A Comprehensive Review and Adequacy Evaluation ...

International Review of Electrical Engineering (IREE.), Vol. 5, N. 2 March-April 2010

A Comprehensive Review and Adequacy Evaluation of Voltage Stability Indices for Modern Distribution Systems Ulas Eminoglu', M. Hakan Hocaoglu^ Saffet Ayasun'

Abstract // is well known that the netn'ork variables and parameters contain sufßcient information to obtain the most sensitive bus or line to the voltage collapse in power systems. Accordingly, there has been several static voltage stability indices developed for identifying the buses or lines on the verge of voltage collapse in the literature. In this paper, various voltage stability indices including those originally developed for transmission systems are first reviewed. Then, their adequacies are quantitatively evaluated on a modern distribution test system which has voltage controlled bus (PV-bu.s). From analyses, it is observed that most of the existing voltage stability indices would be inadequate for assessing the most sensitive bus or line on the verge of voltage collapse in such modern distribution systems. Copyright © 2010 Praise Worthy Prize S.r.t. - All rights reserved. Keywords: Critical Bus, Modern Distribution Networks, Voltage Stability Indices. Voltage Collapse

Nomenclature the f' z^zdj

diagonal

element

of

bus

impedance matrix the impedance of load connected Xo f'' bus

K Vr

R X

z &z.

Ss Sr

P Q X Bus-s Bus-r

magnitude of line sending end voltage magnitude of line receiving end voltage distribution line resistance distribution line reactance distribution line impedance phase angle of the line impedance phase angle of the line sending end voltage phase angle of the line receiving end voltage distribution line receiving end active power distribution line receiving end reactive power Load factor Distribution line sending end bus Distribution line receiving end bus

I.

Introduction

With the increased loading and utilization of the existing power system, the probability of occurrence of voltage collapse has been significantly increased. Therefore, identification of the buses which are prone to the voltage fluctuations have been attracted more attention for the transmission as well as distribution

Manuscript received and revised March 20¡0, accepted April 20¡0

698

systems. For a safe and secure operating of power systems, all insecure operating states must be identified well in advance to facilitate corrective measures to overcome the threat of a possible voltage collapse [I]. The static voltage stability problem has been classically related to the reaching of some maximum admissible load level, beyond which a load flow solution no longer exists [2], In fact, early methods for determining the stability limit relied on solving a sequence of load flow solutions as the system load increases. The static voltage instability condition is said to be reached when the load flow algorithm fails to converge [3]. Divergence of the load flow algorithm, however, may not be necessarily attributed to the instability condition observed at the loading level beyond the critical point. This may be caused by numerical illconditioning close to the voltage stability limit. To mitigate this problem, a continuation load flow which remains well-conditioned at and around the critical point was proposed in [4]. The load flow based methods require solving a large number of load flow problems and, therefore, are computationally expensive. For this reason, many researchers have paid attention to develop the voltage stability indices that could identify the buses or lines on the verge of voltage collapse [2], [5]-[15]. The bus stability index proposed by Chebbo and his co-authors [2] is obtained from calculating the magnitude of the ratio of Thevenin equivalent impedance to the load impedance at a given bus. Another index is derived in [5] by using the minimum singular value of the power flow Jacobian matrix. A voltage stability index is also developed by Kessel et all using bus admittance matrix

Copyright© 2010 Praise Worthy Prize S.rl. - Alt rights reserved

. Eminoglu, M. Hakan Hocaoglu, S. Ayasun

of the system, line transferred power and bus voltages in [6]. However, the mentioned indices of [2], [5]-[6] are generally used for transmission systems and require calculation of the Jacobian [5] and bus admittance matrix [6] which may be singular or not readily available for distribution systems specially radial ones. Different characteristic features of distribution systems such as the radial structure and high R/X ratio make the distribution systems voltage stability analysis quite different and somewhat difficult as compared to the transmission systems. Recently, a number of static methods have been developed in the literature to compute the proximity of buses or lines which is the most sensitive to the voltage collapse in a distribution network [7]-[15]. Most of these indices are developed representing the whole radial distribution network by a single line equivalent [7]-[10] and [15]. The other stability indices, proposed in [11]-[14], are derived by using special formulation of distribution lines i.e.; by using the well-known bi-quadratic equation which relates the voltage magnitude at the receiving end to the voltage magnitude at the sending end and branch power fiow or by using the solution of the line receiving end bus voltage magnitude.

parameters such as bus voltage magnitude, line impedance and line transferred power contain sufficient information to obtain the most sensitive bus or line to the voltage collapse. Using this information, various bus and line stability indices has been proposed for transmission systems which can also be applied to distribution systems [2] and [9] or developed by exploiting special structure of the distribution networks [7]-[8] and [10]-[15]. Chebbo and his co-authors [2] have proposed a stability index obtained from calculating the magnitude of the ratio of Thevenin equivalent impedance to the load impedance at a given bus as follows:

(1) For a secure system, the ratio for al! buses, Eq. (1), is lower than 1.0 and the critical stability limit is reached when the index value of critical bus reaches to the value of 1.0. But as can be seen from equation, the developed stability index requires calculation of bus impedance matrix which may not be easily obtained for radial networks due to the singularity of the bus admittance matrix. Jasmon and Lee developed a line stability index for distribution networks in [7]. In that study, they formulated transferred active and reactive power of the distribution line without using line sending and receiving ends voltages, and developed a line stability index by using a new active power formula transferred from the reduced distribution line of the distribution network shown in Fig. 1. The index is defined as follow:

This paper qualitatively reviews and quantitatively evaluates the adequacy of line [7]-[I0] and bus [2], [11][15] voltage stability indices developed for identifying the bus or line which is the most sensitive to the voltage collapse. Although the indices of [2] and [9] are developed for transmission systems, it can directly be applied for distribution systems without further modification. Meanwhile, the indices proposed in [7]-[8] and [10]-[15] are specially developed for distribution systems. In the adequacy evaluation, a particular distribution test system with a voltage controlled bus (PV-bus) is considered and the effectiveness of these stability indices on the identification of the most critical bus or line are evaluated by comparing their result with that of multiple load flow analysis. For the load fiow analysis, the algorithm presented in [16] is used. It is observed from the analyses that ail static voltage stability indices reviewed in this paper, except for the indices of [2] and [8], would be inadequate for assessing the most sensitive bus or line in such modem distribution networks, ln addition, on the contrary to the suggestion of some static voltage stability studies (i.e., [11]), it is also concluded that the base case load fiow solution with feasible voltage magnitude itself may not be sufficient for assessing the critical bus or line, because the bus that has the minimum voltage magnitude could change with the variation of system's load in modem distribution networks containing voltage controlled bus (P-V bus) and/or Distributed Generation (ÜG) sources.

II.

L = 4[{XP-

RQf+XQ+Rp]

(2)

The developed index is applied to the each reduced distribution line seen from each bus of the network to obtain the critical line which has the highest index value. In this case, the critical line index value is always lower than one for the stable region and the critical point occurs when the index approaches to one. V, P+JQ (Z=R+jX)

Fig. 1. A distribution line model

Another line stability index has been developed in [8] by using line transferred power equations as follows:

Mathematical Formulation of Voltage Stability Indices

L=

(3)

As mentioned in Section I, the network variables and Intemationai Review of Electrical Engineering. Vol. 5. N. 2

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U. Eminoglu, M. Hakan Hocaoglu, S. Ayasun

where, L varies from infinite (no loading) to one (maximum loading). After the load flow calculation the index value (I.) of each line can be calculated using Eq. (3), easily. The line with the lowest L is the weakest line, and its receiving bus is the weakest bus. When its stability index value approaches 1.0, the line, and thus the system, will become critical to lose voltage stability. Moghavvemi and Faruque [9] proposed a line stability index obtained from the solution of the line receiving end voltage magnitude derived by using line reactive power equation, given in Eq. (4a), for transmission as well as distribution networks as follows (4a), (4b):

magnitude (V,.) will exist if the following conditions are satisfied: (6a) or: 4RP

(6b)

Q (14a) ^ > 0 (14b)

Equation !6(a) defines singular point of the Jacobian in terms of bus voltages and, hence, voltage collapse point of the reduced system. It is termed as voltage stability index as given in Eq. I6(b) and applied to the each reduced distribution line which is seen from each bus of the network to obtain the most critical bus which has the lowest index value. The voltage instability occurs when the index value of critical bus iSI„,„) is equal zero, which indicates that the maximum power transfer limit of the line is reached. The general structure and main features of all stability indices reviewed in this paper are tabulated in Table I. In the table. Column II shows the indices which are developed for distribution and/or transmission systems. The line and bus indices are classified in Column III, and their corresponding stability indices are provided in Column IV.

TABLE I GENERAL FEATI[RKS O F STATK- VOLTAGE STABILITY INDICES

Ref. No

System lype Distribution Transmission Systems Systems

Index type Line Bus index index

Index Formula Z ;,

[2] 17]

L = A{{XP - RQY + XQ+RP\ 1/ 2 L—

(8]

2[RP + [9]

L=

[101

L=

Copyright © 2010 Praise Worthy Prize S.rl. - All rights reserved

4XQ

¡ntemational Review of Electrical Engineering. Vot. 5. N. 2

701

Eminoglu, M. Hakan Hocaoglu, S. Ayasun

= K / - A(PX - QRf ~ AV;{PR + QX)

un [12]

113]

V

114] 1151

- -0.5K,+ r,

ni.

Test Cases

To evaluate the effectivetiess of the stability indices, a particular three-bus system of a balanced radial distribution network is considered. One-line diagram, line and load data of the test system are given in Fig. 2. The test system shown in in Fig. 2 could be represented the following case commonly experienced in energy distribution systems; it could be a part of a distribution system in which the voltage magnitude in a specified bus (in our case bus-l) is controlled by a Wind Generator or another Distributed Generation Sources. Moreover, it also could be an equivalent representation of a more complex distribution system with two feeders. In the test system., phase angle of the bus-1 is selected as (5/=0 and the phase angle of the other two buses are measured with respect to the bus-1. Bus voltages are calculated for different loading conditions of the system by using load flow algorithm presented in [16]. Voltage magnitudes of the system could be also obtained analytically by using Eq. (A.5) given in Appendix. In load flow analysis, different loading conditions are considered multiplying the base load by a load factor U) as S^-XSH- The critical bus is identified by evaluating the variation of high and low voltage solutions (feasible and unfeasible voltage magnitudes) of load buses. Figure 3 shows the voltage magnitudes of load buses {Vj and Kj) as the load factor increases, which is known as the nose curve. From this figure, it is clearly seen that the critical loading point of bus-2 and bus-3 is equal to >.^4.582 and 1^4.16, respectively, and the voltage collapse occurs at bus-3 when the system is highly loaded (X.=4.I6). Hence, bus-3 is the most critical bus of the particular test system. In addition, it is also seen from Fig. 3 that the critical bus {bus-3) has lower voltage magnitude for the stable operation point {bus voltage magnitude of load buses

with feasible values located at the upper part of the nose curve) of the system for each loading condition. The same observation could be made from Fig. 4 in which the sensitivities of high and low voltage solutions of load buses with respect to load factor are depicted. The expressions for these sensitivities are given in Eq. (A.6) in Appendix. From Fig. 4, it is clearly seen that absolute values of the sensitivity of high and low voltage magnitude of bus-3 is higher than the sensitivity of voltage magnitude of bus-2 at each load level. Moreover, the difference between these sensitivities is becoming larger as the load factor increases, and the voltage collapse takes place at bus-3 when the slope of its sensitivity curve with respect to load factor reaches to 90°. In this study, secondly, all bus and line stability indices given in Table I are applied to the test system of Fig. 2 for each loading condition of the system. The critical loading point identified by evaluating the value of load factor just before the load flow algorithm diverges. Divergence is assutned when the iteration number of the load flow algorithm reaches to its maximum value, which is selected as fter„ax=500. Figure 5 provides variation of each bus stability index value with respect to the load factor. From this figure, it is seen that all bus stability indices are in close agreement. Moreover, they all correctly estimate the critical bus (bus-3) for each loading condition. Recall that the critical bus is found as bus-3 shown in Fig. 3 and Fig. 4. It is also seen that the critical bus index value obtained from using [11]-[15] decrease as the system load increases, and is close to zero when system's total power is close to the critical loading point, as expected.

Vil - 1 (pu.)

V^^^i

Zi=0.545+j0.15{SÎ)

S2=0.4+j0.25 (MVA)

(Line-l) Z2=0.9+j0.5 (Q) (Line-2)

Sî=O.25+iO.15(MVA)

Fig. 2. A particular distribution test system

International Review of Electrical Engineering, Vol. 5, N. 2

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V. Eminoglu, M Hakan Hocaoglu, S. Ayasun

topologies. In this case, we investigate the performance of the indices when the line resistance, /fj, is chanced to R2=0,78 (Cl). Bus voltages are calculated for different loading conditions of the reconstituted system by using the load flow algorithm given in [16]. Similarly, the critical bus is identified by evaluating the variation of high and low voltage solutions of load buses provided in Fig. 7. From the figure, it is clearly seen that the voltage collapse occurs at bus-2 when the system is highly loaded (1=4.582). Hence, bus-2 is the critical bus of the system. However, it was concluded in some stability studies for example in [II] that voltage collapse occurs at the bus which has the lowest voltage magnitude of the distribution networks. Thus, load flow solution for the base load or any loading condition of radial networks is sufficient for assessing the critical bus. However, the bus which has the minimum voltage magnitude for the base load or any loading condition of the system may change with variation of loads in distribution networks as can be seen from Fig. 8, which illustrates the zoomed part of the nose curve of Fig. 7 for the values of load level (X>3). Although the voltage magnitude of bus-3 is lower than the voltage of bus-2 for the loading levels >.3.2) as can be seen from Fig. 9. On the contrary, it is seen from Fig. 10 that the sensitivity values of bus-2, obtained for unstable operating points (low voltage solutions), with respect to load factor is higher than the sensitivity values of bus-3 for each loading condition of the system l< X il_ \

lil

-

dVL^ca

1

5