Indian Journal of Engineering & Materials Sciences Vol. 9. February 2002 , pp. 20-24

Evaluation of optimum amount of load shedding to prevent dynamic voltage instability in a longitudinal ac power transmission system

G Panda" & A Chakrabartib

"Department of Electrical & Electronics Engineering National Institute of Science & Technology, Berhampur 761008, India

bDeparlment of Electrical Engineering, B. E. College (DU), B. Garden Howrah 711 103, India

Received 28 No vember 2000; accepted 14 December 2001

The present investigations deal with criteria for Optimum Amount of Load Shedding (OALS) using a dynamic load model. The amount of load to be shed has been calculated using establi shed parameters of the dynamic load mode l. In the event of voltage decline, the proposed optimum amount of load shedding criteria can be app lied to calculate the minimum amount of load to be shed at any point of time to avoid a voltage collapse. The effect of shunt compensation on load shed­ding is also investigated. The developed scheme theory has been simulated for a typical longitudinal power transmi ssion system of any urban metropolis of developing countries using digital computer.

The power transmission systems in developing coun­tries are basically radial in configuration, having less number of EHV parallel lines and are easily suscepti­ble to power and voltage instability. The nodal reac­tance being high, these systems are also very much sensitive to reactive power changes and have inherent limitations in real and reactive power loadability.

Literature survey reveals that voltage stability is an important aspect in determining global stabi lity of such weak transmission systems' ·4. TaylorS has re­ported a concept of undervoltage load shedding analogous to under frequency load shedding. Tuan et al. 6 have presented another viable load shedding algo­rithm. The proposed method is based on indications of risk of voltage instability and on sensitivities of these indicators to changes in load to be shed. However, the analysis is based on static models, and the dynamic aspects associated with voltage stabi lity are not taken into account. Tso et al. 7 have presented a load shed­ding scheme taking into account the generator dy­namics. Arnborg et af. 8 also presented a method for under voltage load shedding (UVLS) . The analysis takes into account the generator and load dynamics. Computational attempt in obtaining the stability zones or indexes being conventionaI9

•12, role of load in volt­

age stability study needs its proper modeling' 3. 1). In this paper, an optimum load shedding criterion

has been developed in order to prevent voltage insta­bility of the load bus in the power system in develop-

ing countries. The digital simulations have been con­ducted in order to determine the effect of shUlf't com­pensation on this criterion.

System Modelling A simplified longitudinal line model, with shunt

capacitive compensation at the receiving end, is shown in Fig. 1. The power system has one generator bus and one load bus. There are two identical trans­mission lines connecting the two buses. The voltage phasor diagram of the system (shown in Fig. I) feed­ing a constant power load (induction motor) is shown in Fig. 2, the load being assumed to be operating at UPF due to the compensation.

The instantaneous load power is considered equal to the active power supplied from the transmission system at a given instant of time:

p= EV Sin8= V2C .. . (I ) Xe

where E and V are the magnitudes of generator and load bus voltages respectively, 8 is the power angle,xe

Sending End

EL 8 J Xl:1 YL O

Fig. 1- Two-bus system for load shedding

p . JQ

PANDA & CHAKRABARTI: EVALUATION OF OPTIMUM AMOUNT OF LOAD SHEDDING 21

E Ix.

v

Fig. 2-Phasor diagram of the system

is the equivalent impedance (assumed purely induc­tive) of the system and G, is the load conductance.

From Fig. 2, we get:

£ 2 [ 2 2 = V( I-Bs/,xe) } +( Ixe) ... (2)

where Bsh is the shunt susceptance and I is the current supplied by the system.

Substituting Eq.(l) in relation (2), we get: £2

V 2 = ______ --,---...,--(1 - B,h X e ) 2 + G 2 X;

... (3)

The real and reactive power balance equation can be written as: (VG)xe=£ sin 8 (4)

V( I-BshXe)=£cos8 (5)

From Eq.(5), we get:

£ (l-B,/,xe)=- cos 8 . V

(VG)Xe I . = . x- cos 8 [uslOgEq.(4)]

slO8 V

. cos8 I.e., (l-Bs/,xe)=Gxe-.­

slO8 ... (6)

Model of constant power load (induction motor) is shown in Fig. 3. The load conductance to maintain constant power is G [G is also the reciprocal of (R/ + R/s) or simply reciprocal of R2/s, where R/ & R2 rep­resent of stator and rotor resistances of the induction motor and 's' be the slip, R/ being normally small, it is neglected for simplicity].

The load is approximated to have an exponential recovery when subjected to a step voltage distur­bance '6.

17. A useful model that captures this behavior

is shown in Fig. 4. The dynamic of constant pO'>Jer load can be repre­

sented as:

0) dxp =P/V)-P dl

. . . (7)

P+JQ

~I "I/\/\"\·~l

( l-s)RJ/s :~~ :_--,

Load Bus

Fig. 3-Model of induction motor in load bus (stator resislance neglected)

Vohage(V) Pow~r ,P i

Fig. 4-General dynamic load model

From Fig. 4, the instantaneous load power (P) dur­ing the transient period is given by:

P=XpP/V) :.xp=P/P/V)

(8 ) (9)

where xp is the state variable, Plv) and P,( v) are the transient and steady state load powers respectively, and Tp is the time constant of the load. The load pow­ers are given by following polynomial expressions:

PlV)=C2V2+C/V+CO

Ps(V)=Po(d2 V2+d/ V+do) (10)

.. . (II)

where C2, C/, Co, Po, d2. d/, do are constants and independent of V.

The mismatch between the steady-state load power and the instantaneous load power is the input to the integration block (shown in Fig. 4) which gradually changes the state variable xl' until the error signal fed to the integration is zero.

At this point P = P,(V). Therefore, Eq. (8) gives: Ps(V)

x,,= Pr(V) . . . (12)

The system capability is usually assessed using a P- V curve. The P- V curve is transformed to the xl' - V plane using the relations (9) and (12). The intersection of

22 INDIAN J. ENG. MATER. sel., FEBRUARY 2002

the two curves described by Eqs (9) and (12) are the possible operating points of the power sytem.

Again, substituting Eq. (I) in Eq. (7) [Substituting the value of V2 from equation(3)], we get:

Tp dxp =PlV)-V2C dt

dxp £ 2C T,, - =Pl V)- 2? ? ••• (13)

cit (1- B,hxe ) + C- x;

On simplification relation (13) becomes

P (V) X2[C 2 _ £ 2 C + (1- BShXe)2] , clx p ., 'e X; Ps(V) X; Tp --= 2 C 2 2

cit (1- B,h Xe) + Xe

... (14) Considering the established concepts of load dy­

namics, it becomes evident that the system remains

stable till dx" remains positive, critical stability cll

would be achieved as soon the system is defined by

clxp dx" = 0 and the stability is lost once - becomes

(ll cit negative.

Thus, during the state of dynamical equilibrium, from relation (14), it can be inferred that the stability at the load bus will be maintained if and only if:

P (V)X~ [C 2 _ £ 2 C + (1- B'hXe)2 ]>o .1' X; Ps(V) X;

i.e. , Pl V)x/ d- £2C+(l-B,hXelp,·(V»O .. . (15)

Substitution of (1 - B,h Xe) from Eq . (6) in relation ( 15), we get,

Pl V) (-cos28) >V2C

- V 2C i. e. cos 28> - - ­

p .,(V)

.. s:: I ol(-V 2C] glV1l1g u=-cos 2 P,(V)

at critical stability limit.

... (16)

... (17)

Expression (16) indicates the criterion of stability of the load bus of the LPS line model when the load is composed of induction motor loads. It is further evi­dent from expression (16) that to have stability, the motor active power (V2C) must be less than P,.(V), otherwise with V2C approaching PlV), the ioad bus stability is likely to collapse in order to encounter perturbations, particularly where the load bus voltage decline.

At the stage of critical state of stability,

-V 2C Cos28 = -- =-1

P,(V)

C = Ps(Y) y 2

Assuming G = _1_ , for the induction motor, at R2I ,.

the critical state ,

R2P,(V) Sc= 2 [Sc representing critical slip] . . . (18)

V When a line outage takes place in the transmission

system, the load state xp being still remaining con­tinuous , it would not show a sudden change. How­ever, the reduced line capability, in turn, reduces the load voltage & the instantaneous power being sup­plied to the load. In this instance, it is most likely that there is no intersection between the steady-state load characteristics and new network power capability . The system then may experience a voltage coll apse. In order to prevent the collapse, load should be shed such that a new equilibrium point is reached.

Thus, the minimum amollnt of load to be shed in order to stabilize the system at a given voltage is:

Pshed= P,.(V)-P . . . (19)

where P is the instantaneous power of the load , drawn from the network at the given voltage V. At thi s point

dxp = O. With the load cut amounting to Psherf, the dt

. . I dxp b' . system can run at cntlca state; - e1l1g Just zero. dt

The magnitude of Psher! has been referred as Optimum amount of load shedding (OALS) in any longitudinal load bus predominantly consisting of induction motor drive .

Simulation A longitudinal power transmlSSlOn system con­

nected with an industrial load bus comprising of con­stant power load being simulated, profile of power angle versus constant power load capacity are plotted in Fig. 5. It reveals that with enhancement in constant power load capacity, power angle increases which, in case of higher nominal load bus power (Po), is af­fected adversely (in other words, higher being the in i­ti ai power loading Po, the syste m security marg in gets lower).

PANDA & CHAKRABARTI: EVALUATION OF OPTIMUM AMOUNT OF LOAD SHEDDING 23

40~----~----~----~----~----~----~

00 02 0 4 06 0 8 10 1 2

constant power load capacity (p .u.)

Fig. 5-Profile of power angle vs. constant power load capacity (variation of nominal power Po)

1.0 ,-------------------------------------,

0.8

0.6 C/)"

. Q-Ui 0 4

TI ~

02

00

0.0 0 2 04

\---7 R,=O05 p.u

\ '>. \

'" ~--7 R,=D025 p.u .

- "-'" 'A.X;' ~ R,=O.01 p.u

A

~ 8. - A.....-' _ 6-~ A A 8. A -6- 6.

~ ........ W

0.6 0 8 1 0 12 14

Load bus voltage V(p.u.)

16

Fig. 6-Profile of critical sl ip Scvs. load bus voltage V (variation of rotor resistance, R2)

Fig. 6 shows the profile of critical sl ip (sc) for change in load bus voltage (the critical slip does not increase with increase in voltage, the rotor resistance must be increased in order to have higher critical slip) in the operating region of the load bus.

With increase of nominal power capacity (Po), the situation also improves marginally (Fig. 7). The in­crease of critical slip suggests the higher sustainability of the load bus while encountering critical states .

Fig. 8 reveals the intersection of the two curves (steady state and transient state load characteristics) in xI' - V plane. The point of intersection is marked as points 'a' and 'b'. The nature of curves being ob­tained following expression (12), intersection point 'a' in Fig. 8 indicates the possible operating point at near

C/)"

a. '5i TI :e U

0.7 ,------------------------------------,

0.6

0.5

04

0.3

0.2

01

00

0.0

------7 ' \ Po~p·u. 0 \

\

0.2

\ \ .. 0\ Po=5 p.u.

~ .. 0"- Po=4 P u

}

0.4 0.6 08 1.0

Load bus voltage V(p.u.)

12 14 16

Fig. 7- Profile of critical s lip Sc vs. load bus voltage V (variat ion of nominal power Po)

0-X

10 ,-----------------------------------,

8

6

, \ \

o

line cha'a::teristlCS

(1 line )

s. s. load cha'a::tenstics

(p'eshed)

~ line c:I1a'a:;tenslics ;1 ( 2 lines )

O ~--~---.--~----,_--~--~--~--~

00 0.2 0.4 06 0 8 1 0 12 14 16

Load voltage V (p.u.)

Fig. 8-Voltage collapse in Xp-V plane (load shedding)

nominal voltage (point b is of minimum significance as it exhibits unacceptable voltage magnitude). The system is initially in equilibrium at 'a'. When a line outage takes place in the transmission system, the load state xI' will still be continuous and would not show a sudden change. However, the reduced line capability in turn reduces the load bus vo ltage and the instantaneous power being supplied to the load.

At this instant (for the situation shown in Fig. 8), there is no intersection between the steady-state load characteristics (pre-shed) and the new network capa­bility. Hence, the system experiences a voltage col­lapse. To prevent the voltage collapse, load should be shed such that a new equi librium point ' c' is reached.

24 INDIAN J. ENG. MATER. SCI.. FEBRUARY 2002

2 0

1 8

shunt suscep ta nce , 1 6

~ > 1 •

'" minimum 0> 1 2 ~ load shed

!; 1 0

sh unl susceptance B .~ =O 7 1 P U B.~ = 097 1 pu If'-

~ c> I

8~ ~ ,

08 D

~ minimum 6 lo a d shed

u '" '" 06 a .... 0 '

0 2 i on

1 0 1 5 2 0 2 5 3 0 3 5 4 0

am oun t of load she dding P"'eo (p .u .)

Fig. 9-Load shedding analysis (role of shunt compensation)

Fig. 9 shows the effect of shunt compensation on the amount of load to be shed in order to maintain the stability at a given voltage. Amount of load to be shed decreases with the increase of shunt susceptance. If the minimum amount of load corresponding to the

minimum di stance (p ;~:~~ ) is shed before reaching the

minimum distance, the system could be stabilized. On the other hand, if load shedding is not carried out until afte r the minimum distance is reached, the amount of load required to be shed for stabili zing the system increases monotonically with time.

Conclusions In LPS system, the load density being non-uniform,

constant power loads usually predominate in indus­trial load buses. An optimum amount of load shedding (OA LS ) technique has been developed in the text that could reduce the possible voltage instabilities in those buses during voltage decline due to an outage in the transmission system. Governing parameters that could increase the critical slip being identified, analytical express ions for the optimum amount of load shedding during line trip has been developed and simulated for

a typical LPS line system. Effective role of shunt compensation in order to reduce the magnitude of OLAS has also been highlighted.

References I Chakrabarty A, Kothari D P & Mukhopadhyaya A K, Per­

formance operation and control of EHV power transmission system, (John Wilsey & Sons, Inc ., New York), 1995.

2 Weedy B M & Cox B R, Proc IEEE, (115 ) (1968).

3 Chakrabarty A & Mukhopadhyaya A K, Operating problems in longitudinal power systems, IEEE Int TENCON Con/, No­vember, 1989. India.

4 Gubina F & Strmenik B. IEEE Trans Power Syst, PWRS-1 2, (3) (1997) 11 2 1-1 128.

5 Taylor C W, IEEE Trans Power Syst, PD-7, (2) ( 1992) 480-488.

6 Tuan T Q, Fandino J, Hadjsaid N, Sabonnadiere J C & VU H, IEEE Trans Power Syst. PD-9, ( I) (1994) 34 1-35 I.

7 Tso S K, Zhu T X, Zeng Q Y & Lo K L. IEEE Proc Gener Translll Distrib, 144 (2) ( 1997) 8 1-86.

8 Arnborg S, Andersson C. Hill D J & Hiskens [ A. Int J Elect Power Energy Syst, 19 (2) ( 1997) 141 -149.

9 Barquin J, Gomez T & Pagola F L, IEEE Trans Power Sysr, PWRS-IO, (4) (1995) 1952- 1962.

IO Canizares C A, De Souza A C Z & Quintana. V H. IEEE Trans Power Syst , PWRS- I I, (3) ( 1996) 144 1- 1450.

II Canizares C & Alvarodo F L, Computational experience wirh the point of col/apse method on vel}' large ACIDC sys­tem , 1991 Deep Creek Lake workshop on Vol tage stahili ty and security Aug. 5-7, 1999.

12 Nagao T, Tanaka K & Takenaka K, IEEE Trans Power Syst. PWRS-12, ( I) (1997) 273-281 .

13 Borghetti A, Caldon R, Mari A & Nucci C A, IEEE Trans Power Syst. PWRS-1 2, ( I) (1997) 293-303.

14 Lind R & Karisson D, IEEE Trans Power Syst, PWRS- II , (4) (1996) 1677-1682.

15 Hong Y H, Pan C T & Lin W W. IEEE Trans Power Syst, PWRS-12, (4) (1997) 1555- 1560.

16 Xu W & Mansour Y , IEEE Trans, PS-9. ( I) (1994) 479-493.

17 Hill D J,IEE£ Trans , PS-8. (I) (1993) 166-1 76.