Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy
Conversion of State Feedback Controllers to the Standard AVR+PSS Form A. Elices, L. Rouco, Member, IEEE, H. Bourlès, Senior Member, IEEE, T. Margotin
Abstract—This paper presents a general method to convert State feedback controllers to damp power system electromechanical oscillations to the standard structure formed by an Automatic Voltage Regulator (AVR) and a Power System Stabilizer (PSS). The proposed conversion method is based on least squares fitting in the frequency domain. Index Terms—State Feedback controllers, Automatic Voltage Regulator, Power System Stabilizer, least squares fitting.
I. INTRODUCTION Poorly damped electromechanical oscillations are becoming more likely as the operation of power systems is intensive due to the increase of power exchange among different areas and the lack of new transmission facilities. Electromechanical oscillations are usually classified in two categories: inter-area and local. Inter-area oscillations are typically in the range from 0.2 to 1 Hz and involve large groups of generators oscillating among each other. Local oscillations are in the range from 1 to 2 Hz and correspond to the oscillation of a generator or a small group of generators against the rest of the system [1]. The standard and most widely used solution to damp power system oscillations is the installation of a Power System Stabilizer (PSS) in addition to the Automatic Voltage Regulator (AVR) of the generator. Several methods for PSS design have been proposed. These controllers have been traditionally designed using classical control theory based on the transfer function representation of the system and the frequency response. Electricité de France (EDF) has not followed the standard approach to design controllers for damping electromechanical oscillations. EDF has applied modern control theory instead, based on the state-space representation of the system. The type of solution provided by this theory is a state feedback controller whose control signal is a weighted sum of the state variables of the system. The main objective of this paper is to find a general procedure to covert the state feedback form of the last two controllers developed by EDF to the standard AVR+PSS structure. The first controller is called the Desensitised Four A. Elices and L. Rouco are with Instituto de Investigación Tecnológica (IIT) of Universidad Pontificia Comillas, Alberto Aguilera 23, 28015 Madrid, Spain (e-mail:
[email protected] and
[email protected]). H. Bourlès is with Conservatoire National des Arts et Métiers (CNAM), 21 rue Pinel, 75013 Paris, France (e-mail:
[email protected]). T. Margotin is with Direction des Recherche et Developpement, 1 avenue du Général de Gaulle, 92141Clamart, France.
0-7803-7967-5/03/$17.00 ©2003 IEEE
Loop Regulator (DFLR), aimed at damping local oscillations. This controller arises from the state-space form of the single machine infinite bus system represented by a model with three measurable state variables (terminal voltage, active power and speed deviation). The controller feeds back the three states multiplied by the corresponding gains. An additional integral loop eliminates the steady state error between the reference and the terminal voltage. The gains are calculated according to the desensitisation method [2]-[3] based on optimal control theory, which obtains a robust controller with respect to changes in the generator operating conditions. The second controller is called the Extended Desensitised Four Loop Regulator (EDFLR) and it is aimed at damping both local and inter-area oscillations. It is the result of the application of an extended version of the desensitisation method [4]-[5] to a more complicated design circuit with two generators. The DFLR has already been converted to the standard AVR+PSS structure in [6] and [7]. However, such conversion method cannot be easily extended for the case of the EDFLR. Hence, another general method based on least squares fitting in the frequency domain is proposed for the EDFLR, which is also valid for the DFLR. Section II presents the state feedback structure in detail. Section III presents the proposed conversion method and section IV shows the results obtained for a DFLR and an EDFLR. Section V compares the performance of the original and converted controllers for a simple test system. Section VI provides the conclusions of the paper. II. STRUCTURE OF THE STATE FEEDBACK CONTROLLER. The state feedback structure of the DFLR arises naturally from the application of modern control theory to the design of controller for a generator connected to an infinite bus (Fig. 1). This circuit exhibits a single electromechanical oscillation of the generator against the infinite bus, which corresponds to a local oscillation phenomenon. S1 = 100 MVA G1
V
X
P, q Fig. 1. Design circuit of the DFLR.
Fig. 2 presents a scheme of the state-space model. The
steady state operating conditions are described by the terminal voltage V , the connection reactance X, the active power P and the reactive power q. The control input is the excitation voltage E fd and the measured outputs are the terminal voltage
Vt , speed deviation ω and active power Pe of the generator. To avoid fast and poorly observable dynamics, the system of Fig. 1 is reduced according to the balance realisation technique [8]. This reduction corresponds to the state vector x = (δ ω ψ fd )T with the relevant dynamics of generator
G1 : the rotor mechanical variables δ (rotor angle in p.u.) and
ω (speed deviation in radians per second) and the excitation flux ψ fd . Since a linear system is considered, all variables are supposed to be deviations with respect to the operating point. E fd x x Vt B C + y = Pe ω A
∫
+ Vref
e
−
∫
e
Ke
+
+ −
−
KVt
Vt
E fd
+ −
K Pe
Kω
Pe
ω
Fig. 3. State feedback structure with integral action on the terminal voltage.
S1 = 100 MVA
V1 = 1.0
G1 1
G2
X lo
S 2 V4 = 1.0 4
2
X in
L
V3 = 1.0 3
Fig. 4. Design circuit of the EDFLR. Fig. 2. Design circuit of the DFLR.
As there are as many states as measured variables, the linear transformation H = C−1 provides a new state vector x m equal to y. According to modern control theory, the standard control structure is the state feedback controller (1). This controller obtains the signal E fd as the sum of each of the three measured variables multiplied by a gain.
E fd = − KVt
K Pe
Vt Kω Pe ω
reference voltage Vref . This action is introduced incorporating (2) to the state-space equations of the system (in steady state, e is constant and Vt = Vref ).
In modern control theory, the corresponding standard control structure is the state feedback form with the integral action (2) and an observer. This observer estimates the nonmeasured states. The result is the second order controller in state-space form (3), which obtains the control signal E fd from the measurements y and its internal dynamics ξ . If the
(2) gains are substituted for the transfer functions of E fd over
The resulting state feedback controller with integral action corresponds to the structure of Fig. 3, where the four gains are calculated according to the desensitisation method [2]-[6]. The design circuit of the EDFLR is the two-generator circuit of Fig. 4. This circuit represents two electromechanical oscillations: a local mode of G1 against G2 (mainly controlled by X lo ) and an inter-area mode of G1 and G2 against the infinite bus (primarily controlled by X in ). G1 is represented with a full order generator model whereas G2 is a second order classical model.
represents the relevant dynamics of the full order model of G1
and the classical model of G2 . This model has more states than measured variables, so not every state variable can be measured. ξ = Φ ξ + Γy (3) (1) E fd = −Gξ − Hy
Integral action on the terminal voltage feedback loop is added to eliminate the steady-state error between Vt and the
e = Vref − Vt
The state space structure of this model is the same of Fig. 2 but now, the system is reduced to order 5. This is equivalent to the state vector x = (δ1 ω1 ψ fd 1 δ 2 ω 2 )T , which
each of the inputs y according to (4)-(7), this controller has the structure of Fig. 3. The gains are no longer static but second order transfer functions with the same denominator. KVt ( s) = NVt ( s) D( s) (4) K Pa ( s ) = N Pa ( s ) D ( s )
(5)
Kω ( s) = Nω ( s ) D ( s )
(6)
K e ( s ) = N e ( s ) D( s )
(7)
The order of D( s) is the order of the system (3) and so are the numerators, since the matrix H is never empty. If a DFLR is considered, D( s ) = 1 and the numerators are static gains. For an EDFLR, both D( s) and the numerators are second order polynomials of s. III. CONVERSION TO THE AVR+PSS FORM Fig. 5 presents the standard AVR+PSS structure to which
the controllers are to be converted. The input signal used for the PSS is the accelerating power Pa = Pm − Pe .
Vref
+
−
Vt
E fd
AVR (s )
+ Vs
PSS (s )
Pa
Fig. 5. Standard AVR+PSS structure.
The control signal E fd obtained from the standard structure of
T K (s) + KV ( s ) ( −Vt ( s) ) + E fd = AVR e + 1 s T AVR
(13) − Kω ( s )TPSS + K Pe ( s ) Pa ( s ) 2 H (1 + sTPSS ) From the comparison of (13) and (8), the transfer functions of the AVR and PSS are given respectively by (14) and (15). T K (s) AVR(s) = AVR e + KV ( s) (14) 1 + s TAVR
− Kω ( s )TPSS PSS(s ) = + K Pe ( s ) AVR(s)-1 (15) 2 H (1 + sTPSS ) These transfer functions will be sampled for points s = jωi
Fig. 5 is given by (8), where Vref is zero since it is assumed that no change in the reference voltage occurs. All variables in the frequency range of interest. To select the suitable order of the transfer functions of the AVR and PSS, (4)-(7) are and functions are expressed by their Laplace transform. substituted in (14) and (15), obtaining (16) and (17). E fd ( s ) = AVR( s ) ( −Vt ( s ) ) + AVR( s ) PSS( s) Pa ( s ) (8) TAVR N e ( s ) N (s) + V AVR( s ) = (16) If E fd ( s ) is obtained from the four loop regulator structure of (1 + sTAVR ) D ( s ) D ( s ) Fig. 3, the expression (9) results. The four gains are − Nω ( s ) 1 + sTAVR represented by transfer functions to consider the general case + N Pa ( s ) PSS( s) = × TAVR associated with the EDFLR. The main purpose of this section 2H (17) −1 is to obtain the transfer functions AVR(s) and PSS(s) as a 1 + sTAVR NV ( s) function of these four gains. Hence, all measured variables N e ( s) + TAVR should be expressed as a function of Vt and Pa . Without loss of generality, it is assumed that TAVR = TPSS . E fd ( s ) = − K e ( s )e( s ) − KVt ( s)Vt ( s ) − Kω ( s )ω ( s ) − K Pe ( s ) Pe ( s) (9) This allows cancelling the term (1 + sTAVR ) TAVR with The integration error can be expressed in terms of the terminal voltage according to (10) and the speed deviation is TPSS (1 + sTPSS ) when the PSS(s) transfer function is
obtained. The denominator (1 + sTAVR ) D ( s ) of (16) has the order of controller (3) plus one and so does the numerator. Concerning (17), the numerator is the first term of the product Pm ( s ) = 0 (the speed governor dynamics are slow enough). and the denominator is the second (risen to the power of –1). V ( s) (10) Both terms have the order of (3) plus one. This means that a e( s ) = t s DFLR would need a first order AVR and PSS, whereas an Pa ( s ) 1 (11) EDFLR would need an AVR and PSS of order 3. ω ( s) = ( Pm ( s) − Pe (s ) ) = 2 Hs 2 Hs Once the order of the AVR and PSS are known in advance, As neither the AVR nor the PSS transfer functions have a a suitable transfer function can be proposed. This is crucial for pure integrator, the integrators of (10) and (11) are a good performance of the parameter estimation method. This approximated by the first order transfer function (12). At low method is based on least squares fitting [9]-[10] and will be frequencies, the infinite gain of the integrator is limited to T detailed in the following two subsections for an AVR and PSS but if T is sufficiently high, the crossover frequency is low with three lead-lag networks (third order), which corresponds enough to be out of the frequency range of interest (0.1 Hz – to the EDFLR. For lower order controllers a similar procedure 10 Hz). can be followed. 1 T (12) A. AVR fitting s 1 + sT Fig. 6 presents the suitable AVR transfer function for the Approximating the integrators of (10) and (11) by (12) with EDFLR. It consists of a number of lead-lag networks and a last time constants of respectively TAVR and TPSS and substituting block formed by the static gain K A and the time constant of the resulting expressions in (9), E fd ( s ) is given in terms of the static excitation system TA . The number of lead-lag Vt ( s ) and Pa ( s ) by (13). networks is the order of the system (3) plus one. related to the accelerating power by the swing equation of the rotor (11). In addition, Pe ( s ) − Pa ( s ) , assuming that
AVR
Vt 1 + sTC1 1 + sTB1
+
Vref
+
1 + sTC 2 1 + sTB 2
1 + sTC 3 1 + sTAVR
KA 1 + sTA
xiωi 2TAVR +yiωi y ω 2T -x ω i i AVR i i
Efd
Vs Fig. 6. AVR transfer function to fit for the EDFLR.
K A is chosen to fix the steady-state error (typical values range from 200 to 300) and TA is the thyristor bridge delay (about 3.3 ms) of a static exciter. This time constant corresponds to a frequency of about 50 Hz, which is out of the frequency range of interest (0.1 Hz – 3 Hz). Hence, this time constant will not be considered in the fitting process. Integrator: 20 dB par decade
dB KA
rad/s
1/TAVR
Fig. 7. Selection of TAVR from the AVR gain K A .
According to Fig. 7, if K A is known the integrator is well approximated choosing 1 TAVR as the frequency (in rad/s) where (16) is equal to K A . This means that only TCi and TBi need to be estimated. This is accomplished sampling equation (16) at logarithmic-spaced frequency points ωi (e.g. 200 points) in the frequency range of interest. The corresponding samples xi + jyi are forced to be equal to the transfer function of Fig. 6 evaluated at each ωi in (18). As explained above, the term (1 + sTA ) is not included. Denominators and numerators are respectively at the left and right hand side of (18). ( xi + jyi )(1 + jωi TB1 )(1 + jωi TB 2 )(1 + jω iTAVR ) =
xiω i -xiωi TAVR
-K Aω i
yiωi +xiωi TAVR
0
2
3
2
3
2
(25) K Aωi -ω 3 K A 0
0
xi − yiωi Ta − K A Bi = (26) yi + xiω iTa The matrices Ci and Bi are grouped in C and B for all
samples
so
C = (C1T
that
CT2
CTn )T
and
B = (B B B ) (n is the number of samples). The resulting system Cx = B has far more equations than unknowns so the vector x is calculated according to (27) to minimise the average least square error through the pseudoinverse of C. (27) x = C+ B The searched parameters are found undoing the change of variables (19)-(23). This is accomplished by solving the second and third order equations (28). T 2 − x1T + x2 = 0 ⇒ TB1 , TB 2 (28) 3 2 T − x4T + x3T − x5 = 0 ⇒ TC1 , TC 2 , TC 3 T 1
T 2
T T n
B. PSS fitting Fig. 8 presents the appropriate transfer function form for the EDFLR. It consists of a number of lead-lag networks, the PSS gain K S and a washout filter (last block) to eliminate very slow frequency changes (e.g. due to automatic generation control). The number of lead-lag networks is the order of the system (3) plus one. The washout time constant Tw is usually fixed between 5 and 20 seconds. PSS Pa
1 + sT5 1 + sT6
1 + sT3 1 + sT4
1 + sT1 1 + sT2
KS
sTw 1 + sTw
Vs
(18)
Fig. 8. PSS transfer function to fit for the EDFLR.
(20)
x2' = T2 + T4 + T6
(31)
x3 = TC1TC 2 + TC1TC 3 + TC 2TC 3
(21)
x = T2T4T6
(32)
x4 = TC1 + TC 2 + TC 3
(22)
x = K s TwT1T3T5
(33)
x = K sTw (T1 +T3 + T5 )
(34)
x = K s Tw
(35)
= K A (1 + jω iTC1 )(1 + jω iTC 2 )(1 + jωi TC 3 )
The samples xi' + jyi' are now taken from equation (17) with There are as many equations as samples. Operating the the same criteria as the AVR so that (18) becomes (29). parenthesis, the complex equation is separated into two ( xi' + jyi' )(1 + jω iT2 )(1 + jω iT4 )(1 + jω iT6 )(1 + jω iTw ) = equations (real and imaginary components). These equations (29) = jω iTw K S (1 + jω iT1 )(1 + jωi T3 )(1 + jω iT5 ) are non-linear with respect to the parameters. However, a variable transformation can be found so that the problem The same steps for the AVR are now followed for the PSS. becomes linear in terms of the parameters. Equations (19)-(23) The resulting equations become linear through the change of variables (30)-(36). show this transformation. x1' = T4T6 + T2T6 + T2T4 (30) (19) x1 = TB1 + TB 2
x2 = TB1 TB 2
x5 = TC1 TC 2 TC 3 (23) The resulting two linear equations can be expressed in the matrix form (24), where Ci and Bi are given by respectively
' 3
' 4
' 5
' 6
x = K sTw (T1T3 + T1T5 + T3T5 ) (36) (25) and (26) and x = ( x1 x2 x3 x4 x5 )T . The two rows ' ' ' ' ' Equation (24) becomes Ci x = Bi , where Ci and Bi are given of (25) and (26) correspond to the real and imaginary parts of by (37)-(38) and x ' = ( x1' x7' )T . The elements c11 and (18) after the change of variables. Ci x = B i (24) c12 of the matrix (37) are shown in (39) and (40). ' 7
0 (37) 3 -ωi
c21 = yi'ωi +xi'ωi Tw
(40)
3
' i
2
3
30
−1
0
10
1
10
2
10
10
0 −20 Phase (degrees)
' i
the AVR) in C' and B ' . The solution vector x ' is again obtained through the pseudo-inverse according to (41). x' = (C' ) + B ' (41)
−40 −60 −80 −100 −2 10
If the change of variables is undone, the time constants of Fig. 8 are given by the solutions of the two third order equations (42) and K S by (43).
−1
0
10
1
10 Frequency (Hz)
2
10
10
Fig. 10. DFLR conversion: exact and approximated AVR. Fitting of the PSS from regulator: DFLR 15
T2 , T4 , T6 T1 , T3 , T5
K S = x Tw ' 6
(42)
Exact Fitting
10 Module (dB)
T 3 − x2' T 2 + x1' T − x3' = 0 ⇒ x6' T 3 − x5' T 2 + x7' T − x4' = 0 ⇒
35
20 −2 10
Matrices C and B are grouped for all samples (as with ' i
40
25
(39)
2
Exact Fitting
45
(38)
c11 = x ωi − y ωi Tw ' i
Fitting of the AVR from regulator: DFLR 50
Module (dB)
c11 yi'ωi +xi'ωi 2Tw - xi'ωi 4Tw -yi'ωi 3 ωi 4 -ωi 2 0 c - x'ω +y'ω 2T - y'ω 4T + x'ω 3 0 0 ωi i i i i w i i w i i 21 xi − yiω iTa − K A Bi' = yi + xiωi Ta
(43)
5 0 −5 −10 −2 10
IV. RESULTS The fitting method has been applied to the DFLR described by the four gains (44). Fig. 9 presents the converted DFLR to the AVR+PSS structure. Both the AVR and PSS have a single lead-lag network. The development of subsections III.A and III.B should be repeated for the simpler case of a single compensating network. KV = 31.202 K Pe = 13.417 Kω = −2.387 K e = 12.788 (44)
−1
0
10
1
10
2
10
10
60
Phase (degrees)
40 20 0 −20 −40 −60 −2 10
−1
0
10
1
10 Frequency (Hz)
2
10
10
Fig. 11. DFLR conversion: exact and approximated PSS. AVR
Vref V
1 + 2.2645 s 1 + 14.5151s
AVR
200 E fd 1 + 0.0033s
Vref V
Vs PSS
0.0489s 2 + 0.3581s + 1 0.0154 s 2 + 0.0555s + 1
1 + 2.5038s 1 + 52.7012 s
200 1 + 0.0033s
E fd
Vs 8.1494
1 + 0.2295s 1 + 4.3499 s
5s 1 + 5s
PSS
Pa 3.2851
Fig. 9. Converted DFLR to the structure AVR+PSS.
Fig. 10 and Fig. 11 compare the Bode plots of the AVR and PSS between the approximation obtained with the method and the exact transfer functions when the integrators are not approximated by the first order systems with time constants TAVR and TPSS . The fitting method has also been applied to the EDFLR described by the state matrices (45)-(48). −1.3094 −8.2304 Φ= (45) 7.5285 −2.2972 −3.3250 −11.4512 0.6788 −1.3165 Γ= 4.2433 −3.0916 4.6879 11.3714 G = [12.0158 13.3734]
H = [30.2131 36.8863 −5.7233 12.1232]
0.0206s + 0.0956 s + 1 0.0553s 2 + 0.3870 s + 1 2
5s 1 + 5s
Pa
Fig. 12. Converted EDFLR to the structure AVR+PSS.
Fig. 12 presents the converted EDFLR to the AVR+PSS structure. The AVR has three lead-lag networks which correspond to a second order state-space controller. The presence of complex roots forces the grouping of two lead-lag networks in a second order transfer function. The PSS has only two networks instead of three, because a PSS pole almost cancels with a zero. In this case, the excess of parameters puzzles the fitting method with three networks so a simpler model with two cells gives better results. Again, these two networks are grouped in a second order transfer function due (46) to the presence of complex roots. Fig. 13 and Fig. 14 compare the Bode plots of the AVR and (47) PSS between the approximation obtained with the method and (48) the exact transfer functions.
TABLE I: COMPARISON BETWEEN ACTUAL AND CONVERTED DFLR.
Fitting of the AVR from regulator: EDFLR 40 Exact Fitting Module (dB)
35
30
25
20 −2 10
−1
10
0
10
1
2
10
10
LOCAL MODE | w/r | 1:DFLR | 3:Conversion | n | dam(%) f(Hz) | dam(%) f(Hz) | dam(%) f(Hz) | ---|--------------|--------------|--------------| 1 | 3.58 1.100 | 56.19 1.030 | 59.75 1.030 | 2 | 3.24 1.050 | 54.25 0.929 | 59.09 0.904 | 3 | 2.97 1.000 | 49.19 0.847 | 53.66 0.802 | 4 | 2.76 0.950 | 40.79 0.809 | 42.75 0.777 | 5 | 2.63 0.900 | 33.05 0.792 | 33.86 0.771 | ---|--------------|--------------|--------------|
100
Phase (degrees)
50
0
−50
−100 −2 10
−1
10
0
10 Frequency (Hz)
1
2
10
10
INTER-AREA MODE | w/r | 1:DFLR | 3:Conversion | n | dam(%) f(Hz) | dam(%) f(Hz) | dam(%) f(Hz) | ---|--------------|--------------|--------------| 1 | 1.14 0.450 | 9.29 0.511 | 9.49 0.512 | 2 | 1.27 0.400 | 13.26 0.450 | 13.82 0.451 | 3 | 1.48 0.350 | 16.49 0.381 | 17.35 0.380 | 4 | 1.84 0.300 | 17.33 0.316 | 18.07 0.314 | 5 | 2.50 0.250 | 17.19 0.259 | 17.82 0.257 | ---|--------------|--------------|--------------|
Fig. 13. DFLR conversion: exact and approximated AVR. TABLE II: COMPARISON BETWEEN ACTUAL AND CONVERTED EDFLR. Fitting of the PSS from regulator: EDFLR 15 Exact Fitting Module (dB)
10
5
0
−5 −2 10
−1
10
0
10
1
10
2
10
80
Phase (degrees)
60 40 20 0 −20 −40 −60 −2 10
−1
10
0
10 Frequency (Hz)
1
10
2
10
Fig. 14. DFLR conversion: exact and approximated PSS.
V. PERFORMANCE COMPARISON OF ORIGINAL AND CONVERTED CONTROLLERS
This section compares the performance of the actual and converted controllers for the test system of Fig. 4 for five cases with different operating conditions. The frequency of the local mode is varied from 0.9 Hz to 1.1 Hz and the inter-area mode between 0.25 Hz to 0.45 Hz. Table I compares the performance of the system without regulator (column labeled “w/r”), with the DFLR and with the converted DFLR. Each column presents the frequency and damping ratio of the corresponding mode. Without regulator, it is possible to identify the original frequency of the modes which are poorly damped. When the DFLR is installed the damping ratios improve significantly and the frequencies change slightly. If the damping ratio of the DFLR is compared with that of the conversion, they are very similar. Table II also shows that the performance of the EDFLR and the converted EDFLR are very similar. For an original frequency of 0.3 Hz and 0.25 Hz of the inter-area mode, the conversion deteriorates the performance. This is explained because the PSS fitting at very low frequencies (see Fig. 14) is not as good as for higher frequencies.
LOCAL MODE | w/r | 2:EDFLR | 4:Conversion | n | dam(%) f(Hz) | dam(%) f(Hz) | dam(%) f(Hz) | ---|--------------|--------------|--------------| 1 | 3.58 1.100 | 33.04 0.912 | 31.68 0.920 | 2 | 3.24 1.050 | 32.79 0.893 | 31.40 0.903 | 3 | 2.97 1.000 | 32.83 0.870 | 31.39 0.882 | 4 | 2.76 0.950 | 33.27 0.842 | 31.77 0.857 | 5 | 2.63 0.900 | 34.21 0.808 | 32.65 0.827 | ---|--------------|--------------|--------------| INTER-AREA MODE | w/r | 2:EDFLR | 4:Conversion | n | dam(%) f(Hz) | dam(%) f(Hz) | dam(%) f(Hz) | ---|--------------|--------------|--------------| 1 | 1.14 0.450 | 19.68 0.531 | 18.17 0.542 | 2 | 1.27 0.400 | 26.80 0.460 | 25.00 0.483 | 3 | 1.48 0.350 | 36.91 0.369 | 35.93 0.422 | 4 | 1.84 0.300 | 42.62 0.250 | 38.81 0.197 | 5 | 2.50 0.250 | 29.01 0.198 | 22.28 0.182 | ---|--------------|--------------|--------------|
The comparison of Table I and Table II shows that the EDFLR performs much better than the DFLR for the inter-area mode. Although the DFLR performs better than the EDFLR for the local mode, the EDFLR has still a satisfactory performance. VI. CONCLUSIONS A general method is presented to convert state feedback controllers to damp power system electromechanical oscillations to the standard structure formed by an Automatic Voltage Regulator (AVR) and a Power System Stabilizer (PSS). The state feedback controller structure arises naturally from the third order state-space form of the single machine infinite bus system, represented by three measurable state variables (terminal voltage, active power and speed deviation). The control signal is then obtained as the sum of each measured variable multiplied by a gain. The conversion method has been applied to two controllers developed by Electricité de France. The first one is a state feedback controller called the Desensitised Four Loop Regulator (DFLR) designed to damp local oscillations. The second is a state feedback controller with an observer to estimate the non-measured states. It is called the Extended Desensitised Four Loop Regulator (EDFLR) and it is aimed at damping both local and inter-area oscillations. Although the
DFLR had already been converted to the AVR+PSS structure, the proposed method is general for any state feedback controller including an observer to estimate the non-measured states. The performance of the controllers in the AVR+PSS form is not affected by the conversion process. VII. REFERENCES [1]
H. Breulmann, E. Grebe, M. Lösing, P. Dupuis, M. P. Houry, et al, "Analysis and Damping of Inter-area oscillations in the UCTE:CENTREL Power System", Paper 38-113, Cigré, session 2000, pp. 1050-1060, 2000. [2] A. Heniche, H. Bourlès and M. P. Houry, "A Desensitised Controller for Voltage Regulation of Power Systems", IEEE Transactions on Power Systems, Vol. 10, No. 3, pp. 1461 - 1466, August 1995a. [3] H. Bourlès and A. Heniche, "The Inverse LQR Problem and its Application to Analysis of a Robust Coordinated AVR/PSS", 12th Power Systems Computation Conference, Dresden, pp. 63 - 69, 19th 23rd August 1996. [4] A. Elices, L. Rouco, H. Bourlès and T. Margotin, "An Extended Desensitised Controller for Damping Power System Inter-area Oscillations", Submitted to IEEE Transactions on Control System Technology. [5] A. Elices, L. Rouco, H. Bourlès and T. Margotin, "Design of Robust Controllers for Damping Inter-area Oscillations: Application to the European Power System", Submitted to IEEE Transactions on Power Systems. [6] H. Bourlès, S. Peres, T. Margotin and M. P. Houry, "Analysis and Design of a Robust Coordinated AVR/PSS", IEEE Transactions on Power Systems, Vol. 13, No. 2, pp. 568 - 575, May 1998. [7] H. Quinot, H. Bourlès and T. Margotin, "Robust Coordinated AVR+PSS for Damping Large Scale Power Systems", IEEE Transactions on Power Systems, Vol. 14, No. 4, pp. 1446 - 1451, 2000. [8] A. J. Laub, "Computation of 'Balancing' Transformations", Joint Automatic Control Conference, San Francisco (USA), pp. FA8-E1 FA8-E2, August 1980. [9] L. Ljung, System Identification, Prentice Hall, 1987. [10] J. P. Norton, An Introduction to Identification, Academic Press, 1986.
VIII. BIOGRAPHIES A. Elices received his Electronic Engineer and Ph.D. in Electrical Engineering degrees from Universidad Pontificia Comillas in 1995 and 2001 respectively. He also obtained a Master of Science in Financial Mathematics from the University of Chicago in June 2002. He has mainly worked in the fields of Electromagnetic Transients and Power System Oscillations. L. Rouco received his Electrical Engineer and Ph.D. in Electrical Engineering degrees form Universidad Politécnica de Madrid in 1985 and 1990 respectively. He is Associate Professor and Head of the Departament of Electrical Engineering of the School of Engineering of Universidad Pontificia Comillas, Madrid. He develops his research activities at the Instituto de Investigación Tecnológica. His areas of interest are modelling, analysis, simulation and identification of power system dynamics. H. Bourlès received his Engineering Degree from École Centrale de Paris in 1977, his Ph.D. from Institut National Plytechnique de Grenoble in 1982, and his “habilitation” form Université Paris XI in 1992. He holds the Chair of Automatic Control of Conservatoire National des Arts et Metiers, Paris, since 1997. T. Margotin received his Engineering Degree from Ecole Centrale de Nantes (France) in 1994. He is a Research Engineer at EDF Research Center (Clamart, France). He works on the fields of power system stability, load frequency control and automatic voltage control.
