Power System Voltage Stability Assessment Employing Phasor Measurement Units

H. MesgarnejadElectrical Engineering Department

Iran University of Science and Technology (IUST) Tehran, Iran.

h m es g ar n e j a d @ieee.o r g

S.M. ShahrtashCenter of Excellence for Power System

Automation and Operation at IUST Tehran, Iran.

sh a h rta s h@ i u s t.ac. i r

Abstract—Employing Phasor Measurement Units in wide area monitoring and control of power systems and the availability of phase angles, a new look at the voltage stability indices is performed in this paper. A systematic approach for comparing these indices is presented, where the behavior of different indices to a step load change, for studying long term voltage stability, is compared. This comparison is carried out in two methodologies. First, the accuracy and preciseness of each index, especially in the proximity of voltage collapse point, is evaluated. Then, the effect of error in phase angle measurement on the response of those indices is examined. Finally, an overall ranking for the indices is given that is based on the results of individual comparisons.

Keywords- Voltage stability, Voltage stability indices, Phasor measurement units.

I. INTRODUCTION

A. Overview

In recent years, power systems have been encountered series of blackouts. These system instabilities, intensified by immense loadings, occurred because either of voltage or angle instability or both together. Voltage instability which has caused most of those wide area blackouts has become one of global concerns for system operators. Until now many methods have been developed to moderate this problem. These methods could be classified in two major categories, which are preventive and remedial actions.

In preventive methods, many different system conditions are simulated through an off-line procedure, and effect of different system disturbances are studied. Afterward, based on those simulation results, a list of appropriate actions for preventing system instability, based on pre-disturbance system conditions, is presented [1]. On the other hand, in the second class, predefined remedial actions scheme (RAS) are usually activated based on system data recorded in an on-line way. The trigger signal for those actions either comes from pre-defined system conditions, examined through off-line studies, or according to previously developed stability indices.

decreased due to real-time phase angle measurement in spite of employing traditional phasor estimation techniques [2].

The System Indices for Voltage stability evaluation(SIVs) can be categorized as global or local ones. Global indices usually need full system data such as impedance orJacobean matrix, hence more time consuming than local ones. Therefore, only local indices that only need one PMU data are studied in this paper.

This paper is organized as follows. In the remaining parts of this section definitions and formulations of some voltage stability indices are presented. In section II, the two important issues about system equivalent estimation and the basis of systematic comparison, used in this paper are presented. The dynamic responses of these indices are analyzed in section III. Finally, in the last section, the comparison is carried out based on the selected criteria.

B. SIVs Definitions

In this section, the basis of six voltage stability indices, grouped in four different classes, are presented. These local indices are defined based on a simple two bus system model, illustrated in Fig. 1.

Figure 1. Two Bus System

1) Impedance Matching Indices (ISI, VCPI and VSLBI)

Based on impedance matching theory, fully discussed in [3-5], the maximum power transfer limit, which coincides with the start point of voltage collapse [4], can be represented by the following equation:

U Uˆ k ˆ k After development of Phasor Measurement Units (PMUs)

those stability indices and RAS did not changed, but the accuracy increased and the calculation time, significantly,

Z th I Z k Ik

978-1-4244-5586-7/10/$26.00 C 2010 IEEE

Where Zˆth is the Thevenin’s equivalent impedance of the

system, seen from the load bus, indexed by k.

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2

V

L

Consequently, the Impedance matching Stability Index P P Q Q S S

(ISI) is defined as below: VSI min max , max , max

ISI Zˆ

k Zˆ

th 1 I k

U k

Pmax Qmax Smax

Zˆ k I k U k

3) Power Trasfer Stability Index (PTSI)

The fifth index is based again on the theory of maximum power transfer limit but considerers the load as a constant

In addition, from Eq. 1 it can be inferred that in thevoltage instability point:

impedance element. Therefore, the load power consumption can be calculated as below:

E 2V cos s r 2V cos

i.e., the phasor of load bus voltage will have the same amplitude as the voltage drop through the transmission line (Fig. 2).

2E

S L Z L ZT Z L

Consequently, to calculate the maximum power transfer, it can be written:

S LS

E

Z L 0 L max 2ZT 1 cos

Figure 2. Representation of two bus system voltages phasor diagram at the point of collapse and this index can be presented by [9]:

Therefore, Voltage Collapse Proximity Index (VCPI) and Voltage Stability Load Bus Index (VSLBI) have been defined as below [6][7].

PTSI S L

S L max

2S

L Z

T 1 cos

E 2

VCPI V cos 0.5E 4) Tangent Vector Based Index (TVI)

The last system stability index is considered as the inverse value of system loading curve slope. This index

VSLBI V

V could be calculated as fallow:

TVI

0.5

2) Maximum Loading Indices (VSI)

Second group of indices is based on the calculation of maximum power transfer capacity of the system. The

( )2

!( )2

!

maximum active, reactive and complex power transfer values can be calculated as below [1]:

As the system loading increases and system approachesits stability limit, system states’ derivatives to loading parameter ( ) will approach infinity and this index will show

PmaxQR E 2 R

X 2 X 2

Z L

EE 2 4QX

2 X 2

very small values [10][11].

II. BASIS CONSIDERATIONS

A. System Response

PX Qmax R

E 2 X

2R 2

Z L

EE 2 4PR

2R 2

In this paper, the comparisons are carried out on dynamic responses of the selected indices. The IEEE 14 bus system is used as the context for this research. Dynamic response of

E 2 Z X sin R cos this system is evaluated considering dynamic models of generator and its controllers. In addition, five major loads are

S max 2 X cos R sin 2

Therefore, according to these relations, the total Voltage

modeled with an adaptive voltage dependent model (Eq. 15), introduced in [12], to simulate long term voltage instability of the system.

Stability Index (VSI) can be calculated as [8]:

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V0

The methodology and practical considerations of these

s

Tp X

p P V

0

Pdalgorithms is fully discussed in [13]. However, the effect of the difference in results of the two algorithms has not been

discussed, yet.

t

The estimation error (Eq. 16) is calculated throughP X

V d p V 0

applying two sample data window and is depicted in Fig. 5.As it can be seen, the error is very small and negligible but in

To stimulate the system over its maximum loading limit the step load change is applied to the load on the bus 14 of

the next section it is shown that this small error will affect the preciseness of the stability index in some ways.

the system. Fig. 3 depicts time domain change of static (P0) and dynamic load (Pd) active power. This long term load recovery will cause steady increase in system loading to system voltage stability critical point (Fig.4 Ts1). Afterward, the system load cannot increase further and system voltages will slowly decreases leading to voltage collapse in the system.

Err E (V Z I )

Figure 5. Estimation Error Difference between LS and RLS

(16)

Figure 3. Bus 14 Active and Reactive Load Change

In addition, to evaluated the indices’ responses through the system recovery process, after 51 sec the bus load is decreased to its initial value. Fig. 4 shows all three parts of the process in voltage versus complex load power changes.

Figure 4. Loading Curve of Bus 14

B. Equivalent System Estimation

As mentioned before, some of the selected indices, in both groups, need a locally estimated system equivalent circuit parameter. Therefore, to identify the system equivalent parameters, illustrated in Fig. 1, it is assumed they have slight changes in time. Consequently, those parameters are calculated from sliding window of discrete data samples by using two different identification algorithm, namely least square (LS) and recursive least square (RLS) algorithms.

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On the contrary to the lower error magnitude, the LS method has more oscillations in its transient period starting right after the system recovery (Fig. 5). These oscillations have some contributions on the performance of indices, which is discussed in section III

C. Evaluation Criteria

Before carrying out the comparisons some countable testing parameters should be defined to measure the desirability of each index. Therefore, four criteria are introduced and used.

1) Accuracy in Critical Timings

As is shown in Fig. 6, through the simulation process, there are four critical times points. Two well-known points are maximum load power points which are load complex power (SL) extremes, Ts1, Ts2. Two other, Tc1 and Tc2, are defined from intersection of this curve with maximum calculated complex power (SMAX). Therefore, the first criterion is selected as the accuracy of each index in correctly measuring these four time points.

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Figure 6. Critical Times of Simulated Systems

2) Linearity

As afore-mentioned, the value of each index shows the proximity to voltage collapse point. But different indices show different trends in this process. The more linearity in the trend of an index, the more it is suitable. This feature is selected as the second evaluation criterion. To do this, a linear estimation of each index is found. Then, as illustrated in Fig. 7, two parameters from this process are used for comparison.

R2: as a measure of linearity of index data through simulation time, this must be near 1.0.

b: as a measure of index accuracy to detect the collapse point based on pre-collapse data.

3) Distinguishing Stable and Unstable Regions

Some indices can distinguish stable and unstable regions, in addition to show the proximity to the voltage collapse point. The exhibition of this capability is defined as the third criterion for comparison.

Figure 7. Linear Estimation Parameters

4) Robustness against Error in Phase Angle Estimation

The last attribute which is the most essential one is the robustness to phase estimation error. It is very important, because the accurate phase angle measurement is the most important characteristic of PMUs in comparison to traditional estimation procedures in conjunction with

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SCADAs. It is worth mentioning that, most of the indices have shown very sensitive to the phase angle error. Therefore, it is essential to analyze the effect of any phase error on the behavior of the selected indices. The road map for this analysis is fully discussed in the next section.

III. INDICES EVALUATION

In this section the performances of selected indices under the mentioned system conditions are presented. For each index both LS and RLS based response is calculated. The final comparison and evaluation is given in section V and VI, respectively.

A. ISI

This index can separate the stable and unstable region. However, in the transient period the estimation error cause the unwanted fluctuation which could cause malfunction of control systems, as shown in Fig 8.

Figure 8. ISI Index Response

B. VCPI and VSLBI

These indices have, also, the same trends as ISI index, and only one of them is shown in Fig. 9.

Figure 9. VCPI Index Response

C. VSI and PTSI

The next group of indices cannot distinct the two regions but are as sensitive to estimation outputs as previous ones. Fig. 10 and 11 show the trend of VSI and PTSI, respectively.

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Index Raw Data LinearPolynomial

3Polynomial

4

VSI 0.1122 0.0948 0.0799 0.0887

PTSI 0.067 0.0854 0.0726 0.0753

VSLBI 0.1166 0.1351 0.1192 0.1304

VCPI 0.0187 0.0177 0.0166 0.0171

ISI 0.3249 0.0479 0.1704 0.2913

TVI 3.7213 0.1491 0.8597 3.6818

IV. EFFECT OF PHASE ANGLE ERROR

For the calculation of index estimation error, caused by any inaccuracy in phase angle measurement, a normal distributed error (e) is added to the relevant outputs ( ).

$ # " e %

Considering the fact that the error takes a random behavior value, each index has been calculated for several times (Indexi) and the maximum difference to its ideal value (Index0) is considered as the estimation error (E).

E (Index) max+ Indexi Index0

( Figure 10. VSI Index Response

As is shown in Fig. 10, for the point Tc1, the value of index passes the predefined minimum value. This error is caused by the estimation error, discussed in section II-B. In Addition, Fig. 11 has shown the behavior of PTSI index.

Figure 11. PTSI Index Response

D. TVI

The last, tangent vector based index, also cannot detaches stable and unstable regions, while its performance is shown in Fig. 12.

Figure 12. TVI Index Response

* 'i ) Index0 &

A. Noise Reduction

It is important to mention that the estimation process, used for calculating the equivalent circuit parameters, would reduce the error in estimated parameters. However, because the load side measured values still have some noise with them, this noisy data can cause large errors (Table I). Therefore, the raw phase data is passed through a linear or polynomial least square process to reduce the phase angle error in them. Table I shows those estimation errors employing four different techniques.

TABLE I. EFFECT OF DIFFERENT NOISE REDUCTION TECHNIQUES

As it can be seen, the linear estimator has shown the best results and, thus, it is used in sequel.

B. Effect of Windows Length

Next important issue is the data window length. Figure13 shows the effect of this parameter on the error.

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Index ISI VSI PTSI VCPI VSLBI TVI

Tc1 26.59 26.58 26.59 26.59 26.59 31.29

Tc2 173.4 173.4 173.4 173.4 173.4 118.6

Discussion Maximum Power TransferActualExtreme

Index ISI VSI PTSI VCPI VSLBI TVI

|b|*100 2.78 2.05 1.95 0.76 0.78 22.07

R2*100 94.68 90.03 90.03 94.12 94.13 91.72

Rank 2nd Best Ones

Index ISI VSI PTSI VCPI VSLBI TVI

S1 10.436 12.035 9.935 7.593 9.841 29.654S2 3.533 3.398 1.054 4.651 1.571 2.099

Figure 13. Effect of Length of the Data Window

In view of the fact that the larger data window corresponds to more calculations and, as shown in Fig. 13, for data windows with larger than 20 samples the improvement in estimation is negligible, this window length is selected for testing procedures.

C. Comparison

The comparisons in this section are performed through calculation of each index’s sensitivity to various phase angle errors and different distances to voltage collapse point. This sensitivity criterion is defined as:

max(E ) min(E )

E

,

Index ISI VSI PTSI VCPI VSLBI TVI

E 4.6962 5.4157 4.4707 3.4168 4.4284 13.3443

P 0.45

S1 10.436 12.035 9.935 7.593 9.841 29.654

2) System Loading

The last prominent parameter is the effect of proximity to voltage collapse on the performance of indices. It is very important because as the system approaches its stability limit the index accuracy becomes more vital for maintaining the system security.

TABLE III. INDICES’ SENSITIVITY TO SYSTEM LOADINGS

Index ISI VSI PTSI VCPI VSLBI TVI

E 28.26 27.18 8.43 37.21 12.57 16.79

P 8.00

S2 3.53 3.40 1.05 4.65 1.57 2.09

V. FINAL COMPARISON

A. Numerical

The numerical comparisons are fulfilled in three steps. In first step, from results shown in section III, the accuracy for detecting the voltage stability limit is analyzed. As shown in Table IV, TVI detects the maximum loading point, but otherS

1) Angle Error

i i

Pmax Pmin Pfive indices detect the maximum power transfer.

TABLE IV. INDICES RANKING FOR ACCURACY

The maximum error in phase angle measurement of PMUs is near 0.1 degree [14]. But to study the sensitivity criterion, the performance of each index is calculated for a range of different phase angle errors (Fig. 14).

Figure 14. Effect of Error in Phase Angle Error on each Index

It can be seen in Fig.14 that different indices have shown different robustness, as given in Table II.

TABLE II. INDICES’ SENSITIVITY TO PHASE ANGLE ERROR

Secondly, the indices are compared based on their linearity in estimation. It can be inferred from Table V that the VCPI and VSLBI index have better trends in approaching the voltage stability limit.

TABLE V. INDICES RANKING FOR LINEARITY

A final numerical comparison is based on all sensitivity values calculated in this sub-section, given in Table VI.

TABLE VI. INDICES RANKING FOR ERROR SENSITIVITY

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Rank 1st 3rd 2nd

B. Overall

The overall comparison is accomplished based on three different rankings in numerical step and third feature defined in section II-B.

TABLE VII. ANALYTICAL COMPARISION

Index ISI VSI PTSI VCPI VSLBI TVI

Accuracy 0 0 0 0 0 1Linearity 1 0 0 1 1 0

ErrorSensitivity

0 0 1 0 1 0

Ability of Un/Stable Distinction

1 0 0 1 1 0

Credits 2 0 1 2 3 1

VI. CONCLUSION

In this paper, a systematic comparison between six voltage stability indices is presented. The comparison is conducted on four features of an ideal index and in each one best index or indices are selected. In this way,

- for accuracy: TVI,- for Linearity: ISI, VCPI and VSLBI,- for minimum error sensitivity: PTSI and VSLBI,- and for ability to distinguish between stable and

unstable regions: ISI, VCPI and VSLBI,are chosen as the best ones.

In conclusion, the VSLBI index has more credits and selected as the most informative voltage stability index to detect long-term voltage stability of power system.

REFERENCES

[1] G. Yanfeng, “Development Of an Improved On-Line Voltage Stability Index Using Synchronized Phasor Measurement”, A Dissertation Submitted to the Faculty of Mississippi State University in Partial fulfillment of the Requirements for the Degree of Doctor of Philosophy in Electrical Engineering in the Department of Electrical and Computer Engineering, Mississippi State, Mississippi, December2005.

[2] _, "Wide Area Protection and Emergency Control", IEEE Power Engineering Society – Power System Relaying Committee – System Protection Subcommittee Working Group C-6

[3] K. Vu, M.M. Begovic, D. Novosel, and M. M. Saha, "Use of local measurements to estimate voltage-stability margin," IEEE Transactions on Power Systems, vol. 14, no. 3, pp. 1029-1035, Aug.1999.

[4] I. Smon, G. Verbic, and F. Gubina, "Local voltage-stability index using tellegen's Theorem," IEEE Transactions on Power Systems, vol.21, no. 3, pp. 1267-1275, Aug. 2006.

[5] I. Smon, G. Verbic, and F. Gubina, "Local voltage-stability index using tellegen's Theorem," in IEEE Power Engineering Society General Meeting, 2007.

[6] L. Wang, Y. Liu, and Z. Luan, "Power Transmission Paths Based Voltage Stability Assessment," in Transmission and Distribution Conference and Exhibition: Asia and Pacific, pp. 1-5, 2005.

[7] B.D. Milosevic and M. Begovic, "Voltage-stability protection and control using a wide-area network of phasor measurements," IEEE Transactions on Power Systems, vol. 18, no. 1, pp. 121-127, Feb.2003.

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[8] N. Yanfeng, G. Schulz, and A. Guzman, "Synchrophasor-Based Real- Time Voltage Stability Index," in IEEE PES, Mississippi State Univ., pp. 1029-1036, 2004.

[9] M. Nizam, A. Mohamed, and A. Hussain, "Dynamic Voltage Collapse Prediction In Power Systems Using Power Transfer Stability Index," in PECon'06, pp. 246-250, 2006.

[10] I.L. Lopes, B. Zambroni de Souza, and A.C. Mendes, "Tangent Vector as a tool for voltage collapse analysis Considering a dynamic System Model," in IEEE Porto Power Tech Conference, 2001.

[11] A.C. Valle, G. C. Guimaraes, J.C. Oliverira, and A. J. Morais, "Using Tangent Vector and Eigenvectors in Power System Voltage Collapse Analysis," in IEEE Porto Power Tech Conference, 2001.

[12] Y. G. Zeng, A. Berizzi, and P. Marannino, "Voltage StabilityAnalysis Considering Dynamic Load Model," in Proceedings of the4th International Conference on Advanced Power System Control, Operation and Managment, Hong Kong, pp. 396-401, 1997.

[13] B. Milosevic, M. Begovic, "Voltage-stability protection and control using a wide-area network of phasor measurements", IEEE Transactions on Power Systems, IEEE, Vol. 18, Issue 1, pp. 121- 127, Feb 2003.

[14] IEEE Std C37.118™-2005, "IEEE Standard for Synchrophasors for Power Systems", IEEE Power Engineering Society, IEEE Transactions on Power Delivery, Vol. 13, No. 1, January 1998.

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