Electric Power Systems Research, 7 (1984) 103  108
103
Simultaneous Optimization of the Adjustable Parameters in Multimachine Power Systems G. N. ZHENENKO
Department of Electrical Engineering, Iowa State University, Ames, IA 50011 (U.S.A.) H. B. FARAH
University of Aden (South Yemen) (Received August 1, 1983)
SUMMARY
This paper considers a novel approach to simultaneous optimization o f the adjustable parameters of all the stabilizers in multimachine power systems to obtain the best dynamic performance. It is based on frequency domain analysis using the domainseparation method, Nyquist's stability criterion, and the frequency criterion of quality. With the proposed technique, the condition for stability o f the system and the range o f variation o f the adjustable gains can be determined. In addition, the values o f the adjustable parameters can be selected for the best system dynamic response. The method is especially effective for complex system investigations.
1. I N T R O D U C T I O N
It is well known that the provision for stability and acceptable damping of small oscillations of all normal and after~emergency operations is one of the requirements placed on the automatic excitation control system of generators. It is not possible to define experimentally the required control law and to choose the required adjustment of control channels, since not all the operating conditions of the generator and the system can be covered. Analytical investigations of steady state stability associated with the choice of the law and the optimum adjustment of the stabilizer have thus become urgent [1  9]. Analytical techniques commonly used are the frequency domain method, such as that of Nyquist, and the root locus technique [2  5]. 03787796/84/$3.00
Eigenvalue analysis has also been proposed
[6 8]. The techniques currently in use aim at tuning the parameters of the power system stabilizer of a particular generator to give adequate gain margin for the desired frequency range, i.e., the local mode and the interarea mode. System measurements are required for this technique, and the tuning is done essentially by a 'trial and error' procedure. Settings of other regulators in the system are held constant. The eigenvalue analysis method requires extensive computations, since a detailed model of the system is required. Furthermore, sensitivity analysis techniques are required to obtain the permissible range of variation of the excitation system parameters, or again a 'trial and error' method would be needed. It is more expedient to use a domainseparation (Dseparation)method [ 1 0  1 5 ] which enables direct calculation of the stability areas. However, existing programs using this method determine the optimum values of the adjustable parameters and maximum degree of stability by calculating the areas of equal degree of stability [11 14]. This procedure is rather tedious. The method presented in this paper allows for simultaneous optimization of the adjustable coefficients of all the excitation systems for the best dynamic behavior of the power system. In addition, the method provides the possibility of direct determination of the steady state stability areas using the experimental frequency characteristics. The method is based on: (a) Nyquist's stability criterion and the frequency criterion of quality, © Elsevier Sequoia/Printed in The Netherlands
104 (b) an optimization procedure, proposed by Nelder and Mead, to obtain the o p t i m u m values of the adjustable parameters, and (c) the 'Dseparation' method.
k .A(s)J
= Wi(s)~j(s)
i, j = 1, . . . , n
i =/=j If the adjustable gains Kil and Ki2 of the stabilizers are linearly independent, then the transfer function Wi(s) can be written in the form
Wi(s) = K i l W i l ( 8
) + Ui2Wi2(8 )
G(jco) = tn  g(jco) W(jco)l (1)
I LE,D,,,,(s)j
fEEEFD 1A(S)II _ FD2A(S) FDnA(S)J
:,,(s)] Wl(s)W:(s)"'. Wn(sI fi'~'(sq
~s(s)
(4)
where Wil(s) and Wi:(s) are the transfer functions for the appropriate channel of the stabilizer i. Substituting s = jco, where co is the frequency in (3), and taking (4) into account, we obtain Nyquist's plot (characteristic plot of the system) as a function of the adjustable gains Kil and Ki2 of all the stabilizers:
:A(s) ] = Iila(s)~
...
(3)
V i i ( s ) = 1   Wi(s)'ffii(s )
Under the steady state stability investigation it is very convenient to use the description of the transient responses for small disturbances in the form of a transfer function matrix (frequency characteristics) [16, 17]. The matrix describes both unregulated and regulated excitation systems. Matrix equations describing these relationships are given below:
U,.,,,(s)
V(s) = [ E  ~r(s)W(s)l
where the elements of G(s) for row i and column j have the following form:
2. PROPOSEDSYSTEMMODEL
rTrll(s) 7r12(8)... 7rln(S)1 rEFD1A(Sq lr::(s) ... ~r~"(s)lIEFD2A(s) • ] 1lr21(s) .
Equations (1) and (2) are related to the characteristic determinant by
(2)
A(s)1 where 7ru(S ) = 1ria(s)/EFDi(S) are eigen (i =j) and mutual (i ¢ j) transfer functions of the regulation parameters; EFDj(S) is the deviation of the rotor voltage in generator j (J = 1, 2, .... n); ~riA(s ) is the deviation of condition parameters used for the automatic excitation control; Wi(s) = EFDfA(S)/?riA(S) is a summary of the transfer function for the automatic excitation control (i = 1, 2, ..., n).
(5)
We note that the order of eqn. (5) is equal to the number of stabilizers, which is considerably less than the initial order of the equations describing the system. The frequency characteristics Wil(jco) and Wi2(Jco) are computed on the basis of the transfer functions of the individual elements of the automatic excitation control. The characteristics 7ru(jco) = 7ri~(jco)/ EFDja(jco) are calculated when a harmonic signal EFDia(jco ) is assigned consecutively to each generator. We note that in the capacity of the characteristics lrij(jco), Wil(jco) and Wi2(Jco), we can use the experimental frequency characteristics obtained in an actual power system or physical model of a power system.
3. THEFREQUENCYCRITERIONOF QUALITY For optimization of the adjustable gains it is very important to choose the criterion for the estimate of the quality of the transient responses for small disturbances. Use of the degree of stability ~ [9] (~ is the root of the characteristic equation having the lowest
105 value for its real part) is not convenient, because we must c o m p u t e all the frequency characteristics lrts(s) and Wt(s) for different values of q. It is more convenient to use the frequency criterion of quality [18] and set the connection between the frequency criterion and the maximum degree of stability. We represent the characteristic determinant (3) in the form V(s) 
D(s)

Q(s)
b o ( s   s , ) ... ( s   s , ) q o ( s   p , ) ... ( s   p . )
(6)
where D(s) is the characteristic polynomial of the closed system; s,, . . . , s , , P l , . . . , Pn are, respectively, the roots of the characteristic equation of the closed and nonclosed system. Equation (6) can be written as
d50
tffil
(7)
=
qo f i (s   P t ) ( s   P t * ) tffil
where st, %*, Pt and Pt* are, respectively, the conjugate roots of the closed and nonclosed systems: St, St* = OQ +J50t Pt, Pt* =  & t + J ~_ i
If we substitute s = j50, eqn. (7) can be transf o r m e d into the following form: G(j50) =
Ct f i [((~t2 502)2+ 40/t2502]I/2}
b0 e x p ) j t=l
iffil
= A exp[jCG (50)]
(8)
where A and CG(50)  ~nffil(Ct t~i) are, respectively, the amplitude and phase angle of Nyquist's plot; cSt2 = cot 2 + at 2, tan C t
coi 2 = ~ t 2 + &t2
= 2 0 1 t 5 0 / ( (fOt 2  
502)
tan ~t = 9"o~t50/(05t 2  ¢0 2) The rate of change of Ca(50) is given b y
(9)
We note that (9) is resonancedependent and acquires a maximum value at the frequencies 50 = cSt or 50 = cSt corresponding respectively to the roots of the closed or nonclosed system having the lowest value for its real part. If the imaginary values of the roots cSt and ~ i axe significantly different, we can write eqn. (9) for 50 = ¢5~ or 50 = ~ t in the form dCQ(50)/d50 ~ 1/oq
or
dCo(50)/d50 ~ 1/&t
from which
[dCa(w)/d50]l]max
or
bo l ' I ( s   s t ) ( s   s t * ) G(s)
t=I
_ 2&t [(¢5t2  ¢o2) + 2502] I
O/ira ~ n
t 2at[(¢57 ¢o2) + 2502]
dCa(50) _
&tin ~ [dCo(50)/d50]ll ma~
(10)
Notice that the quality of the transient response is determined b y the value of the degree of stability arm and in only one case Can we obtain a good result, i.e. when ¢5t and ~ t are significantly different. We now examine eqn. (9). If the dominating complex roots of the closed and nonclosed systems coincide, then under the condition 50 = ¢5~ = ~ t the rate of change of Ca(co) is given b y dCa(50)/d50 = (1/ut  1)/&t
(11)
If the real parts of the roots of the closed and nonclosed systems are equal, then eqn. (11) equals zero. Thus, in the c o m m o n case, we cannot use eqn. (9) to estimate the quality o f the transient response in power systems. Since the degree of stability ~tm is determined by the rate of change of the phase angle of D(j50) (first term in eqn. (9)), it is necessary beforehand to obtain the values of the phase angle of D(j50): CD(50) = Ca(50) + CQ(50)
(12)
and dCD(50)/d50 = d[Co(50) + CQ(50)]/d50
(13)
where CQ(50) is the phase angle of the plot of the characteristic polynomials of the nonclosed system. The values of
[email protected](50) we obtain by computing the frequency characteristics ~is(J50). Thus, using (13) it is easy to obtain
106
the rate of change of @v(co) which corresponds to the degree o f stability &rn [d~D(a~)/dc°]l[max
(14)
Application o f the criterion o f quality Let L be the different combinations of adjustable parameters. Let &mi be the minim u m value of the real part of the r o o t (found by (14)) for the combination i of the adjustable parameters, i.e., ~mi =
f(K11, K12,
.., KN2)
5. EXAMPLE
As an example, we consider the steady state stability and tuning of the adjustable coefficients of the fast excitation system for a power system consisting of three generators (Fig. 1). The excitation of the generators is automatically controlled with respect to terminal voltage deviation, first derivative of the terminal voltage, the deviation and first derivative of the terminal voltage frequency and the first derivative of the rotor current.
(15)
then the optimal combination of adjustable gains corresponds to (Xmm = min(0ml, am2, .  . , (~mL)
(16)
We note that (14), (15), and (16) are used with specified values of the parameters Kit. To arrive at an o p t i m u m value (for the best dynamic performance) in (16), an optimization procedure is used. The NelderMead m e t h o d [19], which minimizes a function of N independent variables using the N + 1 vertices of a flexible polyhedron, is suggested for this procedure.
4. C O M P U T A T I O N AREAS
OF
THE
DSEPARATION
After the optimal combination of the adjustable gains of all the stabilizers is defined, the Dseparation areas are calculated. These areas are obtained for any two adjustable parameters, when all the other parameters are fixed. By setting eqn. (5) equal to zero and selecting the adjustable coefficients Kil and Ki2, we obtain expressions defining the boundary of the steady state stability for stabilizer i. The expression defining this b o u n d a r y is G(jco) = Aii(jco)


[KilWil(J(D) + Ki2Wi2(jco)] ×
X ~ ~fii(jco)Aij(jo9) = 0
(17)
i=1
where Aij(jco) = (1)~+JMi#(jco), M~j(jco) is a determinant matrix of G(jco) corresponding to row i and column j. We note that the Dseparation areas are c o m p u t e d for each stabilizer when the other stabilizers' adjustable gains are optimum.
Z6V Fig. 1. System circuit.
The problem is to find the steady state stability areas in the plane of adjustable coefficients o f deviation K~I and first derivative of frequency of the terminal voltage K~2 and the optimal combination K n and Ki2 for all generators corresponding to the best degree of stability. The adjustments of the voltage deviation and derivative channels and derivative of the rotor current were taken to be constant in the calculations. The voltage deviation channel gain was assumed to be Koy = 50 un.ex.nom./un.st.volt., KIv = 4.3 un.ex.nom./un.st.volt, s for the voltage derivative channel and KI =   1 . 5 un.ex.nom./ un.rot.cur, s for the current derivative channel. The results of optimization of the adjustable coefficients of all the stabilizers with the help of the NelderMead technique are given in Table 1. The stability areas in the plane Kil and K~2 for the generators 1, 2 and 3 are shown in Figs. 2, 3 and 4, respectively. These areas were c o m p u t e d under conditions when the adjustable gains w e r e optimum. The points A, B and C in Figs. 2  4 correspond to the o p t i m u m values of these gains. The numbers in Figs. 2  4 show the frequencies of swing in rad/s. The results of the investigations show that with joint control of G1, G2 and G3, there is
107 6. CONCLUSIONS
TABLE 1 Optimum values of adjustable gains G1
G2
G3
K11
KI~
K21
K22
K31
17
0.2
8
1.6
5
8
I
6
,
K32 0.4
The technique presented allows direct determination of the permissible ranges of variation of the adjustable parameters of the automatic excitation control and calculation of the optimum values of these parameters satisfying the best dynamic performance. The results shown in the paper illustrate the efficiency of the new method in determining the adjustable parameters of the regulators in multimachine power systems.
4
12 2
REFERENCES A. 20
40
2 Fig. 2. Stability area for generator G1.
15~
131I
B° 10
z0 j
21
0i 1 Fig. 3. Stability area for generator G2. i0
,
_~,=
2 C
2o
0 2
L
40
[ 
Fig. 4. Stability area for generator G3.
a sharp improvement in system performance. The degree of stability increased to ~ = 1.41 sl. We notice that without control of G2, the system will be unstable.
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