voltage stability analysis

Voltage stability analysis of radial distribution networks

M. Chakravorty, D. Das*

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

Received 2 December 1999; revised 17 April 2000; accepted 19 May 2000

Abstract

The paper presents voltage stability analysis of radial distribution networks. A new voltage stability index is proposed for identifying the

node, which is most sensitive to voltage collapse. Composite load modelling is considered for the purpose of voltage stability analysis. It is

also shown that the load ¯ow solution of radial distribution networks is unique. q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Radial distribution networks; Voltage stability analysis

1. Introduction

A power system is an interconnected system composed of

generating stations, which convert fuel energy into electri-

city, substations that distribute power to loads (consumers),

and transmission lines that tie the generating stations and

distribution substations together. According to voltage

levels, an electric power system can be viewed as consisting

of a generating system, a transmission system and a distri-

bution system.

The transmission system is distinctly different, in both its

operation and characteristics, from the distribution system.

Whereas the latter draws power from a single source and

transmits it to individual loads, the transmission system not

only handles the largest blocks of power but also the system.

The main difference between the transmission system and

the distribution system shows up in the network structure.

The former tends to be a loop structure and the latter gener-

ally, a radial structure.

The modern power distribution network is constantly

being faced with an ever-growing load demand. Distri-

bution networks experience distinct change from a low

to high load level everyday. In certain industrial areas,

it has been observed that under certain critical loading

conditions, the distribution system experience voltage

collapse. Brownell and Clarke [1] have reported the

actual recordings of this phenomenon in which system

voltage collapses periodically and urgent reactive

compensation needs to be supplied to avoid repeated

voltage collapse.

Literature survey shows that a lot of work has been done

on the voltage stability analysis of transmission systems [2],

but hardly any work has been done on the voltage stability

analysis of radial distribution networks. Jasmon and Lee [3]

and Gubina and Strmchnik [4] have studied the voltage

stability analysis of radial networks. They have represented

the whole network by a single line equivalent. The single

line equivalent derived by these authors [3,4] is valid

only at the operating point at which it is derived. It can

be used for small load changes around this point.

However, since the power ¯ow equations are highly

nonlinear, even in a simple radial system, the equivalent

would be inadequate for assessing the voltage stability

limit. Also their techniques [3,4] do not allow for the

changing of the loading pattern of the various nodes

which would greatly affect the collapse point.

In this paper, a new voltage stability index for all the

nodes is proposed for radial distribution networks. It is

shown that the node at which the value of voltage stability

index is minimum, is more sensitive to voltage collapse.

Composite load modelling is considered for voltage stability

analysis. It is also shown that the load ¯ow solution with

feasible voltage magnitude for radial distribution networks

is unique.

2. Methodology

In Ref. [5], a simple load ¯ow technique for solving radial

distribution networks has been proposed. For the purpose of

Electrical Power and Energy Systems 23 (2001) 129±135

0142-0615/01/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.

PII: S0142-0615(00)00040-5

www.elsevier.com/locate/ijepes

* Corresponding author. Tel.: 191-3222-83042; fax: 191-3222-55303.

E-mail address: [email protected] (D. Das).

deriving the voltage stability index of radial distribution

networks, this load ¯ow technique [5] will be explained in

brief:

Fig. 1 shows a 15-node sample radial distribution network

and Fig. 2 shows the electrical equivalent of Fig. 1.

From Fig. 2, the following equations can be written:

I� jj� � V�m1�2 V�m2�r� jj�1 jx� jj� ; �1�

P�m2�2 jQ�m2� � Vp�m2�I� jj� �2�

where

jj� branch number,

m1� branch end node� IS( jj),

m2� receiving end node� IR( jj),

I( jj)� current of branch jj,

V(m1)� voltage of node m1,

V(m2)� voltage of node m2,

P(m2)� total real power load fed through node m2,

Q(m2)� total reactive power load fed through node m2.

From Eqs. (1) and (2), we get

uV�m2�u4 2 {uV�m1�u2 2 2P�m2�r� jj�

22Q�m2�x� jj�}uV�m2�u2

1{P2�m2�1 Q2�m2�}{r2� jj�1 x2� jj�} � 0: �3�

Let,

b� jj� � uV�m1�u2 2 2P�m2�r� jj�2 2Q�m2�x� jj�}; �4�

c� jj� � {P2�m2�1 Q2�m2�}{r2� jj�1 x2� jj�}: �5�

From Eqs. (3)±(5) we get,

uV�m2�u4 2 b� jj�uV�m2�u2 1 c� jj� � 0: �6�

From Eq. (6), it is seen that the receiving end voltage

uV�m2�u has four solutions and these solutions are:

1. 0:707�b� jj�2 {b2� jj�2 4c� jj�}1=2�1=2;2. 20:707�b� jj�2 {b2� jj�2 4c� jj�}1=2�1=2;3. 20:707�b� jj�1 {b2� jj�2 4c� jj�}1=2�1=2;4. 0:707�b� jj�1 {b2� jj�2 4c� jj�}1=2�1=2:

Now, for realistic data, when P, Q, r, x and V are

expressed in per unit, b( jj) is always positive because

the term 2{P�m2�r� jj�1 Q�m2�x�jj�} is very small as

compared to uV�m1�u2 and also the term 4c(jj) is very

small as compared to b2(jj). Therefore {b2�jj�24c�jj�}1=2 is nearly equal to b(jj) and hence the ®rst

two solutions of uV�m2�u are nearly equal to zero and

not feasible. The third solution is negative and so not

feasible The fourth solution of uV�m2�u is positive and

feasible. Therefore, the solution of Eq. (6) is unique.

That is

uV�m2�u � 0:707�b� jj�1 {b2� jj�2 4:0c� jj�}1=2�1=2: �7�

M. Chakravorty, D. Das / Electrical Power and Energy Systems 23 (2001) 129±135130

Nomenclature

NB total number of nodes

LN1 total number of branches

TPL total real power load

TQL total reactive power load

jj branch number

IS( jj) sending end node

IR( jj) receiving end node

r( jj) resistance of branch jj

x( jj) reactance of branch jj

Fig. 1. Single line diagram of a radial distribution feeder. Fig. 2. Electrical equivalent of Fig. 1.

Actually,

P(m2)� sum of the real power loads of all the nodes

beyond node m2 plus the real power load of node m2

itself plus the sum of the real power losses of all the

branches beyond node m2.

Q(m2)� sum of the reactive power loads of all the

nodes beyond node m2 plus the reactive power load

of node m2 itself plus the sum of the reactive power

losses of all the branches beyond node m2.

The complete algorithm and ¯owchart for calculating

the total real and reactive power loads fed through

node m2 {i.e. P(m2) and Q(m2), for m2 � 2; 3;¼;NB}

have been given in Ref. [5].

The real and reactive power losses in branch jj are given

by:

LP� jj� � r� jj�{P2�m2�1 Q2�m2�}uV�m2�u2 ; �8�

M. Chakravorty, D. Das / Electrical Power and Energy Systems 23 (2001) 129±135 131

Fig. 3. 69-Node radial distribution network.

Fig. 4. Plot of Vmin vs. TPL. Fig. 5. Plot of Vmin vs. TQL.

LQ� jj� � x� jj�{P2�m2�1 Q2�m2�}uV�m2�u2 : �9�

Note that the substation voltage is known, i.e. uV�1�u � Vs:

Initially, LP( jj) and LQ( jj) are set to zero for all jj, then the

initial estimate of P(m2) and Q(m2) will be the sum of the

loads of all the nodes beyond node m2 plus the load of node

m2 itself.

For all the branches jj � 1; 2;¼;LN1; set m1 � IS� jj�and m2 � IR� jj� and compute P(m2) and Q(m2) using the

algorithm given in Ref. [5], then compute b( jj) and c( jj)

using Eqs. (4) and (5), and after that compute uV�m2�u using

Eq. (7) and compute the losses using Eqs. (8) and (9). This

will complete one iteration. Then, update the loads and

repeat the same process. The details have been given in

Ref. [5].

The convergence criterion of this load ¯ow method [5] is

that if, in successive iterations, the difference of real power

and reactive power delivered from the substations is less

than 0.10 kW and 0.10 kV Ar, respectively, the solution

has then converged.

3. Voltage stability index

From Eq. (7), it is seen that, a feasible load ¯ow solution


Fig. 6. Plot of SImin vs. TPL. Fig. 7. Plot of SImin vs. TQL.

Table 1

Critical loading condition for different types of load

Load type Substation voltage (pu) Critical loading condition

TPL (MW) TQL (MV Ar) SImin� SI65 (pu) Vmin� V65 (pu)

Constant power (CP) 1.0 12.212 08.654 0.0491 0.4708

1.025 12.767 09.047 0.0728 0.5194

1.05 13.422 09.537 0.0756 0.5244

Constant current (CI) 1.0 15.051 10.656 0.1044 0.5028

1.025 15.812 11.199 0.1152 0.5826

1.05 16.594 11.752 0.1269 0.5969

Constant impedance (CZ) 1.0 14.055 09.954 0.2195 0.6845

1.025 14.764 10.458 0.2423 0.7016

1.05 15.492 10.974 0.2669 0.7188

Composite load (40% CP,

30% CI and 30% CZ)

1.0 14.651 10.377 0.0745 0.5224

1.025 15.468 10.956 0.0818 0.5349

1.05 16.244 11.506 0.0929 0.5521

of radial distribution networks will exist if

b2� jj�2 4:0c� jj� $ 0: �10�From Eqs. (4), (5) and (10) we get

{uV�m1�u2 2 2P�m2�r� jj�2 2Q�m2�x� jj�}2

24:0{P2�m2�1 Q2�m2�}{r2� jj�1 x2� jj�} $ 0:

After simpli®cation we get

uV�m1�u4 2 4:0{P�m2�x� jj�2 Q�m2�r� jj�}2

24:0{P�m2�r� jj�1 Q�m2�x� jj�}uV�m1�u2 $ 0: �11�Let

SI�m2� � {uV�m1�u4 2 4:0{P�m2�x� jj�2 Q�m2�r� jj�}2

2 4:0{P�m2�r� jj�1 Q�m2�x� jj�}uV�m1�u2 �12�where

SI(m2)� voltage stability index of node m2 �m2 �2; 3;¼;NB�:For stable operation of the radial distribution networks,

SI�m2� $ 0; for m2 � 2; 3;¼;NB:

By using this voltage stability index, one can measure the

level of stability of radial distribution networks and thereby

appropriate action may be taken if the index indicates a poor

level of stability.

After the load ¯ow study, the voltages of all the nodes are

known, the branch currents are known, therefore P(m2) and

Q(m2) for m2 � 2; 3;¼;NB can easily be calculated using

Eq. (2) and hence one can easily calculate the voltage stabi-

lity index of each node �m2 � 2; 3;¼;NB�: The node at

which the value of the stability index is minimum, is more

sensitive to the voltage collapse.

4. Load modelling

For the purpose of voltage stability analysis of radial

distribution networks, composite load modelling is consid-

ered. The real and reactive power loads of node `i' is given

as:

PL�i� � fPLo�i��c1 1 c2uV�i�u 1 c3uV�i�u2�; �13�

QL�i� � fQLo�i��d1 1 d2uV�i�u 1 d3uV�i�u2�: �14�In Eqs. (13) and (14), f is a scaling factor and f is varied from

zero to a critical value at which voltage collapse takes place,

i.e. loads are gradually increased at every node. Constants

(c1, d1), (c2, d2) and (c3, d3) are the compositions of constant

power, constant current and constant impedance loads,

respectively.

5. Analysis

To demonstrate the effectiveness of the proposed method,

a 69-node radial distribution network [6] is considered. Fig.

3 shows a 69-node radial distribution network. Line data and

nominal load data (i.e. r, x, PLo and QLo) are given in

Appendix A. In the present work, a composition of 40%

constant power �c1 � d1 � 0:4�; 30% of constant current

�c2 � d2 � 0:3� and 30% of constant impedance �c3 � d3 �0:3� loads are considered.

For this network, when the load was increased gradually,

it was found that the minimum value of voltage stability

index is occurring at node 65. Therefore, node 65 is more

sensitive to voltage collapse. It was also observed that node

65 has the minimum voltage.

Figs. 4 and 5 show the plots of V(65) vs. TPL and

V(65) vs. TQL for different substation voltages. Points

A, B and C indicate the critical loading point beyond

which a small increment of load causes the voltage

collapse.

Figs. 6 and 7 show the plots of SI(65) vs. TPL and

SI(65) vs. TQL for different substation voltages. Points

A, B and C indicate the critical loading point beyond

which a small increment of loading causes the voltage

collapse.

Analysis was also carried out for constant power load (i.e.

c1 � d1 � 1; c2 � c3 � d2 � d3 � 0), constant current load

(i.e. c2 � d2 � 1; c1 � c3 � d1 � d3 � 0) and constant

impedance load (i.e. c3 � d3 � 1; c1 � c2 � d1 � d2 � 0).

Table 1 indicates the critical loading conditions for different

types of load and different values of substation voltage.

From Table 1, it is seen that the critical loading for constant

current load is the maximum and that for constant power

load is minimum. The critical loading for constant impe-

dance lies between these two and that for the composite load

solely depends on the percentage composition of the three

loads.

The stability index and consequently the voltage are

minimum for constant power load and maximum for

constant impedance load and that for constant current load

is in between these two. Similarly, the composition of loads

governs the position of the stability index for the composite

load.

6. Conclusions

It has been shown that the load ¯ow solutions of radial

distribution networks is unique. A new voltage stability

index has been proposed for radial distribution networks.

Using this voltage stability index, it is possible to compute

the stability index value at every node and the node at which

the value of the voltage stability index is minimum is most

sensitive to voltage collapse. The effectiveness of the

proposed technique has been demonstrated through a 69-

node radial distribution network.


Appendix A

Line data and nominal load data are given in Table A1.


Table A1

Line data and nominal load data of 69-node radial distribution network

Br. no. ( jj) Sending end node IS( jj) Receiving end node IR( jj) Branch resistance (V) Branch reactance (V) Nominal load

Receiving end node

PLo (kW) QLo (kV Ar)

1 1 2 0.0005 0.0012 0.0 0.0

2 2 3 0.0005 0.0012 0.0 0.0

3 3 4 0.0015 0.0036 0.0 0.0

4 4 5 0.0251 0.0294 0.0 0.0

5 5 6 0.3660 0.1864 2.60 2.20

6 6 7 0.3811 0.1941 40.40 30.00

7 7 8 0.0922 0.0470 75.00 54.00

8 8 9 0.0493 0.0251 30.00 22.00

9 9 10 0.8190 0.2707 28.00 19.00

10 10 11 0.1872 0.0619 145.00 104.00

11 11 12 0.7114 0.2351 145.00 104.00

12 12 13 1.0300 0.3400 8.00 5.00

13 13 14 1.0440 0.3450 8.00 5.50

14 14 15 1.0580 0.3496 0.0 0.0

15 15 16 0.1966 0.0650 45.50 30.00

16 16 17 0.3744 0.1238 60.00 35.00

17 17 18 0.0047 0.0016 60.00 35.00

18 18 19 0.3276 0.1083 0.0 0.0

19 19 20 0.2106 0.0690 1.00 0.60

20 20 21 0.3416 0.1129 114.00 81.00

21 21 22 0.0140 0.0046 5.00 3.50

22 22 23 0.1591 0.0526 0.0 0.0

23 23 24 0.3463 0.1145 28.0 20.0

24 24 25 0.7488 0.2475 0.0 0.0

25 25 26 0.3089 0.1021 14.0 10.0

26 26 27 0.1732 0.0572 14.0 10.0

27 3 28 0.0044 0.0108 26.0 18.6

28 28 29 0.0640 0.1565 26.0 18.6

29 29 30 0.3978 0.1315 0.0 0.0

30 30 31 0.0702 0.0232 0.0 0.0

31 31 32 0.3510 0.1160 0.0 0.0

32 32 33 0.8390 0.2816 14.0 10.0

33 33 34 1.7080 0.5646 9.50 14.00

34 34 35 1.4740 0.4873 6.00 4.00

35 3 36 0.0044 0.0108 26.0 18.55

36 36 37 0.0640 0.1565 26.0 18.55

37 37 38 0.1053 0.1230 0.0 0.0

38 38 39 0.0304 0.0355 24.0 17.0

39 39 40 0.0018 0.0021 24.0 17.0

40 40 41 0.7283 0.8509 1.20 1.0

41 41 42 0.3100 0.3623 0.0 0.0

42 42 43 0.0410 0.0478 6.0 4.30

43 43 44 0.0092 0.0116 0.0 0.0

44 44 45 0.1089 0.1373 39.22 26.30

45 45 46 0.0009 0.0012 39.22 26.30

46 4 47 0.0034 0.0084 0.0 0.0

47 47 48 0.0851 0.2083 79.0 56.40

48 48 49 0.2898 0.7091 384.70 274.50

49 49 50 0.0822 0.2011 384.70 274.50

50 8 51 0.0928 0.0473 40.50 28.30

51 51 52 0.3319 0.1114 3.60 2.70

52 9 53 0.1740 0.0886 4.35 3.50

53 53 54 0.2030 0.1034 26.40 19.00

54 54 55 0.2842 0.1447 24.0 17.20

References

[1] Brownell G, Clarke H. Analysis and solutions for bulk system voltage

instability. IEEE Computer Applications in Power 1989;2(3):31±5.

[2] Ajjarapu V, Lee B. Bibliography on voltage stability. IEEE Transac-

tions on Power Systems 1998;13(1):115±25.

[3] Jasmon GB, Lee LHCC. Distribution network reduction for voltage

stability analysis and load ¯ow calculations. International Journal of

Electrical Power and Energy Systems 1991;13(1):9±13.

[4] Gubina F, Strmcnik B. A simple approach to voltage stability assess-

ment in radial networks. IEEE Transactions on Power Systems

1997;12(3):1121±8.

[5] Das D, Kothari DP, Kalam A. A simple and ef®cient method for load

¯ow solution of radial distribution networks. International Journal of

Electrical Power and Energy Systems 1995;17(5):335±46.

[6] Mesut EB, Wu FF. Optimal capacitor placement on radial distribution

systems. IEEE Transactions on Power Delivery 1989;4(1):725±34.


Table A1 (continued)

Br. no. ( jj) Sending end node IS( jj) Receiving end node IR( jj) Branch resistance (V) Branch reactance (V) Nominal load

Receiving end node

PLo (kW) QLo (kV Ar)

55 55 56 0.2813 0.1433 0.0 0.0

56 56 57 1.5900 0.5337 0.0 0.0

57 57 58 0.7837 0.2630 0.0 0.0

58 58 59 0.3042 0.1006 100.0 72.0

59 59 60 0.3861 0.1172 0.0 0.0

60 60 61 0.5075 0.2585 1244.0 888.0

61 61 62 0.0974 0.0496 32.0 23.0

62 62 63 0.1450 0.0738 0.0 0.0

63 63 64 0.7105 0.3619 227.0 162.0

64 64 65 1.0410 0.5302 59.0 42.0

65 11 66 0.2012 0.0611 18.0 13.0

66 66 67 0.0047 0.0014 18.0 13.0

67 12 68 0.7394 0.2444 28.0 20.0

68 68 69 0.0047 0.0016 28.0 20.0

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