Modelling of Out of Step Conditions in Power Systems

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IEEE Transactions on Power Systems,Vol. 11. No. 2, May 1996 839

Modeling of Out-of-Step Conditions in Power Systems’

N.V. Kosterev, V.P. YanovskyDepartment of Electrical Power Engineering

Kiev Polytechnic Institutz, Kiev, Ukraine

Ab st ra c t: The out-of-step protective relays are em-ployed for detecting out-of-step conditions in power sys-

tems [l, , 3 ] . The relay parameter setting requires simula-

tions of expected out-of-step conditions. Here, the out-of-

step conditions are modeled as a periodic motion, and are

represented analytically by the Fourier series. The Fourier

coefficients are computed using numerical iterations. Atwo-level iteration process is developed to obtain periodic

bus-voltage dynamics. The computed bus voltages are

used next to derive and to plot apparent impedance and

line power loci needed for the out-of-step relay parame-

ter setting [l, 2, 31. The proposed techniques are illus-

trated using an example of the National Power Company

of Ukraine.

Keywords: out-of-step relaying, power system modeling,

transient stability, power system protection.

1. IntroductionThe out-of-step OS) conditions when occur in power sys-

tems may lead to severe outages in the transmission sys-

tem, create large stress on electrical equipment, and inter-

rupt manufacturing process of energy consumers [l 3 41.The OS conditions result from the system instability

caused by large faults, stressed operating conditions, se-

quences of disturbance contingencies, etc. In most cases,

the stability loss starts so-called two-frequency OS condi-

tions, when we say that the system can be approximately

divided in two subsystems (accelerating and decelerating

generator groups) according to their average frequencies

[3,4]. An imaginary line separating the generator groups is

called the out-of-step cut-plane. The out-of-step relays areemployed to detect the OS conditions in power systems.

Typical out-of-step relays used in the CIS (former Soviet

Union) countries are briefly described in Appendix A.

When the OS conditions are detected, appropriate actions

have to be taken t o terminate them. In complex modern

power systems, sustained OS conditions are not allowed,

and generator groups going out-of-step are required to be

separated in the first OS slip cycle. The controlled sepa-

ration is done along the OS cut-plane at pre-selected lo-

cations in such a way that generation and load in the sep-

arated subsystems are reasonably balanced [l ,2, 31.

95 SM 540-5 PWRS A paper recommended and approvedby the IEEE Power System Engineering Committee of the

IEEE Power Engineering Society for presentation at

the 1995 IEEE/PES Summer Meeting July 23-27, 1995

Portland OR. Manuscript submitted August 1 1994;

made available for printing June 5, 1995.

D .N.KosterevDepartment of Electrical Engineering

Oregon State University, USA

A transmission system planning engineer determines con-trolled OS cut-planes in order to place the out-of-step

relays effectively. Next, a transmission system planning

engineer has to provide a protective relay engineer with

quantitative information needed for the relay parameter

setting. This information includes apparent impedance

and line power loci for (i) the out-of-step conditions in the

controlled cut-plane, (ii) stable swings, (iii) the OS con-

ditions in adjacent (other tha n controlled) cut-planes. To

this end, extensive simulations of power system transients

for various power system contingencies are required.

The traditional approach for computing the OS conditions

includes simulations of electro-mechanical transients using

a transient stabili ty program. Typically, a severe single-

contingency disturbance, e.g. a three-phase fault, is ap-plied t o the system long enough to result the system in-

stability. This approach usually requires large computa-

tional time and computer storage of simulated transients.

Engineers at the National Power Company of Ukraine also

indicated that it was not always easy to obtain OS con-

ditions in a specified cut-plane by applying single or even

double contingency disturbances. To break the system in

the specified areas, complex stressed operating conditions

had to be created and multiple-contingency disturbances

had t o be applied. Thi s increases complexity of the pro-

cess, and also can make thus generated data be sensitive

to time and sequence of disturbance occurrences.

The joint effort of a power-engineering research group at

the Kiev Polytechnic Institute and the National PowerCompany of Ukraine has been put t o develop an advanced

approach for modeling the OS conditions. This modeling

approach has to be used primarily for the out-of-step relay

setting. The ultimate performance of the method is to be

judged by qualitative and quantitative representation ofapparent impedance and line power plots during the OSslip cycles. Also, using this approach, an engineer should

be able to generate the OS conditions in a specified cut-

plane by defining the cut-plane explicitly. This allows to

obtain a variety of the OS conditions in controlled and ad-

jacent cut-planes which is essential for ensuring both, high

sensitivity and selectivity of the relay settings. Another

expected advantage of new method has to be computa-

tional time savings when compared with numerical inte-gration. The output data should be arranged in compact

and easily accessible data bases used later by interactive

computer packages assisting an engineer in the relay set-

ting.

A computationally effective method for modeling the OSconditions in power systems has been developed, and is

0885+8950/96/ 05.00 Q 1995 IEEE



840

presented here. The proposed method models the out-of-

step conditions in power systems as periodic motion repre-

sented analytically by the Fourier series [6, 71. The Fourier

coefficients are computed using numerical iteration . This

method is implemented in a computer package, and has

been used by the NPCU and other power companies in

the former Soviet Union since 1985. Years of operating

experience confirmed efficiency and reliability of the pro-

posed approach for the out-of-step relay setting.

2. Modeling Out-of-Step Conditions in

Power SystemsConsider a power system model [4] where:

synchronous machines are described by the classical

models E = const;

composite electrical loads are given by shunts with

constant admittances;

transmission network is represented by one-line di-

agram, and its elements are given by ?r equivalents

with constant parameters.

Then, the power system model can be described by the

following set of algebraic and differential equations:

synchronous generators:

power transmission system:

0 = G E y o ;

where:

6 - generator rotor angle;

Pm mechanical power;

M - inertia constant;

E = E'(cos 6 + j sin 6) - generator transient EMF

x a generator transient reactance;

y - network admittance matrix;

G - diagonal mat rix consisting of admi ttan ces between thegenerator terminals and the generator transient EMFs E .

Consider a power system with n generators and m trans-

mission network buses. First , a pre-disturbance steady

sta te is computed using a powerflow program. Init ial con-

ditions for E' are determined for each generator from the

steady state and are held constant through the entire com-

putations.

Consider two subsystems, L 1 and L a , going out-of-step.

Denote average frequencies of both subsystems asw1 and

w2 respectively. Assume that w > 0 2 , i.e. the genera-

tor group L 1 is accelerating, and L2 is decelerating. A

periodic solution for system 1) is sought in the reference

frame rotating with the angular frequency w2.

To increase the algori thm computational efficiency, th e fol-

lowing properties of power system model are used 1):

s - generator rotor speed;

Pd - damping power;

U = u d + U q - bus voltages;

the generator model is linear in parameter 0These observations help to reduce the problem dimension-

ality by designing a two-level iteration process consisting

of inner and outer loops. The outer iteration loop com-

putes bus voltages in the transmission sy stem , while the

inner iteration loop determines generator out-of-step mo-

tion. The outer loop consists of the following steps:

A. Assign an analytical expression for generator rotor an-gle dynamics in terms of the truncated Fourier series.

Then, according to the generator membership to a par-

ticular subsystem, ,ye have:

6 ( t ) = 6; + c(6, COS kwt + f k Sin k w t ) , 1 E L i ,

k = l

r

6' ( t ) = 6; +wt +E(& OS kw't + al l Sin k w t ) ,

where: w = w1 w2 (may be specified by an engineer,

or estimated using various well-known methods [3]). This

completes an initial step of the outer iteration. At t = 0,the expressions for 6 t) assume their steady-state values.

B. For a given rotor angle 6 ( t ) , the generator currentj ( t ) = GE(t) injected to the power network is written

E L2, ( 2 )k = l

as:G E ( t ) = G (E' cos 6 ( t )+ E' sin 6 ( t )

r

= f ( t ) = I,d + jI: +c(l, kOS kwt + :k Sin k w t )

k = l

+ jE(Izkos kwt + I:k sin k w t ) . 3)k = l

To compute f t), sine and cosine functions of the Fourier

series for 6 t) are to be taken. Either Bessel functions

or power series approx imation can be used. For the lat-ter, the n-th order polynomial, f 6 ) = ao6, + a 1 6 " - ' +. + a , - l 6 + a , has to be computed. To this end, theGorner scheme can be employed by representing the poly-

nomial as f 6 ) = (. . . a06+ a l + a 2 ) . . )6 + a,). Then,

to compute f 6 ) , we need only n 1) multiplications of

Fourier coefficients, and n additions of Fourier coefficients

and a constant. This algorithm is very effective numer-

ically. The truncated Taylor series expansion is used to

approximate sine and cosine functions. Then , sin 6 ( t ) and

cos 6 ( t )are computed by matching appropriate power se-

ries coefficients.

C. Specify analytic expressions for bus voltage dynamics:

r

0(t)= u,d jui + ( u f k COS kwt + u f k Sin k w t )

k = l

r

+ j ( u 2 k COS kwt + u : k Sin k w t ) . 4)k = l



841

D. Next, the voltages a(t) re applied to the synchronous

generators. U ( t )= Ud(t)+j U,(t) are substituted in (1) for

the corresponding generators as their terminal voltages.

Next, the iteration process described in Appendix B is

applied to the generator models to find the out-of-step

motion of generator rotor angle S t ) in term s of the Fourier

series. This step forms the inner iteration loop.

Conditions for terminating the outer iteration loop are

similar to those for Newton’s method. The conditions are

checked here, and if they are satisfied the outer iteration

loop is quit. Otherwise, the iteration proceeds to step B.

The following advantages of the two-level iteration process

are to be noticed:

Bus voltage computations use the same admittance

matrix Y - l at every iteration step.

Dimensions of the system are reduced significantly.

A t each outer iteration step, m-order sets of linear al-

gebraic equations (5) in complex variables are solved.

At each inner iteration loop, n second order equa-

tions (1) are iterated.

For the inner iteration loop, the choice of matrixA as the Jacobian provides high (almost quadratic)

convergence rate.

The presented method takes into account intra-area gen-

erator swings in the subsystems L and Lz. The further

approximation of the OS conditions can be made by ne-

glecting intra-area oscillations in the subsystems:

6‘ = 6; ,1 E L1, bm = 6,m +w t , m E La.

Then, the problem is reduced to solvipg sets of linear al-

gebraic equations 5 ) .

3. Validation Studies

The approach was developed in early 1980’s, and its

performance was evaluated using a benchmark model of

Kazahstan power system prepared by the Institute of Elec-

trodynamics (electric power research inst itute in Ukraine),

Figure 1. The system is represented in a transient stabil-

ity program (TSP) with 37 buses, 8 generation stat ions of

total capacity 3900MW. Generators are modeled in de-

tail with excitation and governor controls. Considered

disturbance is tripping 75% of generation at power sta-

tion 264. This disturbance results in instability, where

stations { 238 , 239 } are accelerating, and the rest of

the system is decelerating. The out-of-step cut-plane is

shown by a dotted line. The electr ical center is located

on line 257-256, 20kV line with impedance 22+ 96Q.

Impedance and power loci are plotted for this line in Fig-

ure 2 by solid lines, and used as a reference for this study.Impedance axis are marked in steps of 100R, and power

axis are marked in steps of 100MVA.

Next, the out-of-step conditions are computed using theproposed modeli ng approach. The OS cut-plane is spec-

ified explicitly by defining accelerating and decelerating

generator groups. The Fourier coefficients for bus voltages

during the OS slip cycles are computed and used to derive

the apparent impedance and power plots for line 257-256,

shown in Figure 2 by broken lines. Apparent impedance

Figure 1: benchmark model: equivalent model of

Kazahstan power system

and line power plots computed using the proposed mod-

eling approach are very similar to those computed by the

TSP, both qualitatively and quantitatively. This observa-

tion also is confirmed by other studies and practical expe-

rience. Studies show tha t apparent impedance plots de-

pend mainly on out-of-step cut-plane, transmission system

state (open lines and lines in service), and power loading

of the protected line and in the system. At the same time,

apparent impedance plots are somewhat insensitive to th e

speed of the OS slip cycle, and consequently values of w

chosen in the algorithm.

Figure 2: apparent impedance and line power

The proposed methods assumes so-called two-frequency

out-of-step conditions. This is consistent with presently

used out-of-step relays which are designed to detect two-

frequency OS conditions [3]. There are systems where the

initial disturbance can cause instability resulting in sev-

eral generator groups going out-of-step already in the first

OS slip cycle (called multi-frequency OS conditions). Sus-

tained long enough, the two-frequency OS conditions also

may develop in multi-frequency regime due to the sec-

ondary stability loss. If there is a possibility of having

multi-frequency conditions in the system, they should be

simulated and their effect on the OS relay performance

should be analyzed. The proposed modeling approach

does not characterize such conditions, and numerical in-

tegration is preferred here. The syBt,ems where mult i-

frequency OS conditions can occur in th e first OS slip cycleare not very common, and fast detection of two-frequency

OS conditions and controlled system separation in the first

OS slip cycle prevent the instability spread.

4. An Illustrating Example of the Algo-rithm Application for Out-of-Step RelaySettingThe developed algorithm was implemented in a computer

package and has been used by the National (formerly Uni-



842

....................... D C Slyutich Nep

Khmelnitsk

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

__

south

UkraineNFSLadmk PS

I .

Figure 3: One-line diagram of the National Power

Company of Ukraine, Kiev area

fied) Power Company of Ukraine (NPCU) for the out-of-step relay parameter setting since 1985. Out-of-step

relay settings are updated quarterly by the NPCU engi-

neers. Typically, two engineers are involved in this pro-

cess. Occasionally, additional studies are conducted when

requested by other power companies in the CIS.

We present an example of using the algorithm for out-of-

step relay setting in Kiev area. This example is prepared

by engineers at the planning department at the NPCU.

The northern service area of the NPCU is depicted in Fig-

ure 3. The figure shows 330kV and 75OkV(thick lines)

transmission lines and major generation and load centers

in the considered area. The Kiev area includes concen-

tration of large industrial and residential loads as well as

large generation capacities: Tripolsk, Chernobyl, TEC-5 and TEC-6. These generation station serve area loads

and export power to industrial centers of Vinnica, Zitomir,

West Ukraine. Disturbances tend to accelerate the Kiev

area with respect to the rest of NPCU system. To pr+

vide higher power transfers through the interties outgoing

from the area, they should be protected against the OSconditions. The following OS cut-planes are considered:

A) Kiev - the rest of Ukraine 1; B) Kiev - the rest of

Ukraine 2 ; C) Vinnica - West Ukraine; D) Vinnica and

Moldova - West Ukraine. The controlled OS cut-planes

are shown by dotted lines.

The OS conditions in the controlled cut-plane are com-

puted for the normal case and various contingencies in-

cluding open lines in the cut-plane and lines adjacent tothe cut-plane. The OS cut-plane is defined explicitly by

specifying the lines in the cut-plsne. Once the OS condi-

tions are computed for the nominal system, they can be

used a starting point for computing the OS conditions for

various contingencies. This batch-type process increases

the algorithm computat ional efficiency. The computed OSconditions are stored in a da ta base in terms of the Fourier

coefficients representing the periodic bus-voltage dynam-

ics at the selected buses of interest.

I1

This information includes bus volt

nals for (i) the OS conditio

belongs to the controlled OS

erate), and (ii) stable swingsadjacent cut-planes OS relay s

Apparent impedance and line power

the Fourier expressions for the bus

has a feature which allows to de

electrical centers in the system.

relaying engineer, who performs the

setting (see Appendix A ) .

5 . Conclusions

ing the out-of-step conditions in pow

motion. The method is designed

the out-of-step relay parameter

tion is described analytically in

and the Fourier coefficients are comp

iterations. The two-level iteration

realization, fast algorithm convergence can be achieved

compared with the numerical i

tation of the mod

National Power Company of Ukraine.

stitute and was

companies in the

the National Pow



843

t

Figure 4:out-of-step relay, the first stage

Appendix A: Typical Out-of-step Relay used in theCIS Countries

We briefly describe a typical out-of-step relay manufactured

and used by most power companies in the CIS countries. Themanufactured relay consists of three stages and can be cus-tomized to specific applications by appropriate switches [3].

The first stage detects the OS conditions in the first OS slipcycle. This stage is implemented using two mho relays, and

its operation is similar to that of the out-of-step relays usedin the North America [I 21, 01s shown in Figure 4. The trip-

ping time difference between outer (A) and inner B) relaysshould exceed the timer setting required to discriminate the

out-of-step relaying against line faults [l 2, 31. If the OS is

detected, then the protected line is opened. The inner mhorelay should operate for all expected out-of-step conditions in

the cut-planes which include the protected line, and should bediscriminated against stable swings and out-of-step conditionsin adjacent cut-planes. The outer mho relay should be dis-

criminated against the maximum line loading conditions underwhich the steady-state stability is maintained.

The second stage reflects previous trends in OS protections,

when power systems had a relatively simple structure, the OSconditions could be tolerated temporarily, assuming that suc-

cessful resynchronization was possible after a fewOS slip cycleswithout separating the system [3]. The system separation wasenforced only after a pre-specified number of OS slip cycles.This is still reflected in the OS relay design [3]. The secondstage determines the existence of the out-of-step conditions af-ter a pre-selected time interval. This stage is implementedusing a combination of an mho relay an$ a directional power

relay, as shown in Figure 5. If the power relay either oper-ates or restrains when the impedance loci is within the mho

relay operating region, the out-of-step half-cycle is registered.

Next, when the mho relay restrains, the full OS cycle is regis-

tered, assuming that a previous half-cycle has been recorded.A counting sequence example is shown in Figure 5: 0-initialpoint, A I - 1/2 OS cycle, B1- 1 OS cycle, Az- 3/2 cycle, Bz- 2cycles, . (up to 4 cycles). When the mho relay restrains, the

timer is started, and if there is no next cycle registered within

the timer setting, the out-of-step relay is reset. The mho relaysettings are often the same as those for the outer relay in thefirst stage. At the end of the second stage, the electrical cen-ter is detected using an mho relay. The requirements for this

relay settings are the same as those for the inner mho relayin the first stage. If the OS conditions are sustained at the

end of the second stage and the electrical center is detected,the protected line is disconnected. The third stage duplicatesthe second stage with a time delay. In modern systems, fasttermination of OS conditions is required in the first OS cycle,and the second stageis often used as a backup of the first stage

and separate the system in the secondOS slip cycle.

Figure 5: out-of-step relay, the second stage

The relay has three functional blocks:(i) first OS cycle detector (first stage),(ii) OS cycle counter (second stage),iii) electrical center detector (second stage).

Typically, three mho relays are used, each carries two functions.A directional power relay is used in the transient cycle counterand is coordinated with the corresponding mho relays.

An illustrating example of the OS relay settings at theChernobyl-Vinnica line (see section 4) is prepared by a pro-

tective relaying engineer at the NPCU. Table 1 presents therelay functions and parameter settings. Figure 6 shows mhorelay characteristics (solid),aswell as expected OS conditionsalong the protected line (dashed) and stable swings (dotted

lines). Zr is the impedance vector representing the heaviestloading conditions at the line, under which the steady state

stability is maintained.

Table 1: Set tings and functions of th e out-of-step relay.

2 maximum impedance reach;8 - angle of the maximum impedance reach;S - relay characteristic offset in the 3rd quadrant;b/a - ratio of minor and major axis for the elliptic characteris-

tic (= 1 for a circle).Functions (i)-(iii) correspond to the relay functional units de-scribed above.

. .-50

100 -50 0 50 1 15 ,ApparentResistance [Ohm]

Figure 6: out-of-step relay characteristics



844

Appendix B: Computing Periodic Motion

Consider the following differential equation:

k = F ( x , ) , z to) = 20 (62

where F ( x , ) s a continuously differentiable in x, 2a periodicin t , bounded function ( if the function F ( z , ) has a period T

different from 2x, then it can be transformed to a 2a periodic

function by scaling time by a factor 2 x / T ) . A periodic solution

x ( t ) = z ( t + 27r) of 6 ) is represented in terms of the Fourierseries:

x ( t ) = O +Z M ~os k t + ~k sin k t j , (7)

k = l

where:

r - number of considered harmonics,

Mk k - k- th harmonic coefficients,M O constant component.

To find the Fourier coefficients, the following iteration procy

dure is used. The function F ( x , t ) is linearized at a point dand the following linear time-varying non-homogeneous system

is constructed:

= A(zt , ) z+ @(z , t ) , 8)

where the function @ ( x , t ) is evaluated at the linearization

point x and is 2x periodic in t :

t )= O +C ( B ~osk t + c k sin kt). 9)k = 1

The computed periodic solution of system (8) is taken as a next

solution ( 2 ) estimate:

k c t l = A ( x t , ) x t t 1 + @(xi, ) . 10)

By substituting Fourier polynomials 7) and (9) into left andright sides of equation 10) respectively and equating the cor-

responding harmonic coefficients, the system of linear algebraicequations is solved to compute Mit1, Mit1,NLtl, k = 1.. .F).

Then, the iteration is proceeded by linearizing system 1) at

the next point xctl(t). The iteration process terminates whenthe following conditions are satisfied:

M; M i I [ E , Nit N i II E

MA I [ E , ~k = 1 . . r, 11)

where E is a specified tolerance.

If matrix A = A( ' )s independent on t and nonsingular, thenthe system has the unique 27r periodic solution with Coefficients:

Mi'' = -(A2 + k2E)-'(AB; +kc:),N; = ( A 2+ k2E)-l (-ACL +kB;), 12)

= A-1B6+1 k = 1, . . r ,

where E is an identity matrix.

Performance of the presented iteration process depends upon

the choice of matrix A ( z , t ) [9]. In general, Jacobian can berecommended as a good candidate. Since the computationaltime and memory storage are much less for a time-invariantmatrix, a time-invariant matrix A is used. To this end, Jaco-bian is evaluated only for an absolute value of ~ ( t )t t = 0:

.

The above iteration process corresponds to the Newton (gradi-ent) method for solving a set of nonlinear equations. Various

modifications of this technique can be used to achieve compu-

tational efficiency and to secure algorithm convergence [9].

Acknowledgement

The authors would like to thank engineers in

partment at the National Power Company of

ing to prepare this paper.

References

Applied Protective Relayzng,Corporation, 1979.

E.W.Kimbark, Power SystemCircuit Breakers and P

1950.

Ya. Gonik, S.Iglitskiy, Protectivenating t h e Out-of-Step1988 [in Russian].

V.A.Venikov, Electro-Mech an cPower Systems, Mir Publishers, 1

V.A.Venikov, TranssentPow w Sys tems, Pergamon

D.Lika, Yu.Ryabov, IteratsEquations in Nonlinear O s

Shtiintza, 1974 [in Russian].

Power Stations Under PowerVisha Shkola, Kiev, 1986 [in R

Nickolay V. Kosterev received his EDegrees in Power Engineering from Kievin 1963 and 1969 respectively. He was

Degree from the Ukrainian Acad

rently he is Professor of Power

been a Principal Investigator of aincluding development of

computer simulators for power engineerelated to the leading Bower and indust

institutes and power stations in the CI

member of Doctoral Committee at the UScience. He published over 1 1 0 articles an

a book.

Valentin P. Yanovsky receDegrees in Power Engineering

in 1972 and 1985 respectively.fessor at KPI, Power Engineeringcurrent research include pmodeling and simulations.

Dmitry N. Kosterev r

State University in 1992 . He was a stud

He developed a computer package

ter setting implemented by power

of interest include power system

measurements and signal processing.

Modelling of Out of Step Conditions in Power Systems

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