Output controllability analysis of fixed speed wind turbine

PHYSCON 2011, Le´ on, Spain, September, 5–September, 8 2011 OUTPUT CONTROLLABILITY AND STEADY-OUTPUT CONTROLLABILITY ANALYSIS OF FIXED SPEED WIND TURBINE J.L. Dom´ ınguez-Garc´ ıa Electrical Engineering Area IREC Spain [email protected] M.I. Garc´ ıa-Planas Matem` atica Aplicada I Universitat Polit` ecnica de Catalunya Spain [email protected] Abstract This paper deals with the concepts of output controllability and steady output controllability, it demonstrates that they are not equivalent of concepts. A linear system has been calculated from the nonlinear equations of the squirrel cage induction generator, supposing it connected directly to the grid and assuming a steady state operating point. The study of output controllability and steady-output controllability concepts of the introduced system is done. Key words Output Controllability, Steady-Output Controllability, Squirrel Cage Induction Generator, Linear System. 1 Introduction In the theory of continuous linear time-invariant dynamical control systems the most popular and the most frequently used mathematical model is given by the following differential state equation and algebraic output equations ˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t)(1) where xis the state vector, yis the output vector, u is the input (or control) vector, A∈Mn(R)is the state matrix, B∈Mn×m(R)is the input matrix, C∈ Mp×n(C)is the output matrix, and D∈Mp×m(C)is the feedthrough (or feedforward) matrix. Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control (see [1], [4] for example). Systematic study of controllability was started in the mid 20 century and theory of controllability is based on the mathematical description of the dynamical system. Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied. In the literature there are many different definitions of controllability which depend on the type of dynamical control system. In this paper it is considered the output controllability. Output controllability is the related notion for the output of the system, the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. A controllable system is not necessarily output controllable, and an output controllable system is not necessarily controllable. On the other hand, it is well known the recent increasing of wind power in the electrical network. Since, it can be interesting study and ensure the outputcontrollability of Fixed-Speed Wind Turbines (FSWT), which can affect directly the behavior of power systems. This paper is organized as follows. In Section 2, it is introduced the concepts of controllability and outputcontrollability. The steady output-controllability is defined in Section 3. In the section 4, two examples are developed. In section 5, the system under study is presented, and linearized to obtain the linear system. The output controllability and steady output-controllability of the system is calculated in section 6. Finally, the conclusions are summarized in Section 7. 2 Controllability and output Controllability The most frequently used fundamental definition of controllability for linear control systems with constant coefficients is the following. Definition 2.1. Dynamical system (1) is said to be controllable if for every initial condition x(0) and every vector x1∈Rn, there exist a finite time t1and control u(t)∈Rm,t∈[0, t1], such that x(t1) = x1. This definition requires only that any initial state x(0) can be steered to any final state x1at time t1. However, the trajectory of the dynamical system between 0 and t1is not specified. Furthermore, there is no constraints posed on the control vector u(t)and the state vector x(t). Controllability can be easily computed by means of the following algebraic criteria: the system is controllable if and only if the matrix presented in the equation 2 has full rank. C=B AB A2B . . . An−1B(2) This matrix is called controllability matrix. Theorem 2.1. Dynamical system (1) is controllable if and only if rank C=n. Similar to the state controllability of dynamical control system, it is possible to define the so-called output controllability for the output vector y(t)∈Rpof dynamical system. Although these two concepts are quite similar, it should be mentioned that the state controllability is a property of the differential state equation , whereas the output controllability is a property both of the state equation and algebraic output equation. Definition 2.2. Dynamical system (1) is said to be output controllable if for every y(0) and every vector y1∈ Rp, there exist a finite time t1and control u1(t)∈Rm, that transfers the output from y(0) to y1=y(t1). Therefore, output controllability generally means, that we can steer output of dynamical system independently of its state vector. For a linear continuous-time system, like (1), described by matrices A,B,C, and D, it is defined the output controllability matrix oC =CB CAB . . . CAn−1B D(3) and it is obtained the following result. Theorem 2.2. Dynamical system (1) is output controllable if and only if rank oC =p. It should be pointed out, that the state controllability is defined only for the linear differential state equation, whereas the output controllability is defined for the input-output description i.e., it depends also on the linear algebraic output equation. Therefore, these two concepts are not necessarily related. Theorem 2.3. The output controllability character is invariant under feedback. Proof. Let Fbe a matrix in Mm×n(R)Considering (A+BF, B, C, D), it is easy to compute (A+BF )k obtaining C(A+BF )kB= CAkB+P0≤`≤k−1CAk−`−1BF (A+BF )`B Making the following column elementary transformations cj+cj−1F B+cj−2F(A+BF )B+. . .+c1F(A+BF )j−2B where c`indicates the `column of the output controllability matrix of (A, B, C, D), it is obtained the output controllability matrix for (A+BF, B, C, D). 3 Steady-output Controllability Within the linear systems theory, it is often asked about the possibility that the state-steady outputs converge to a constant value. In order to be able to analyze this concept, it is given the following definition. Definition 3.1. A vector is called constant steady-state output controllable if there exists an input constant vector usuch that lim t→∞ y(t) = K(4) where Kis a p×1constant output vector. Taking Laplace transforms to the system ˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t),(5) reformulating this definition in the following manner. Proposition 3.1. A constant output vector Kis steadyoutput controllable if there exists an input u(s) = k s such that lim t→∞ y(t) = lim s→0sy(s) = K(6) Clearly a necessary condition for constant steadyoutput controllability of the system is that the system be stable. The concept of stability is very important in systems theory. Remember that a system is stable if and only if rank s0In−A B C0=rank sIn−A B C0,∀s0∈R+. Proposition 3.2 ([5]). A necessary and sufficient condition for constant steady-state output controllability of a stable system is rank A B C0=n+min {m, p}(7) No all systems are stable but some times, it is possible to stabilize them by means a feedback or/and an output injection, concretely, it can be said that a system as (5) is stabilizable if and only if there exist a feedback F∈ Mm×n(R)or/and output injection J∈Mn×p(R)such that the system close loop system ˙x(t) = (A+BF +JC)x(t) + Bu(t) y(t) = Cx(t)(8) is stable. Now, it can be analyzed the conditions for constant steady-state output controllability of a stabilizable system, having the following result. Proposition 3.3. A necessary and sufficient condition for constant steady-state output controllability of a stabilizable system is rank A B C0=n+min {m, p} Proof. rank A+BF +JC B C0= rank InJ 0IpA B C0In0 F Im= rank A B C0 Then rank A B C0=n+min {m, p} if and only if rank A+BF +JC B C0=n+min {m, p} 4 Output controllability vs. Steady-output controllability In order to demonstrate that both output controllability and steady output controllability are not equivalent concepts, two different examples are developed. Example 4.1. Let            ˙x1 ˙x2=0 1 0 0x1 x2+0 1u y=0 1x1 x2 rank CB CAB=rank 1 0=1=p Then the system is output-controllable. rank A B C0=rank   010 001 010 = 2 < n+min (m, p)=3 Therefore, the system is not steady-output controllable. Example 4.2. Let            ˙x1 ˙x2=0 0 1 0x1 x2+0 1u y1 y2=1 0 0 1x1 x2 rank CB CAB=rank 0 0 1 0= 1 < p = 2 Then the system is not output-controllable. The system is stable because rank     s0 0 −1s1 1 0 0 0 1 0    = 3 ∀s∈C and rank A B C0=rank     000 101 100 010    =3=n+min (m, p) Then, the system is steady-output controllable. 5 Modeling of FSWT The global analyzed system is a wind power generator connected directly to the grid. The controllability condition for the system described in this section can be found in [2]. The linear system is defined by means of the squirrel cage induction generator differential equations. The differential equations of the generator are time dependant. Its inputs are the voltage of the grid. The outputs are the active and reactive power delivered by the wind power generator. Supposing the system to be in steady state. This hypothesi implies constant slip. Therefore, the system can be described as: d dt     ∆isq ∆isd ∆irq ∆ird     | {z } ˙ X = −1 LsLr−M2α1α2α3α4 −α2α1−α4α3 α5α6α7α8 −α6α5−α8α7 | {z } A     ∆isq ∆isd ∆irq ∆ird     | {z } X +1 LsLr−M2 Lr0 0Lr −M0 0−M! | {z } B ∆vsq ∆vsd | {z } U (9) where α1=Lrrs α2=M2˙ θr+ (LsLr−M2)˙ θ α3=−Mrr α4=MLr˙ θr α5=−Mrs α6=−MLs˙ θr α7=Lsrr α8= (LsLr−M2)˙ θ−LsLr ˙ θr The αiparameters have constant value. They are dependant of the machine parameters such as stator and rotor impedance. Moreover ∆indicates a little variation of the selected operating point. 5.1 Linearizing the System The desired output signals, both active and reactive power are nonlinear functions, described as: Qs=3 2(vsdisq −vsqisd) Ps=3 2(vsdisd +vsqisq)(10) Then, it is necessary linearize these equations to obtain the linear system of the outputs. Hence, applying Taylor’s approximation around the steady state operating point to these equations. Qss =3 2((vsd0isq0−vsq0isd0) | {z } Qss0 +(vsd0∆isq −vsq0∆isd +isq0∆vsd −isd0∆vsq)) Pss =3 2((vsd0isd0+vsq0isq0) | {z } Pss0 +(vsd0∆isd +vsq0∆isq +isd0∆vsd +isq0∆vsq)) (11) where the values with the 0-subscript are the constant values corresponding to the steady state operating point. To simplify the calculations, it is used to linearize the system a small variation in the power values. ∆Qss =Qss −Qss0 ∆Pss =Pss −Pss0 (12) Then, the output system described as Y = CX + DU can be written as follows: ∆Qss ∆Pss =vsd0−vsq00 0 vsq0vsd00 0    ∆isq ∆isd ∆irq ∆ird     +−isd0isq0 isq0isd0∆vsq ∆vsd  (13) 6 Output controllability and steady output controllability of FSWT In the following section, it is studied output controllability and steady-output controllability of FSWT. Applying the theorem 2.1 in the linearized system, it can be computed rank oC. Let S∈Gl(n;R)be such that C1=SC =I20it is easy to prove that rank oC =rank C1B C1AB C1A2B C1A3B D= 2. Therefore, the system is output controllable. With respect steady-output controllability, it is computed rank (A B C0)in order to apply proposition 3.3. From the parameters of the generator, it can be guarantied M26=LrLs, thus the matrix Bhas also full rank. Hence, rank A B C0= 6 = n+min {m, p} and the system is steady-output controllable. 7 Conclusion This paper has presented the concepts of output controllability and steady output controllability. Moreover, by means of two different examples have been demonstrate the non equivalence of both concepts. Also, a linear system has been calculated from the nonlinear equations of the squirrel cage induction generator. Output controllability and steady output controllability have been demonstrated using the A,B,C matrices. Moreover, the demonstration is made with a generic system. Therefore, it can be ensured not only for an example. Due to the output controllability condition, it can be concluded that any output can be reached regulating the voltage inputs. On the other hand, steady output controllability condition can ensure the output controllability on a long term. References C.T. Chen, “Introduction to Linear System Theory”. Holt, Rinehart and Winston Inc, New York, (1970). J.L. Dom´ ınguez-Garc´ ıa (2010), “ Modeling and control of squirrel cage induction generator with full power converter applied to windmills”. Master Thesis, Oulu Katsuhiko Ogata Modern Control Engineering (3rd ed.), Ed. Prentice-Hall, NJ, (1997). J. Klamka, Controllability of dynamical systems-a survey. Archives of Control Sciences 2, pp. 281-307, (1993). D. Qiu, Q. Wang, Y. Zhou, Steady-state output controllability and output controllability of linear systems, Computational Intelligence and Industrial Applications, IEEExplore (2009). On page(s): 147 - 150