fast transient stability evaluation by estimation of...

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TitleFast Transient Stability Evaluation by Estimation of Electrical Power Out

put

Author(s) Ishigame, Atsushi; Taniguchi, Tsuneo

Editor(s)

CitationBulletin of University of Osaka Prefecture. Series A, Engineering and nat

ural sciences. 1989, 37(2), p.87-97

Issue Date 1989-03-31

URL http://hdl.handle.net/10466/8467

Rights

87

Fast Transient StabMty Evaluation by Estimation

of Mectrieal Power Output

Atsushi ISHIGAME" and Tst;rieo TANIGUCHI**

(Received November 15, 1988)

fhis paper describes a new approach fbr fast evaluation of power system tran:

sient stabdity, taking.into account the volSage dependence of system logd. [he

method is based on the. direct method using energy function. In the mpthoa ･we pr"pose tb calculate the' potentia1 enetgy by polyrioniial expression of ele6uical power''

geUntgrUat{orfsOrsytshteemrff.fll,:3tiOn Of fast eValUatiOn･ The 'method 'is applied to a3 and ag

IThe time required fbr calculation of the proposed method is considerably reducedthan that of the usual direCt method, and' in rrtost cases, the accurticy Qf evaluation is

'also satisfactory. ･' '

1. Introduction

It is a very important problem to construct the reliable power system that prevents

interruption of power supply which is catised by large disturbances. One of the impor-

tant items which should be investigated at this juncture is transient stability, and fast

analysing method for transient stabMty is indispensable for its on-lme evaluation.

Recently, the voltage dependence of system load rnarkedly affects the (lynamic

behavior of a power system. However, on-line evaluation for power system transient

stability with consideration of load characteristics may not be realized, because much

computing time is required, even if the usual direct method is used. ,

in this paper, a new approach is presented, which is based on the direct method

using energy function. In the method, it is proposed to calculate the potential energy by

polynomial expression of electrical power ourtppt, taking into account the valtage

dependence ofsystemload. '' '' / The above method is applied to a 3 machine 9 buses systdrni) and ito a 9 machine

25 buses system2) as numerical examples, and the time for'required calculation of the

proposed method is compared with that of the usual direct method.

2. VbltageDependenceofCompositeLoad

For several years, much effbrt has been devoted to the modeling ofload 6haracter-

istic and some load models have been proposed･fbr simulation studies. Among these

models, a V"-type model is the most fundamental. In this pape.r, the following d,y,namic lg. a.d I,p. ode13),.,is employed.

tt .,, , ,, t.. ,. ,/' t./ ,.1 '4 .t.t . t- tt *

**

Graduate Stu'derit, Department of Elecuical Engineering, Coliege of Engineering.

Department of Electripal E#gineering, College Qf Engipeeringt

'

88 Atsushi ISHIGAME and Tsuneo TANIGUCHI

p(k) = po I (EE(,k))"P + Ap i.Zki

e(k) = k, + (e, - ko)Eo"q ( C

(E･(i)-E(i-i)N

L E<i-i) Je

(E.U,'YP( EU) - Eij- 1)

E(k)

Eo

gy(kz)At,}

J

)nq+Aq

E9-i)

,g,(EEU,)

)e- kT-p "' 1 (1)

)nq.

(2)

where

Po, eo, Eo : mitial values of active power, reactive power and load voltage, respectively,

P(k), e(kl, E(k): values at tk = (k - 1)･At of active power, reactive power and load

voltage, respectively,

np, nq: load parameters of active and reactive pbwer respectively,

71p, 71i : time constants of active and reactive power respectively,

At: time interval.

Specially, this model is static load model atAp =Aq = O, and

np = nq = 2 : constant-impedance load,

np =nq = 1 : constant-currentload,

' np = nq = O: constant-power load.

ln the Eq. (1), when load voltage changes from Eo to E at step response, the change

ofPis shown in Fig. 1. The characteristic ofe in the Eq. (2) is also like Fig. 1.

E1.5

<p,u,)

1,.O

O.5

o.o

2.

P (p,u,)

5

2.0

1.5

1.0

O.5

o.oo.o O.2 O.4 O.6 e.e 1.o

t (sec)

Ftg. 1 Power response of load with voltage change

Egst 7hansient Stability Evaiaation by Estimation ofElectrical Power Output 89

3. MathematicalFormulatiQn

3.l. Systemequation

The fbilowing assumptions are made for multimachine systems.

(1) Each synchronous machine is represented by a constant voltage behind its tran-

sient reactance.

(2) Mechanical power input is constant and the governor action is not taken into

account.(3) Each synchronous machine is a round-rotor machine.

Under the above assumptions, the motion of the ith machine is generally described

by the foilowing differential equation.

Mlrf2,2ii.D,(lfii/ii-g6,o)

' =P>ni-IEV 2 Gb･ -

- Pini - "Plei

ni.i (Clij Sin 6e' + Dij cos 6ij)

(i = 1, ".". n) 2, 3,

･(3)

n : number of machine in the system,

Di : dampingcoethcientofithmachne,Plgi : elecuical power output ofith machine,

6i : rotor angle in the reference frame,

6ij :6im6i, n n6o : centerofangle,6o= iZ.iMlr6ili.iM}

M} : inertia constant of ith machine,

P.i : mechanical power input ofith machne,

Ei : internal voltage behind transient reactance ofith machlne.

The case of constant impedance load, Clr, Dij are constants, but under considering

load characteristics, Clv, Dij･ change with load voltage, and resultant electrical power

output varies. For this reason, in solving the system's differential equations, sequential

calculations for loads are required.

3.2. Loadcalculation

The nodal equation which gives the current-voltage relation of a multi-machine

system is written approximately at the (k)th interval as follows:

IG (k) EG (k) IL(k) (= o) =y(k-i) EL(k) (4) i(k) (= o) E(k)

where

EG (k), EIL(k), E(k) : internal voltage vector of generators, load-bus voltage vector and

floating-bus voltage vector, respectivelyIG (k),IL(k),I(k): generator current vector, load-bus current vector, fioating-bus current

90･ Atsushi ISHIGAME and Tsuneo TANIGUCHI

vector, respectivelyYfa-i): modified nodal admittance matrix involvingload admttance YL pt-i)

By elmination of floating buses, '

IG(k) YGG YGL. EG(k) I]L(mp(=O) " Y]LG yLL(k-!) ,E,(k) (5)

and Eq.v(5) is written in the following hybrid form4)

[i,',10kil,1,l,ll-l[:it')l2i3Z.1'1,],!,tsi?,/o,),,]. (6'

hi2(k-1) = YGL (yLL(k-1))-i

h,,(k-1)=-(yLL(k-1))-i YLG

h22 (k-1) = (ILa (k-1)) -1

Then, admittance representation of ith load at (k)th interval is given by Eq. (7).

yLi(k)=<?i(mp-iei(k))/(ELi(k))2 ' (7) ' ' ' ' We now perform the matrix reduction on the Y matrix at the generator-buses as

.ib""kl '; lllfif2iiiZ`,k('bPoj' 2iie.M8.",Flilfi .!IYi,,YG`k' matri?c be denoted by yb(k){ and, qefine

By repeating this process for each interval, the loads are calculated with every tme.

3.3. Energy type Lyapunov function and stability evaluation

The transient energy function V, which is always defined fbr the post fault system,

can be given in Eq. (g)5,6).

V- Vk+% ' n =,F, Mlr dii2 /2 , ... vk + iS, pt (eiO - ei) +:.t-,i i.#.i Bijopi (cos e#o - cos eij)

+,#, i£.n, .6ist.leojt, GijEiE7 cosetid(ei+ei) ...px}, (g)

where

Yb = Gij + iBv,

ei = 6i - 6o: new rotor angle with respect to center of angle 6.,

eiO: initial stable point ofei ' ' ' '' '' ' '

' eij = ei - ej

]F;7st 7}zxnsient Stabtiity Evaluation by Estimation ofElectrieal Power Output

'wi: angularvelocityofithmachne, .,, ,.. ,,. ' '''Wo=d6oldt, ' '. 1. tt t tt --vei = wi = wi - wo,

vk: knetic energy dueto the relative angu1ar velociti6s, .. ･

Vb : potential energy due to the deviations ofangles from their stable points.

, ,t .. /. . ' 4'

3

2

Vcr

1

o

-- 1

vVk

.-----...-- "-'----r" "'`'-"r'"::

:'

:::t

Vp

Ftg. 2

Peij(p,u,)

1.4

1.2

1.0

O.8

Q.6

O.4

O.2

o.o

-6.2

t cr O.5 1.0t (sec)

Concept of the decision for eritical clearing tirne

fi-.:･-. 7

u][2 ]'

[3]

/-

Q.5 1.0

Fig. 3 Curve ofglectrical power outppt

.1.5e ij(rad)

9t


The value of energy function gradually increases during the fault, and by finding

the time where this value equals to the critical value Vb,, the critical clearing time for

stabihty can be determined.7)

This procedure is as follows:

(1) Construct an energy function V(D = V)t(t) + Vb (t),

(2) Find the time t" where dVb(t)/dt = O,

(3) Set V},(t") = Vbr,

(4) Find the critical clearing time t., by finding the time where Vb, = V(t).

4. PreposedMethod

4.1. Esimationofelectricalpoweroutput

in this section, we investigate the estmation of electrical power output of gener-

ator which consider the nonlinearky of the load. Figure (3) shows an example of the

electrical power output curve along the sustained fault trajectory in a multmachne

system.

Curve [1] is the power-angle considering the constants load. Curve [2] is the

power-angle considering voltage dependence of the load. Curve [3] is AP.ij (eij) which

shows curve [1 ] -curve [2] . Hence, using this electrical power output 4i of ith machine

can be identified by the forms of Eq. (10) and Eq. (1 1). The curve identified by Eq.(10)

is ptustratedadotted line in Fig.3. '

APdj(eij･) L- Kl (eij - eije)i/3

+ K2(eij - eijO) + K3 (eg - eije)3

n"Plei = i Peij'

pbij = ctr sin eij + Do･ cos eij･

-{Kl (eij･ - etie)i/3 + K2 (eij - eij･e) + K3 (eij - e ijO)3}

(1O)

(1 1)

[he first and second term of right hand side in the Eq. (1!) are the power con-

sidering constant impedance load, and the third term indicates the effect of the voltage

dependence of load. The Constants Kl, K2, K3 are identified with the method ofleast

squares.

'Ihe merit of Eq. (1 1) is as foilows:

(1) The effect of the voltage dependence of load can be dealt with correctively, because

the nonlmearity ofelectrical power output is almost expressed by Eq. (10).

(2) Energy function is analytically calculated, because electrical power output is

expressed by polynomial of relative rotor arigle of generators.

(3) Approximate value ofcritical energy is easily obtained.

4.2. Approximationofswing

in the case, considering the voltage dependence ofload, sequential computation of

system load is required for the process of calcuiation of the sustained fault trajectory.

So we propose an approximate solution of the swing equation, fbr the fast determina-

tion of the sustained fault trajectory and for the estimation of electrical power output.

jF2zst 77ansient Stabtiity Evaluation by Estimetion ofE:lectrical fower Ouiput 93

The approximate solution derives as follows :

(1) Estimate the values of a4iO, Bi and 7} of ith machne, by solving the Eq. (3) for a

few steps after the fault occurred. ･'(2) Using these values,

d6i,wi = dt

= att, ale= qie +k¥1

-k*AtZ 6ie Ti

(12)

(3) Making use of the above wi,

6i --- 6i O+Icoidt (13)

where

ori: angular acceleration ofith machine,

6i: correction constant ofangular acceleration ofith machine,

11r: tme constant of ith machine,

6iO : initial rotor angle of ith machine,

m=tlAt.

Figure 4 shows 6 and (D fbr the sustained fault. Each process ofsimulation,

and proposed method in this paper is Mustrated in Fig. 5. Double line of Fig. 5

part that the sequential computation of load is required.

direct

is the

6 (rad)

6

4

2

o

: Simulation

- --:Approximation

f

-

4

-

7

-

7

'

7/

'

. 4621

/

'

/

-

63

61

o.o O.2 O.4 O.6 O.8

t (sec)

Fig. 4-1 Approximation of･6i for a 3 machine system

94 Al sushi ISHIGAME and Tsuneo TANIGUCHI

w14

12

10

B

6

4

2

o

.(rad!sec)

:

-- T':

4q

-

SimulationApproximati6ri.

77

fr

7

Z

7

7

f

;--

f

7!-

f

-

q

-

7 Z

-

/

-

/

!

/

!

'

ul 2

tu 3

'

Simulation

method

Direct

method

Propesed

method

o.o

Ftg. 4-2

fau1t

oqcurred

O.2 O.4 O.6 O. t (see)

'Approximatigp of wi for a 3 mp, , chine systegi'

.--------- to 1 e

rred

Trajectoryealculation<sustainedfault>

"Regularfalsi"

'eriticalTrajectorycalculation'"

<sustainedfault>

.iteratio

elearin

ealculationofV,Vp

alongthetrajectory'

'Approxilnatetrajectory

-H-- t----"--nt-t--+---T---L-r--- .--------------

ealculateV,VpusingconstantKl,k2,K3'

tt

-･. .'. . Vp' max

nv- :The sequential computation of load

Fig. 5 - -The process ofestiipation for manst'ent stability

for

time

fust 71,unsient Stability'Evaluati'bn by Elstirnation ofE:lectrical Power Ouiput 95

5. Simulation

The dsual direct and proposed method under considering the voltage d6pendence

of load are applied tg the translgnt stability analysis of two model systems which con-

sist of 3 gen. erators 9 buses and 9 generators 25 buses.-The assumed disturbahee atFis

the 3-phase short circuit which ,occurs one of two lines, and the faulted 1ine is cleared by

opening the circuit breaker of faulted line after a certain lapse of time.

Table.1 shows the value of load parameter used in this analysis, and Table 2 shows

the critical clearing time in each method. The comparison of the ttme required to calcu-

late is demonstrated by the ratio, based on the tme required for the usual direct

method and the resuks are written in brackets of Table 2.

Further on the proposed method(1), first 5 steps are used for calculating the

trajectory and every S steps sampling data are used for calculating the electrical power

output. On the proposed method (2), fitst 3 steps for the trajectory and every 10 steps

for the electrical power output.

G22 F.2

7

Fl.s

8

F3

1

9

Gl

4

3

6

G3

(a). 3 machine system

F6.FoFF4Gl G91

'19

.1311,

17

20

12

21 22

14

23.

15

lg

24

1625

10

G 22

3G3G4 4

55G'

78G7G8G6

6

Fig.6

(b) / 9. ..rn,a c/ h. i･ n e. s y･ ,F t･e rtt -.

Condguration of a 3 apd a 9 maphinp systemS

Table 1 Load parqmeter

..Pcharactesist.ics Qqharacteristics

np Tp Ap nq Tq. ,Aq

@ 2.0-･ -,e･ 2.0

- o

Q 1.27 .O18' -- 1.17･ 1-.19- .O17 1.20

@ -1.061 .o2ei 1.00 -1:29- .025･ 'O.89

@ O,95' -le25' i･is8i 'l.S3' ･.032''1'132'

(!l) O.81i .O10' '' i.7'7' ･'oLgl .040' 1:74


1

Table 2 Estimation of the critical clearing time (sec) for a 3 and a 9 machine systems

System FaultLoad

Parameter

simulation

methoddirect

methodProposedmethod

(1)(2)Fl o O.44-O.45 O.42 O.38O.37

@ O.40-O.41 O.40 O.38O.38@ O.40-O.41 O.39 O.36O.38@ e.37-O.38 O.41 O.39O.41@ O.38-O.39 O.37 O.36O.39

[1] [O.141IO.09]

3F2 ¢ O.26-027 O.24 O.MO.24

machine' @ e.21-022 O.23 O.22O.21@ O.21-O.22 O.22 o.2eo.2o

[1] [O.18][O.12],

F3 @ O.42-O.43 O.42 O.39O.39@ O.44-O.45 O.46 O.44O.44..@ O.47-O.48 O.49 O.47O.46

[11 [O.12]fO.08]

F4 o O.3-O.33 O.35 O.33O.32@ O.31-O.32 O.33 O.32O.33@ O.29-O.30 O.32 O.31O.32@ O.27-e.28 O.29 O.30O.31

[1] [O.15][O.09]F5 @ O.23-O.24' O.19 O.18O.18

9 @ O.22-O.23 O.17 O.17O.18rnachine @ O.18-O.19 O.16 O.16O.16

@ O.16-O.17 O.16 e.15O.16[1] [O.181[O.10]

F6 @ Oi4-O.15 O.14 O.15O.15@ o.lg-o.2e O.15 OJ5O.15@ O.17-O.18 O.15 O.14O.14

[1]' [O.16]{O.10]

From the results of table 2, it is proved that the proposed method gives the good

approximation of the direct method, and that the calculating time of proposed method

is about 1O% of that of the'usual direct method.

6. Conclusion

in this paper, we demonstrated the method for fbst evaluation of transient stability,

constdering the voltage dependence ofload. , From the result, the time required for calculation is considerably reduced than that

of the usual direct method and the accuracy is satisfactory in most cases. In the future,

it is expected that the power system might be largor and system load characteristics

might be more various. In the sense, the proposed method seems to be very suitable and

adaptable for fbst evaluation of the developed power system.

I72zst 7)ransient Stabtiity Evalaation bj, Elstimation ofETectrical JPbwer Ouiput 97

1)2)3)4)

5)

6)

7)

Refetences

M. Musatizi, et aL, IEEE [Ittans., VoL EC-1, No. 4, pp34-38, (1986)

N. Kakimoto, et al., PE87-60, I.E.E. of JapanY. Kojima, et al., 'Lecture 982 in annual Meeting of I.E.E.- of Japan (1988)

M. Gotou, et aL, Trans.'l.EEJ., 52-B85 (1977)

T. Athay, et aL, IEEE Trans., Vol. PAS-98, No. 2, pp.573-584, (1979)

K. Maruyama, et at, PE-87-61, LE.E. of Japan

N. Kakimoto, et al., Trans. I.E.E. of Japan, Vol. 98, No. 5/6

f

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