Aggregation Method for Dynamic Models in Power System

R. S. Baheti

3120

Equations (3),

(12) and (18) yield

d 2 dt 4 (t) = -u (t) A{4 (t) - ~ a lu(t)] } k

(19)

De fine e rror function by

(20) From Equations (1), (19) and (20)

(2 1)

From Equation (9) T

~

(22)

A ~k

Fig. 2.

case is considered in the following to determine integral absolute error of single motor dynamics approximated by two basis models .

Equations (9) - (22) yield

Consider a step change in bus vo lta ge from steady-

state .

2

2 d X (t)= -~ Xk( t) + u (t) ~T[A - ~I] dt k 'k 'k where I is identity matrix. :f.k (t) by

y,

(t)

-

T

k

lk(t)

Vo for t

d dt Y2 (t)

T

~k E.k(t).

2 _ u (t) Tl

'"

Tl < Tk < T2'

2 u (t) [ Y (t) 2 T2

t

(1 -

'k

-

b

2k

ak[u(t)]}

(29)

Error funct i on is defined by

Let

i=l

[Yl (t) - b lk ak[u(t)]}

A motor model given by Equation (1) is approximated by basis models in Equation (29) . It is assumed that

dt ~(t)

m -IT

0

(24)

The error model can be written as

where

<

(28)

d dt Yl (t)

~

u(t)

Let t he basis models be given by

Def ine (mxl) vectors

-k

d

Let

(23)

"i<

2 _u (t) A{ z (t) - ea [u(t)J} From Equations (19) and (24)

Dynamics of a ppro xima tion error.

X (t) - Y (t) - Y ( t ) k 2 l

I dt

(30)

(25) - (27) 2 V '\, 2 d = ..l. X (t) + V £ k[ak(v ) - a (V 1 ) I '" dt Xk(t) O k l 'k k

From Equations

T1J

(31)

(26)

where

Equation (25) can be written as J

'"

dt Xk (t)

2 u

,

(t)

(T 2 - 'k) (Tl- 'k)

Xk (t)

(27)

k,

A block diagram of approximation error is shown in Figure 2. The notation BLS['k] represents bilinear system in Figure 1 with time constant 'k' The notation BLS [Tj] represents bilinear system with time constant Tj.

(32 )

(T - T ) 'k 2 2 l ~

From Equation (31)

it is seen that error function

Xk(t) does not change sign for (31) yield

t~O.

Equations (28) -

As expected it is easy to see that when motor time constant Tk is equal to ~asis time constant Tj, there

is no approximation and Xk(t)=O for

t~O.

When Tk is not equal to T " the model can be used to analyze approximation errorJand provides an insight in the selection of basis models. VI.

SELECTION OF BASIS TIME CONSTANTS

The selection of time constants for basis models in Equation (2) is application dependent. A special

'k

The integral absolute error J[1/ 1 k] IT 1 T 2'

is maximum at

The generalization of above results when number of basis is more than two i s not apparent. Further work is needed to unders tand the selection of basis models and their influence on aggregation.