fig. 12. Rotational speed of the synchronous generator.
Fig. 13. Percentage control error of the synchronous generator rotational speed.
sented by the small spikes in Fig. 13 (Dx% < 2:5%) due to the instantaneous variation of the disturbance variables. As far the output voltage is concerned, Fig. 14 presents the output voltage, V, while Fig. 15 shows the corresponding percentage error, DV %. Also, for the present disturbance pattern, the actual and the target output voltages are practically coincident for most of the considered time interval (Fig. 14), with the exception of the control errors observed during the starting period.
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Fig. 14. Voltage of the synchronous generator.
Fig. 15. Percentage control error of the synchronous generator voltage.
Figs. 16 and 17 show the time variations of the control variables, i.e. the gear box transmission ratio, s, and the rotor voltage, Vf . Again, these variables vary in accordance with the disturbances in order to counteract their effects. The previous applications have proven the adaptive nature of the EAFL control technique, which automatically tunes itself in order to take into account all the non-linearities and time variances of the system under control. In particular, it adaptively annihilates the time delays of the
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Fig. 16. Time history of the transmission ratio (control variable).
Fig. 17. Time history of the rotor voltage (control variable).
system components. Therefore, we can state that the EAFL control system has the property of forcing a physical system characterized by inherently relevant time delays to react promptly to external disturbances. In order to compare the performances of the EAFL control technique with those of a classic control algorithm, a PI controller has been applied to the wind system. It is noteworthy to observe that the PI controller is a linear deterministic control algorithm. This means that it is necessary to
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Fig. 18. Time history of the disturbance changes.
know the wind system transfer function in order to design the PI controller. Moreover, such a PI controller presents a good behavior only in the neighborhood of the design working condition. For such a reason, the PI controller parameters need to be scheduled in order to extend their validity to the whole range of the wind system working conditions. The disturbances that perturb the wind system are sketched in Fig. 18. Figs. 19 and 20 show the synchronous generator rotational speed, xg , and the corresponding percentage control error, Dx%, respectively. In the same figures, the synchronous generator rotational speed, xg , and the corresponding percentage control error, Dx%, obtained by using the EAFL algorithm, are also reported. Comparing the results of these two algorithm, it can be noticed that, although the PI controller produces a smaller max-
Fig. 19. Rotational speed of the synchronous generator.
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Fig. 20. Percentage control error of the synchronous generator voltage.
imum control error, it retains such a control error. On the other hand, the EAFL control system tunes itself in order to annihilate the control error. It is important to notice that if the disturbance pattern or the working conditions were different, the PI controller performances would be different. On the contrary, as demonstrated by the two previous test cases, the EAFL algorithm retunes itself to minimize the control error. In Fig. 21, the time history of the gear transmission ratios corresponding to the PI controller and to the EAFL control system are reported.
Fig. 21. Time history of the transmission ratio (control variable).
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7. Conclusions In the present paper, an EAFL control technique is proposed. The EAFL controller is used to control a wind power plant operating as an isolated electric power generation station. The wind system is composed of a horizontal axis three blade wind turbine connected to an electrical generator through a gear box. The wind power plant configuration is composed of a three blade horizontal axis wind turbine connected to a three phase synchronous generator by means of a gear box with variable transmission ratio. Instead of a real wind system, a numerically simulated wind power plant is considered, and the mathematical models of its components are presented. After a brief review of the fuzzy logic system, the EAFL controller is introduced and described. Its main features are the one step ahead estimation of the control error and the online optimization of the fuzzy rule base parameters. It is noteworthy that no a priori deterministic model of the controlled system is required. The EAFL controller is applied to the control of a wind power plant starting from rest and reaching a steady state condition, which is then perturbed by abrupt changes of the wind speed and the electric load. The proposed EAFL technique controls the output voltage and the electrical generator rotational speed. The EAFL controller applied to such a wind system has been tested with two different disturbance action patterns. In all the considered control cases, the EAFL control system proves its ability to control the output variables adequately under the actions of relevant and abrupt disturbances. The control system automatically tunes itself in order to counterbalance all the non-linearities and time variances of the system under control, thus proving its adaptive characteristic. In particular, it has the property to force a physical system characterized by inherently relevant time delays to react promptly to external disturbances. Moreover, a comparison between the performances of a PI controller and those obtained by the EAFL control system has been performed. The comparison shows that the PI controller retains a residual control error, and its performances depend on the disturbance pattern and the working conditions. On the contrary, the EAFL control system tunes itself in order to annihilate the control error for each of the disturbance patterns and for all the working conditions.
Acknowledgements This research has been supported by the Italian Agencies CNR (Consiglio Nazionale delle Ricerche) and MURST (Ministero dell’Universita e della Ricerca Scientifica e Tecnologica). The authors are grateful to the Reviewers for some very important suggestions.
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Estimator based adaptive fuzzy logic control technique for a wind turbine–generator system Andrea Dadone, Lorenzo Dambrosio * DIMeG, Sezione di Macchine ed Energetica, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy Received 28 June 2001; accepted 26 November 2001
Abstract The control of a wind power plant, operating as an isolated power source, is analyzed. The plant consists of a wind turbine and a three phase synchronous electric generator, connected by means of a gear box. The mathematical models of the wind turbine and of the electrical generator are indicated. The use of an estimator based adaptive fuzzy logic control technique to govern the system is proposed. The results of a control test case are shown in order to demonstrate the reliability of the proposed control technique. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Wind turbine; Electric generator; Adaptive control; Fuzzy logic
1. Introduction Wind power plants are generally used to convert wind energy into electrical energy. These plants consist of a wind turbine and an electrical generator connected by means of a gear box. The wind turbine converts wind kinetic energy into mechanical energy, which is transformed into electrical energy by the electrical generator. These plants are controlled by an appropriate control system. The flow field about a wind turbine is unsteady and three dimensional. Mathematical models for such complex flows are available in the literature. They may be divided into momentum, vortex and finite difference models. The momentum models rely on actuator surfaces to approximate wind turbine effects and subdivide the flow domain into a finite number of stream tubes [1,2]. Vortex models simulate the wind turbine blades by means of bounded distributed vortices *
Corresponding author. Tel.: +39-80-5963406; fax: +39-80-5963411. E-mail address: [email protected] (L. Dambrosio).
0196-8904/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 1 9 9 - 6
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[3]. Finally, finite difference models solve the fluid dynamic governing equations by means of finite volume or finite difference methods [4]. The wind turbine drives a synchronous three phase electric generator. Since the stator and rotor windings of the electric generator together with the external load are balanced, a per-phase analysis can be adopted [5]. The wind power plant is controlled by an appropriate control system, which must yield an asymptotic disturbance reduction in order to minimize the differences between the actual and the reference values of the output voltage and frequency. Open loop [6] and closed loop [7,8] classical control techniques can be used to control wind power plants. In the present paper, we introduce the estimator based adaptive fuzzy logic (EAFL) control, which represents an innovative type of adaptive fuzzy logic control system [9,10]. The EAFL controller continuously optimizes the internal parameters of the fuzzy logic rule base, in order to adapt these parameters to the system dynamic behavior. Indeed, starting from a first glance fuzzy logic rule base, the EAFL minimizes an appropriate cost function in order to adapt the fuzzy controller parameters to the controlled wind system. This control technique employs a gradient descent algorithm to minimize the cost function, which is based on the control error estimated one step ahead. Such errors are estimated by using the least square algorithm (LSA) in recursive form [11–13]. The EAFL requires the observability of the controlled plant, while it does not require any a priori knowledge of its deterministic model. Nonetheless, the mathematical models of the wind turbine and of the synchronous electrical generator are given in the present paper because the proposed EAFL control technique is here applied to a numerically simulated wind system instead of being used to control a real wind plant. In particular, a rather simple momentum model is adopted for the wind turbine, while a complete dynamic model of the rotor and stator windings of the electrical generator is employed. Taking into account that the power plant is assumed to be an isolated power source, the disturbances here considered are the wind speed variability as well as the external load changes. Obviously, the EAFL control has to counterbalance such disturbances in order to control the system outputs represented by the voltage and the rotational speed of the electric generator. The required control is obtained by acting on two appropriate control variables. The first one is the transmission ratio of the gear box. The second one is the voltage of the generator rotor windings of the synchronous generator configuration. In the next sections, the mathematical models of the horizontal wind turbine as well as of the electrical generator are first outlined. Then, the fuzzy logic system and the estimator based adaptive fuzzy logic control technique are described. Finally, some results pertaining to the application of the EAFL control algorithm to the numerically modelled wind system are presented. For comparison, the results of a PI controller applied to the wind system that undergoes the same disturbance pattern have been reported.
2. Mathematical model of the horizontal axis wind turbine The mathematical model of the horizontal axis wind turbine here considered is based on stream tube discretization, which relies on momentum conservation in both the axial and tangential
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directions. The aerodynamic characteristics of the wind turbine blades are represented by the lift (CL ) and drag (CD ) coefficients, which depend on the inflow angle. According to [1], the thrust and torque acting on a blade element dr, at a distance r from the center of the rotor, are given by cos / CD 2 2 1þ tan / dr; dT ¼ qprrð1 aÞ Vw CL 2 CL sin / ð1Þ sin / CD 1 2 2 0 2 1 dQ ¼ qpr rð1 þ a Þ ðXrÞ CL dr; cos2 / CL tan / where r ¼ Ab =A is the solidity ratio, while A and Ab are the rotor area and the total blade area, respectively. Moreover, q, a, a0 and / represent the air density, the axial and tangential induced velocity coefficients and the inflow angle, respectively. Finally, X is the rotor angular speed and Vw is the undisturbed wind speed. Following Eqs. (1), the wind turbine torque, QT , and power, P , are given by Z Rmax dQ; ð2Þ QT ¼ Rmin
P ¼ QT X;
ð3Þ
where Rmin and Rmax represent the internal and external rotor radii, respectively. The wind turbine power coefficient, CP , is given by: CP ¼
P : ð1=2ÞqAVw3
ð4Þ
In the present paper, we consider a three blade horizontal axis wind turbine. Its external Rmax and internal Rmin rotor radii are equal to 5 m and 0.5 m, respectively. The considered standard conditions are: Vw ¼ 10 m/s; P ¼ 22 kW; tip speed ratio k ¼ 7, with k ¼ XRmax =Vw . The maximum power and maximum airfoil efficiency criteria are adopted to design the outer section of the rotor blade, while the constant chord criterion is used for its inner section. An NACA 0012 airfoil profile is selected for the rotor blade, and the analytical expressions for the lift and drag coefficients are taken from [14]. The corresponding power coefficient CP versus tip speed ratio k is shown in Fig. 1. 3. Mathematical model of the three phase synchronous generator Based on the usual hypothesis of balanced external load as well as balanced stator and rotor windings, a fixed axis reference system dq can be used again to determine the governing equations of both the stator and the rotor windings of a synchronous electrical generator [5]: dif 3 did þ Laf ; dt 2 dt dif did Ld þ Lq iq x; vd ¼ ra id þ Laf þ dt dt diq vq ¼ ra iq Lq þ ðLaf if Ld id Þx; dt vf ¼ rf if þ Lff
ð5Þ
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Fig. 1. Power coefficient vs. tip speed ratio.
where ra and rf are the stator and rotor internal resistances, respectively, while Ld and Lq are the synchronous inductances with respect to the d and q axes. Moreover, Laf represents the mutual inductance between the stator and the rotor windings, Lff is the rotor self-inductance, x is the angular frequency, vf and if are the voltage and current of the rotor windings. Finally, the variables vd , vq , id and iq are related to the phase voltages and currents. The external load is represented by an inductance, LL , and a resistance, RL , so that the corresponding governing equations in the dq reference system are v d ¼ LL
did ; dt
v q ¼ LL
diq ; dt
vd ¼ RL ðid iLd Þ;
vq ¼ RL ðiq iLq Þ;
ð6Þ
where iLd and iLq are the inductance external load currents in the d–q system. The dynamic equilibrium of the rotating parts for both the wind turbine and the synchronous generator can be expressed as dxg QT QT 3 ð7Þ ¼ QE ¼ p½ðLaf if þ Ld id Þ Lq iq id ; J 4 dt s s where J , s and t represent the moment of inertia of all the rotating parts (referring to the electrical generator shaft), the transmission ratio of the gear box and the time, respectively. Moreover, QE is the rotor electromagnetic torque, QT the wind turbine torque computed by means of Eq. (2), x the shaft rotational speed and p the number of polar pairs. Eqs. (5)–(7) represent a set of eight equations which can be solved with respect to the eight unknown variables id , iq , if , vd , vq , iLd , iLq and x by means of a Runge–Kutta iterative method. Finally, the phase voltages va , vb and vc can be computed from vd and vq by means of the following relations:
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va ¼ vd cosðpxtÞ vq sinðpxtÞ; vb ¼ vd cosðpxt 2=3pÞ vq sinðpxt 2=3pÞ;
ð8Þ
vc ¼ vd cosðpxt þ 2=3pÞ vq sinðpxt þ 2=3pÞ;
4. Fuzzy logic system The fuzzy logic system (FLS) employs a set of N fuzzy linguistic rules. These rules may be provided by experts or can be extracted from numerical data. In either case, the engineering rules in the FLS are expressed as a collection of IF–THEN statements. Therefore, a fuzzy rule base R containing N fuzzy rules can be expressed as R ¼ ½Rule1 ; Rule2 ; . . . ; Rulei ; . . . ; RuleN ;
ð9Þ
where the ith rule is ~ THEN ½uðkÞ is bi ; Rulei : IF ½zðkÞ is A
ð10Þ
where k refers to the variable values at time t ¼ kDt. Moreover, the vector T
zðkÞ ¼ ½z1 ðkÞ; . . . ; zl ðkÞ
ð11Þ
represents all the l fuzzy inputs to the FLS. On the other hand, uðkÞ and bi represent the fuzzy output of the FLS and its fuzzy set, respectively. In the antecedent of the ith rule, the term ~ ¼ ½A~1 ; . . . ; A~l T A i i
ð12Þ
represents the vector of the fuzzy sets referring to the input fuzzy vector zðkÞ. The membership ~ of the fuzzy sets are Gaussian and assume functions of both the input vector zðkÞ and the vector A the following expressions: lzj ðkÞ ¼ expð1=2½ðzj ðkÞ ^zj Þ=rzj 2 Þ;
lA~ij ðkÞ ¼ expð1=2½ðzj ðkÞ A^ij Þ=rA~ij 2 Þ;
ð13Þ
where ^zj and rzj are the mean value and the variance of the Gaussian membership function of the jth input, zj ðkÞ. Likewise, A^ij and rA~ij are the mean value and the variance of the Gaussian membership function of the jth fuzzy set referring to the ith fuzzy rule, A^ij . The terms ^zj and rzj are known constants, while A^ij and rA~ij represent the unknown parameters of the FLS. As it will be shown, these parameters will be adapted to the controlled wind system by minimizing an appropriate cost function. The output of the fuzzy controller, uðkÞ, assumes the following expression [9]: PN
Q bi lj¼1 lQij ½zj;max ðkÞ ; uðkÞ ¼ PN Ql i¼1 j¼1 lQij ½zj;max ðkÞ i¼1
ð14Þ
where lQij ½zj ðkÞ ¼ lzj ðkÞlA~ij ðkÞ :
ð15Þ
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Fig. 2. Layout of the fuzzy control system.
Moreover, zj;max ðkÞ ¼
^zj r2zj þ A^ij r2A~i
j
r2zj þ r2A~i
ð16Þ
j
is the value of the jth input that maximizes Eq. (15). The maximization of Eq. (15) represents the supremum operation in the sup–star composition of the ith rule [9]. This fuzzy controller appears to be parameterized by hðkÞ ¼ f A^ij ðkÞ; rA~ij ðkÞ; bi ðkÞ; i ¼ 1; 2; . . . ; N ; j ¼ 1; 2; . . . ; lg:
ð17Þ
In the next section, a procedure that allows an online adaptation of the parameters hðkÞ to the controlled wind system will be introduced. The fuzzy logic control system adopted in the present paper is represented in Fig. 2. The fuzzy input vector is defined as zðkÞ ¼ ½yðk 1Þ; rðkÞ; uðk 1ÞT ;
ð18Þ
where yðkÞ is the output of the plant (controlled variable), uðkÞ is the control variable (output of the fuzzy controller) and rðkÞ represents a reference signal for yðkÞ. 5. Estimator based adaptive fuzzy logic In general, an adaptive fuzzy logic (AFL) control starts from an initially assumed set of parameters hð0Þ, whose only requirement is to stabilize the plant. Then, at each time step, the AFL control adapts the set of parameters hðkÞ in order to minimize the cost function: 1 J ðkÞ ¼ e2y ðkÞ; 2 where ey ðkÞ is the control error defined as ey ðkÞ ¼ rðkÞ yðkÞ:
ð19Þ
ð20Þ
The control error ey ðkÞ can be determined only if a deterministic model of the controlled system is available. In the present paper, we suppose that no a priori deterministic model of the controlled system is available. The EAFL control here suggested allows solving this class of problems. Indeed, instead
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of deriving the appropriate change in each internal parameter from the control error ey ðkÞ, the EAFL control refers to an approximate estimation of the control error e^y ðkÞ ¼ rðkÞ y^ðkÞ
ð21Þ
and to the corresponding cost function 1 J^ðkÞ ¼ e^2y ðkÞ: 2
ð22Þ
In Eq. (21), y^ðkÞ represents the estimated value of the output at the time k to be evaluated. The present system can be expressed in an auto regressive moving average (ARMA) as follows [11]: yðkÞ ¼ ak yðk 1Þ þ bk uðk 1Þ;
ð23Þ
where ak and bk represent the time varying coefficients of model (23). If the controlled plant is observable, then Eq. (23) represents its model in state space notation. In such a case, the model coefficients ak and bk are unknown. These coefficients can be estimated online by applying the LSA in recursive form [11–13]. As a consequence, the basic scheme of the fuzzy control system has to be modified as shown in Fig. 3, where the LSA estimator evaluates the coefficients a^k and b^k . Assuming that such coefficients do not change from the time k to the time k þ 1, the estimated model of the controlled system one step ahead, i.e. at time k þ 1, assumes the following expression: y^ðk þ 1Þ ¼ a^k yðkÞ þ b^k uðkÞ;
ð24Þ
which is the output of the LSA parameter estimator (Fig. 3). The signal y^ðk þ 1Þ is compared to the reference signal rðk þ 1Þ, and the difference determines the modification of the fuzzy controller parameters hðkÞ. This is implemented by rewriting the cost function J^ at time k þ 1 as 1 2 ak yðkÞ þ b^k uðkÞg : ð25Þ J^ðk þ 1Þ ¼ frðk þ 1Þ ½^ 2 The minimization of the cost function J^ðk þ 1Þ can be easily accomplished by using the gradient descent algorithm as follows:
Fig. 3. Layout of the estimator based adaptive fuzzy logic control system.
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hðkÞ ¼ hðk 1Þ g
oJ^ðk þ 1Þ ; oh
ð26Þ
where the sensitivity derivatives of J^ðk þ 1Þ with respect to h (refer to Eq. (17)) are given by Ql oJ^ðk þ 1Þ j¼1 wij ðkÞ ^ ; ¼ bk e^y ðk þ 1Þ PN Ql obi i¼1 j¼1 wij ðkÞ Ql PN vij ðkÞ i¼1 cij j¼1 wij ðkÞ½bi uðkÞ oJ^ðk þ 1Þ ^ ; ¼ bk e^y ðk þ 1Þ PN Ql ð27Þ o A^ij i¼1 j¼1 wij ðkÞ Q P rA~ij v2ij ðkÞ Ni¼1 cij lj¼1 wij ðkÞ½bi uðkÞ oJ^ðk þ 1Þ ; ¼ b^k e^y ðk þ 1Þ PN Ql orA~ij i¼1 j¼1 wij ðkÞ where A^ij ^zj vij ðkÞ ¼ 2 ; rA~i þ r2zj
wij ðkÞ ¼ exp½1=2ð A^ij ^zj Þ2 =ðr2A~i þ r2zj Þ: j
ð28Þ
j
The coefficient g is the rate of descent which can be chosen arbitrarily. Moreover, cij is equal to 1 if the ith rule is dependent on the jth input, otherwise it is equal to 0.
6. Results The wind power plant configuration considered to test the control system is composed of a three blade horizontal axis wind turbine connected to a three phase synchronous generator by means of a gear box with variable transmission ratio. In the present application, such a component has been simulated by means of the following second order transfer function: GðsÞ ¼
x2n ; s2 þ 2dxn s þ x2n
ð29Þ
where xn ¼ 10 and d ¼ 4. Such a second order transfer function refers to a gear box actuator. It is characterized by a large rising time Ts ¼ 1:73. In the computed applications, we have considered a controlled system starting from rest. We have also assumed that the power plant reaches a steady state condition after a starting time interval. During the steady state time interval, we have hypothesized that appropriate disturbances perturb the system status. Such disturbances are represented by the instantaneous changes of the wind speed (Vw ) and the resistance load (RL ), shown in Fig. 4. In particular, first, the disturbances act separately: the wind speed perturbation is followed by a load perturbation occurring after a separating time interval. Then, we analyze a more severe situation corresponding to disturbances acting in a rapid sequence: the wind speed perturbation ends exactly at the beginning of the load perturbation. During the starting time interval, the control system has to guarantee the required time sequence of the system status. During the steady state time interval, the control system has to
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Fig. 4. Time history of the disturbance changes.
counteract the disturbance effects in order to preserve the desired steady state conditions. The aimed target is to control the output voltage and the rotational speed of the electric generator. The required control is obtained by acting on two appropriate control variables. The first one is the transmission ratio, s, of the gear box. The second one is the voltage of the generator rotor windings, Vf , for the synchronous generator configuration [5]. Fig. 5 shows the time history of the rotational speed, xg , of the synchronous generator, while Fig. 6 presents the corresponding percentage control error, Dx%. In Fig. 5, dotted and solid lines represent the target and the actual rotational speed, respectively. A quick glance at these figures allows us to state that the EAFL control technique tunes itself in order to maintain a small
Fig. 5. Rotational speed of the synchronous generator.
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Fig. 6. Percentage control error of the synchronous generator rotational speed.
percentage control error. Except for the very beginning of the starting period, where small absolute errors lead to large percentage errors, the only evidence of the disturbance actions is represented by the small spikes in Fig. 6 (Dx% < 2:5%) due to the instantaneous variation of the disturbance variables. Fig. 7 presents the output voltage, V, while Fig. 8 shows the corresponding percentage error, DV %. Again, the actual and the target output voltages are practically coincident for most of the considered time interval (Fig. 7), with the exception of the control errors observed during the starting period, which are due to the low excitation of the synchronous generator. Moreover,
Fig. 7. Voltage of the synchronous generator.
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Fig. 8. Percentage control error of the synchronous generator voltage.
Fig. 8 shows no significant spikes during the steady state time interval, therefore indicating no practical effect of the instantaneous changes of the resistance load and the wind speed. Finally, Figs. 9 and 10 show the time variations of the control variables, i.e. the gear box transmission ratio, s, and the rotor voltage, Vf . Such control variables vary in order to counterbalance the disturbance effects. In particular, Fig. 9 reports not only the actual value of the transmission ratio but also the signal computed by the EAFL controller. Comparing the actual
Fig. 9. Time history of the transmission ratio (control variable).
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Fig. 10. Time history of the rotor voltage (control variable).
value and the signal of the transmission ratio, it can be concluded that the EAFL controller takes into account the time delay of the gear box. Though the previous test case does span the whole map of the wind system (coefficient of power vs. tip speed ratio), a second test case characterized by a different pattern of disturbance variables has also been considered (Fig. 11). Figs. 12 and 13 show the synchronous generator rotational speed, xg , and the corresponding percentage control error, Dx%, respectively. Again, a quick glance at these figures allows us to state that the EAFL control technique retunes itself in order to maintain a small percentage control error: the only evidence of the disturbance actions is repre-
Fig. 11. Time history of the disturbance changes.
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Fig. 12. Rotational speed of the synchronous generator.
Fig. 13. Percentage control error of the synchronous generator rotational speed.
sented by the small spikes in Fig. 13 (Dx% < 2:5%) due to the instantaneous variation of the disturbance variables. As far the output voltage is concerned, Fig. 14 presents the output voltage, V, while Fig. 15 shows the corresponding percentage error, DV %. Also, for the present disturbance pattern, the actual and the target output voltages are practically coincident for most of the considered time interval (Fig. 14), with the exception of the control errors observed during the starting period.
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Fig. 14. Voltage of the synchronous generator.
Fig. 15. Percentage control error of the synchronous generator voltage.
Figs. 16 and 17 show the time variations of the control variables, i.e. the gear box transmission ratio, s, and the rotor voltage, Vf . Again, these variables vary in accordance with the disturbances in order to counteract their effects. The previous applications have proven the adaptive nature of the EAFL control technique, which automatically tunes itself in order to take into account all the non-linearities and time variances of the system under control. In particular, it adaptively annihilates the time delays of the
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Fig. 16. Time history of the transmission ratio (control variable).
Fig. 17. Time history of the rotor voltage (control variable).
system components. Therefore, we can state that the EAFL control system has the property of forcing a physical system characterized by inherently relevant time delays to react promptly to external disturbances. In order to compare the performances of the EAFL control technique with those of a classic control algorithm, a PI controller has been applied to the wind system. It is noteworthy to observe that the PI controller is a linear deterministic control algorithm. This means that it is necessary to
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Fig. 18. Time history of the disturbance changes.
know the wind system transfer function in order to design the PI controller. Moreover, such a PI controller presents a good behavior only in the neighborhood of the design working condition. For such a reason, the PI controller parameters need to be scheduled in order to extend their validity to the whole range of the wind system working conditions. The disturbances that perturb the wind system are sketched in Fig. 18. Figs. 19 and 20 show the synchronous generator rotational speed, xg , and the corresponding percentage control error, Dx%, respectively. In the same figures, the synchronous generator rotational speed, xg , and the corresponding percentage control error, Dx%, obtained by using the EAFL algorithm, are also reported. Comparing the results of these two algorithm, it can be noticed that, although the PI controller produces a smaller max-
Fig. 19. Rotational speed of the synchronous generator.
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Fig. 20. Percentage control error of the synchronous generator voltage.
imum control error, it retains such a control error. On the other hand, the EAFL control system tunes itself in order to annihilate the control error. It is important to notice that if the disturbance pattern or the working conditions were different, the PI controller performances would be different. On the contrary, as demonstrated by the two previous test cases, the EAFL algorithm retunes itself to minimize the control error. In Fig. 21, the time history of the gear transmission ratios corresponding to the PI controller and to the EAFL control system are reported.
Fig. 21. Time history of the transmission ratio (control variable).
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7. Conclusions In the present paper, an EAFL control technique is proposed. The EAFL controller is used to control a wind power plant operating as an isolated electric power generation station. The wind system is composed of a horizontal axis three blade wind turbine connected to an electrical generator through a gear box. The wind power plant configuration is composed of a three blade horizontal axis wind turbine connected to a three phase synchronous generator by means of a gear box with variable transmission ratio. Instead of a real wind system, a numerically simulated wind power plant is considered, and the mathematical models of its components are presented. After a brief review of the fuzzy logic system, the EAFL controller is introduced and described. Its main features are the one step ahead estimation of the control error and the online optimization of the fuzzy rule base parameters. It is noteworthy that no a priori deterministic model of the controlled system is required. The EAFL controller is applied to the control of a wind power plant starting from rest and reaching a steady state condition, which is then perturbed by abrupt changes of the wind speed and the electric load. The proposed EAFL technique controls the output voltage and the electrical generator rotational speed. The EAFL controller applied to such a wind system has been tested with two different disturbance action patterns. In all the considered control cases, the EAFL control system proves its ability to control the output variables adequately under the actions of relevant and abrupt disturbances. The control system automatically tunes itself in order to counterbalance all the non-linearities and time variances of the system under control, thus proving its adaptive characteristic. In particular, it has the property to force a physical system characterized by inherently relevant time delays to react promptly to external disturbances. Moreover, a comparison between the performances of a PI controller and those obtained by the EAFL control system has been performed. The comparison shows that the PI controller retains a residual control error, and its performances depend on the disturbance pattern and the working conditions. On the contrary, the EAFL control system tunes itself in order to annihilate the control error for each of the disturbance patterns and for all the working conditions.
Acknowledgements This research has been supported by the Italian Agencies CNR (Consiglio Nazionale delle Ricerche) and MURST (Ministero dell’Universita e della Ricerca Scientifica e Tecnologica). The authors are grateful to the Reviewers for some very important suggestions.
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