Two-Level Techniques for Non-Linear Optimal Control Problems : Their Application to a Synchronous Machine

TWO-LEVEL TECHNIQUES FOR NON-LINEAR OPTIMAL CONTROL PROBLEMS

THEIR

APPLICATION TO A SYNCHRONOUS MACHINE. A. TITLl ~

J. GALY ~

The application of "classical" optimization methods of optimal control to large scale dynamic systems produces various problems because of the amount and complexity of the required calculation. An efficient means of getting round these difficulties is certainly the introduction of multilevel dynamic optimization techniques. Three possible methods are presented from the calculus of variations, for interconnected subsystems. The limitations of these techniques are given and suggestions are made in order to avoid this problem. A study of the optimal control of a synchronous machine is given to illustrate the application of this kind of method".

INTRODUCTION Research into the optimal control of complex dynamical processes, using "classical" optimisation techniques. poses a great many problems. These problems are due particularly to the number of calculations necessary and to their complexity. An efficient means of getting round these difficulties would appear to be the introduction of control systems on several levels or of hierarchical control. Since the time of the first work on these new control concepts ((1) to ((6) , a lot of research has been done in this area. The following references, without being exhaustive, nevertheless provide the principal results ((7) to ((20).

Two notions fundamental to hierarchical control are decomposition (division of work) and coordination. - decomposition consists of two aspects : vertical decomposition which enables one to divide the overall control function into several levels, and horizontal decomposition which allows the process to be divided into sub-processes. - coordination

enables all the sub-problems to reach the overall solution.

If attention is limited to dynamic optimisation problems, then only one of the levels of the vertical decomposition is used, so that the problem is no larger strictly a hierarchical control problem but is rather a hierarchical calculation problem, a decomposition coordination using the calculus of variations. ~

Laboratoire d'Automatique et d'Analyse des Systemes du C.N.R.S. B.P. 4036. 31055 Toulouse Cedex. France.

530

The control which result from this hierarchical calculation can be used only if the rate of convergence of the coordination algorithms is substantially faster than the system dynamics. It is essentially to the use of hierarchical calculation procedures for realistic problems that this paper addresses itself. On the first part of the paper, methods of dynamical hierarchical optimisation are explained and compared. In the second part, the most useful of the methods is used to solve a significant practical problem, i.e. the control of a system of interconnected synchronous machines. The problem of the optimal control of interconnected systems Consider a dynamic process which can be represented by N interconnected sub-systems (Fig. 1). Each of these sub-processes has associated with it a criterion function

tf

tJ (\'

R.

J.

Xi' MJdt

+

gi (Yi(t o ), Yi(t f ))

(1 )

State equation

(2)

o

and the following equations and constraints

,

Y. = f. (Y . , Xi' M , t) J. J. J. Q. J.

i (Y., X., M. , t).? 0 J.

J.

(3)

J.

X., M., t) dt J.

J.

~

0

Constraints

(4)

k. [Y.(t ))

0

(5)

li[Yi(t f )]

0

(6 )

J.

J.

0

T. (Y . , X. , M. ) J.

J.

J.

J.

- z.J.

0

Model equation

(7)

Interconnection equation

(8)

N

X.J.

~

j =1

C .. Z. J.J

J

The overall problem therefore is N

max

L

i=1

R.J.

(9)

subject to equations (2) to (8) for i = 1, 2, ... N where to' t

f

are fixed.

Since such problems are in practice of high dimension, it is necessary to use methods of decomposition-coordination to solve them. Three decomposition-coordination methbds using the calculus of variations In order to solve this optimisation problem, it is necessary to use the calculus of variations. There are essentially three methods which can be used and these methods are explained briefly below :

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Model-coordination method (feasible method, TITLI (10).) The method consists of solving N independant sub-problems on the lower level ; these are defined by : i = 1, 2, •.• N [max Ri . subject to

Z1

for g i v e n Z (2)

to

( 10)

(S)

ZN

which leads to an imbalance on the model equation (7). The coordination between the sub-problems is done by improving the coordination variable Z on the higher level, using, for example, the gradient algorithm or the Newton algorithm given below k+1

Z.

1

Zk i

~

~

+

K [- eX...

-

1

Z +1 = Z - c 1 1

~

( 11 )

~

t

[~dZ ( - 0(,.1- .

J=

1

C .. T J1

t.

fj.~J -1 [- d..1- .

J=

1

Cj . T A.. .] 1 J'"' J

( 12 )

< <2.

where k is the iteration inde x of the coordination, K is a constant> 0, 0 C and ,)..J 4 1. are Lagrange parameters associated with constraints (7) and (8).

et 1.

Method of coordination by the Criterion Function (infeasible method, PEARSON (8)) On the lower level, for a givenj3 , the following sub-problems are solved independently ( 13) i

1, 2, •• N

• subject to (2) to (7) This method gets its name from the fact that it is the criterion function which is modified. The coordination variable, using the following algorithms :

fl>~+1 =fi~1 - K[X.1 1

t

j =1

fo '

can be improved on the higher level

C1J ..

or

fi>~+1 = fl~ - c[~

(Xi -

t

z.J-1

C. . j=11JJ

t

Z.]

[X. C .. 1 j=11JJ

(15 )

Here, two formulations are possible. Mixed method of coordination (TITLI (10~ For the first, the sub-problem is identical to problem (13), but it is necessary to solve it using a given)) and Z, and this modifies the coordination task, which can now be represented by : (11)

+

(14)

(gradient type)

or

(12)

+

(15)

(Newton type).

For the second formulation, the coordination method remains the same, and the sub-problems can be written as :

532

I tf(

max i=1,2, ... N for a given and Z

t) (Y i' X.1 , M.1 ,

0

fo

+fi/

N [ Xi - . Ci·Z j ]] dt J=1 J

(16 )

+ gi [\(t o ), Yi(t f )]

subject to (2) to (7)

Comparison of the advantages and disadvantages of these methods. Whereas, generally speaking, method of coordination by the criterion function is always applicable, model-coordination method and mixed method of coordination have conditions of applicability which are, respectively : mM i (m

w

~

( 17)

mZ i

mMi + mXi ~ mZ i = number of components of the vector w ).

(18)

Condition (17) constitutes an important limitation to the application of the Method of Coordination by the Model. In addition, in order to use it on the coordination level, the calculation of both ~ and~ is necessary for each subproblem, whereas for the mixed method of coordination, onlyoC is required. If precautions are not taken at the decomposition level, the method of decomposition-coordination by the Criterion Function generates singular subproblems which are difficult to solve. On the other hand, for a coupling using state variables (linear) it is the only possible method, and this coupling type of problem is to be found in a great many applications. As for the coordination it can be shown (10) that when it is applica ble, the Newton algorithm is always convergent, whereas the gradient algorithm, although always applicable, has conditions of convergence. On the other hand, although a Newton type coordination involves a smaller number of iterations than a gradient type coordination, it nevertheless requires more calculations for its implementation. Finally, the present methods of coordination require that, for this kind of problem, the horizon be fixed. However, this limitation can be circumvented by treating the final time as an additional coordination parameter. Application of decomposition-coordination to a synchronous machine The overall problem : The problem of the optimal regulation, around a given operating point, of a synchronous machine connected to an infinite network (21) can be formulated as follows T min [A (Y _Y )2+ (19 ) Y1 1 1p M 0

f

o

subject to

~1 ~2

Y 3

=

Y2

(20)

B1 - A1 Y2 - A2 Y3 sin M - C Y + C cos Y 1 3 2 1

(21 )

T: Ai' B , Ci ' Ay., Yi ' i = 1, 2, glven. i 1 p

(22)

Mp are constants and Y (0), Y (0), Y (0) are 1 2 3

The overall problem was solved on the I.B.M. 370/165 computer at L.A.A.S. Convergence to the optimum took place in 3 iterations of a quasi-

533

linearisation procedure which has been perfected by the authors. Fig. 2 shows the optimum state control trajectories. These are identical to those in reference(Mukhopadhyay, (21)). Remarks : this problem has three state equations and three adjoint equations even for a single machine. For realistic multi-machine problems the number of state-costate equations could easily become enormous. This would make the on-line resolution of the associated two-point boundary value problem virtually impossible. If decomposition-coordination is used, then since on the lowest level only a low-order problem needs to be solved, the resolution of the corresponding two-point boundary value problem is very easy, and such methods could then be feasible for realistic multi-machine systems. With this in mind, in the following section. a decomposition-coordination technique is applied to a machine and to a multi-machine situation. Decomposition of the problem. This decomposition is carried out by assuming that each state equation represents a sUb-system (which has obviously, in this case, no physical reality). It is a question. therefore, of a mathematical decomposition method from a state variable model. and this decomposition leads to a state variable coupling so that only the Method of Coordination using the Criterion Function will be applicable. Since the overall criterion function is already in a separable form, only the interactions between the sUb-systems shown in Fig. 3 need to be taken into account. Several approaches are therefore possible : a) define one sub-problem per state variable b) put two state variables together in one sub-problem leaving the third for a second sub-problem. Since. at the present time, there are no criteria for choosing between these possibilities, a careful study of each of them has been made. From this, the following points have emerged : - the isolation of the Y variable in a sub-problem can 1 discontinuities for the coordination variable;

result in general

- the second order conditions are difficult to use except in the case of Fig. 4. and this was finally chosen. If it is assumed that : (23)

the problem is then formulated in terms of interconnected sUb-systems. as follows : (24)

(25) (26) (27)

XI - YII = D

(28)

XII - YI

(29)

Y (0), I

= 0

YI~(O) being given

suppose~l and~~ to be Lagrange parameters

2

534

associated with constraints (28) and (29). In the method of coordination by the criterion function, these parameters remain coordination variables, and the subproblems are written as : ~~g:[lrQgl~!!1_1

~~nljT [A Y1 (Y'l- Y1p)2 Ay,v,~ +J3,X1-}rry,J dt] +

(30)

fl fl

for given I' n subject to (25) and (26) and for given Y (D), Y (0). I I 1

mi~ [;r

X II

II

T

[Ay (Y II -Y 3 )2+ AM (M II -M 3 P P

0

for given

2

)2+~IIXII-;BIYII]

dtJ (31)

fi I' fi II

subject to (27) for given YII(O) These sub-problems are solved by a quasi-linearisation method. The coordination task is : foI ]k+1 [ fln

(32)

[foI] k

= flII

However, the sub-problem I is singular in relation to the pseudo control variable XI' To avoid this, it is possible : -either, as in 8auman (22), to modify the coupling equations: 2 2 XI - Y = 0 II

2 =0 I1 which has the drawback of producing an erroneous solution - or, taking into account (28), which should be satisfied at the overall solution, to replace YII by X in the criterion function. 1 X II

2

- y

(33) (34)

Coupled Machines. Whereas the last problem produced a purely mathematical decomposition, this one leads to a physical decomposition in which each subsystem corresponds to a machine : Machine 1 Equations (19), (20), (21), (22) Machine 2 : Y' Y' (35) 1

2

8

Y'

2

8' -A' Y' -A' Y' sin Y' sin 2Y' 11223 12 1 2 Y' M' - C' Y' + C' cos Y' 31321 min

M'

+

A'

+

A'

Y 3

(37)

(Y' _Y'

3

(M' -M'

M

(36)

P

3p

)2

(38)

) 2 ] dt

coupling Y1 = Y'1' The overall problem is one of minimising the sum of the partial criterion functions (19) and (38) subject to equations (20) to (21), (35) to (37) and (39). If is the parameter associated with (39), the first partial cri terion function is modified by fi Y1 and the second by - foy' 1 ; these two criterion functions are minimised subject to the state equations using a quasilinearisation procedure which converges in three or four iterations.

J3

535

The initial trajectories for this procedure correspond to constant values on (0. T) and are equal to the boundary conditions for the two-point boundary value problem. 2

The constant K of the coordinator is equal to 10- • Additional results comparing the above method with the overall solution will be presented at the same time as the paper. Conclusion The decomposition-coordination methods which can be obtained from the calculus of variations for dynamic problems of high dimension have been briefly presented. The most frequently used of these methods has been illustrated on a practical example. that of the optimal regulation of a synchronous machine. It has been pointed out that the decomposition-coordination approach does not in fact trackle the practical control problem since the resulting control is open loop. In fact. these methods can only be used as control methods if the disturbances are known and measured. and if the convergence time of the two-level structure is small in relation both to the evolution time of the system dynamics and to the average period of the disturbances requiring predictive control action. This limitation has been realised by a great many research workers who are at the present time developing "on-line" coordination algorithms which would allow effective control to be carried out.

Acknowledgement The authors are indebted to Dr. M.G. SINGH for his hepful comments.

Symbols used dY.

o

Y.1

state variable for sub-process number i (Y.

M.

control variable

Xi

coupling input

Z.

coupling onput

1

1

1

(to' t f ) : time interval N : number of sub-processes integral criterion terminal criterion

cL 1. • /flj

. :

1

lagrangian parameters

k. c : iteration index

536

1

--)

dt

REFERENCES 1. Macko. D•• 1967. "General systems theory-approach to multilevel systems". Systems Research Center Report. Cleveland. SRC 106-A-67-44. 2. Mesarovic. M.D •• Macko. D•• and Takahara. 1968. "Structuring of multilevel systems". Proc. IFAC Symposium on multivariable systems. Dusseldorf. 3. Mesarovic. M.D .• Macko. D•• and Takahara. 1969. "Two coordination principles and their application in large scale systems control". IV IFAC Congress. Warsaw. Poland. 4. Lasdon. L •• and Schoeffler. J.D .• 1965. "A multilevel technique for optimization". J.A.C.C. Procedings. Troy. New-York. 5. Brosilow. C.B .• Lasdon. L .• and Pearson. J.D .• 1965. "Feasible optimization methods for interconnected systems". J.A.C.C. Proceedings. Troy. New-York. 6. Mesarovic. M.D •• Macko. D.• Takahara. Y.• 1970. "Theory of hierarchical multilevel systems". Academic Press. New-York and London. 7. Lasdon. L.S •• 1970. "Optimization theory for large scale systems". Mac Millan series for operations research. New-York. 8. Wismer. D.A •• Editor. 1971. "Optimization methods for large-scale systems". Mc Graw Hill. 9. Himmalblau. D.M .• Editor. 1973. "Decomposition of large-scale problems". North-holland/American Elsevier. 10. Titli. A.• 1972. "Contribution a l'etude des structures de commande hierarchisee en vue de l'optimisation des systemes complexes". Thesis. Universite Paul Sabatier. Toulouse. 11. Cheneveaux. B•• 1972. "Contribution a l'optimisation hierarchisee des systemes dynamiques". Thesis. Universite de Nantes. 12. Galy. J •• 1973. "Optimisation dynamique par quasi-linearisation et commande hierarchisee". Thesis. Universite Paul Sabatier. Toulouse. 13. Singh. M.G •• 1972. "Some applications of hierarchical control". Thesis. Control Engineering group. University of Cambridge. G.B. 14. Findeisen. W.• 1968. IEEE Trans. Syst. Sci. Cybern .• SSC-5. 155. 15. Pallai. I.M •• Almasy. G.A .• Veress. G.E •• 1972. "On the optimal strategy for optimization of chemical plants". V IFAC Congress. Paris. 16. Imbert. N.• Clique. M.• Fossard. A•• 1971. "Optimisation hierarchisee des systemes dynamiques. Convention 70-7-2506. D.G.R.S.T. 17. Benveniste. A.• Bernhard. P .• Cohen. G•• 1972. RAIRO. 14-71. 18. Grateloup. G.• Titli. A•• 1973. Int. J. Systems Sci.. 4. 577. 19. Titli. A.• Lefevre. T .• Richetin. M•• Int. J. Systems Sci •• 4. 865.

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20. Guegen, C.J., Manich Mayol, F., 1973, Information Sciences, 6, 235. 21. Mukhopadhyay, B.K., 1972, Proc. LE.E., 119. 22. Bauman, J. 1968, "Multilevel optimization techniques with application to trajectory decomposition", in Advances in control Systems, Academic Press New-York.

538

coupling input Xj

state variable

S.P. noj

z·I

Mj

coupling output

YI1

sub-problem

Y I1

ne. 1 Y I2

(Y1 ,Y 2 )

Xn

sub-problem

XI

control variable Fig. 1 - Sub-system nO i

I

Y1

I

M

--!.. I

I I I I

Y3 I

I I----

I I

Y1 Y2

nO 2

Y

n

Y2 _ _ _ :....JI

Y I

(Y3)

Mn Fig. 3 - Decomposition in two sub-problems

~7S~-------------r--------------r-------------~------------~

Y

1p - - - - - -

t

~70~------------~----------------------------~------------~---o 1 2 Fig. 4 - Optimal state variable

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