METHOD AND SYSTEM OF SUBSYNCHRONOUS OSCILLATIONS AND INTERACTIONS DAMPING

Abstract

A method and system of subsynchronous oscillations and interactions damping integrated in in a rotor converter based on an adaptive state feedback controller with two spinning vectors, and a Kalman filter whose parameters are optimized by minimizing maximum sensitivity under a constraint of positive damping for a plurality of sensible scenarios is provided. The damping signal generated by the damping module is applied either to a power proportional integer controller or to a current proportional integer controller.

Claims

1. A system of subsynchronous oscillations and interactions damping comprising: a measuring means configured to measure an input power measurement; at least one damping module integrated in a rotor converter, the at least one damping module configured to generate an output damping signal from an input power measurement; at least one proportional integer controller integrated in the rotor converter, the output damping signal incorporated to an input of the at least one proportional integer controller; and a control means configured to apply an output of the at least one proportional integer controller to a machine rotor; wherein the at least one damping module further comprises a Kalman filter and a state feedback controller with a first spinning vector for subsynchronous modes, and a second spinning vector for supersynchronous modes.

2. The system according to claim 1, wherein the proportional integer controller in whose input the damping signal is incorporated is a power proportional integer controller.

3. The system according to claim 2 further comprising a subtraction module at the input of the power proportional integer controller which subtracts the output damping signal from a set-point value for an active power.

4. The system according to claim 2 further comprising a subtraction module at the input of the power proportional integer controller which subtracts the output damping signal from a set-point value for a reactive power.

5. The system according to claim 1, wherein the proportional integer controller in whose input the output damping signal is incorporated is a current proportional integer controller.

6. The system according to claim 5 further comprising a multiplier module which multiplies the output damping signal by a relation between an induction generator magnetizing inductance and a stator inductance; and a subtraction module at the input of the current proportional integer controller which subtracts the multiplied damping signal from the set-point d-axis rotor current components.

7. The system according to claim 5 wherein the set-point d-axis rotor current components is computed at a power proportional integer controller using as inputs the active power and set-point active power.

8. The system according to claim 5 wherein the set-point d-axis rotor current components is computed at a power proportional integer controller using as inputs the active power and set-point reactive power.

9. The system according to claim 1, wherein the input power measurement is selected between a wind turbine active power and a common coupling voltage.

10. A method of subsynchronous oscillations and interactions damping comprising: measuring an input power measurement; generating an output damping signal from an input power measurement; and incorporating the output damping signal to an input of the at least one proportional integer controller; applying an output of the at least one proportional integer controller to a machine rotor; wherein the step of generating the output damping signal further comprises applying a Kalman filter and a state feedback controller with a first spinning vector for subsynchronous modes, and a second spinning vector for supersynchronous modes.

11. The method according to claim 10 wherein variables of the first spinning vector comprise a first frequency, a first gain and a second gain; and variables of the second spinning vector comprise a second frequency, a third gain and a fourth gain.

12. The method according to claim 10, wherein the Kalman filter is applied through a discrete-time state-space model with a sampling time.

13. The method according to claim 10 further comprising optimizing the variables of the first spinning vector and the second spinning vector by minimizing maximum sensitivity under a constraint of positive damping for a plurality of wind turbine generation plant scenarios.

14. The method according to claim 13 wherein the plurality of wind turbine generation plant scenarios comprise variations of at least one parameter selected from wind speed, grid reactance, compensation factor and reactive power.

15. A computer program comprising computer program code means configured to perform the method according to claim 10, when the program is run on a computer, a digital signal processor, a field-programmable gate array, an application-specific integrated circuit, a micro-processor, a micro-controller, or any other form of programmable hardware.

Description

BRIEF DESCRIPTION

[0043] Some of the embodiments will be described in detail, with reference to the following figures, wherein like designations denote like members, wherein:

[0044] FIG. 1 depicts a first embodiment of the SSI damping system of the present invention wherein the damping signal is applied to the current PI controller;

[0045] FIG. 2 depicts a second embodiment of the SSI damping system of the present invention wherein the damping signal is applied to the power PI controller;

[0046] FIG. 3 depicts in further detail the components of the damping module, according to an exemplary embodiment of the present invention; and

[0047] FIG. 4 depicts a flow chart of the process of computing the Kalman filter settings, according to an exemplary embodiment of the present invention.

DETAILED DESCRIPTION

[0048] The matters defined in this detailed description are provided to assist in a comprehensive understanding of the present invention. Accordingly, those of ordinary skill in the art will recognize that variation changes and modifications of the embodiments described herein can be made without departing from the scope and spirit of the present invention. Also, description of well-known functions and elements are omitted for clarity and conciseness.

[0049] Note that in this text, the term “comprises” and its derivations (such as “comprising”, etc.) should not be understood in an excluding sense, that is, these terms should not be interpreted as excluding the possibility that what is described and defined may include further elements, steps, etc.

[0050] FIG. 1 shows a first exemplary embodiment of the system of the present invention, where a damping module 130 implements the steps of a first exemplary embodiment of the method and computer program of the present invention. The damping module 130 is a supplementary controller integrated in a rotor converter 100 which comprises at least a power proportional integer (PI) controller 110 and a current PI controller 120. Notice that in case the rotor converter 100 does not provide enough capabilities to implement the damping module 130, said damping module 130 may be implemented in any additional device in such as an external static synchronous compensator (STATCOM).

[0051] The input variables of the damping module 130 are power measurements 200 selected from wind turbine active power (P) or the common coupling voltage (Vpcc). Only one of these two measurements 200, wind turbine active power or common coupling voltage, is required in the damping module 130. The optimal measurement 200 may be selected by a previous mathematical analysis once the power system configuration is defined. The output of the damping module is a damping signal 210 that is subtracted from the desired or set-point value of the reactive power Q.sub.ref at subtraction module 140. The resulting damped signal 220 is introduced in the power PI controller 110 along with the reactive power supplied through the generator stator Q. The output is a set-point q-axis rotor current component iq.sub.ref in a reference frame moving in synchronism with the stator voltage space vector. The reference frame d-axis is aligned with the stator voltage space vector whereas the q-axis is in quadrature with the stator voltage space vector. In the event that stator flux is employed as a reference, the output is a set-point d-axis rotor current component.

[0052] The set-points for d-axis and q-axis rotor current components (id.sub.ref and iq.sub.ref) are introduced in the current PI controller 120 along with the measured d-axis and q-axis rotor current components (id and iq), generating output signal 230. The output signal 230 comprises the d-axis and q-axis components of the voltage space vector applied by the rotor converter to the machine rotor. These two signals are used in the pulse-width modulation (PWM) generator of the rotor converter.

[0053] Notice that the present invention can be implemented with a single damping module 130 which can be applied either to the set-point active power P.sub.ref or to the set-point reactive power Q.sub.ref supplied through the generator stator. In order to determine which of the two options provides a more optimal SSI damping, a previous mathematical analysis may be carried out.

[0054] FIG. 2 shows a second exemplary embodiment of the system of the present invention, where a damping module 130 implements the steps of a second exemplary embodiment of the method and computer program of the present invention. In this case, the damping module is applied to the input of the current PI controller 120, as a subtraction of the set-point d-axis rotor current component id.sub.ref. That is, inputs of the first power PI controller 110 are the active power P computed from measured rotor speed and estimated motor torque and set-point reactive power Q.sub.ref; inputs of the second power PI controller 110 are reactive power Q and set-point reactive power Q.sub.ref; and inputs of the current PI controllers 120 are damped signal 220 and the set-point d-axis rotor current component id.sub.ref. Notice that, before subtracting from the set-point d-axis rotor current component id.sub.ref to generate the damped signal 220, the damping signal 210 is multiplied at multiplier 150—or at the damping module itself—by the relation between the induction generator (IG) magnetizing inductance (Lm) and the IG stator inductance (Ls). The stator inductance (Ls) is computed as the sum of the magnetizing inductance (Lm) and the stator leakage inductance.

[0055] As in the previous case, notice that the present invention can be implemented with a single damping module 130 which can be applied either to the set-point active power P.sub.ref or to the set-point reactive power Q.sub.ref supplied through the generator stator. In order to determine which of the two options provides a more optimal SSI damping, a previous mathematical analysis may be carried out.

[0056] FIG. 3 shows in greater detail the components of an exemplary embodiment of the damping module 130, namely a state feedback controller 132 and a Kalman filter 131. The Kalman filter 131 estimates the state variables of a state-space signal model, consisting of two spinning space vectors rotating at constant frequencies f.sub.1 and f.sub.2 and a direct current (DC) value. Typically, one spinning vector rotates clockwise and the other one rotates anticlockwise and, therefore, frequencies f.sub.1 and f.sub.2 have opposite signs.

[0057] The state variables of this model are the real and imaginary parts of each space vector and the DC value. If y(t) represents the measured signal, it can be estimated, according to the proposed model, as:

[00001]$\begin{array}{c}y\left(t\right)=A_o+A_1\mathrm{real}\left(e^{j2\pi f_{1}t}\right)+\\ =A_2\mathrm{real}\left(e^{j2\pi f_{2}t}\right)=\\ =A_o+A_1\cos \left(2\pi f_{1}t\right)+A_2\cos \left(2\pi f_{2}t\right)\end{array}$

where A.sub.o is the dc component and A.sub.1 and A.sub.2 are the magnitudes of the spinning vectors used to reprent the signal y(t). Defining the state variables as:

x.sub.1(t)=A.sub.1 cos(2πf.sub.1t),x.sub.2(t)=A.sub.1 sin(2πf.sub.1t),

x.sub.3(t)=A.sub.2 cos(2πf.sub.2t),x.sub.4(t)=A.sub.2 sin(2πf.sub.2t),

x.sub.5(t)=A.sub.o

and, the state-space model is:

[00002]$\frac{d}{\mathrm{dt}}\stackrel{\stackrel{X\left(t\right)}{︷}}{\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\\ x_4\left(t\right)\\ x_5\left(t\right)\end{array}\right]}=\stackrel{\stackrel{A}{︷}}{\left[\begin{array}{ccccc}0& -2\pi f_1& 0& 0& 0\\ 2\pi f_1& 0& 0& 0& 0\\ 0& 0& 0& -2\pi f_2& 0\\ 0& 0& 2\pi f_2& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]}\stackrel{\stackrel{X\left(t\right)}{︷}}{\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\\ x_4\left(t\right)\\ x_5\left(t\right)\end{array}\right]}+\stackrel{\stackrel{W\left(t\right)}{︷}}{\left[\begin{array}{c}{w}_1\left(t\right)\\ w_2\left(t\right)\\ w_3\left(t\right)\\ w_4\left(t\right)\\ w_5\left(t\right)\end{array}\right]}$

where X(t) is the state vector, A is the state matrix and W(t) is the process noise vector.

[0058] The estimated output ŷ(t) is represented by the sum of the real parts of the spinning vectors (x.sub.1 and x.sub.3) and the dc component (x.sub.5):

[00003]$\hat{y}\left(t\right)=\stackrel{\stackrel{C}{︷}}{\left[\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\right]}\stackrel{\stackrel{X\left(t\right)}{︷}}{\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\\ x_4\left(t\right)\\ x_5\left(t\right)\end{array}\right]}+v\left(t\right)$

where C is the output matrix and v(t) is the measurement noise.

[0059] The variances of the process and measurement noises are previously defined by trial-and-error to achieve a satisfactory match between the measured and the estimated output of an example. The optimal state estimation, according to Kalman filter formulation, is calculated as follows:

[00004]$\frac{d\hat{X}\left(t\right)}{\mathrm{dt}}=A\hat{X}\left(t\right)+K_e\left(y\left(t\right)-\hat{y}\left(t\right)\right)=\left(A-K_{e}C\right)\hat{X}\left(t\right)+\mathrm{Gy}\left(t\right)$

where K.sub.e is the optimal Kalman gain computed from the noise variances and the model matrices. The estimated state vector ({circumflex over (X)}(t)) is obtained integrating the above differential equation.

[0060] Then, the state feedback controller 132 is applied using a linear combination of the real and imaginary parts of the model space vectors as the control variable u(t):

[00005]$u\left(t\right)=-K_c\hat{X}\left(t\right)=-\stackrel{\stackrel{K_c}{︷}}{\left[\begin{array}{ccccc}{k}_{c1}& k_{c2}& k_{c3}& k_{c4}& 0\end{array}\right]}\stackrel{\stackrel{\hat{X}\left(t\right)}{︷}}{\left[\begin{array}{c}{\hat{x}}_1\left(t\right)\\ {\hat{x}}_2\left(t\right)\\ {\hat{x}}_3\left(t\right)\\ {\hat{x}}_4\left(t\right)\\ {\hat{x}}_5\left(t\right)\end{array}\right]}$

[0061] The DC value is not used in the control variable computation to guarantee a controller with DC gain equal to 0. Frequencies f.sub.1 and f.sub.2 and gains k.sub.c1, k.sub.c2, k.sub.c3 and k.sub.c4 applied to the state variables are the six design parameters of the damping module 130 stabilizer. The final state-space model of the damping module 130 module is expressed as follows:

[00006]$\frac{d\hat{X}\left(t\right)}{dt}=E\hat{X}\left(t\right)+Gy\left(t\right)u\left(t\right)=H\hat{X}\left(t\right)$

where E=A−K.sub.eC, G=K.sub.e and H=−K.sub.c. This model is single-input single-output (SISO), linear and time-invariant (LTI). Therefore, an equivalent fifth-order transfer function can be calculated to represent exactly this SISO and LTI model. The structure of this transfer function can be written as:

[00007]$\begin{array}{c}F\left(s\right)=\frac{U\left(s\right)}{y\left(s\right)}\\ =\frac{K\left(1+T_{1}s\right)}{\left(1+T_{2}s\right)}\frac{s\left(s^2+2{\rho }_3{w}_{n3}s+w_{n3}^2\right)}{\left(s^2+2{\rho }_1{w}_{n1}s+w_{n1}^2\right)\left(s^2+2{\rho }_2{w}_{n2}s+w_{n2}^2\right)}\end{array}$

where ρ.sub.1 and w.sub.n1 are, respectively, the damping factor and the natural frequency (rad/s) of the lower-frequency second-order component in the filter transfer function; ρ.sub.2 and w.sub.n2 are, respectively, the damping factor and the natural frequency (rad/s) of the upper-frequency second-order component in the filter transfer function; ρ.sub.3 and w.sub.n3 are, respectively, the damping factor and the natural frequency (rad/s) of the second-order zeroes in the filter transfer function; K is an additional gain of the filter transfer function; T.sub.1 is the lead time constant in the lead-lag first-order compensator; and T.sub.2 is the lag time constant in the lead-lag first-order compensator.

[0062] Notice that the damping module 130 is not applied as a transfer function because of the lower robustness of this kind of implementation. However, the transfer function is useful to understand the effect of the damping module 130 in the frequency domain. F(s) shows that the resulting damping module 130 is a double band-pass filter synchronized at two different frequencies, w.sub.n1 and w.sub.n2 in rad/s, together with a first-order lead-lag compensator to achieve an acceptable stability margin. The weaker the grid connected to the generator is, the greater the separation required between w.sub.n1 and w.sub.n2 is. Splitting the band-pass filter in two (around frequencies w.sub.n1 and w.sub.n2) is very convenient when dealing with weak grids.

[0063] The most robust real-time implementation is obtained by a discrete-time state-space model with a sampling time T.sub.s and the following formulation:

{circumflex over (X)}[k+1]=M{circumflex over (X)}[k]+Gy[k]

u[k]=H{circumflex over (X)}[k]

being M=A.sub.d−K.sub.eC and

[00008]$\begin{array}{c}{A}_d=e^{{\mathrm{AT}}_s}\\ =\left[\begin{array}{ccccc}\cos \left(2\pi f_1{T}_s\right)& -\sin \left(2\pi f_1{T}_s\right)& 0& 0& 0\\ \sin \left(2\pi f_1{T}_s\right)& \cos \left(2\pi f_1{T}_s\right)& 0& 0& 0\\ 0& 0& \cos \left(2\pi f_2{T}_s\right)& -\sin \left(2\pi f_2{T}_s\right)& 0\\ 0& 0& \sin \left(2\pi f_2{T}_s\right)& \cos \left(2\pi f_2{T}_s\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]\end{array}$

[0064] FIG. 4 presents an exemplary embodiment of the off-line tuning of the Kalman filter and state-feedback controller settings for robust operation. The sensible operations are first defined 310, then the set of plants is computed 320, an optimization process is undertaken 330 and, as a result, Kalman filter and state-feedback controller settings are defined 340. Kalman filter and state-feedback controller require six tuning parameters corresponding to the frequencies of the two spinning vectors (Kalman filter) and the four gains (state-feedback controller) applied to sin and cosine components of the two spinning vectors. These parameters are tuned for obtaining a robust solution for the whole operational scenarios implying different wind speed, reactive power level and short-circuit impedance. The stability criteria employed in the optimization are minimum damping and maximum sensitivity. The objective function is the minimization of the maximum sensitivity of all possible scenarios guaranteeing that the system is stable. This means that the minimum damping of the whole system (i.e. machine controllers, converters and grid) in all scenarios considered should be positive.

[0065] Tuning the damping module 130 involves an optimization carried out using a set of feasible plants. This set of plants is defined by linearization of a power system non-linear model at different operating points and critical parameter values. For example, different values of wind speed, grid reactance, compensation factor or reactive power are considered when defining the set of plants for the damping module 130 tuning purposes. The optimization parameters are frequencies f.sub.1 and f.sub.2 and gains k.sub.c1, k.sub.c2, k.sub.c3 and k.sub.c4 in the damping module 130, whereas the noise Kalman filter variances are set as constant values. The optimization cost function is the maximum sensitivity of the system with a chosen damping module 130. This optimization cost function is only considered valid if the stability of the power system is guaranteed, i.e. the minimum damping calculated by modal analysis must be greater than zero. This optimization cost function is a standard selection in robust control theory.

[0066] Although the present invention has been disclosed in the form of preferred embodiments and variations thereon, it will be understood that numerous additional modifications and variations could be made thereto without departing from the scope of the invention.

[0067] For the sake of clarity, it is to be understood that the use of “a” or “an” throughout this application does not exclude a plurality, and “comprising” does not exclude other steps or elements.