Transient Stability Assessment and Parametric Study Using Back Propagation Algorithm

Copyright @ IFAC Power Plants and Power Systems Control, Brussels, Belgium, 2000

TRANSIENT STABILITY ASSESSMENT AND PARAMETRIC STUDY USING BACK PROPAGATION ALGORITHM

Toshihisa Funabashi

Meidensha Corporation, 36-2, Nihonbashi- Hakozakicho, Chuo-Ku, Tokyo, 103-8515, Japan

Abstract: This paper proposes a new method using back propagation algorithm and equal area criterion for fast transient stability assessment. The basic concept of the proposed method is shown for a one-machine with infinite-bus system. Then, examples of transient stability assessment are shown to demonstrate the powerfulness of the method in three categories, i.e., stability assessment, stability limits and parametric study. Finally, the example of stability limits calculation using back propagation algorithm is described in detail. Copyright©2000 1FAC Keywords: Transient stability, Transient stability analysis, Stability limits, Back propagation, Back propagation algorithm

controller, this shortcoming results in inability of online simulation or delay in response of user interface. Stability limits and parametric study include the same problem, because they have transient stability assessment in their procedures. To develop a fast transient stability assessment method, attentions have been paid to applications of conventional equal area criterion (Xue, et al.,1922; Zhang, et al., 1998; Khan, et al., 1998) and of artificial neural networks (Aboytes, et al., 1996; Aggarwal, et al., 1994; Fitton, et al., 1996).

1. INTRODUCTION To secure high reliability and economical operation it is important for the planner and operator of a power system to maintain and improve transient stability of the power system. Thus, a method is necessary to assess transient stability precisely and efficiently. The conventional approach of transient stability assessment utilizes solving the differential equations by numerical integration in time domain and observing the generator's phase angle to assess the system's stability. "Stability limits" such as critical clearing time, critical reclosing time and maximum safe transmitted power can be calculated by changing parameter's value in stability assessment and searching the limit between stability and instability. "Parametric study" can be done by changing the value of one of the parameters and execl1ting "stability assessment" or "stability limits" repeatedly. One of the shortcomings of conventional approach of transient stability assessment is that it takes much time for the real bulk power systems. When the transient stability assessment is implemented as an option in a power system's dynamic simulator or in a power system's

This paper proposes a new method using equal area criterion and back propagation algorithm for fast transient stability assessment. The basic equations are derived and the concept of the proposed method is shown for a one-machine with infinite-bus system. Then, examples of transient stability assessment are shown to demonstrate the powerfulness of the method in three categories, i.e., stability assessment, stability limits and parametric study. Finally, the example of stability limits calculation using back propagation algorithm (Werbos, 1990)is described in detail.

265

(1) fault 1 10

2. THE CONCEPT OF THE PROPOSED METHOD

=

P = Pmax

•

Pm .. =

sin8 ViVb

----x--

= 8 0 = sin- t {Po (Xd' + Xt + Xe) / Vi Vb } (3)

8

As an example to show the new method, a onemachine with infinite-bus system is shown in Fig.I. The proposed method is developed under the following conditions. (1) Field voltage is constant. (2) Impedance values of direct-axis circuit and quadrature-axis circuit are equal to each other. Based on the conditions, equations are derived for describing the problem. For electrical systems generator's electrical output is represented using the phase angle between the generator's internal voltage and infinite-bus.

=

(i)

=0

Wo

where, OJ

= do

/ dt

(4)

(2) fault-elearing 1

= 11

During faults d 2 t5

M (J)

""""j1(

n

(5)

Po

=

(6)

(1)

(7)

. s1no

X = Xd' + Xt + Xe where, phase angle of generator's internal voltage Vi : magnitude of generator's internal voltage Vb : magnitude of infinite-bus voltage Xd ' : generator's transient reactance Xl : transformer reactance Xe : line reactance

(8)

o:

(9)

(3) reclosing During reclosing d 20 - P0 2 dt -

M OJn

(10)

For mechanical system, a swing equation can be described as follows. M

(2)

OJn

where,

M : generator's inertia constant generator's rated angular frequency : generator's mechanical input power

())1l :

Po

P : generator's electrical output power A chart in Fig.3 shows the concept of the proposed method applied to the example power system-I shown in Fig.I. Inputs for this system are

According to the time sequence offault and reclosing, the generator's power is represented using the phase angle.

Po,

,

~,

~,

Xd

,

A;, .' M OJ,{ant), too t, ad ~, ~, (Oz' crd 8<'2'

(1) Pre-fault

1<10 P = Po (2) Fault (3 phases to ground) 10 <1<1 1 Vb = 0 :.p = 0 (3) Post-fault 11 <1<1 2 Vb = 0 :.p = 0 (4) Post-reclosing (3phase reclosing) 1>1 2 P = Pmax • sino

Output of this system are

02'

OJ 2

,

and

0,2.

The relationship between inputs and outputs are describrd by five equations, (3), (8), (9), (14) and (15). From these equations variables 00' OJ, and t5, can be deleted. But if they are deleted commonness of the system description is lost from these equations and the powerfulness of the method can not be shown. So, they are not deleted here.

The power versus angle curve is shown in Fig.2. The phase angle and the angular velocity at the time of disturbance are described as follows.

266

using equal area criterion. In Fig.2, the area 51 equals to the area 52 when CS 2 equals to CS c2 (critical reclosing angle). SI = PO (oc 2 S2

-

00 )

= Pm ... (COSO o +cos0c 2 ) - po(,,-Oo

-Oc 2 )

Fig.l An example power system-l COSOC 2 =

p

Po / Pm ... • (20 0 + 11")- coso o

Because

i

Po / Pm ... = sino o Oc 2 = COS-I {(20 0 + ,,)sin 0 0

ill I :

Fig.2 Power versus angle curve for the system-l

t

I

(15)

(8)->(14)

01 : (9)->(14) (3) Compare the CS c2 calculated from procedure (1) and the CS 2 calculated from the procedure (2). For examples, CS 2 < CS c2 means the system being stable.

final disturbance

t

COSO o }

(2) Give the final disturbance time i.e., reclosing time t2 in Fig.2 and calculate the phase angle CS 2 at the final disturbance. 00 : (3)->(9)

lOO -00 lOO ~o

Inputs

-

2

Inputs

Po..-..:··········_···············: Vi ..-..: Vb - . . :

Xd'~: X, ..-..: X.~:

M

3.2 POST-FAULT STABIliTY ASSESSMENT (NO RECLOSING) 0"

............

..-..: : outputs w. - . . : ••••.••••••••••••.. - •••••• (const)

Because the sample power system includes singlecircuit line, stability assessment with no reclosing is not meaningful. So, the method of post-fault stability assessment for the power system with double-eircuits line shown in Fig.4 is described here. The final disturbance in this case is fault clearing at the time of t=t1.

Fig.3 Conceptual chart for the system-l

3. TRANSIENT STABILITY ASSESSMENT The basic procedure of the proposed method is described here. Transient stability assessment is executed in three steps.

(1) Calculate the critical clearing angle CS cl by using equal area criterion. In Fig.S, the area SI equals the area S2 when CS I equals to CS cl (critical clearing angle). (2) Give the final disturbance time i.e., fault clearing time tl in Fig.S and calculate the phase angle CS 1 at the final disturbance. The concept of the method for this example is shown in Fig.6. (3) Compare the CS cl calculated from procedure (1) and the CS I calculated from the procedure (2). CS 1 < CS cI means the system being stable.

(1) Let inputs other than the final disturbance time be constant. Calculate the critical phase angle CS c by using equal area criterion. (2) Let all the inputs be constant. Calculate the phase angle at the time of the final disturbance CS from the equation at the time of the final disturbance. (3) Compare CS c and CS to assess the stability of the system.

3. J POST-RECLOSING TRANSENT STABIliTY ASSESSMENT The procedure of the assessment is shown.

post-reclosing

stability

(1) Calculate the critical reclosing angle CS c2 by

Fig.4 An example power system-2

267

in the same way as section 3.2. (3) Compare 5 cl with 5 1 and modify the value oftl to get the 5 cl and 5 1 to be closer with each other than before. If the difference between 5 cl and 5 I is greater than the threshold value, then go to (2), otherwise tcl=tl.

p

i

0,

As commonly known, the critical clearing time can be calculated directly from the relationship between the final disturbance time and the critical phase angle 5 c, after the critical clearing angle is calculated by using equal area criterion. In this paper the procedure is generalized for the critical clearing time to be calculated by using the same way as other stability limits such as maximum safe transmitted power.

lOO ~o

Fig.5 Power versus angle curve for the system-2 inputs 1,

t=t o

inputs Po~:

.

V,~:

Vb~: Xd'~:

Xt~: X.~:

M

- ~

-

-_

~

4.2 CRITICAL RECLOSING TIME

~~~Fputs ~

The procedure of calculating the critical reclosing time tc2 is shown.

~,

~:

Wn~'

(const)

.

~

(1) Let inputs other than reclosing time 12 be constant and give the initial value 120 to 12. (2) Calculate the critical reclosing angle 5 c2 and the phase angle 5 2 at the time of reclosing (t=12) in the same way as section 3.1 . (3) Compare 5 c2 with 5 2 and modify the value of 12 to get the 5 2and 5 c2 to be closer with each other than before. If the difference between 5 2and 5 c2 is greater than the threshold value, then go to

~

: outputs .

Fig.6 Conceptual chart for the system-2

4. STABILITY LIMITS

(2).

Transient stability limits such as critical clearing time is calculated in three steps. 4.3

(1) Let inputs other than one input parameter be constant and give the initial value to this parameter. (2) Calculate the critical phase angle 5 c by using equal area criterion and calculate the critical clearing angle 5 c and phase angle 5 at the time of final disturbance in the same way as the previous chapter. (3) Compare 5 c with 5 and modify the parameter's value to get the 5 c and 5 to be closer with each other than before. If the difference between 5 c and 5 is greater than the threshold value, then go to (2) otherwise stop calculations.

POST-FAULT MAXIMUM TRANSMITTED POWER

SAFE

The procedure of calculating the post-fault maximum safe transmitted power is shown. (1) Let inputs other than pre-fault power flow PO be constant and give the initial value POO to PO. (2) Calculate the critical clearing angle 5 cl and the phase angle 5 1 at the time of fault clearing (t=tl) in the same way as section 3.2. (3) Compare 5 cl with 5 I and modify the value of PO to get the 5 1 and 5 cl to be closer with each other than before. If the difference between 5 1 and 5 cl is greater than the threshold value, then go to

4. J CRITICAL CLEARING TIME

(2).

The procedure of calculating the critical clearing time tc1 is shown.

4.4

( I) Let inputs other than clearing time t 1 be constant and give the initial value tlO to t1. (2) Calculate the critical clearing angle 5 cl and the phase angle 5 I at the time offault clearing (t=tl)

POST-RECLOSING MAXIMUM TRANSMITTED POWER

SAFE

The procedure of calculating the post-reclosing maximum safe transmitted power is shown.

268

(1) Give the initial value Xemin to the parameter Xe. (2) Calculate the critical reclosing time tc2 in the same way as section 4.2. (3) IfXe is greater than Xemax then stop calculation, otherwise increment Xe by .6.Xep and go to (2).

(1) Let inputs other than pre-fault power flow PO be

constant and give the initial value POO to PO. (2) Calculate the critical clearing angle 5 c2 and the phase angle 5 2 at the time of reclosing (t=t2) in the same way as section 3.1. (3) Compare 5 c2 with 5 2 and modify the value of PO to get the 5 2 and 5 c2 to be closer with each other than before. If the difference between 5 2 and 5 c2 is greater than the threshold value, then go to (2).

Where, Xemin and Xemax are pre-set minimum and maximum of Xe respectively. .6.Xep is a pre-set parameter increment for parametric study. The relationship between Xe and tc2 demonstrates the influence of line reactance on the critical reclosing time.

5.3 INFLUENCE OF GENERATOR'S INERTIA ON MAXIMUM SAFE CONSTANT TRANSMITTED POWER

5. PARAMETRIC STUDY Influence of a parameter's value on stability limits

can be analyzed by the proposed method. This analysis can be done by changing parameter's value

In this case the valuable is the pre-fault power flow PO and the parameter is the generator's inertia constant M. Influence of M on PO can be analyzed by repeating the calculation of the maximum safe transmitted power POc with changing the value of the parameter M. The procedure is as follows.

and repeating stability assessment procedure described in chapter 3 or stability limits procedure described in chapter 4.

5.1 INFLUENCE OF PRE-FAULT POWER FLOW ON CRITICAL CLEARING TIME

(1) Give the initial value Mmin to the parameter M. (2) Calculate the maximum safe tranmitted power POc in the same way as section 4.3. (3) If M is greater than Mmax then stop calculation, otherwise increment M by .6.Mp and go to (2).

In this case the valuable is the critical clearing time tcl and the parameter is the pre-fault power flow PO. Influence of pre-fault power flow PO on the critical clearing time tc 1 is analyzect by repeating the calculation of the critical clearing time by changing the value of PO. The procedure is as follows.

Where, Mmin and Mmax are pre-set minimum and maximum of M respectively. .6.Mp is a pre-set parameter increment for parametric study. The relationship between M and POc demonstrates the influence of the generator's inertia contant on the maximum safe transmitted power.

(1) Give the initial value POmin to the parameter PO.

(2) Calculate the critical clearing time tcl in the same way as section 4.1. (3) If PO is greater than POmax then stop calculation, otherwise increment PO by .6.POp and go to (2).

6. CALCULATION OF STABILITY LIMITS BY USING BACK PROPAGATION ALGORITHM

Where, Prnin and Pmax are pre-set minimum and maximum of PO respectively. .6.POp is a pre-set parameter increment for parametric study. Calculation is executed from PO=POmin to PO=POmax by incrementing PO by .6. POp. The relationship between PO and tcl demonstrates the influence of pre-fault power flow on the critical clearing time.

In the modification of variables in stability limits calculations described in chapter 4, an amount of modification is constant. Alternatively, it can be done so as to minimize the error function.

(16)

The next example is reclosing. Because pre-fault stability limits is not meaningful in the case of the power system with one circuit line. Fig.7 is a system for calculating stability limits, which is gained by putting into concrete the concept of the system shown in Fig.3. For examples, the procedure for calculating maximum safe transmitted power POc is described. The relationship between the pre-fault power flow PO and the error function E is shown in Fig.8. From equation (16) the equations (17) to (19) are described

5.2 INFLUENCE OF liNE REACTANCE ON CRITICAL RECLOSING TIME In this case the valuable is the critical clearing time tc2 and the parameter is the line reactance Xe. Influence of Xe on tc2 is analyzed by repeating the calculation of the critical reclosing time tc2 with changing the parameter's value. The procedure is as follows.

269

as follows.

Fig.8Relationship between Po and E where,

oE

7. CONCLUSIONS

oJ z

In this paper, a new method for fast transient stability assessment is proposed based on a one-machine with infinite-bus system. The method utilizes equal area criterion and back propagation algorithm. The back propagation algorithm described in the paper does not need any kind of learning algorithm. It can be solved directly using a algebraic equations. In practical applications for a multi-machine system, however, the assistance of the learning method such as artificial neural networks might be effective. In the future work, simulations to verify the typical solution times and the precision of the method for practical systems are planned.

oJ 2 = 1 o§ \

0§2

op.

=~(t -tlf 2M

2

00 1 = ~ (t OP. 2M I 00 2 - - = t 2 - t, OW

_

I

)' 0

I

oJ\ oJ 0

=1

OWl = am

op.

(t _ t )

M

I

•

(19)

Then, in the place of constant increment .0.POp in the procedure (3) of section 4.4, the new modification· can be utilized as follows. P On ...

=

P Oold

_ 0E E 0P old

REFERENCES Aboytes, F., and R Ramirez (1996). Transient Stability Assessment in Longitudinal Power Systems Using Artificial Neural Networks, IEEE Trans. on Power Systems, Vol.ll, No.4, pp.2003-201O

(20)

o

Aggarwal, RK., A.T. Johns, Y.H. Song, R W. Dunn and D.S. Fitton (1994). Neural-network Based Adaptive Single-pole Autoreclosure Technique for EHV Transmission Systems, lEE Proc. Gener. Transm. Distrib., VoU41, No.2, pp.l55-160

time

operating conditions

Fitton, D.S., R W. Dunn, RK. Aggarwal, AT. Johns and A Bennett (1996). Design and Implementation of an Adaptive Single Pole Autoreclosure Technique for Transmission Lines using Artificial Neural Networks", IEEE Trans. on Power Delivery, Vol.ll, No.2, pp.748-755

circuit parameters

Fig.7 Network design for back propagation

Khan, AZ., and F. Shahzad (1998). A PC Based Software Package for the Equal Area Criterion of Power System Transient Stability, IEEE Trans. on Power Systems, VoU3, No.l, pp.2126

WerOOs, P.l. (1990) Backpropagation Through Time:

270

What It Does and How to Do It, Proceedings of the IEEE, Vo1.78, No.lO, pp1550-1560 Xue, Y., L. Wehenkel, E. Euxibie, and B.Heilbronn (1992). Extended Equal Area Criterion Revised, IEEE Trans. on Power Systems, Vol.?, No.3, pp.lOI2-1022 Zhang, Y., L. Wehenkel and M. Pavella (1998). SIME : A Comprehensive Approach to Fast Transient Stability Assessment, Trans. lEE Japan, Vo1.118-B, No.2, pp.I27-132

271