Application of Optimal Control Theory and Root Locus Method to the Design of Linear Feedback Controllers for Synchronous Machines

3126

M. Ara ki, K. Kob a shi and B. Kondo

EXAtlPLE As an example, we studied the system with the values of constants given in Table 4. The prefault equilibrium is

°

if 01 = 1.56 p.u. 01 = 0.99 rad By the method described in §3, we obtained the next values of k.: I k1=8.37 k2=5. 25 k =-17.63 k4=-3 1.62 3 The corresponding values of q.I were ql=1000, q2=1000, q3=30.0 and the response of the sytem for the rectangular disturbance 6Tm =

1o.~o

otherwise is as shown in Fig.3. For the above value of ki' the postfault equilibrium is 002 1.29 rad if 02= 1.72 p.u. where z02 depends on h · From this we obtain 5 G(s) 0.073s(s 2+0 . 31s+23.17) The root loci of G(s)is as shown in Fig.4 . To obtain a suitable damping property, we required all the poles of the imaginary system to satisfy 1 tan- ! 1m( so) ! / ! Re (s o) !<45° i.e. to be in the sector shown by the dotted lines in Fig.4. Thus, we choose h5=-50 . 0 The values of the feedback coefficients are h =-8.78, h2=5. 25, h =-4. 30, 1 3 h4=-3 1.62, h =-50.0 5 The transient responses in the postfault state is satisfactory as shown in Fig.5. CONCLUSION We proposed a method to design a linear feedback controller for the synchronous machine based on the double criteria: (a) good voltage regulation in the prefault state (b) and good transient behavior in the postfault state So long as the example is concerned, the designed system exhibits satisfactory performance both in the prefault and postfault states. However, the following points are left for further studies. First, we have some arbitrariness in choosing the weights q.I of the performance inde x, and the choice of qi have fairl y large influence on the postfault behavior . Actually , we observed , in the study of examples, that there are sevpral sets of weights which result in similar

response in the prefault state but produce fairly different behavior in the postfault state. Thus, it seems important to clarify the influence of the weights q.I on the postfault stability property. This influence can be decomposed into two relations: i.e. the relation between the weights q.I and the feedback coefficients k.I , and the relation between the k.I amd the best location of the eigenvalues o! A2 (which is obtained from the root loci of G(s)). The first relation is not simple but can be estimated from the frequency domain equation for the LQ optimal problem [lJ. Concerning the second relation, we are in a lucky situation. Since k i appear only in the third row of A, the numerator of G(s) does not include k.I and the denominator depends linearly on k .I . (N ote that the equilibrium values 002 andi f02 do not depends on k., fortunately.) So, it is not difficult I ~ to tell how the root lo ci of G(s) de pend on k . . Thu s , we hav e good chance to establish a s~stematic way to search for the weights which produce nice behavior in the both states. Second, we onl y used the eigenvalues of A2 as the criterion to evaluate th e po stfault behavior. But, the behavior ot the system during the emergency state also influences the postfault stability and a consideration on this point is expected. Third, we considered only two extreme states: i.e. the prefault state with a fi xed operating condition and the corresponding postfault state. But the controller is required to exhibit satisfactory performance under variety of operating conditions. Some consideration on this point seems necessary. One positive way to attack this aspect wo uld be the introduction of piecewi se linear controller. Forth, consideration of the multimachine ca se must be made. Acknowledgements: The auth ors are thankful to Dr. J. Matsuki and Mr. Y. Osa wa of Dept . Electrical Eng., Kyoto University f or their helpful suggestions. REFERENCES [lJ And er son, 3 . D. O. and J . B. Moo re (1971). Linear Optimal Control. Prentice-Hall, nglewood Cliffs. pp.7S-S0. [ 2J And ers on, J. H. (1971). The control of a synchronous mac hi ne usin g op t imal co ntrol theory. Proc. IEEE, 59, pp.25- 35. [3J Dani els, A.R .-,--b--:- H.Davis and 11. K. Pal (1975). Linear and nonlinear optimization for power system performance. IEEE Trans. Dower Apparatus & Sys tems, vol. PAS-94, pp. S10-S19.

3128

M. Araki, K. Kobashi and B. Kondo

Fig. 3 Responses of the system for the rectangular change of the mechanica l torque (Simu l ated by the 7-th order Park model.)

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Fi g. 5 Responses of the system in the postfault state (Simulated by the 7-th order Park model.)

For Discuss ion see page 3135

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