I.  INTRODUCTION

 

Voltage collapse problem has been one of the major problems facing the electric power utilities in many countries. The problem is also a main concern in power system operation and planning. It can be characterized by a continuous decrease of the system voltage. In the initial stage the decrease of the system voltage starts gradually and then decreases rapidly. Stressed power system; i.e. high active power loading in the system. In bulk transmission network to avoid the cost of building new lines and generation facilities. When a bulk transmission network is operated close to the voltage instability limit, it becomes difficult to control the reactive power margin for that system. As a result the system stability becomes one of the major concerns and an appropriate way must be found to monitor the system and avoid system collapse. One of the major reasons of voltage collapse is the heavy loading of the power system, which is comprised of long transmission lines. The system appears unable to supply the reactive power demand. Producing the demanded reactive power through synchronous generators, synchronous condensers or static capacitors can overtake the problem [1]. Another solution is to build transmission lines to the weakest nodes. Voltage collapse may occur due to a major disturbance in the system such as generators outage or lines outage.

 

In many algorithms have been proposed in the literature for voltage stability analysis. Most of the utilities have a tendency

 

 

to depend regularly on conventional load flows for such analysis. Some of the proposed methods are concerned with voltage instability analysis under small perturbations in system load parameters.

 

II.  POWER FLOW PROBLEM

 

The solution of power flow predicts what the electrical state of the network will be when it is subject to a specified loading condition. The result of the power flow is the voltage magnitude and the angle at each of the system nodes. These bus voltage magnitudes and angles are defined as the system state variables [2]. That is because they allow all other system quantities to be computed such as real and reactive power flows, current flows, voltage drops, power losses etc., Power flow solution is closely associated with voltage stability analysis. It is an essential tool for voltage stability evaluation. Much of the research on voltage stability deals with the power-flow computation method. The power-flow problem solves the complex matrix equation

 

                                                        (1)

 

                                                             (2) 

The Newton-Raphson method is the most general and reliable algorithm to solve the power-flow problem. It involves iterations based on successive linearization using the first term of Taylor expansion of the equation to be solved. From Equation (1), we can write the equation for node k (bus k) as

 

                                                      (3)

 

                     (4)

 

    (5)

 

,           (6)

 

                                                                             (7)

 

 

 

 

 

                           =                            (8)

 

                                              (9)

 

III.  PERFORMANCE EIGEN VALUE ANALYSIS METHOD

 

It can predict voltage collapse in complex power system networks. It involves mainly the computing of the smallest Eigen values and associated eigenvectors of the reduced Jacobin matrix obtained from the load flow solution [3]. The Eigen values are associated with a mode of voltage and reactive power variation, which can provide a relative measure of proximity to voltage instability. Then, the participation factor can be used effectively to find out the weakest nodes or buses in the system

 

A.  Effect of Load Modeling

It is important to have an analytical method to predict the voltage collapse in the power system, particularly with a complex and large one. The modal analysis or Eigen value analysis can be used effectively as a powerful analytical tool to verify both proximity and mechanism of voltage instability [4]. It involves the calculation of a small number of Eigen values and related eigenvectors of a reduced Jacobin matrix. The stability margin or distance to voltage collapse can be estimated by generating the Q-V curves for that particular bus the steady state induction machine load model is considered in this study.

                   (10)

              (11)

                                                          (12)

                                                             (13)

                                                   (14)

                                                  (15)

                                       (16)

                                  (17)

Then

                                           (18)

 

B.  Modal Analysis & Q – V Curve

 

The modal analysis mainly depends on the power-flow Jacobin matrix. The voltage-reactive power curves are generated by series of power flow simulation. They plot the voltage at a test bus or critical bus versus reactive power at the same bus. The bus is considered to be a PV bus, where the reactive output power is plotted versus scheduled voltage. Most of the time these curves are termed Q–V curves rather than V–Q curves. Scheduling reactive load rather than voltage produces Q–V curves. These curves are a more general method of assessing voltage stability [5]. They are used by utilities as a workhorse for voltage stability analysis to determine the proximity to voltage collapse and to establish system design criteria based on Q and V margins determined from the curves. Operators may use the curves to check whether the voltage stability of the system can be maintained or not and take suitable control actions. The sensitivity and variation of bus voltages with respect to the reactive power injection can be observed clearly. The main drawback with Q–V curves is that it is generally not known previously at which buses the curves should be generated. In normal operating condition, an operator will attempt to correct the low voltage condition by increasing the terminal voltage.

 

C.  Effect of Load Modeling

 

The load representation can play an important factor in the power system stability. The load characteristics can be divided into two categories, static characteristics and dynamic characteristics. The effect of the static characteristics is discussed in this section. Recently, the load representation has become more important in power system stability studies. In the previous analysis, the load was represented by considering the active power and reactive power. Both were represented by combination of constant impedance (resistance or reactance), constant current and constant power (active or reactive) elements. This kind of load modeling has been used in many of the power system steady state analyses. The effect of the static load modeling on voltage stability is presented in this section. A voltage dependent load model is proposed. The new load model is used instead of the constant load used previously. A significant change in the stability limit or distance to voltage collapse should be noticed clearly [6, 7].

 

 

 

D.  Voltage Dependent Loads

 

Hp

Volts

Rpm

Torque

(N.m)

I

(A)

rs

(ohm)

X1S

(ohm)

Xm

(ohm)

X1r

(ohm)

rr

(ohm)

J

Kg.m2

500

2300

1773

1980

93.6

0.262

1.206

54.02

1.206

1.187

11.06

2250

2300

1786

8900

421.2

0.029

0.226

13.04

0.226

0.022

63.87

Voltage dependency of reactive power affects the steady state stability of power system. This effect primarily appears on voltages, which in turn affect the active power. It is well known that the stability improves and the system becomes voltage stable by installing static reactive power compensators or synchronous condensers. The active and reactive proposed static load model for a particular load bus in this study is an exponent function bus voltage as shown in the following equations:

                                                           (19)

                                                         (20)

Then the load flow equation (2.6) at load bus k can be written as

 

         (21)

         (22)

E.  Effect of Induction Motor Load

 

Induction machine motor is one of the most popular loads in the power system. About 50-70% of all generated power is consumed by electric motors with about 90% of this being used by induction motors. Therefore, it is considered an important part of the power system load and a significant attention regarding this type of load has been taken for both dynamic and steady state analysis. In this research, the induction machine load is considered using the steady state model analysis.

 

IV.  PROBLEM FORMULATION

 

The Modal analysis method has been successfully applied to two different electric power systems. The Q-V cures are generated for selected buses in order to monitor the voltage stability margin. Different voltage dependent load and Induction machine load models are simulated. A power flow program based on Mat lab is developed to,

 

A.  Analyses with constant impedance Load

 

The modal analysis method is applied to the three suggested test systems. The voltage profile of the buses is presented from the load flow simulation. Then, the minimum Eigen value of the reduced Jacobin matrix is calculated. After that, computing the participating factors identifies the weakest load buses, which are subject to voltage collapse.

 

 

 

 

B.  Analysis considering effect of induction machine load

 

The modal analysis including the induction machine load is performed for the three suggested test systems. The induction machine load can be connected to any bus in the tested system. In this study two-induction machine loads with different ratings have been selected for the analysis. The machines data are shown in Table 1.

 

TABLE .1.  MACHINE PARAMETER

 

The voltage profile of the buses is presented from the load flow solution. Then, the minimum Eigen value of the reduced Jacobin matrix is calculated. After that, computing the participating factors identifies the weakest load buses, which are subject to voltage collapse [8, 9].

 

C.  The IEEE 14 Bus System

 

Table.2 shows the voltage profiles of all buses of the IEEE 14 Bus system as obtained from the load flow including induction machine load model 1 & 2.

 

TABLE. 2.   VOLTAGE PROFILES OF IEEE 14 BUS SYSTEM

 

BUS NO

CONSTANT LOAD MODEL

IMPEDANCE LOAD MODEL 1

IMPEDANCE LOAD

MODEL 2

1

1.060

1.060

1.060

2

1.040

1.040

1.040

3

1.010

1.010

1.010

4

0.979

0.983

0.983

5

0.983

0.986

0.987

6

1.070

1.070

1.070

7

1.046

1.049

1.050

8

1.080

1.080

1.080

9

1.050

1.055

1.056

10

1.049

1.053

1.053

11

1.056

1.058

1.058

12

1.024

1.027

1.027

13

1.044

1.049

1.050

14

1.029

1.050

1.053

 

The result shows the effect of both induction machine load and the constant load. It can be seen that all the bus voltages are within the acceptable level. In general, the lowest voltage compared to the other buses can be noticed at bus number 4 in all cases. Table.3 shows the Eigen values of all buses of the IEEE 14 Bus system as obtained from the load flow including induction machine load model 1 & 2.

 

TABLE .3.  EIGEN VALUES OF IEEE 14 BUS SYSTEM

 

S.No

CONSTANT LOAD MODEL

IMPEDANCE LOAD MODEL 1

IMPEDANCE LOAD

MODEL 2

1

   62.5497

   62.7566

   62.7774

2

   40.0075

   40.1996

   40.2196

3

   21.5587

   21.6384

   21.6466

4

   18.7197

   18.8205

   18.8311

5

   15.7882

   15.8638

   15.8714

6

   11.1479

   11.2021

   11.2077

7

    2.7811

    2.8274

    2.8321

8

    5.4925

    5.5355

    5.5399

9

    7.5246   

    7.6189

    7.6290

 

Table .3. shows the participation factors of all buses of the IEEE 14 Bus system as obtained from the load flow including induction machine load model 1 & 2.

 

TABLE. 4.  PARTICIPATION FACTORS OF IEEE 14 BUS SYSTEM

BUS

NO

CONSTANT

LOAD

 MODEL

IMPEDANCE

 LOAD

MODEL 1

IMPEDANCE LOAD

MODEL 2

4

0.0091

0.0092

0.0092

5

0.0045

0.0046

0.0046

7

0.0691

0.0704

0.0706

9

0.1912

0.1939

0.1942

10

0.2319

0.2376

0.2382

11

0.1095

0.1136

0.1140

12

0.0225

0.0226

0.0226

13

0.0351

0.0346

0.0345

14

0.3270

0.3135

0.3121

 

 

D.  The IEEE 30 Bus System

 

Table.5. shows the voltage profiles of all buses of the IEEE 30 Bus system as obtained from the load flow including induction machine loads at bus 30.

 

TABLE .5 VOLTAGE PROFILES OF IEEE 30 BUS SYSTEM

 

BUS NO

CONSTANT LOAD MODEL

IMPEDANCE

LOAD MODEL

1

IMPEDANCE

LOAD MODEL

 2

1

1.060

1.060

1.060

2

1.043

1.043

1.043

3

1.019

1.020

1.020

4

1.010

1.011

1.011

5

1.010

1.010

1.010

6

1.009

1.010

1.011

7

1.001

1.002

1.002

8

1.010

1.010

1.010

9

1.048

1.049

1.049

10

1.040

1.040

1.041

11

1.082

1.082

1.082

12

1.054

1.055

1.055

13

1.071

1.071

1.071

14

1.038

1.039

1.039

15

1.033

1.034

1.034

16

1.041

1.042

1.042

17

1.035

1.035

1.036

18

1.023

1.024

1.024

19

1.020

1.021

1.021

20

1.024

1.025

1.025

21

1.025

1.027

1.027

22

1.025

1.027

1.027

23

1.018

1.020

1.020

24

1.006

1.010

1.011

25

0.983

0.991

0.993

26

0.964

0.973

0.975

27

0.977

0.988

0.991

28

1.008

1.011

1.011

29

0.956

0.979

0.984

30

0.944

0.979

0.986

 

The result shows the effect of both induction machines load and the constant load. It can be seen that all the bus voltages are within the acceptable level except buses 29 and 30. In general, the lowest voltage compared to the other buses can be noticed at bus number 30 in all cases [10]. Table.6 shows the Eigen values of all buses of the IEEE 30 Bus system as obtained from the load flow including induction machine load model 1 & 2.

 

TABLE .6.   EIGEN VALUES OF IEEE 30 BUS SYSTEM

 

S.NO

CONSTANT LOAD MODEL

IMPEDANCE LOAD MODEL 1

IMPEDANCE LOAD MODEL 2

1

110.2056

110.3383

110.3615

2

100.6465

100.7790

100.8104

3

65.9541

66.0366

66.0507

4

59.5431

59.5990

59.6125

5

37.8188

37.8559

37.8646

6

35.3863

35.4126

35.4185

7

23.4238

23.4500

23.4558

8

23.0739

23.1397

23.1521

9

19.1258

19.1603

19.1676

10

19.7817

19.7989

19.8026

11

18.0785

18.1123

18.1192

12

16.3753

16.4800

16.5022

13

13.7279

13.7888

13.8023

14

13.6334

13.6568

13.6612

15

11.0447

11.0704

11.0750

16

0.5060

0.5211

0.5240

17

1.0238

1.0355

1.0380

18

1.7267

1.7555

1.7618

19

8.7857

8.7949

8.7970

20

7.4360

3.5873

3.5887

21

3.5808

4.0554

4.0564

22

4.0507

7.5141

7.5303

23

6.0207

5.4839

5.4898

24

5.4527

6.1933

6.2299

 

 

Table.7 shows the participation factors of all buses of the IEEE 30 Bus system as obtained from the load flow including induction machine load model 1 & 2.The simulation results of voltage profile and participation factor of IEEE 14 & 30 bus systems are presented as shown in the Fig. 1 to 4 respectively.

 

TABLE .7 PARTICIPATION FACTORS OF IEEE 30 BUS SYSTEM

 

S.NO

CONSTANT LOAD MODEL

IMPEDANCE

LOAD MODEL

1

IMPEDANCE

LOAD  MODEL

 2

1

0.0004

0.0004

0.0004

2

0.0005

0.0005

0.0005

3

0.0005

0.0006

0.0006

4

0.0002

0.0002

0.0002

5

0.0037

0.0040

0.0041

6

0.0121

0.0130

0.0132

7

0.0037

0.0041

0.0041

8

0.0081

0.0088

0.0090

9

0.0111

0.0120

0.0122

10

0.0079

0.0087

0.0088

11

0.0115

0.0125

0.0127

12

0.0165

0.0181

0.0184

13

0.0179

0.0196

0.0200

14

0.0172

0.0189

0.0192

15

0.0176

0.0189

0.0191

16

0.0189

0.0203

0.0206

17

0.0238

0.0255

0.0258

18

0.0395

0.0414

0.0419

19

0.1055

0.1070

0.1073

20

0.1729

0.1770

0.1778

21

0.1028

0.1015

0.1013

22

0.0025

0.0026

0.0026

23

0.1934

0.1858

0.1842

24

0.2118

0.1988

0.1961

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V.  CONCLUSION

In this paper, the voltage collapse problem is studied. The Modal analysis technique is applied to investigate the stability of two well-known power systems. The method computes the smallest Eigen value and the associated Eigen vectors of the reduced Jacobin matrix using the steady state system model. The magnitude of the smallest Eigen value gives us a measure of how close the system is to the voltage collapse. Then, the participating factor can be used to identify the weakest node or bus in the system associated to the minimum Eigen value.

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 Fig.1.   Voltage profile of IEEE 14 bus system

 

 

 

 

 

 

 

 

 

 

                Fig. 2.  Participation factor of IEEE 14 bus system

 

 

 

                Fig. 3.  Voltage profile of IEEE 30 bus system                

 

 

 

 

 

      

 

 

 

 

 

Fig. 4.   Participation factor of IEEE 30 bus system

 

VI .  REFERENCES

 

[1] C. W. Taylor, “Power System Voltage Stability.” New York: MaHraw-       Hill, 2000.

[2] Sauer, Peter W. and Pai, M. A. “Power System Dynamics and Stability”        New Jersey Prenitice Hall, 2002.

[3] Machowski, Bialek and Bumby “Power System Dynamics and Stability”        John Wiley & Sons Ltd, 2002.

[4]  Sirisuth, Piya “Voltage Instability analysis using the Sensitivity of        Minimum Singular Value of Load Flow Jacobian” 2004.

[5] Ajjarapu, V. and Lee, B. “Bibliography on Voltage Stability” IEEE Trans.        on Power Systems, vol. 13, pp. 115-125, 2006.

[6] C. Counan, M. Trotignon, E. Corride, G. Bortoni, M. Stubbe, and J.        Deuse, “Major incidents on the French electric system- Potentiality and        curative measures,” IEEE Trans. on Power Systems, vol. 8, pp.879-886,        Aug.2005.

[7] R. DÕAquila, N. W. Miller, K. M. Jimma, M. T. Shehan, and G. L.        Comegys, “Voltage stability of the Puget Sound System under        Abnormally Cold Weather Conditions,” IEEE Trans. on Power Systems,        vol. 8, pp. 1133-1142, Aug. 2006.

[8] F. D. Galiana and Z. C. Zeng, “Analysis of the Load Behavior near        Jacobian Singularity,” IEEE Trans. On Power Systems, vol. 7, pp. 1529-       1542, Nov. 2003.

[9] P. Kessel and H. Glavitsch, “Estimating the Voltage Stability of a Power        System,” IEEE Trans. on Power Delivery, vol. 1, pp. 346-353, July 2005.

[10] Y. Tamura, H. Mori, and S. Iwamoto, “Relationship between Voltage        Stability and Multiple Load Flow Solutions in Electric Systems,” IEEE        Trans. on Power Apparatus and Systems, vol. PAS-102, pp. 1115 – 1123,        May 2004.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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