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Robust nonlinear coordinated control of power systems
Pergamon
Aufomorica. Vol. 32, No. 4, pp. 611-618, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved coos1098/% SlS.cKJ + 0.00
ooo5-1098(95)00186-7
Brief Paper
Robust Nonlinear Coordinated
Control of Power
Systems* YOUYI WANG? and DAVID Key Words-Robust transient stability.
J. HILLS
control; feedback linearization; nonlinear control systems; power system control;
KGKT = 1;
Abstract-To prevent an electric power system losing synchronism under a large sudden fault and to achieve voltage regulation are major objectives in power system design. This paper applies the Riccati equation approach, together with the direct feedback linearization (DFL) technique, to design robust nonlinear controllers for transient stability enhancement and voltage regulation of power systems under a symmetrical three-phase short circuit fault. A DFL excitation controller and a DFL coordinated controller (excitation and fast valving controller) are proposed. The simulation results show that a power system can keep transiently stable, even when a large sudden fault occurs at the generator terminal. The robust nonlinear DFL controllers proposed here can greatly improve transient stability and achieve voltage regulation.
X& = XT + :xL + x*; xc = XT+ $XL+ x:; xT reactance of transformer: x,_, direct axis reactance: x,, direct axis transient reactance; XI. reactance of transmission line; x, = XT + ix,; X,d mutual reactance between excitation coil and stator coil: V, infinite bus voltage. 1. Introduction In recent years, a great deal of attention has been paid to the application of feedback linearization approaches to power systems (see e.g. Marino, 1984; Lu and Sun, 1989; Mielczarski and Zajaczkowski, 1989, 1990, Ilic and Mak, 1989; Chapman and Ilic 1993). In our previous paper (Wang et al., 1993), we examined the direct feedback linearization (DFL) (Gao er al., 1992) technique aimed at achieving transient stability enhancement and voltage regulation of power systems. Implicit in that paper is the assumption that many parameters in the power system are known, a priori. In many cases (e.g. inertia and time constants) this assumption since these parameters are is completely reasonable, essentially constant, and can be determined from power system design information. Key situations where this assumption proves restrictive for transient stability analysis are when it is also assumed that the transmission line reactance xL and the infinite bus voltage V, are known in the transient period. These parameters will not, in practice, be constant. and are poorly known. In these cases, parameter adaptation (see Goodwin and Sin, 1984; Pierre, 1987) can be smployed to deal with uncertainties in .rL and V, when a fault occurs. In Wang et al. (1994), a nonlinear adaptive controller was proposed to achieve transient stability enhancement and voltage regulation. But when a large fault occurs, especially very close to the generator terminal, reactances of transmission lines in a power system change a lot, and the configuration of the electric power system varies very quickly. It is very difficult for an adaptive controller to adapt to parameter variations. Wang et al. (1994) showed that for the example system, when a fault occurs at one-tenth of the transmission line from the generator terminal, the system using an adaptive controller can no longer maintain transient stability. More recently, a few papers have considered oscillatory stability enhancement of power systems using robust control techniques (Chow, 1988; Othman et al,, 1989; Pai an Sauer, 1989). Rather than the approximate linearization model used in those papers, here a nonlinear classical model of the power system is considered. This is consistent with normal power system modelling practice (Anderson and Fouad, 1977; Bergen, 1986), where more detailed linear models are used for oscillatory stability and simplified nonlinear models for large disturbance stability, i.e. transient and voltage stability. In this paper, we shall use a robust control technique based on the Riccati equation approach, together
List of symbols 6(r) power angle;
w(t) relative speed; P,(t) mechanical input power; P<(t) active power delivered to bus; w,) synchronous machine speed, % = 27&i D per-unit damping constant; inertia constant (in s); transient EMF in the quadrature axis; E;(Z EMF in the quadrature axis; W) equivalent EMF in the excitation coil; GO) Td0 direct axis transient short-circuit time constant; steam valve opening: X,(r) input of power control system; 4(r) excitation current: 40) quadrature axis current; I,(t) reactive power; Y$:,’ generator terminal voltage; input of the SCR amplifier of the generator; 40 k, gain of excitation amplifier: TT time constant of turbine, with typical numerical values of 0.2-2.0 s; KT gain of turbine; constant (per unit), typically R = R regulation 0.05 p.u.; TG time constant of speed governor, typically around 0.2 s; KG gain of speed governor;
-
*Received 13 Auril 1993: revised 11 Mav 1994: revised 11 April 1995; recieied in final form 27 SepteAber 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hassan Khalil under the direction of Editor Tamer Bagar. Corresponding author Dr Youyi Wang. Tel. +65 7991423; Fax +65 7912687; E-mail [email protected]. t School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263. $ Department of Electrical Engineering, The Univeristy of Sydney, NSW 2006, Australia. 611
Brief Papers
612
excitation controller and a new robust nonlinear coordinates controller (excitation and fast valving controller) to enhance transient stability and to achieve voltage regulation of an example power system under a symmetrical three-phase short-circuit fault. 2. Dynamical model A simplified dynamical model of a single machine-infinite bus power system is considered in this paper. The model consists of a single generator connected through two parallel transmission lines to a very large network approximated by an infinite bus, and is shown in Fig. I. A model for the generator with both excitation and power control loops can be written as follows (Bergen. 1986: Anderson and Fouad. 1977): mechanical equations d(f) = w(t).
0)
&J(l) = - &I) L
+$[P”,(t)
-- P,(t)]:
Stage 3: The fault is removed by opening the breakers of the faulted line at t = 0.25 s. Stage 4: The transmission lines are restored at I = 1.4 s. Srage 5: The system is in a postfault state. Fault sequence 2 (permanent fault). Stage 1: The system is in a prefault steady state. Stage 2: A fault occurs at t = 0.1 s. Stage 3: The fault is removed by opening the breakers faulted line at t = 0.25 s. Stage 4: The system is in a postfault state.
3. Design of a robust nonlinear excitation controller Since we only consider the excitation loop, P,,, is constant, and the plant can be modelled by (l)-(3). From the results given in Wang et al. (1993). if the parameters in the power system are known. we can design a DFL control law
(2)
u&r(r),
II =
z
generator electrical clynamics E;(t) = [Et(t) -~ E,(t)] ,f;: d,,
of the
{W- T’[Q,(t) +$]4)) T&lJ(t) - pew+pITI”.
xyc
(3) where
(13)
x&r)
i+(t) is the new input and
turbine dynamic5 P”,(f)
=
(4)
; P,(l) + $h X,(t): I I
We now linearize
turbine value control X,(t)
= -;
(8
X,(f)
+9
P,(r) ~ &
w(t)
1
the model
(l)-(3)
AS(t) = w(t),
(5)
electrical equations
b(t) = -&w(t)
E,(t) =2
\
E;(t) ~ xd--\;
-6, EXI) = k,u,(t).
V, cos 6(t).
APJt)
(6) (7) (8)
as follows:
- gAP,(t),
= ~ +, AP,(t) + $, “t(t)
Remark 3.1. The compensating law (13) is simple and practically realizable, the linearization is valid over the whole working region (O”< s(t) < 180”), and the linearized model is independent of the operating point of the system.
(9) (10) (II) v(r) =
$
\
[x$E$t) + Vi.rf
+ 2x-,.r,*‘,,P,(t) cot 6(t)] 7
(12)
The fault we consider in this paper is a symmetrical three-phase short circuit fault on one of the transmission lines. The total reactance of the transmission line is x, (= 0.4853). The parameter A is the fraction of the faulted line to the left of the fault. If A = 0. the fault is at the generator bus, A = 0.5 puts the fault in the middle of the line, and so on. Two different fault sequences are as follows. Fault sequence I (temporary fiurlt). Stage 1: The system is in a prefault steady Stage 2: A fault occurs at t = 0.1 s.
From the above disussion, we can see that the compensating law (13) contains xL and V,; if we know xL and V,. a DFL control law can be easily designed. However, in practice, a fault can occur at any time and anywhere on the transmission line. When a large sudden fault occurs, the reactance of the transmission line xL and the infinite-bus voltage V, change a lot, and it is impossible to know XJ_ and V, during the transient period. In this paper, the changes in I, and v, are treated as parametric uncertainties. Considering the uncertainties in xL and V,, we have
+(T’ + +~;o(% -4)$$ w(t) + (7’ + 1,
AT’)Q,(t)w(t)
ad f
AT’)o(t)
state. where
~ P,,,,,
M(t) = 8(t) ~ 6,,, AP,(t) = P,(t) - P,,,,, T’ + AT’ =*
Generator and AV, denote respectively, and
At, Fig. I. Generator
connected through infinite bus.
transmission
lines to an
the
uncertainties
0%
I in
T;,,.
xl_ and
V,
613
Brief Papers Using the control law (13) gives the linearized follows:
model as
A&) = w(r) G(r) = -&o(t)
-ZAP.(r),
Ak(r) = -[ $, + c(r)]Ap.(r) + [+ +
(14)
+ [ $, + &)]uXr)
d~~]A~fQ&)o(~)
+ [++ r(r)]Ai+o(r). Define XT(r)= [As(r) w(r) Ape(r)]. model (14) can be rewritten as
i(r) = (A where
ro
Then
+ AA)x(r) + (I3 +
the
linearized
AB)t+(r),
(15)
1
= -k,AiS(r)
01 0
)
’
0
B=[O
AB=[O 0
+ p(r)][AF’
--CL(f) 0 1
u,-(r) = -k,A$(r)
j&f
0
p = [$
[
AA=00 0 P
p(r)lT,
+ AT’Q.(r)].
llT,
E, =
i-l 00
E*= [O 0
l]T,
F(r) =
0
0 & 0
0
1
-p(r)
,
1,
Solution of the design problem of a nonlinear robust control law u&r(r), r] to transiently stabilize the power system model (l)-(3) is equivalent to solving the robust control problem for the linearized plant (15) which involves solving the following algebraic Riccati equation: + P(A - BR-‘ET&) + P(a2DDT - BR-‘BT)P + ET(I - E,R-‘E:)E,
= 0,
(16)
Theorem 3.1. The power system (l)-(3) under a symmetrical three-phase short circuit fault is transiently stable via the DFL nonlinear control law (13), and + E~E,)x(r)
(17)
if there exists a positive-definite solution P > 0 of the Riccati equation (16). If the power system (l)-(3) is transiently stable via the control law (13) and (17) then the power system under a symmetrical three-phase short circuit fault can avoid the loss of synchronism. Moreover, in the postfault period, lim lAS(r)l = 0, ,--rcO
lim Iw(r)l = 0, r--r-
= constant.
The control procedure is as follows. Step 1. The fault occurs at r = to and the robust control law is r&,,(r), rl, (13) where u,,(r) = -k,AA(r) - k,w(r) - k,AP,(t). Srep 2. At r = r,, the feedback law is switched to ur[un(r), r], where
r+&) = -k,,AV,(r)
where R = ETE, >O (Khargonekar et al., 1990, Xie er al., 1992). Our main ‘stability’ result is as follows.
u&) = -R-‘(BTP
lim IAs ,%+r
lim IAV,(r)l= 0. I--+X
5 azZ (CY> 0), y is a constant and lpi 5 p.
(A - BR-‘E;E,)TP
(18)
Thus, in many cases, the generator terminal voltage is not the same in the postfault state as in the prefault state, which is undesirable in practice. To achieve both transient stability and voltage regulation, we employ the above robust nonlinear controller in the DFL excitation control scheme proposed in Wang et al. (1993). Then, we achieve voltage regulation, i.e.
where D, E, and E2 are constant matrices, with
D = [O 0
- k,AP,(r),
but only that
&I, 10
- k,w(r)
f\i~ IAS(r)l = 0,
AT’ and AT’ are bounded; so is p. The matrices AA and BE can be expressed as
ABI= WW,
- kpAP,(r).
where ha(r) = d(r) - aO. Pwing to the difference in initial values between s(r) and s(r), it is clear that by introducing s(r) in the control law (18) we can no longer obtain
p(r),
[AA
+ EIE,)x(r) - k,w(r)
Since thepower angle s(r) cannot be measured, the estimate of s(r), s(r), can be obtained by using a S detector (see Wang et al., 1992) s(r) = o(r). Then udr) becomes
)
-& -2 0 -+
L
FT(r)F(r)
In the robust control scheme proposed in this section, the robust DFL nonlinear control law is udut(r), t], (13) where t+(r) = -R-‘(BTP
00
A=
fault is equivalent to designing a robust linear control law the linerized plane (15) with parametric uncertainties. Using the results in Khargonekar et al. (1990) and Xie et al. (1992), we know that a linearized plant model in the form (15) is asymptotically stable if there exists a stabilizing solution P>O of the Riccati equation (16). Moreover, a suitable feedback law is given by (17). Since designing a nonlinear robust control law udr) to stabilize the power system model (l)-(3) is equivalent to designing a robust control law q(r) to stabilize the linearized plant (15) we can conclude that the nonlinear control law (13) and (17) can transiently stabilize the power system (l)-(3) for all allowed 17 changes in x,_ and Vs. The proof is complete.
udr) to stabilibe
lim IAP,(r)l = 0. ,-=
Proof By using the DFL technique, the power system (l)-(3) can be linearized as (14) or (15). To design a robust nonlinear control law r+(I) to transiently stabilize the power system (l)-(3) under a symmetrical three-phase short-circuit
- k,,o(r)
-k,,&(r).
The design procedure for kV1, k,, and kp, can be found in Wang et al. (1993). The switching time I, is determined by trial and error for a given system. 4. Simulation results I In our example, we can obtain, from (15) that A=[;
-:.625
&;J9,l,
-_[i
B = [0 0 0.516351T, AR = [0 0
& jrj, Il(r)
B=[$+&r)][AT’+AT’Q,(r)]. We obtain that_ Ip( s 0.2347, (AT’150.6053, 3.09050 and IpI 5 /3 = 3.23. Choosing 100 E,=
o;Po,
E2 = [O 0
l]T,
L-1001 D=[O
0
119
F(r) = [0
Y!
p(r)
1,
IAT’\ =
614
Brief Papers
y =0.2347 and a’ = 2 X (0.2347)‘>0, we have R = E:E, = 1 > 0 and FT(t)F(t) 5 &I. Solving the Riccati equation (16) gives [k,
k,
k,] = j-1.31
- IS.78
61,631.
The robust nonlinear control law is ur[+(r), t], (13) where
(A = 0.05) the excitation controller cannot keep the system transiently stable. In this section we shall discuss the design of a robust nonlinear coordinated controller. The plant considered here can be described by (l)-(5). By using the robust DFL nonlinear compensating law (13). we can linearize the plant as follows:
i(t) = [A + AA(t)]x(t) + [B + AB(t)]u(t), where and
Ap,(l) = J’,(r) - P,,I. V,,(f) = 1.31[&) - S,,] + 157&J(r)
u(t) = [u&j
- 61.63[P,(r) - P,,,J,
x(t) = [As(t)
u,2(r) = -47.03[V,(t) - V,,,]+ 6.‘)3~(t) - 28.60[P,(r) - f,,,,,].
0,,=314.159,
D=5.0. T,=O.2s,
H=4.0s,
T;,,=6.Ys,
k,-I.
KC;-1,
K,=l.
Tr=2.0s,
M’,(t)
M’,,,(t) AA’,(t)]‘,
1
0
0
0
0
~- D 2H
_J!!L 2H
3 2H
0
O
1 -0 T’
0
0
~- 1 T,
0
0
0 _K,_L r,; Rwo
max ik,udt)j = 1.8 p.u.
5. Design of robust nonlinear coordinated controller In the previous section, we have seen that when a large sudden fault occurs close to the generator terminal
o(t)
0
The operating point is 6,) = 72”, P,,,()= 0.9 p.u. and V,,, = 1.Op.u. The physical constraint on the excitation voltage is
Firstly. we show the responses of the power angle under different fault locations (A = 0.5, 0.1, 0.055 and 0.05) in Fig. 2 (t, = 0.7 s). From the results, we can see that when a fault occurs close to the generator terminal (A = 0.05) the system, using only the excitation controller (with the excitation limit), cannot maintain synchronism. Next we test our new robust nonlinear excitation controller (A = 0.5, t, =0.7s) under the two different fault sequences discussed in Section 2. Figure 3 shows the [es&s. The system responses at a different operating point (a,, = 52”) So = 47”. I’,,,,,= 0.45 p.u. and V,,, = 1.003 p.u., are shown in Fig. 4. From the results, we can see that, using the new robust nonlinear excitation controller proposed in Section 3, the system can achieve both transient stability enhancement and voltage regulation.
&(t)lT. Pdt) = P&J -Pm,,,
A=0
xd = 1.863, x; = 0.257. xr = 0.127. x,‘, = 1.712.
- XEO,
0
The parameters of the system under consideration are R = 0.05 p.u..
Ax,(r) = x,(t)
00
0
0
-$
00 0 0
1s; 1 0
-CL(I) 0 0
AA=
5 Tr
L
00 00
0 0
, AB=
1
00 00
11 0 ? 0 CL(t) 0
and F(t) and p are as defined in (15). Because of the mechanical behaviour of the steam valve, we expect it to come into action only in the transient period. To ensure the stability of the overall system after the power control loop switches off, the feedback control law is expected to be
The feedback linear-quadratic follows. Step 1. Assume constant. Solve robust feedback
gain matrix can be obtained by using optimal theory. The design procedure is as that the mechanical input power P,,,(t) is the Riccati equation (16), and obtain the control law for the linearized model
q(t) = -k,,Aa(t)
- k,,o(t)
- kp,APe(t);
the control law for the real plant is t+[uAt), t], (13). Step 2. Apply the control law (13) to the plant, and find the closed-loop system through the excitation loop. Then find the feedback control law for the plant through the power control loop PXr) = pc(t) - P,” = -k,,A6(t)
- kwZW(t) - kmAPr(t)
- km,APm(t) - kxzAXE(t).
i Time (set) Fig. 2. The power angle for different fault locations.
In order to achieve voltage regulation, we introduce generator terminal voltage feedback through the excitation loop. The control sequence becomes the following.
615
Brief Papers /sit
sequence 2
r
Fault sequence t
0.6 k 0
1
3 2 Time (see)
3
,
5
Fig. 3. The responses under different fault sequences.
Step I. The fault occurs at 8 =r,, &@%r(rXfl? and
the feedback
law is
using a S-detector (Wang ef cxl..,1992). The switching times t, and tZ can be determined by trial and error in simulation, and f, can be obtained from the circuit breaker data. Remark 5.1. The reasoning behind this design of the control
where
sequence is as follows. t+,(t) = -&Ad(t)
- k,,w(l)
- kPIAP&)
Srep 2. At t = t,, the feedback law switches to tl&r~(i)~ t], and P&t) = Pmo, where nt&f is as above. Srep3. At t = fa, the feedback faw switches to z+&&), i], and P,(r) = !$,a, where u&) = -k,,Av,(r)
- kw34)
- kmAPeW,
and the estimate of the power angle S(l) can be obtained by
1. In Step 1, we introduce power angle, s(r), control (and so refax the generator terminal voltage V,(f) controT) to improve the transient stability of the power system. 2. When the system is transiently stable, the governor input is switched off to reduce the control cost in PC(t). 3. After ta, the system is in the post-transient
period. The
Fault sequence 2
1 Time (set) Fig. 4. The responses at different operating points.
2
1
Time (set)
4
t,
Brief Papers generator terminal voltage Vt(t) control is introduced to relax the power angle, 6(f), control and to achieve the voltage regulation. 6. Simulation results II In this section, we shall show, through simulation results, that transient stability improvement of the system can be achieved by employing the new nonlinear coordinated controller in different cases. Consider the plant as described by (l)-(12). The system parameters are given in Section 4. and the operating point is S, = 72”, Pm0= 0.9 p.u., V,,, = 1.0 p.u. The physical constraint on the excitation voltage is the same as in Section 4, and the physical constraint on the steam valve opening is 0 5 X,(t) 5 1.0 p.u. The control sequence is as follows. 1. The fault occurs at I = 0.1 s. the control @r,(t), 4 and
Step
law is
PC(t)= 71.86A@r) - 60.47w(r) + 32.39Af’,(r) - 4185AP,,,(r) - 99.42AX,(t) + P,,,, where t+,(t) = 1.3111&r) + 1578w(r) ~ 61.63AP,(t).
ii TIME
(me.)
Step 2. At
t = 1.2s. the feedback law switches to udt+,(t)] and P,(t) = P,,,,,. Step 3. At t = 1.5 s. the feedback law switches to ur[u,(r), rj and P,(t) = f,,,,,, where u&f) = -47.03AV,(t) + 6.93w(r) - 28.6Af,(r). The simulations are reformed for the folloiwng cases. Case 1. First we compare the responses from the new robust
nonlinear coordinated controller and the nonlinear excitation controller (Wang et al., 1993) under fault sequence 1. The initial values of the S detector (see Wang et al., 1992) is 79”, i.e. I& = 7Y, and the location of the fault is 0.1, i.e. A = 0.1. Figure 5 shows the responses of the power angle s(r) and the generator terminal voltage Vt(t). We can see from the simulation results that for the example system, the new robust nonlinear coordinated controller can achieve better results than the nonlinear excitation controller (Wang et al., 1993). Case 2. The responses of the power angle S(f) under the different fault locations A = 0.5, 0.1, 0.05 and 0.01 (S,, = 79”)
Fig. 6. The power angle responses.
are shown in Fig. 6. Comparing with the results in Wang et al. (1993), we can see that, using the nonlinear coordinated
controller, the system loses synchronism when A =O.Ol. When using the robust nonlinear coordinated controller, the system can still keep transient stability even when A = 0.01; i.e.. using the robust nonlinear coordinated controller, the system can keep transiently stable even in the case where a large sudden fault occurs close to the generator terminal. Case 3. We test the system under different faulj sequences. The simulation results are shown in Fig. 7 (6,) = 79” and A = 0.1). The simulation results show that the robust nonlinear coordinated controller can effectively improve the transient stability of the example system under differnt fault sequences. The simulation results also show that the robust nonlinear coordinated controller can achieve both transient stability enhancement and voltage regulation under different fault sequences.
onlinear
1 TIME
(sec.)
Fig. 5. The responses for different controllers.
DFL
csa&dkI
1 TIME
1 (see.)
1
Ii
617
Brief Papers
a TIME (~4 Fig. 7. The
TIME
&ee.)
responses under different fat& sequences.
Case 4. We now test the system responses under different fault sequences (A =O.l) at a different operating point: 6, = 42”, Pm0= 0.45 p.u., V, = 1.Op.u. The fault sequences and the control sequence are as before. Figure g shows the responses of the Rower angle 6(f) and the generator terminaf vottage Vt(t) (So = 477. The simulation results show that the robust no&near coordinated controller can achieve both transient stability enhancement and voltage regulation, regardless of operating points. Remark 6.1. Comparing with the results in Section 4 and in Wang ei at. (1993), we can conclude the following from the simtdation rest&s. 1. The robust nonlinear coordinated controller proposed in this paper can effectively improve transient stability of the power system. The system can keep transiently stable even in the case where a large sudden fault occurs close to the generator terminal.
2. The robust nonlinear coordinated controller can achieve good transient stability results, irrespective of the operating point of the system and the fault sequence. 3. The robust nonhenar coordinated control can achieve both transient stability enhancement and voltage regulation in the presence of ail admissible changes in .rL and V’. 7. Conclusions This paper has applied a Riccati equation approach, combined with the so-called direct feedback linearization (DFL) technique, to design robust nonlinear controflers for power systems, A new robust nonlinear excitation controller and a new robust nonlinear coordinated controller have been proposed to achieve both transient stability enhancement and voltage regulation in the presence of all admissible changes in the transmission line reactance xL and the infinite bus voltage V,. Design procedures have been developed for both types of controllers. ‘The robust nonlinear controllers were
P
4
k
TIME (sec.) Fig. 8. The responses at different operating
4
lb
618
Brief Papers
tested through an example system in different cases. and the controllers have been compared with the nonlinear DFL controller in Wang et al. (1993). The simulation results show that transient stability of power systems can be effectively improved by using the proposed robust controllers, regardless of the system operating point, the fault location and the fault sequences. The robust nonlinear coordinated controller can maintain transient stability of the system even when a large sudden fault occurs close to the generator terminals.
References Anderson, P. M. and A. A. Fouad (1977). Power System Control and Stability. Iowa State University Press, Ames. IA. Bergen, A. R. (1986). Power Systems Analvsis. Prentice-Hall, Englewood Cliffs. NJ. Chaoman. J. W. and M. D. Ilic (1993). Stabilizing a multimachine power system via decentralized feedback linearizing excitation control. lEEE Trans. Power Syst.. PWRS-8,830-839. Chow, J. H. (1988). A pole-placement design approach for systems with multiple operating conditions. In Proc. 27th IEEE Conf on Decision and Control. Austin. TX, pp. 1272-1277. Gao, L.. L. Chen. Y. Fan and H. Ma (1992). A nonlinear control design for power systems. Automatica, 28, 975-979. Goodwin, G. C. and K. S. Sin (1984). Adaptlue Filtering, Predicting and Control. Prentice-Hall, Englewood Cliffs, NJ. Ilic, M. D. and F. K. Mak (1989). A new class of fast nonlinear voltage controllers and their impact on improved transmission capacity. In Proc. American Control Conf, Pittsburgh, PA, pp. 1246-1251. Khargonekar, P. P.. 1. P. Petersen and K. Zhou (1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and H” control theory. IEEE Trans. Autom.
Control, AC-35, 356-361. Lu. 0. and Y. Z. Sun (1989). Nonlinear stabilizine control of multimachine systems. IEEE Trans. Power Sys’l PWRS-4, 236-241. Marino, R. (1984). An example of a nonlinear regulator. IEEE Trans. Autom. Control, AC-29, 276-279. Mielczarski. W. and A. Zajaczkowski (1989). Nonlinear controller for synchronous generator. In Proc. IFAC Nonlinear Control Syustem Design Symp. 89, Capri. Mielczarski, W. and A. Zajaczkowski (1990). Nonlinear stabilization for synchronous generator. In Proc. 11th IFAC World Congress, Tallin. Othman, H.. J. J. Sanchez-Gasca, M. A. Kale and J. H. Chow (1989). On the design of robust power system stabilizers. In Proc. 28th IEEE Conf: on Decision and Control, Tampa, FL, pp. 1853-1857. ’ Pai, M. A. and P. W. Sauer (1989). A framework for application of generalized Kharitonov’s theorem in the robust stability analysis of power systems. In Proc. 28th IEEE Conf: on Decision and Control, Tampa, FL, pp. 1818-1821. Pierre. D. A. (1987). A perspective on adaptive control of power systems. IEEE Trans. Power Syst.. PWRS-2, 387-396. Wang, Y.. L. Ciao and D. J. Hill (1992). On the design of a new nonlinear excitation controller for power systems. In Proc. International Conf: on Automation, Robotics and Computer Vision 92, Singapore, CO-10-4. Wang, Y., D. J. Hill, L. Gao and R. H. Middleton (1993). Transient stability enhancement and voltage regulation of power systems. IEEE Trans. Power Syst. PWRS-8, 620-627. Wang, Y., D. J. Hill, R. H. Middleton and L. Gao (1994). Transient stabilization of power systems with an adaptive control law. Aufomatica, 30, 1409-1413. Xie, L., M. Fu and C. E. de Souza (1992). H, control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Trans. Autom. Control. AC-37, 1253-1256.
Aufomorica. Vol. 32, No. 4, pp. 611-618, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved coos1098/% SlS.cKJ + 0.00
ooo5-1098(95)00186-7
Brief Paper
Robust Nonlinear Coordinated
Control of Power
Systems* YOUYI WANG? and DAVID Key Words-Robust transient stability.
J. HILLS
control; feedback linearization; nonlinear control systems; power system control;
KGKT = 1;
Abstract-To prevent an electric power system losing synchronism under a large sudden fault and to achieve voltage regulation are major objectives in power system design. This paper applies the Riccati equation approach, together with the direct feedback linearization (DFL) technique, to design robust nonlinear controllers for transient stability enhancement and voltage regulation of power systems under a symmetrical three-phase short circuit fault. A DFL excitation controller and a DFL coordinated controller (excitation and fast valving controller) are proposed. The simulation results show that a power system can keep transiently stable, even when a large sudden fault occurs at the generator terminal. The robust nonlinear DFL controllers proposed here can greatly improve transient stability and achieve voltage regulation.
X& = XT + :xL + x*; xc = XT+ $XL+ x:; xT reactance of transformer: x,_, direct axis reactance: x,, direct axis transient reactance; XI. reactance of transmission line; x, = XT + ix,; X,d mutual reactance between excitation coil and stator coil: V, infinite bus voltage. 1. Introduction In recent years, a great deal of attention has been paid to the application of feedback linearization approaches to power systems (see e.g. Marino, 1984; Lu and Sun, 1989; Mielczarski and Zajaczkowski, 1989, 1990, Ilic and Mak, 1989; Chapman and Ilic 1993). In our previous paper (Wang et al., 1993), we examined the direct feedback linearization (DFL) (Gao er al., 1992) technique aimed at achieving transient stability enhancement and voltage regulation of power systems. Implicit in that paper is the assumption that many parameters in the power system are known, a priori. In many cases (e.g. inertia and time constants) this assumption since these parameters are is completely reasonable, essentially constant, and can be determined from power system design information. Key situations where this assumption proves restrictive for transient stability analysis are when it is also assumed that the transmission line reactance xL and the infinite bus voltage V, are known in the transient period. These parameters will not, in practice, be constant. and are poorly known. In these cases, parameter adaptation (see Goodwin and Sin, 1984; Pierre, 1987) can be smployed to deal with uncertainties in .rL and V, when a fault occurs. In Wang et al. (1994), a nonlinear adaptive controller was proposed to achieve transient stability enhancement and voltage regulation. But when a large fault occurs, especially very close to the generator terminal, reactances of transmission lines in a power system change a lot, and the configuration of the electric power system varies very quickly. It is very difficult for an adaptive controller to adapt to parameter variations. Wang et al. (1994) showed that for the example system, when a fault occurs at one-tenth of the transmission line from the generator terminal, the system using an adaptive controller can no longer maintain transient stability. More recently, a few papers have considered oscillatory stability enhancement of power systems using robust control techniques (Chow, 1988; Othman et al,, 1989; Pai an Sauer, 1989). Rather than the approximate linearization model used in those papers, here a nonlinear classical model of the power system is considered. This is consistent with normal power system modelling practice (Anderson and Fouad, 1977; Bergen, 1986), where more detailed linear models are used for oscillatory stability and simplified nonlinear models for large disturbance stability, i.e. transient and voltage stability. In this paper, we shall use a robust control technique based on the Riccati equation approach, together
List of symbols 6(r) power angle;
w(t) relative speed; P,(t) mechanical input power; P<(t) active power delivered to bus; w,) synchronous machine speed, % = 27&i D per-unit damping constant; inertia constant (in s); transient EMF in the quadrature axis; E;(Z EMF in the quadrature axis; W) equivalent EMF in the excitation coil; GO) Td0 direct axis transient short-circuit time constant; steam valve opening: X,(r) input of power control system; 4(r) excitation current: 40) quadrature axis current; I,(t) reactive power; Y$:,’ generator terminal voltage; input of the SCR amplifier of the generator; 40 k, gain of excitation amplifier: TT time constant of turbine, with typical numerical values of 0.2-2.0 s; KT gain of turbine; constant (per unit), typically R = R regulation 0.05 p.u.; TG time constant of speed governor, typically around 0.2 s; KG gain of speed governor;
-
*Received 13 Auril 1993: revised 11 Mav 1994: revised 11 April 1995; recieied in final form 27 SepteAber 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hassan Khalil under the direction of Editor Tamer Bagar. Corresponding author Dr Youyi Wang. Tel. +65 7991423; Fax +65 7912687; E-mail [email protected]. t School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263. $ Department of Electrical Engineering, The Univeristy of Sydney, NSW 2006, Australia. 611
Brief Papers
612
excitation controller and a new robust nonlinear coordinates controller (excitation and fast valving controller) to enhance transient stability and to achieve voltage regulation of an example power system under a symmetrical three-phase short-circuit fault. 2. Dynamical model A simplified dynamical model of a single machine-infinite bus power system is considered in this paper. The model consists of a single generator connected through two parallel transmission lines to a very large network approximated by an infinite bus, and is shown in Fig. I. A model for the generator with both excitation and power control loops can be written as follows (Bergen. 1986: Anderson and Fouad. 1977): mechanical equations d(f) = w(t).
0)
&J(l) = - &I) L
+$[P”,(t)
-- P,(t)]:
Stage 3: The fault is removed by opening the breakers of the faulted line at t = 0.25 s. Stage 4: The transmission lines are restored at I = 1.4 s. Srage 5: The system is in a postfault state. Fault sequence 2 (permanent fault). Stage 1: The system is in a prefault steady state. Stage 2: A fault occurs at t = 0.1 s. Stage 3: The fault is removed by opening the breakers faulted line at t = 0.25 s. Stage 4: The system is in a postfault state.
3. Design of a robust nonlinear excitation controller Since we only consider the excitation loop, P,,, is constant, and the plant can be modelled by (l)-(3). From the results given in Wang et al. (1993). if the parameters in the power system are known. we can design a DFL control law
(2)
u&r(r),
II =
z
generator electrical clynamics E;(t) = [Et(t) -~ E,(t)] ,f;: d,,
of the
{W- T’[Q,(t) +$]4)) T&lJ(t) - pew+pITI”.
xyc
(3) where
(13)
x&r)
i+(t) is the new input and
turbine dynamic5 P”,(f)
=
(4)
; P,(l) + $h X,(t): I I
We now linearize
turbine value control X,(t)
= -;
(8
X,(f)
+9
P,(r) ~ &
w(t)
1
the model
(l)-(3)
AS(t) = w(t),
(5)
electrical equations
b(t) = -&w(t)
E,(t) =2
\
E;(t) ~ xd--\;
-6, EXI) = k,u,(t).
V, cos 6(t).
APJt)
(6) (7) (8)
as follows:
- gAP,(t),
= ~ +, AP,(t) + $, “t(t)
Remark 3.1. The compensating law (13) is simple and practically realizable, the linearization is valid over the whole working region (O”< s(t) < 180”), and the linearized model is independent of the operating point of the system.
(9) (10) (II) v(r) =
$
\
[x$E$t) + Vi.rf
+ 2x-,.r,*‘,,P,(t) cot 6(t)] 7
(12)
The fault we consider in this paper is a symmetrical three-phase short circuit fault on one of the transmission lines. The total reactance of the transmission line is x, (= 0.4853). The parameter A is the fraction of the faulted line to the left of the fault. If A = 0. the fault is at the generator bus, A = 0.5 puts the fault in the middle of the line, and so on. Two different fault sequences are as follows. Fault sequence I (temporary fiurlt). Stage 1: The system is in a prefault steady Stage 2: A fault occurs at t = 0.1 s.
From the above disussion, we can see that the compensating law (13) contains xL and V,; if we know xL and V,. a DFL control law can be easily designed. However, in practice, a fault can occur at any time and anywhere on the transmission line. When a large sudden fault occurs, the reactance of the transmission line xL and the infinite-bus voltage V, change a lot, and it is impossible to know XJ_ and V, during the transient period. In this paper, the changes in I, and v, are treated as parametric uncertainties. Considering the uncertainties in xL and V,, we have
+(T’ + +~;o(% -4)$$ w(t) + (7’ + 1,
AT’)Q,(t)w(t)
ad f
AT’)o(t)
state. where
~ P,,,,,
M(t) = 8(t) ~ 6,,, AP,(t) = P,(t) - P,,,,, T’ + AT’ =*
Generator and AV, denote respectively, and
At, Fig. I. Generator
connected through infinite bus.
transmission
lines to an
the
uncertainties
0%
I in
T;,,.
xl_ and
V,
613
Brief Papers Using the control law (13) gives the linearized follows:
model as
A&) = w(r) G(r) = -&o(t)
-ZAP.(r),
Ak(r) = -[ $, + c(r)]Ap.(r) + [+ +
(14)
+ [ $, + &)]uXr)
d~~]A~fQ&)o(~)
+ [++ r(r)]Ai+o(r). Define XT(r)= [As(r) w(r) Ape(r)]. model (14) can be rewritten as
i(r) = (A where
ro
Then
+ AA)x(r) + (I3 +
the
linearized
AB)t+(r),
(15)
1
= -k,AiS(r)
01 0
)
’
0
B=[O
AB=[O 0
+ p(r)][AF’
--CL(f) 0 1
u,-(r) = -k,A$(r)
j&f
0
p = [$
[
AA=00 0 P
p(r)lT,
+ AT’Q.(r)].
llT,
E, =
i-l 00
E*= [O 0
l]T,
F(r) =
0
0 & 0
0
1
-p(r)
,
1,
Solution of the design problem of a nonlinear robust control law u&r(r), r] to transiently stabilize the power system model (l)-(3) is equivalent to solving the robust control problem for the linearized plant (15) which involves solving the following algebraic Riccati equation: + P(A - BR-‘ET&) + P(a2DDT - BR-‘BT)P + ET(I - E,R-‘E:)E,
= 0,
(16)
Theorem 3.1. The power system (l)-(3) under a symmetrical three-phase short circuit fault is transiently stable via the DFL nonlinear control law (13), and + E~E,)x(r)
(17)
if there exists a positive-definite solution P > 0 of the Riccati equation (16). If the power system (l)-(3) is transiently stable via the control law (13) and (17) then the power system under a symmetrical three-phase short circuit fault can avoid the loss of synchronism. Moreover, in the postfault period, lim lAS(r)l = 0, ,--rcO
lim Iw(r)l = 0, r--r-
= constant.
The control procedure is as follows. Step 1. The fault occurs at r = to and the robust control law is r&,,(r), rl, (13) where u,,(r) = -k,AA(r) - k,w(r) - k,AP,(t). Srep 2. At r = r,, the feedback law is switched to ur[un(r), r], where
r+&) = -k,,AV,(r)
where R = ETE, >O (Khargonekar et al., 1990, Xie er al., 1992). Our main ‘stability’ result is as follows.
u&) = -R-‘(BTP
lim IAs ,%+r
lim IAV,(r)l= 0. I--+X
5 azZ (CY> 0), y is a constant and lpi 5 p.
(A - BR-‘E;E,)TP
(18)
Thus, in many cases, the generator terminal voltage is not the same in the postfault state as in the prefault state, which is undesirable in practice. To achieve both transient stability and voltage regulation, we employ the above robust nonlinear controller in the DFL excitation control scheme proposed in Wang et al. (1993). Then, we achieve voltage regulation, i.e.
where D, E, and E2 are constant matrices, with
D = [O 0
- k,AP,(r),
but only that
&I, 10
- k,w(r)
f\i~ IAS(r)l = 0,
AT’ and AT’ are bounded; so is p. The matrices AA and BE can be expressed as
ABI= WW,
- kpAP,(r).
where ha(r) = d(r) - aO. Pwing to the difference in initial values between s(r) and s(r), it is clear that by introducing s(r) in the control law (18) we can no longer obtain
p(r),
[AA
+ EIE,)x(r) - k,w(r)
Since thepower angle s(r) cannot be measured, the estimate of s(r), s(r), can be obtained by using a S detector (see Wang et al., 1992) s(r) = o(r). Then udr) becomes
)
-& -2 0 -+
L
FT(r)F(r)
In the robust control scheme proposed in this section, the robust DFL nonlinear control law is udut(r), t], (13) where t+(r) = -R-‘(BTP
00
A=
fault is equivalent to designing a robust linear control law the linerized plane (15) with parametric uncertainties. Using the results in Khargonekar et al. (1990) and Xie et al. (1992), we know that a linearized plant model in the form (15) is asymptotically stable if there exists a stabilizing solution P>O of the Riccati equation (16). Moreover, a suitable feedback law is given by (17). Since designing a nonlinear robust control law udr) to stabilize the power system model (l)-(3) is equivalent to designing a robust control law q(r) to stabilize the linearized plant (15) we can conclude that the nonlinear control law (13) and (17) can transiently stabilize the power system (l)-(3) for all allowed 17 changes in x,_ and Vs. The proof is complete.
udr) to stabilibe
lim IAP,(r)l = 0. ,-=
Proof By using the DFL technique, the power system (l)-(3) can be linearized as (14) or (15). To design a robust nonlinear control law r+(I) to transiently stabilize the power system (l)-(3) under a symmetrical three-phase short-circuit
- k,,o(r)
-k,,&(r).
The design procedure for kV1, k,, and kp, can be found in Wang et al. (1993). The switching time I, is determined by trial and error for a given system. 4. Simulation results I In our example, we can obtain, from (15) that A=[;
-:.625
&;J9,l,
-_[i
B = [0 0 0.516351T, AR = [0 0
& jrj, Il(r)
B=[$+&r)][AT’+AT’Q,(r)]. We obtain that_ Ip( s 0.2347, (AT’150.6053, 3.09050 and IpI 5 /3 = 3.23. Choosing 100 E,=
o;Po,
E2 = [O 0
l]T,
L-1001 D=[O
0
119
F(r) = [0
Y!
p(r)
1,
IAT’\ =
614
Brief Papers
y =0.2347 and a’ = 2 X (0.2347)‘>0, we have R = E:E, = 1 > 0 and FT(t)F(t) 5 &I. Solving the Riccati equation (16) gives [k,
k,
k,] = j-1.31
- IS.78
61,631.
The robust nonlinear control law is ur[+(r), t], (13) where
(A = 0.05) the excitation controller cannot keep the system transiently stable. In this section we shall discuss the design of a robust nonlinear coordinated controller. The plant considered here can be described by (l)-(5). By using the robust DFL nonlinear compensating law (13). we can linearize the plant as follows:
i(t) = [A + AA(t)]x(t) + [B + AB(t)]u(t), where and
Ap,(l) = J’,(r) - P,,I. V,,(f) = 1.31[&) - S,,] + 157&J(r)
u(t) = [u&j
- 61.63[P,(r) - P,,,J,
x(t) = [As(t)
u,2(r) = -47.03[V,(t) - V,,,]+ 6.‘)3~(t) - 28.60[P,(r) - f,,,,,].
0,,=314.159,
D=5.0. T,=O.2s,
H=4.0s,
T;,,=6.Ys,
k,-I.
KC;-1,
K,=l.
Tr=2.0s,
M’,(t)
M’,,,(t) AA’,(t)]‘,
1
0
0
0
0
~- D 2H
_J!!L 2H
3 2H
0
O
1 -0 T’
0
0
~- 1 T,
0
0
0 _K,_L r,; Rwo
max ik,udt)j = 1.8 p.u.
5. Design of robust nonlinear coordinated controller In the previous section, we have seen that when a large sudden fault occurs close to the generator terminal
o(t)
0
The operating point is 6,) = 72”, P,,,()= 0.9 p.u. and V,,, = 1.Op.u. The physical constraint on the excitation voltage is
Firstly. we show the responses of the power angle under different fault locations (A = 0.5, 0.1, 0.055 and 0.05) in Fig. 2 (t, = 0.7 s). From the results, we can see that when a fault occurs close to the generator terminal (A = 0.05) the system, using only the excitation controller (with the excitation limit), cannot maintain synchronism. Next we test our new robust nonlinear excitation controller (A = 0.5, t, =0.7s) under the two different fault sequences discussed in Section 2. Figure 3 shows the [es&s. The system responses at a different operating point (a,, = 52”) So = 47”. I’,,,,,= 0.45 p.u. and V,,, = 1.003 p.u., are shown in Fig. 4. From the results, we can see that, using the new robust nonlinear excitation controller proposed in Section 3, the system can achieve both transient stability enhancement and voltage regulation.
&(t)lT. Pdt) = P&J -Pm,,,
A=0
xd = 1.863, x; = 0.257. xr = 0.127. x,‘, = 1.712.
- XEO,
0
The parameters of the system under consideration are R = 0.05 p.u..
Ax,(r) = x,(t)
00
0
0
-$
00 0 0
1s; 1 0
-CL(I) 0 0
AA=
5 Tr
L
00 00
0 0
, AB=
1
00 00
11 0 ? 0 CL(t) 0
and F(t) and p are as defined in (15). Because of the mechanical behaviour of the steam valve, we expect it to come into action only in the transient period. To ensure the stability of the overall system after the power control loop switches off, the feedback control law is expected to be
The feedback linear-quadratic follows. Step 1. Assume constant. Solve robust feedback
gain matrix can be obtained by using optimal theory. The design procedure is as that the mechanical input power P,,,(t) is the Riccati equation (16), and obtain the control law for the linearized model
q(t) = -k,,Aa(t)
- k,,o(t)
- kp,APe(t);
the control law for the real plant is t+[uAt), t], (13). Step 2. Apply the control law (13) to the plant, and find the closed-loop system through the excitation loop. Then find the feedback control law for the plant through the power control loop PXr) = pc(t) - P,” = -k,,A6(t)
- kwZW(t) - kmAPr(t)
- km,APm(t) - kxzAXE(t).
i Time (set) Fig. 2. The power angle for different fault locations.
In order to achieve voltage regulation, we introduce generator terminal voltage feedback through the excitation loop. The control sequence becomes the following.
615
Brief Papers /sit
sequence 2
r
Fault sequence t
0.6 k 0
1
3 2 Time (see)
3
,
5
Fig. 3. The responses under different fault sequences.
Step I. The fault occurs at 8 =r,, &@%r(rXfl? and
the feedback
law is
using a S-detector (Wang ef cxl..,1992). The switching times t, and tZ can be determined by trial and error in simulation, and f, can be obtained from the circuit breaker data. Remark 5.1. The reasoning behind this design of the control
where
sequence is as follows. t+,(t) = -&Ad(t)
- k,,w(l)
- kPIAP&)
Srep 2. At t = t,, the feedback law switches to tl&r~(i)~ t], and P&t) = Pmo, where nt&f is as above. Srep3. At t = fa, the feedback faw switches to z+&&), i], and P,(r) = !$,a, where u&) = -k,,Av,(r)
- kw34)
- kmAPeW,
and the estimate of the power angle S(l) can be obtained by
1. In Step 1, we introduce power angle, s(r), control (and so refax the generator terminal voltage V,(f) controT) to improve the transient stability of the power system. 2. When the system is transiently stable, the governor input is switched off to reduce the control cost in PC(t). 3. After ta, the system is in the post-transient
period. The
Fault sequence 2
1 Time (set) Fig. 4. The responses at different operating points.
2
1
Time (set)
4
t,
Brief Papers generator terminal voltage Vt(t) control is introduced to relax the power angle, 6(f), control and to achieve the voltage regulation. 6. Simulation results II In this section, we shall show, through simulation results, that transient stability improvement of the system can be achieved by employing the new nonlinear coordinated controller in different cases. Consider the plant as described by (l)-(12). The system parameters are given in Section 4. and the operating point is S, = 72”, Pm0= 0.9 p.u., V,,, = 1.0 p.u. The physical constraint on the excitation voltage is the same as in Section 4, and the physical constraint on the steam valve opening is 0 5 X,(t) 5 1.0 p.u. The control sequence is as follows. 1. The fault occurs at I = 0.1 s. the control @r,(t), 4 and
Step
law is
PC(t)= 71.86A@r) - 60.47w(r) + 32.39Af’,(r) - 4185AP,,,(r) - 99.42AX,(t) + P,,,, where t+,(t) = 1.3111&r) + 1578w(r) ~ 61.63AP,(t).
ii TIME
(me.)
Step 2. At
t = 1.2s. the feedback law switches to udt+,(t)] and P,(t) = P,,,,,. Step 3. At t = 1.5 s. the feedback law switches to ur[u,(r), rj and P,(t) = f,,,,,, where u&f) = -47.03AV,(t) + 6.93w(r) - 28.6Af,(r). The simulations are reformed for the folloiwng cases. Case 1. First we compare the responses from the new robust
nonlinear coordinated controller and the nonlinear excitation controller (Wang et al., 1993) under fault sequence 1. The initial values of the S detector (see Wang et al., 1992) is 79”, i.e. I& = 7Y, and the location of the fault is 0.1, i.e. A = 0.1. Figure 5 shows the responses of the power angle s(r) and the generator terminal voltage Vt(t). We can see from the simulation results that for the example system, the new robust nonlinear coordinated controller can achieve better results than the nonlinear excitation controller (Wang et al., 1993). Case 2. The responses of the power angle S(f) under the different fault locations A = 0.5, 0.1, 0.05 and 0.01 (S,, = 79”)
Fig. 6. The power angle responses.
are shown in Fig. 6. Comparing with the results in Wang et al. (1993), we can see that, using the nonlinear coordinated
controller, the system loses synchronism when A =O.Ol. When using the robust nonlinear coordinated controller, the system can still keep transient stability even when A = 0.01; i.e.. using the robust nonlinear coordinated controller, the system can keep transiently stable even in the case where a large sudden fault occurs close to the generator terminal. Case 3. We test the system under different faulj sequences. The simulation results are shown in Fig. 7 (6,) = 79” and A = 0.1). The simulation results show that the robust nonlinear coordinated controller can effectively improve the transient stability of the example system under differnt fault sequences. The simulation results also show that the robust nonlinear coordinated controller can achieve both transient stability enhancement and voltage regulation under different fault sequences.
onlinear
1 TIME
(sec.)
Fig. 5. The responses for different controllers.
DFL
csa&dkI
1 TIME
1 (see.)
1
Ii
617
Brief Papers
a TIME (~4 Fig. 7. The
TIME
&ee.)
responses under different fat& sequences.
Case 4. We now test the system responses under different fault sequences (A =O.l) at a different operating point: 6, = 42”, Pm0= 0.45 p.u., V, = 1.Op.u. The fault sequences and the control sequence are as before. Figure g shows the responses of the Rower angle 6(f) and the generator terminaf vottage Vt(t) (So = 477. The simulation results show that the robust no&near coordinated controller can achieve both transient stability enhancement and voltage regulation, regardless of operating points. Remark 6.1. Comparing with the results in Section 4 and in Wang ei at. (1993), we can conclude the following from the simtdation rest&s. 1. The robust nonlinear coordinated controller proposed in this paper can effectively improve transient stability of the power system. The system can keep transiently stable even in the case where a large sudden fault occurs close to the generator terminal.
2. The robust nonlinear coordinated controller can achieve good transient stability results, irrespective of the operating point of the system and the fault sequence. 3. The robust nonhenar coordinated control can achieve both transient stability enhancement and voltage regulation in the presence of ail admissible changes in .rL and V’. 7. Conclusions This paper has applied a Riccati equation approach, combined with the so-called direct feedback linearization (DFL) technique, to design robust nonlinear controflers for power systems, A new robust nonlinear excitation controller and a new robust nonlinear coordinated controller have been proposed to achieve both transient stability enhancement and voltage regulation in the presence of all admissible changes in the transmission line reactance xL and the infinite bus voltage V,. Design procedures have been developed for both types of controllers. ‘The robust nonlinear controllers were
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TIME (sec.) Fig. 8. The responses at different operating
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tested through an example system in different cases. and the controllers have been compared with the nonlinear DFL controller in Wang et al. (1993). The simulation results show that transient stability of power systems can be effectively improved by using the proposed robust controllers, regardless of the system operating point, the fault location and the fault sequences. The robust nonlinear coordinated controller can maintain transient stability of the system even when a large sudden fault occurs close to the generator terminals.
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