Power System Voltage Stability Index

Power System Voltage Stability Index

Atef A. Elemary1, M.M.R Ahmed2 Department of Electrical Engineering Jazan University

Jazan, Saudi Arabia

Ebtisam M. Saied3, Mohamed A. Hamdy4

Department of Electrical Engineering Benha University

Cairo, Egypt

Abstract After some damaging blackouts, voltage instability and collapse have become worldwide concern problems. Voltage instability is a nonlinear local phenomenon, which occurs due to insufficient reactive power production or as the inability to transfer it to consumers. In most dangerous cases, it can result in voltage collapse. Therefore, early identification of the process is essential. A power system is decomposed into selected radial paths regarding reactive power flow. Next, two bus equivalents

its simple algorithm and clear physical meaning. Banasila [9] used it in the optimal reactive power dispatch and expert system.

power system stability

for these radial transmission paths are determined. The analytical proof of maximal stable power transfer for two bus equivalents is available. However, each candidate critical load bus may be fed via more transmission paths. It is essential that a set of power sources, feeding the particular candidates critical reactive power drain bus, has sufficient reactive power reverse. Therefore, the proposed voltage collapse assessment procedure combines the voltage collapse proximity generators index and an available reactive power reserve of the involved generators. The methods properties and numerical results are presented.

Keywords Power System, voltage stability, indicator

rotor angle

stability

transient stability

frequency

stability

short term

short term

long term

large disturbance

long term

voltage

small disturbance

stability

short term

small disturbance

1. INTRODUCTION

   In recent years, economical and environmental reasons have forced power systems to be operated closer to their security limits. Traditionally, their limits are associated with thermal and stability limitations have become less restrictive as nowadays most of electric utilities use faster relays to reduce the fault clearing time, fast response excitation systems and FAC,TS devices to reduce the amplitude of the disturbance. As a result, the power systems world is increasingly concerned with voltage stability and collapse problems, especially after blackouts of some world (New York, London,,), some of which have been reported to be associated with voltage collapse [1,3, 11-15].

   Recently IEEE/CIGRE task recommended numerous definitions in regarding to power system stability including voltage stability. Figure 1 summarizes these explanations. In general terms voltage index is applied to study voltage stability [1].

   In order to prevent voltage collapse, fast and accurate voltage stability index is needed to help in monitoring the operating condition. In past articles, some useful voltage stability indices have been presented, such methods of testing the Jacobian, the voltage stability assessment using optimization, the point of collapse method ( POC), the voltage proximity indicator (VCPI), fast voltage stability index (FVSI) [6]. Methods of Thevennins impedance indicator, identifying the weak elements of a power system and the multiple load flow solutions (MLF)as in which, the voltage collapse indicators (VCPI) or (L-index) [2, 16-21] has raised especial regards for

   Fig. 1. Classifications of power system stability.

2. VOLTAGE STABILITY

   The voltage stability can be described with different definitions according to the points of views. Its the system ability to keep its voltage within assured specified limits at all loading conditions all over the hour. From the load point of view it can be defined as the load ability or (P-V curve) to give more power as the load point of view it can be described as the load ability to supply more power as the loading is increased without voltage dips outside the limits as shown in Figure (2).

   Fig. 2. P-V curve for the simple

   The most common technique of analyzing the transfer capacity of a system against voltage stability is to create a so- called P-V curve of the bus bar with load or production. P-V curves are often respecting to as nose curves due to their shape [22-25]. The nose point (generally raised to as the critical point) of the curve provides the maximum power which can be supplied to the load.

   An alternative common technique of analyzing the transfer capacity of a system respecting voltage is to gso-called so called Q-V curve for a bus bar [4, 9]. Figure 3, shows the reactive power which must be inserted into a bus bar to keep a specific voltage. These curves are plots of the critical bus voltage against the reactive power of the same bus. To obtain Q-V curve of a bus a fictitious synchronous capacitor is represented at that precise bus and the bus is supposed to be a voltage-controlled bus without reactive bounds.

   where:

   Lj: voltage collapse proximity indicators

   l: set of load buses

   G: set of generators buses

   Cji: elements of matrix C determined by:

   L- indicator varies in the range between 0 (no load system) and 1 (voltage collapse).

   B. Fast Voltage Stability Index (VVSI)

   The stability index of this method is defined by the following formula [7]

   where:

   Z: line impedance

   X: line reactance

   Qj : receiving end reactive power at load bus j Vi : complex voltage at generator bus i

   Such that the stability index range is defined by:

   (2)

   Fig. 3. Q curve and reactive power margin

3. VOLTAGE INSTABILITY

   As the disturbance increases in load or system changes this causes quick drop in voltage or drift downward, and operates and automatic system controls fail to halt the decay.

   Causes of voltage instability can be summarized in the following:

   1. Stressed power system (high active power loading Pr, and/ or also high reactive power loading Qr, in the system).

   2. Low power factor of loads of loads, which results higher

      values of Q.

   3. Load characteristics at low voltage magnitudes different from their designed values.

4. STEADY STATE VOLTAGE STABILITY ASSESSMENT

   There are different methods used to steady state voltage stability assessment. These methods are basically depends on one of the following approaches:

   A. Voltage Collapse Proximity Indicators (VCPI)

   The criteria of this indicator is defined as follows [7,9].

   (1)

   1. Thevennins Impedance Indicator

      To calculate this indicator it is required to calculate the equivalent Thevennins impedance (Zth) for each load bus in the system. The ratio of this impedance to the equivalent load impedance (ZL) at the same bus is taken as the indicator. Zth of a load bus is calculated by removing the load at this bus, replacing all loads and generators in the system by an equivalent impedances with appropriate signs, forming Ybus and inverting it to obtain Zbus , and ZTh is equal to the diagonal elements of Zbus corresponding to the bus required. These steps repeated at each bus as well as load flow is to be carried out for each case to determine the equivalent Thevennins voltage. The certerion for voltage stability by this indicator is given by:

5. MATHEMATICAL DERVIATION

   In this section a proposed method is made to calculate the indicator based on the change of reactive and actie powers. The L- indicator [7,9] depend on the calculation of load flow in each once. To overcome this problem a proposed method is investigated. Figure 4 shows two buses connected by transmission line to derive a new mathematical formula of change voltage and its angle based on change of active and reactive power angle. Figure 5, shows phasor diagram of the system shown in figure 4.

   where:

   (12)

   Fig. 4. Two bus radial system

   For incremental change of , we get Therefore, equation 12 can be put in the form

   Vs

   I' X

   Vs

   Rewrite equation 13 gives:

   (13)

   2

   1

   Vr

   ' I R

   I

   V

   Vx

   (14)

   From equations 9 and 14, we can get the following relation

   Fig. 5. Phasor diagram

   From figure 5 the following relation are calculated [10]

   (3)

   (4)

   (15)

   We can assume that therefore equation 15 yields to:

   (16)

   Rewrite equation 6 yields to:

   (5)

   (6)

   Equation 16 gives a new formula to calculate the change of voltage and its angle based on change of active and reactive power of loads. Also, this equation depends on the base case load flow and network data. Therefore, the repeated load flow in each case of change active and reactive power is not required.

   (7)

   For small change of angle , the following assumption are made:

   1, and Equation 7, can be put in the form

   (8)

   Equation 8 can be put in the form:

6. CASE STUDY

   To test the capability of the proposed method the following applications are made.

   1. Two bus system

   The pu data of the system under studying are given in pu:

   Vs = 1.05

   P = 0.3

   Q = 0.2

   R = 0.083

   X = 0.31

   In this system the following applications are made

   such that

   Equation 11 can be put in the form:

   (9)

   (10)

   (11)

   1. System under Condition of Power Factor4

      In this case the active and reactive power are change to get constant power factor. The test results are demonstrated as follows

      1. L- indicator (VCPI)

   From Fig. 6 it is noted that the system under study is stable when the active power P and reactive power Q reach the values 1.38 pu and 0.92 pu. Respectively. Otherwise the system collapse to unstable region when the active power P and reactive power Q greater than the above values.

   Fig. 6 Active and reactive power in pu relation with L-inductor at constant pf

   1. FVSI indicator

      It can be noted that from figure 7 the system under study is stable when the active power P is vary up to the value of.2 pu and the reactive power Q is vary up to the value of 0.8 pu. The system behavior is changed to unstable when the active power P and reactive power Q greater than 1.2and 0.8 pu

      respectively.

      Fig 7 Active and reactive power in pu relation with FVSI-indicator at constant pf

   2. Thevennins impedance indicator

   In this case it is noted that from figure 8 the system under study is stable when the active P is varying up to the value of

   1.2 pu and the reactive power Q is varying up to the value of

   0.8 pu, otherwise the system is unstable. The system behavior is changed to unstable when the active power P reaches 1.21 pu and the reactive power Q reaches 0.8067pu.

   Fig. 8 Active and reactive power in pu relation with ZTh-inductor at constant pf

   2). System under Condition of Constant Active Power P

   In this case P = 0, and the applications are made as shown below.

   1. L-indicator (VCPI)

      Figure 9 show that the system under study is stable when the reactive power Q is vary up to the value of 1.1196 pu and the angle Ø is vary up to 1.309 rad., the system behavior is changed to the unstable when the reactive Q reaches 1.7014 pu and the angle Ø reaches 1.3963 rad.

      Fig. 9 Reactive power relation with L-inductor at constant P

   2. FVSI indicator

      It is noted that from figure 10, the system under study is stable when the reactive power Q is varying up to the value of 0.8242 pu and the angle Ø is vary up to 1.2217. The system behavior is changed to the unstable when the reactive power Q reaches 1.1196 pu and the angle Ø reaches 1.309 pu.

      Fig. 10 (a) Variation of reactive power with FVSI inductor

      (b) Variation pf angle with FVSI inductor

   3. Thevennins impedance indicator

   Figure 11 shows that the system under study is stable when the reactive power Q is vary up to the value of 0.8242 pu and the angle is vary up to 1.2217, the system behavior is changed to the unstable when the reactive power Q reaches 1.1196 pu and the angle Ø reaches 1.309 rad.

   Fig. 11(a) Change of reactive power with ZTh-inductor at constant power

   (b) Change of Power Factor angle ZTh-inductor at constant power

   3) System under Condition of Constant Reactive Power Q

   In this case Q = 0, and the applications are made as shown below

   a) L-indicator (VCPI)

   The system under study shown in figure 12 is stable when the active power P is varying up to the value of 2.286 pu and the angle Ø is varying up to 0.0873. the system behavior is changed to the unstable when the reactive power P reaches 5.7273 pu and the angle Ø reaches 0.0349 rad.

   Fig. 12 (a) Change of reactive power with ZTh-inductor at constant power

   (b) Change of Power Factor angle ZTh-inductor at constant power

   b) FVSI indicator

   We can note that the function between active power and FVSI indicator is constant function because the FVSI indicator depends on the reactive power Q, such that Q = 0 and there is the same function between angle Ø and FVSI indicator

   c). Thevennins impedance indicator

   From figure 13 it is noted that

   1. The system under study is stable when the active power P is varying up to the value of 2.286 pu and the power factor angle is varying up to 0.0873 rad.

   2. The system behavior is changed to the unstable when the active power P reaches 2.8601 pu and the power factor angle Ø reaches to 0.698 rad.

      Fig. 13 (a) Active power as a function with ZTh-inductor at constant Q

      (b) Power Factor as a function with ZTh-inductor at constant Q

7. APPLICATION ON IEEE 6-BUS POWER SYSTEM

   NETWORK

   Figure 14 shows the configuration of IEEE 6-bus network power system. This figure includes base case load flow. In this section the stability index under base case condition is made and the effect of load change on the stability index is also investigated.

   Fig. 14 IEEE 6=bus power system network

   1. Base Case results

      TABLE 1 SHOWS THE SYSTEM IS STABLE BASED ON THE BASE CASE

      Bus number	L-indicator	ZTh- indicator
      4	0.093	0.0692
      5	0.1143	0.0793
      6	0.0956	0.0824

      1. Change in Bus 4

         The effect of load change on bus 4 at constant P, Q and power factor on the stability are studied and shown in Table 2. It is noted that the system is stable at all condition

         TABLE 2: L AND ZTH INDICATORS AT CONSTANT P, Q AND POWER FACTOR WITH CHANGES AT BUS 4

         Constant active power P							Constant reactive power Q							Constant power factor (pf)
         Q	L-indicator			ZTh indicator			P	L-indicator			ZTh indicator			P	Q	L-indicator			ZTh indicator
         	buses			buses				buses			buses			buses			buses
         	4	5	6	4	5	6		4	5	6	4	5	6			4	5	6	4	5	6
         0.8	0.108			0.075			0.8	0.087			0.099			0.8	0.496	0.055			0.107
         1	0.138			0.088			1	0.077			0.091			1	0.619	0.064			0.138
         1.2	0.168			0.108			1.2	0.066			0.082			1.2	0.744	0.073			0.17
         1.4	0.198			0.123			1.4	0.055			0.073			1.4	0.868	0.082			0.204
         1.6	0.228			0.142			1.6	0.044			0.063			1.6	0.991	0.091			0.24
         1.8	0.261			0.169			1.8	0.033			0.055			1.8	1.11	0.099			0.278
         2	0.295			0.209			2	0.021			0.047			2	1.24	0.109			0.319

      2. Change in bus 5

         The effect of load change on bus 5 at constant P, Q and power factor on the stability are studied and shown in Table 3. It is noted that the system is stable at all condition of studies.

         TABLE 3: L AND ZTH INDICATORS AT CONSTANT P, Q AND POWER FACTOR WITH CHANGES AT BUS 5

         Constant active power P							Constant reactive power Q							Constant power factor (pf)
         Q	L-indicator			ZTh indicator			P	L-indicator			ZTh indicator			P	Q	L-indicator			ZTh indicator
         	buses			buses				buses			buses					buses			Buses
         	4	5	6	4	5	6		4	5	6	4	5	6			4	5	6	4	5	6
         0.8		0.13			0.086		0.8		0.108			0.112		0.8	0.496		0.074			0.121
         1		0.162			0.102		1		0.097			0.097		1	0.619		0.084			0.156
         1.2		0.195			0.126		1.2		0.086			0.088		1.2	0.744		0.094			0.193
         1.4		0.228			0.143		1.4		0.074			0.079		1.4	0.868		0.103			0.232
         1.6		0.263			0.167		1.6		0.062			0.068		1.6	0.991		0.112			0.274
         1.8		0.302			0.199		1.8		0.05			0.059		1.8	1.11		0.121			0.319
         2		0.349			0.249		2		0.037			0.05		2	1.24		0.131			0.368

      3. Change in Bus 6

   The effect of load change on bus 6 at constant P,Q and power factor on the stability are studied and shown in Table 4. It is noted that the system is stable at all condition of studied

   TABLE 4: L AND ZTH INDICATORS AT CONSTANT P, Q AND POWER FACTOR WITH CHANGES AT BUS 6

   Constant active power P							Constant reactive power Q							Constant power factor (pf)
   Q	L-indicator			ZTh indicator			P	L-indicator			ZTh indicator			P	Q	L-indicator			ZTh indicator
   	buses			buses				buses			Buses					buses			Buses
   	4	5	6	4	5	6		4	5	6	4	5	6			4	5	6	4	5	6
   0.8			0.108			0.09	0.8			0.091			0.133	0.8	0.496			0.061			0.123
   1			0.135			0.106	1			0.082			0.096	1	0.619			0.07			0.157
   1.2			0.161			0.131	1.2			0.072			0.089	1.2	0.744			0.078			0.193
   1.4			0.189			0.15	1.4			0.062			0.079	1.4	0.868			0.086			0.232
   1.6			0.222			0.174	1.6			0.051			0.071	1.6	0.991			0.094			0.273
   1.8			0.264			0.209	1.8			0.04			0.064	1.8	1.11			0.101			0.316
   2			0.343			0.262	2			0.029			0.055	2	1.24			0.109			0.363

   It is noted that from Table 2,3, and 4 the changes in active and reactive power at any conditions the voltage stability indicators L-indicator and ZTh indicator are increase and the stability of the system are decrease.

8. CONCLUSION

A method for computation static voltage collapse proximity indicator VCPI, FVSI and ZTh indicator are presented. Some case studies conducted on typical power networks have also been presented. These networks represented by two buses connected by transmission line and IEEE 6-buses system. This paper presented a new formula to get change in voltage and its angles based on the change on active and reactive powers.

This formula is used to calculate L-index, FVSI- index and ZTh- index directly without need to compute load flow in each time. Therefore, this formula depends only to compute base case load flow for one time to find the initial steady state condition. This study is important to prevent a possible voltage collapse with simple calculations of voltage stability indicator.

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Generator	P (Mw)	Q (Mvar)
1	107.9	16
2	60	74.4
3	60	89.6

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6	70	70

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Bus	V (kV)
1	o 241.5 0
2	o 241.6 3.1
3	246.1 3.3o
4	227.6 4.2o
5	o 226.7 5.3
6	o 231 5.9

APPENDIX