Automatically determining control parameters for a voltage regulator of a synchronous machine

Abstract

A synchronous machine includes a stator with stator windings connected with stator terminals to an electrical grid and a rotor with rotor windings rotatable mounted in the stator, wherein a voltage regulator of the synchronous machine is adapted for outputting an excitation signal to adjust a current in the rotor windings for controlling the synchronous machine. A method for determining control parameters for the voltage regulator includes (i) receiving a first time series of values of the excitation signal and a second time series of measurement values of the terminal voltage in the stator terminals, (ii) determining coefficients of a system transfer function of the synchronous machine, and (iii) determining the control parameters for the voltage regulator from the coefficients of the system transfer function.

Claims

1. A method for determining control parameters for a voltage regulator of a synchronous machine, the synchronous machine comprising a stator with stator windings connected with stator terminals to an electrical grid, and a rotor with rotor windings rotatably mounted in the stator, wherein the voltage regulator is adapted for outputting an excitation signal to adjust a current in the rotor windings for controlling the synchronous machine, the method comprising: receiving a first time series of values of the excitation signal and a second time series of measurement values of the terminal voltage in the stator terminals, wherein the first time series and the second time series are acquired over a time interval; determining coefficients of a system transfer function of the synchronous machine, wherein the system transfer function is a rational function, wherein the coefficients of the system transfer function are determined recursively with a regression analysis with instrumental variables based on the first time series as system input and the second time series as system output; and determining the control parameters for the voltage regulator from the coefficients of the system transfer function by comparing a closed loop transfer function formed of a controller transfer function of the voltage regulator and the system transfer function with a desired closed loop transfer function.

2. The method of claim 1, wherein in the regression analysis with instrumental variables, the coefficients of the system transfer function are recursively determined by: calculating an actual step excitation signal by filtering the previous step excitation signal with a denominator of the system transfer function with actual step coefficients; calculating an actual step terminal voltage by filtering a previous step terminal voltage with the denominator of the system transfer function with the actual step coefficients; calculating an actual step response by filtering the first time series of measurement values of the excitation signal with the system transfer function with actual step coefficients; and determining next step coefficients by minimizing a difference between a n.sup.th derivative of the actual step terminal voltage and a product of a coefficient vector and an instrumental variable vector, wherein n is the order of denominator of the system transfer function, and wherein the coefficient vector is formed of the next step coefficients, and the instrumental variable vector is formed from derivatives of the actual step response and the actual step excitation signal.

3. The method of claim 1, further comprising: forming an initial step excitation signal by filtering the first time series of measurement values of the excitation signal with a high pass filter; and forming an initial step terminal voltage by filtering the second time series of measurement values of the terminal voltage with the high pass filter.

4. The method of claim 1, further comprising: determining initial step coefficients for the system transfer function by minimizing a n.sup.th derivative of an initial step terminal voltage and a product of a coefficient vector and a regression vector, wherein the coefficient vector is formed of the initial step coefficients, and the regression vector is formed from derivatives of the initial step terminal voltage and an initial step excitation signal.

5. The method of claim 1, wherein the closed loop transfer function is determined from a product of the controller transfer function and the system transfer function; and/or wherein the desired closed loop transfer function is a second order transfer function with a desired settling time and/or a desired overshoot.

6. The method of claim 1, wherein the controller transfer function is a transfer function of a PID controller; and wherein the control parameters set factors for a proportional controller part, an integral control part, and a differential controller part.

7. The method of claim 1, wherein a desired steady state error is set by comparing the closed loop transfer function and the controller transfer function.

8. The method of claim 1, wherein the controller transfer function is a transfer function of a lead-lag controller.

9. The method of claim 1, further comprising: generating a voltage to be applied to the rotor windings by generating the first time series of values of the excitation signal and applying the first time series to the synchronous machine; and measuring the second time series of the terminal voltage.

10. The method of claim 1, wherein the control parameters are determined during a commissioning of the synchronous machine; and/or wherein the control parameters are determined periodically during a continuous operation of the synchronous machine.

11. A method for controlling a synchronous machine, the method comprising: determining control parameters according to the method of claim 1; applying the determined control parameters to the voltage regulator of the synchronous machine; and controlling the synchronous machine with the voltage regulator.

12. A computer program for determining control parameters for a voltage regulator, wherein, when being executed on a processor, the computer program is adapted for performing the method of claim 1.

13. A non-transitory computer readable medium for determining control parameters for a voltage regulator, wherein the computer program according to claim 12 is stored on the non-transitory computer readable medium.

14. A controller of a voltage regulator of an synchronous machine, wherein the controller is adapted for performing the method of claim 1.

15. A synchronous machine system, comprising: a synchronous machine with a stator having stator windings connected via terminals to an electrical grid, and a rotor having rotor windings, wherein the rotor is rotatably mounted in the stator; and a voltage regulator for supplying a voltage to the rotor windings; wherein the voltage regulator comprises an electrical converter, wherein the voltage regulator is adapted for generating the voltage for the rotor windings; and wherein the voltage regulator comprises a controller for controlling the converter and for performing the method of claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The subject matter of the disclosure will be explained in more detail in the following text with reference to exemplary embodiments which are illustrated in the attached drawings.

(2) FIG. 1 schematically shows a synchronous machine system according to an embodiment of the disclosure.

(3) FIG. 2 shows a diagram with a response function used for controlling the system of FIG. 1.

(4) FIG. 3 schematically shows a block diagram of parts of the system of FIG. 1 including a controller according to an embodiment of the disclosure.

(5) FIG. 4 shows a flow diagram for a method for determining control parameters for a voltage regulator according to an embodiment of the disclosure.

(6) The reference symbols used in the drawings, and their meanings, are listed in summary form in the list of reference symbols. In principle, identical parts are provided with the same reference symbols in the figures.

DETAILED DESCRIPTION OF THE DISCLOSURE

(7) FIG. 1 shows a synchronous machine system 10 with a synchronous machine 12 and a voltage regulator 14. The synchronous machine 12, such as a synchronous generator and/or synchronous motor, is connected to an electrical grid 16 via its stator windings 18 and in particular the stator terminal 20. The stator windings are mounted to the stator 22 of the synchronous machine 12 and a rotor 24 of the synchronous machine 12 is rotatable mounted in the stator 20 and carries rotor windings 26, which are supplied by the voltage regulator 14.

(8) As shown in FIG. 1, the rotor windings 26 may be supplied by an exciter system 28, which includes exciter windings 30 rotating with the rotor 24 and supplied by static voltage regulator windings 32 electrically connected to the voltage regulator 14. A voltage from the voltage regulator 14 induces a voltage in the exciter windings 30, which is rectified with a exciter rectifier 34 and supplied to the rotor windings 26.

(9) The voltage regulator 14 includes a converter 36, which is supplied via a transformer 38 connected to the electrical grid 16. A controller 40, which measures a terminal voltage y and a terminal current i in the terminals 20 of the synchronous machine 12 controls the converter 36.

(10) For example, a synchronous generator 12 may contribute to an electrical power in the electrical grid 16 by transforming mechanical energy to electrical energy. The mechanical energy may be supplied by steam turbines, gas turbines or water turbines. The turbines may power the rotor 24 of the synchronous generator 12, which holds a magnetic field, which is generated by a current that flows through the rotor winding 26, such as a coil of wires attached to the rotor 24. The rotational motion of the magnetic field on the rotor 24 relative to the stator windings 18, such as coils of wires that are mounted on the stator 22, induces a voltage in the stator windings 18. Usually, three stator windings 18 are geometrically mounted on the stator 22 so that under steady state conditions the induced voltages y measured at the terminals 20 of the stator windings 18 result in three sinusoidal waveforms with identical amplitude and frequency, phase-shifted by 120 degrees. The frequency and the amplitude of the terminal voltages y is mainly influenced by the angular velocity of the rotor 24 and the intensity of the field current, respectively.

(11) A stable operation of the electrical grid 16 may require that the magnitude of the (usually three-phase) terminal voltage y of the synchronous generator 12 is almost constant and deviates only little from a given desired value. However, the terminal voltage y of a grid-connected generator 12 may non-locally depend on all other entities present in the electrical grid 16, such as electrical loads, renewable power sources, other generators, and may deviate substantially from its desired waveform. Hence, it may be necessary to regulate the magnitude and frequency of the terminal voltage y to a desired values by adequately adjusting the field current and rotor velocity, respectively. Often, the current in the rotor windings 26 is provided by an exciter system 28, which may be seen as second generator 30, 32. Since the windings 30 of the exciter system 28 may be connected through the rectifier 34 to the rotor windings 26 of the synchronous generator 12, the magnitude of the terminal voltage y of the synchronous generator 12 may be regulated through the manipulation of field current of the exciter system 28.

(12) The (automatic) voltage regulator (AVR) 14 is used to regulate the actual magnitude of the terminal voltage y(t) of the synchronous machine 12. In general, its output is either connected directly to the rotor windings 26 of the synchronous machine 12, or to the windings 32 of the exciter system 28. The output signal u.sub.avr(t) of the voltage regulator 14 may be determined based on the error e(t)=r(t)−y(t) between the desired magnitude of the terminal voltage r(t) and the measured magnitude of the terminal voltage y(t).

(13) For example, in the case of a PID controller, the output signal may include three terms

(14) $u_{\mathrm{avr}}\left(t\right)=k_{p}e\left(t\right)+k_i{\int }_0^{t}e\left(s\right)\mathrm{ds}+k_d\frac{d}{\mathrm{dt}}e\left(t\right)$

(15) The output of the voltage regulator 14 may be formed through a linear combination of the error, the integral of the error and derivative of the error. The constant coefficients k.sub.p, k.sub.i and k.sub.d of the linear combination may be called controller parameters or controller gains. The control parameters influence the performance of a closed-loop formed by the dynamical system including the voltage regulator 14, the synchronous machine 12 possibly including an exciter system 28.

(16) FIG. 2 shows a diagram with an example of the actual magnitude of the terminal voltage y.sub.m(t) during a step from an initial desired value r.sub.i to a final desired value r.sub.f under the control of the controller 40, which is usually called step response.

(17) Furthermore, FIG. 2 shows performance indices for the step response. In general, it may be desirable that the closed-loop reacts quickly to changes in the reference signal so that the actual magnitude of the terminal voltage y(t) approaches the new desired value r.sub.f as fast as possible. The rise time T.sub.r, the settling time T.sub.s, and the overshoot OS associated with a step response may be used as controller performance indices. For example, the rise time T.sub.r may refer to the time interval in that the magnitude of the terminal voltage y(t) requires to increase from 10% to 90% of the reference step between the initial desired value r.sub.i and the final desired value r.sub.f. The settling time T.sub.s may refer to the time interval from the moment when the reference step occurs until y(t) stays within the +2% of r(t). The overshoot OS may refer to the maximal excess of y(t) over the final desired value r.sub.f and/or may be stated in %, such as OS=.sub.t>0.sup.max(y(t)−r(t))/r(t).

(18) The form of the curve of terminal voltage y(t) with the settling time T.sub.s and the overshoot OS is typical for a second order system. Below, it will be assumed that the closed loop composed of the controller 40 and the synchronous machine 12 is a second order system for setting the control parameters of the controller 30 to achieve specific values of the settling time T.sub.s and the overshoot OS.

(19) FIG. 3 shows a block diagram with controller parts 42, 44, 44, 48 of the controller 40 and a block 50 illustrating the parts of the system 10, such as the synchronous machine 12 and the exciter system 28 reacting on the exciter signal u(t) of the controller 40. FIG. 4 shows a flow diagram for a method that may be performed by the controller parts 42, 44, 44, 48.

(20) In step S10, the system including synchronous machine 12 and optionally the exciter system 28, is excited over a certain time interval [0, T] and two time series of signals, the exciter signal u.sub.c, such as a voltage applied to the windings 32 or 26, and the magnitude of the terminal voltage y.sub.m, may be recorded with a certain sampling time h.

(21) For example, the excitation signal u may be generated by a signal generator 42 and/or may be either a step-signal or a pseudo-random binary signal. The signal generator 42 may generate a voltage to be applied to the rotor windings by generating the first time series of values of the excitation signal u and applying them to the synchronous machine 12.

(22) However, it also may be that the excitation signal u.sub.c is generated by the controller 48, i.e. during operation of the synchronous machine 12.

(23) The second time series of may be determined by measuring the terminal voltage y.sub.m.

(24) The first time series of values of the excitation signal u and a second time series of measurement values of the terminal voltage y may then be received in the system identification block 40.

(25) In step S12, the system identificator 44 determines coefficients 52 of a system transfer function G(s) of the system indicated by block 50. The system trans transfer function G(s) is a rational function in the Laplace variable s.

(26) $G\left(s\right)=\frac{b_0{s}^m+b_1{s}^{m-1}+.Math.+b_m}{s^n+a_1{s}^{n-1}+.Math.+a_n},n\ge m$

(27) In general, the coefficients b.sub.0, . . . , b.sub.mb.sub.0, . . . , b.sub.m, a.sub.1, . . . , a.sub.n (indicated as reference numeral 52 in FIGS. 3 and 4) of the system transfer function G(s) are determined recursively with a regression analysis with instrumental variables based on the first time series as system input and the second time series as system output, as will be explained in the following.

(28) Depending on the particular application the model structure (n, m), i.e., the order of the numerator m and the order of the denominator n, may vary. In a typical application, in which the windings 26 of the synchronous machine 12 is energized by an exciter system 28 and the small signal behaviour of the terminal voltage y while the synchronous machine 12 is disconnected from the electrical grid 16, is of interest, the structure may be fixed to (2,1). By increasing the denominator order, it also may be possible to take the dynamics of a sensor into account in order to obtain a better fitting model. Similarly, (n, m) may be be adapted to different situations, e.g. when the winding 26 of the synchronous machine 12 is energized by a static excitation system.

(29) Simply Refined Instrumental Variable Method

(30) The simply refined instrumental variable (SRIV) approach may be used to compute the coefficients 52, b.sub.0, . . . , b.sub.m, a.sub.1, . . . , a.sub.n of the system transfer function G(s) to model the small-signal behaviour of the system 50 including the synchronous machine 12 and possibly the exciter system 28.

(31) The basis of the computation is the data set

(32) (u, y)

(33) where u and y are the discrete-time signals that map from a discrete interval {1,2, . . . , T} to the real numbers. Here, y is the measured magnitude of the terminal voltage and u is the voltage of the winding 26, 32 either of the synchronous machine 12 or of the exciter system 28. The constant h is the sampling time with which the signals are recorded. The data can be collected either by exciting the system by a reference step or by applying a pseudo-random binary signal to the field winding voltage.

(34) Subsequently, we refer to the coefficients 52, by coefficient vector
θ=(a.sub.1, . . . , a.sub.n,b.sub.0, . . . , b.sub.m).sup.T

(35) and use u.sub.c and y.sub.c to denote the continuous-time signals obtained from a zero-order hold interpolation of the signals u and y, respectively. The parameters θ are computed iteratively.

(36) Initial Parameter Computation

(37) Note that in the following all signals have been Laplace transformed. Initially, the signals u.sub.c and y.sub.c are filtered with a filter with cut-off frequency λ=10/(2πh)

(38) $\frac{{\lambda }^{n+1}}{{\left(s+\lambda \right)}^{n+1}}.$

(39) Specifically, the following filter responses are computed for i∈{1, . . . , n}, j∈{0, . . . , m} with y.sub.f=y.sub.c and u.sub.f=u.sub.c

(40) $u_f^{\left(j\right)}=s^j.Math.\frac{{\lambda }^{n+1}}{{\left(s+\lambda \right)}^{n+1}}{u}_f{y}_f^{\left(i\right)}=s^i.Math.\frac{{\lambda }^{n+1}}{{\left(s+\lambda \right)}^{n+1}}{y}_f$

(41) An initial step excitation signal u.sub.f may be formed by filtering the first time series of measurement values of the excitation signal u with a high pass filter and/or an initial step terminal voltage yf may be formed by filtering the second time series of measurement values of the terminal voltage y with the high pass filter.

(42) If y.sub.f.sup.(i) were the derivative of y.sub.f and y.sub.f was the response of G(s) to u.sub.f, then the following equation would be satisfied for all t=∈[0, h(T−1)]
y.sub.f.sup.(n)(t)=−a.sub.1y.sub.f.sup.(n−1)(t)− . . . −a.sub.ny.sub.f(t)+b.sub.0u.sub.f.sup.(m)+ . . . +b.sub.mu.sub.f(t)y.sub.f.sup.(n)(t)

(43) Using the regression vector
φ.sup.T(t)=(−y.sub.f.sup.(n−1)(t), . . . , −y.sub.f(t),u.sub.f.sup.(m), . . . u.sub.f(t))

(44) It follows that y.sub.f.sup.(n)(t)=φ.sup.T(t)θ. The initial parameter estimate follows by minimizing the deviation of y.sub.f.sup.(n)(t) from φ.sup.T(t)θ summarized over the sampling times t.sub.i=(i−1)h with i∈{1, . . . , T}. The objective function follows by
Σ.sub.i=1.sup.T|y.sub.f.sup.(n)(t.sub.i)−φ.sup.T(t.sub.i)θ|.sup.2

(45) In general, initial step coefficients θ.sub.0 for the system transfer function G(s) may be obtained by minimizing the n.sup.th derivative of the initial step terminal voltage y.sub.f and a product of the coefficient vector θ.sub.0 and the regression vector φ.sup.T, wherein the coefficient vector is formed of the initial step coefficients and regression vector is formed from derivatives of the initial step terminal voltage y and an initial step excitation signal u.

(46) The initial step coefficients θ.sub.0 may be explicitly computed by
θ.sub.0=(Σ.sub.i=1.sup.Tφ(t.sub.i)φ.sup.T(t.sub.i)).sup.−1Σ.sub.i=1.sup.Tφ(t.sub.i)y.sub.f.sup.(n)(t.sub.i).
Parameter Update

(47) In the parameter update procedure, the initialization procedure is essentially repeated and the coefficients θ are determined reclusively. However, there are two significant differences. First, the filter to compute the higher-order derivatives is given by the denominator of the system transfer function

(48) $G\left(s\right)=\frac{B\left(s,{\theta }_k\right)}{A\left(s,{\theta }_k\right)}=\frac{b_0^k{s}^m+b_1^k{s}^{m-1}+.Math.+b_m}{s^n+a_1^k{s}^{n-1}+.Math.+a_n^k}.$

(49) Here,
θ.sub.k=(a.sub.1.sup.k, . . . a.sub.n.sup.k,b.sub.0.sup.k, . . . , b.sub.m.sup.k).sup.T

(50) are the coefficients 52 in the actual iteration k. Now, the filter responses are computed by

(51) $u_f^{\left(j\right)}=\frac{s^j}{A\left(s,{\theta }_k\right)}{u}_f{y}_f^{\left(i\right)}=\frac{s^i}{A(s,{\theta }_k}{y}_f.$

(52) The actual step excitation signal u.sub.f is calculated by filtering the previous step excitation signal u.sub.f with the denominator A of the system transfer function G(s) with actual step coefficients θ.sub.k. Analogously, the actual step terminal voltage y.sub.f is calculated by by filtering the previous step terminal voltage y.sub.f with the denominator A with the actual step coefficients θ.sub.k.

(53) The second difference stems from the use of instrumental variables in the computation of the coefficient update. To this end, the system transfer function G(s) is used to compute a noise-free response

(54) $x\left(t\right)=\frac{B\left(s,{\theta }_k\right)}{A\left(s,{\theta }_k\right)}{u}_c\left(t\right).$

(55) The actual step response xis calculated ny by filtering the first time series of measurement values u.sub.c of the excitation signal with the system transfer function G(s) with actual step coefficients θ.sub.k.

(56) Subsequently, the derivatives i∈{1, . . . , n−1} of x are computed by

(57) $x_f^{\left(i\right)}=\frac{s^i}{A\left(s,{\theta }_k\right)}x.$

(58) and the instrumental variable vector
ζ.sup.T(t)=(−x.sub.f.sup.(n−1)(t), . . . , −x.sub.f(t),u.sub.f.sup.(m)(t), . . . , u.sub.f(t))

(59) is defined.

(60) The next step coefficients θ.sub.k+1 may be determined by minimizing a difference between a n.sup.th derivative of the actual step terminal voltage y.sub.f and a product of the coefficient vector θ.sub.k and the instrumental variable vector ζ.sup.T. The instrumental variable vector ζ.sub.T is formed from derivatives of the actual step response x and the actual step excitation signal u.sub.f.

(61) In particular, the next step coefficients θ.sub.k+1 can be computed by
θ.sub.k+1=(Σ.sub.i=1.sup.Tζ(t.sub.i)φ.sup.T(t.sub.i)).sup.−1Σ.sub.i=1.sup.Tζ(t.sub.i)y.sub.f.sup.(n−1)(t.sub.i).

(62) The iteration may be repeated until |θ.sub.k+1−θ.sub.k| is sufficiently small and/or smaller as a threshold, e.g. less than 0.01, or the number of iterations exceed a specific number, such as 10.

(63) Note that the method is tuning free, i.e., given the data set (u, y), there are essentially no parameters that influence the outcome of the computation.

(64) Moreover, the filter responses y.sub.f.sup.(i), i∈{1, . . . n,} are computed at once for all i∈{1, . . . , n} by solving one initial value problem. A similar comment applies to u.sub.f.sup.(j) and x.sub.f.sup.(i). Hence, to u.sub.f.sup.(j) and x.sub.f.sup.(i). Hence, to compute the filter responses, two initial value problems may have to be solved in the initial parameter computation and three initial value problems may have to be solved in each coefficient update iteration. For example, a fourth order Runge-Kutta scheme with fixed step size may be used to solve the initial value problems.

(65) Computation of Control Parameters

(66) In step S14, the parameter computer 46 uses the coefficients 52 of the model G(s) together with further specifications 56 used to determine the control parameters 54.

(67) In general, there exist a variety of methods to design a controller 48 with a controller transfer function C(s). In the following it is assumed that the system transfer model function has a structure with n=2 and m=1 and is of the form

(68) $G\left(s\right)=\frac{K_p}{\left(T_{1}s+1\right)\left(T_{2}s+1\right)}.$

(69) It may happen that the identified transfer function G(s) has two complex conjugate poles and it is not possible to express G(s) with two real poles. In this case, we modify the coefficient a.sub.1 of the denominator polynomial of G(s) by â.sub.1=√{square root over (1.5.Math.4.Math.a.sub.2)} so that the modified transfer function Ĝ(s)=b.sub.0/(s.sup.2+â.sub.1s+a.sub.2) is guaranteed to be representable with two real poles.

(70) In particular, as explained below, a closed loop transfer function formed of a controller transfer function C(s) of the controller 48 and the system transfer function G(s) is compared with a desired closed loop transfer function, such as shown in FIG. 2, to determine the controller parameters 54.

(71) In the following two embodiments for control parameters computation for two different controllers 48 are described. The first controller is a so-called series/cascaded PID controller. The second controller is a double lead-lag controller.

(72) PID Controller Parameter Computation

(73) The specifications 56 to compute the parameters of the PID controller are a desired settling time T.sub.s and a desired overshoot OS (see FIG. 2).

(74) The transfer function of the PID controller 48 is given by

(75) 0$C\left(s\right)=V_p\frac{1+{\mathrm{sT}}_a}{{\mathrm{sT}}_a}.Math.\frac{K_b+\left(1+K_b\right)T_{b}s}{K_b+T_{b}s}.$

(76) Note that the the control parameters T.sub.a, T.sub.b and K.sub.b may be used to set the factors k.sub.p, k.sub.i and k.sub.d for a proportional controller part, an integral control part and a differential controller part as described above.

(77) The control parameters 54, T.sub.a, T.sub.b and K.sub.b are determined so that the open loop transfer function, which is the product of the controller transfer function C(s) and the system transfer function G(s), results in

(78) $C\left(s\right)G\left(s\right)=\frac{V_p{K}_p}{{\mathrm{sT}}_1\left(s\alpha T_2+1\right)}$

(79) where α∈]0,1[ is a parameter to be determined in a second step. Explicitly, the control parameters 54 are given by
T.sub.a=T.sub.1,T.sub.b=T.sub.2(1−α), and K.sub.b=(1−α)/α.

(80) From the open loop transfer function, the closed-loop transfer function follows by

(81) $\frac{V_p{K}_p}{s^2{T}_1{T}_2\alpha +{\mathrm{sT}}_1+V_p{K}_p}.$

(82) This closed-loop transfer function is compared with a second order transfer function (see FIG. 2), where the standard form of the second order transfer function may be provided by

(83) $\frac{{\omega }^2}{{\omega }^2+2\mathrm{\vartheta \omega }s+s^2}$

(84) for which the heuristic

(85) $\mathrm{\vartheta \omega }\approx \{\begin{array}{cc}4T_s,& \vartheta \in ]0,1[\\ 6T_s,& \vartheta \ge 1\end{array}$

(86) accurately relates the damping coefficient with the settling time T.sub.s. By comparing the coefficients of the closed-loop and the standard form the second order transfer function, the parameters

(87) $\alpha =\{\begin{array}{cc}{T}_s/\left(8T_2\right),& \vartheta \in ]0,1[\\ T_s/12T_2),& \vartheta \ge 1\end{array}\mathrm{and}{V}_p=\frac{T_1}{4{\vartheta }^2\alpha T_2}.Math.\frac{1}{K_p}.$

(88) cab be obtained. The damping coefficient may be derived from the overshoot specification OS. For OS>0, the damping coefficient follows by

(89) $\vartheta =\frac{\ln \left(\mathrm{OS}/100\right)}{\sqrt{{\pi }^2+{\ln }^2\left(\mathrm{OS}/100\right)}}$

(90) and =1 for OS=0. Note that the parameter a is restricted to the interval]0,1[. However, if the desired settling time T.sub.s is too large a might result in being greater than one. In this case, the desired settling time T.sub.s may be simply reduced so that α∈]0,1[.

(91) Lead-Lag Controller Parameter Computation

(92) The specifications 56 to compute the control parameters 54 of a lead-lag controller 48 are a desired settling time T.sub.s, a desired overshoot OS and a desired steady state error e.sub.ss.

(93) The controller transfer function C(s) of the controller 48 is given by

(94) $C\left(s\right)=K_R\frac{1+{\mathrm{sT}}_{C2}}{1+{\mathrm{sT}}_{B2}}.Math.\frac{1+{\mathrm{sT}}_{C1}}{1+{\mathrm{sT}}_{B1}}.$

(95) The numerator time constants of the controller are set to
T.sub.C1=T.sub.1,T.sub.C2=T.sub.2

(96) so that the open loop transfer function results in

(97) $C\left(s\right)G\left(s\right)=\frac{K_R{K}_p}{\left(1+{\mathrm{sT}}_{B1}\right)\left(1+{\mathrm{sT}}_{B2}\right)}.$

(98) Let T.sub.B1=T and T.sub.B2=cT, then the closed-loop follows by

(99) $\frac{K_R{K}_p}{K_R{K}_p+1+s\left(c\right)T+s^2{\mathrm{cT}}^2}$

(100) The gain K.sub.R is chosen, so that desired steady state error e.sub.ss is enforced in the closed-loop, i.e.,

(101) 0$\frac{K_R{K}_p}{K_R{K}_p+1}\le e_{\mathrm{ss}}/100.$

(102) The remaining parameters T.sub.B1 and T.sub.B2 (or equivalently c and T) may be chosen to enforce the desired settling time and overshoot in the transfer function

(103) $\frac{K_R{K}_p}{K_R{K}_p+s\left(1+c\right)T+s^2{\mathrm{cT}}^2}$

(104) which closely approximates the actual closed-loop transfer function. In a first step, the damping coefficient is derived from the overshoot specification like the one of the PID controller. By comparing the coefficients with the standard form of the second order transfer function

(105) $\frac{{\omega }^2}{{\omega }^2+2\mathrm{\vartheta \omega }s+s^2}$

(106) It results that

(107) $\mathrm{\vartheta \omega }=\frac{1+c}{\mathrm{cT}},{\omega }^2=\frac{K_R{K}_p}{{\mathrm{cT}}^2}$

(108) By solving the equations for c, it follows that

(109) $c=\frac{4K_R{K}_p{\vartheta }^2-1}{2}+\frac{{\left({\left(4K_R{K}_p{\vartheta }^2-1\right)}^2-4\right)}^{\frac{1}{2}}}{2}$

(110) In case that c would result in a complex number, the gain K.sub.R may be increased to ensure that c is real. The time constant T is determined by the settling time heuristic

(111) $\frac{1+c}{\mathrm{cT}}=\mathrm{\vartheta \omega }\approx \{\begin{array}{cc}4T_s,& \vartheta \in ]0,1[\\ 6T_s,& \vartheta \ge 1\end{array}$

(112) which results in

(113) $T_{B2}=\frac{1+c}{c}.Math.\{\begin{array}{cc}{T}_s/8,& \vartheta \in ]0,1[\\ T_s/12,& \vartheta \ge 1\end{array}$

(114) and
T.sub.B1=cT.sub.B2.
Control of Synchronous Machine

(115) The three-step procedure for steps S10, S12, S14 may be applied during the commissioning of the voltage regulator 14 to determine the controller parameters 54 and/or periodically to update the controller parameters 54, while the system 10 is in operation. In this case, the data in step S10 may be obtained by measuring the desired signals u, y while the system 10 is operating.

(116) After the steps S10, S12, S14, the control parameters 54 may be applied to the controller 48 of the voltage regulator 14.

(117) In step S16, the controller 48 then may regulate the terminal voltage y towards a possibly time dependent reference signal r(t). The controller 48 may receive the reference signal r(t) and may control the magnitude of the terminal voltage y.sub.m(t) towards this reference signal r(t).

(118) While the disclosure has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the disclosure is not limited to the disclosed embodiments. Other variations to the disclosed embodiments can be understood and effected by those skilled in the art and practising the claimed disclosure, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single processor or controller or other unit may fulfil the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims should not be construed as limiting the scope.

LIST OF REFERENCE SYMBOLS

(119) 10 synchronous machine system

(120) 12 synchronous machine

(121) 14 voltage regulator

(122) 16 electrical grid

(123) 18 stator windings

(124) 20 stator terminal

(125) 22 stator

(126) 24 rotor

(127) 26 rotor windings

(128) 28 exciter system

(129) 30 exciter windings

(130) 32 voltage regulator windings

(131) 34 exciter rectifier

(132) 36 converter

(133) 38 transformer

(134) 40 controller

(135) 42 signal generator

(136) 44 system identification

(137) 46 parameter computation

(138) 48 voltage controller

(139) 50 block representing system

(140) t time

(141) y, y(t) magnitude of terminal voltage

(142) T.sub.r rise time

(143) T.sub.s settling time

(144) OS overshoot

(145) G(s) system transfer function

(146) 52 system transfer model coefficients

(147) C(s) controller transfer function

(148) 54 control parameters

(149) u, u(t) excitation signal

(150) r(t) reference signal

(151) 56 specifications