INTRODUCTION
Electric Power Systems (EPS) have become very complex from the operation, control and stability maintenance standpoints. The voltage stability problem deserves special attention, since power systems have been operating dangerously close to their stability limits. Voltage collapse and energy rationing occurrences have been reported worldwide.
Special care with transmission capacity expansion and the development of efficient
operation techniques to best use the system’s capabilities is crucial.
The power industry restructuring process has introduced a number of factors
that have increased the possible sources of system disturbances, leading to
a less robust, more unpredictable system as far as the operation is concerned
(Morison et al., 2004).
The lack of new transmission facilities, cutbacks in system maintenance, research
force downsizing, unpredicted power flow patterns, just to name a few are some
of the important factors that affect the security of power systems. The mainstream
philosophy of the restructured sector is to minimize investments, minimize costs
and maximize the equipment’s utilization. The regulatory agencies usually
define minimum voltage stability security margins for both normal operating
conditions and contingency situations. Under normal operating conditions, the
minimum margin should be a bit larger depending on the demand. Figure
1 shows the idea of the Voltage Stability Margin (VSM) in a very simple
way based on the wellknown PV curve.
 Fig. 1:  Illustration of Voltage Stability Margin (VSM) 

Consider the load flow equations given by:
where, superscripts s and c stand for scheduled (generation minus consumption)
and calculated quantities. In this study a constant power factor model is used
for the loads. None the less, the inclusion of voltage dependent load models
is straightforward. λ^{bc} and λ^{max} correspond,
respectively to the current and the maximum loading factors both for normal
operating conditions (base case).
 Fig. 2:  Impact
of contingencies on VSM 

VSM is computed from both loading levels, being a measure of the distance between
them. For instance, VSM can be given by:
The occurrence of contingencies results in variation of the VSM as shown in
Fig. 2. Contingency 1 results in a smaller VSM as compared
to the base case condition. Contingency 2 presents a more severe impact on VSM
since its postcontingency maximum loading is less than the base case loading.
This constitutes an infeasible contingency case. Of course this analysis is
valid in case the load and generation patterns and control settings remain unchanged
after the contingency occurs. A wellaccepted contingency analysis procedure
is based on splitting the process in different stages. The first stage is usually
referred to as contingency ranking and contingencies from a predefined list
are analyzed and ranked by using a simple, computationally fast method. Then,
the topranked (most severe) contingencies are selected and analyzed in a latter
stage now using more complete, powerful and accurate methods. This stage is
usually referred to as contingency analysis or evaluation. For instance, the
voltage stability condition of the most severe contingencies may be evaluated
by continuation methods (Ajjarapu and Christy, 1992).
Finally, preventive and/or corrective control strategies must be obtained to
deal with contingencies that result in insecure or emergency operating situations.
The procedure just described has been discussed by Stott
et al. (1987) for ranking and analyzing contingencies for MW flow
overloads and voltage magnitude violations. This study tackles the contingency
ranking problem that is the one of correctly ranking contingencies regarding
their impacts on VSM. Contingencies with the smallest VSM are the most severe
and ranked at the top of the list. The evaluation of the impact of contingencies
on the VSM has been dealt with in several research works found in the literature.
A crucial point related to this problem is that a proposed ranking method must
be efficient from the computational effort standpoint while keeping acceptable
accuracy. Also, it should be suited for use in many power system analysis process
environments, from operation planning to real time operation. Finally an appropriate
treatment for the infeasible contingency cases is also proposed.
Many research works on contingency ranking for voltage stability can be found in the literature assume that the operating state at base case is known for both normal operation and maximum loading condition. The idea here is to propose a method for ranking contingencies based on information from the base case and the postcontingency operating states. Thus, only one load flow is required for each contingency. Therefore, the method is intended to be computationally fast and suited for real time operation application where decisions must be taken within a small time frame. The ranking method is based on the computation of performance indices for each contingency. In their turn, the performance indices are based on the postcontingency operating state. Some of these indices include the computation of voltage stability indices which are in turn obtained from the equations of power flows through branches (transmission lines and transformers). These voltage stability indices are multiplied by weight factors which are based on the outage branch precontingency apparent power flow and on the voltage (magnitude and phase angle) variations.
The high order performance index methods have drawback of more computation
time. The performance index methods also have masking phenomena. Many attempts
were made to use fuzzy logic techniques in contingency analysis (Hsu
and Kuo, 1992; Lo and Abdelall, 2000; Ozdemir
and Singh, 2001) which is based on separate consideration of state variables
like line flows and voltage deviation. But it may give many misrankings if
we take only one variable based rank list. In the study a fuzzy logic based
combined rank list is developed considering the performance indices for each
contingency for different single outage cases. They are combined with other
performance indices based on the relationship between the branch current and
maximum apparent power flows.
THEORETICAL BACK GROUND
Voltage stability index: Figure 3 shows a branch that connects the branch i and j. The real and reactive power flows through branch ij are:
 Fig. 3:  Branch
ij of a power system 

where, θ^{eq}_{ij} = θ_{i}θ_{j}+φ_{ij} is the angle spread at branch ij . In the case of a transmission line; a_{ij} =1, φ_{ij} = 0. For transformer; b^{sh}_{ij} = 0, φ_{ij} = 0. For pure phase shifters; b^{sh}_{ij} = 0, a_{ij} = 1 = For phase shifters; b^{sh}_{ij} = 0. From Eq. 3 and 4 one gets:
Where:
Equation 5 has real solutions when b^{2}4ac≥0;
Finally, a voltage stability index is defined as follows:
VSI tends to zero as the load increases and the system approaches its voltage stability limit. Even though VSI was originally derived for a radial system, it will be meshed systems since it provides a good approximation of the system’s voltage stability condition.
Performance indices: One of the main points discussed in this study is related to the definition of appropriate Performance Indices (PI) that reflect the actual postcontingency operating conditions regarding voltage stability.
Five PI have been chosen and were used in the proposed ranking method. The
choice of using more than one PI is based on the following ideas. First, no
voltage stability index alone is able to reflect the actual postcontingency
VSM in an accurate way, due to the nonlinearities of the problem. All voltage
stability indices, VSI included present some degree of inaccuracy due to the
simplified assumptions they are based on. Therefore, it is expected that a PI
defined in terms of voltage stability indices would also carry some degree of
inaccuracy. It is important that the voltage stability index be associated to
other quantities to compensate for such inaccuracies and minimize errors. Secondly,
it was also found that each definition of a PI favors the identification of
certain severe contingencies.
One of the contributions of this study is to show that each one of the different PI may be able to identify a number of severe contingencies and the union of the PI may result in almost all severe contingencies identified. The definition of PI and their association to other quantities were based on exhaustive tests. In this study, the following five performance indices were defined:
Weighting factors i were added to the first three performance indices to improve their accuracy. VSI_{min} is the smallest postcontingency voltage stability index among the branches connected to the bus with the smallest base case voltage magnitude.
Weight α_{1 }is the base case apparent power through outage branch km. Weight α_{2} is the largest nodal phase angle variation from the base case to the contingency case.
Weight α_{3} is the largest nodal voltage magnitude variation from the base case to the contingency case. Note that factors i are based on relevant system physical quantities and their variations are closely related to the voltage stability phenomenon. S^{l}_{max} is calculated by:
where, branch l connects buses i and j and φ = <(V^{2}_{i}/S^{*}_{ij})
(Albuquerque and Castro, 2003). Obviously, subscript
l represents all branches in the system but the outage branch km.
PROPOSED FUZZY MODEL FOR COMBINED RANKING
The contingencies causing line flow overloads may not necessarily cause bus
voltage problems (Ozdemir and Singh, 2001) and vice
versa, line flow problems and voltage limit violations problems must be dealt
separately by associated performance indices.
 Fig. 4:  Fuzzy
surface of proposed model for IEEE14 bus system 

 Fig. 5:  Fuzzy
surface of proposed model for practical indian 75 bus system 

In the proposed approach, a new method is proposed to combine the five different
rankings using fuzzy inference.
In this method five rankings are taken as inputs to the fuzzy tool box (available in MATLAB) and fuzzy coefficient are generated based on some predefined ifthen rules as a output. The fuzzy model used in this approach is tested in MATLAB 7.0 Fuzzy tool box.
Fuzzy Interference Structure (FIS) for combined performance indices ranking: The FIS structure given below is tested in MATLAB 7.0. Fuzzy Toolbox. Figure 4 shows the fuzzy surface of the proposed model for the IEEE14 bus system.
Type: Mamdani, No. of inputs: 5, No. of outputs: 1, No. of rules: 7 and Method = min Or Method = max, Imp Method = min, Agg Method = max, Defuzzification Method = centroid.
Figure 5 shows the fuzzy surface of the proposed model for
the practical Indian 75bus system. For practical Indian75 bus system. Type:
Mamdani, No. of inputs: 5, No. of outputs: 1, No. of rules: 17 and Method =
min, OrMethod = max, ImpMethod = min, AggMethod = max, Defuzzification Method
= centroid.
Table 1:  Fuzzy
rules for IEEE14 bus system



Fuzzy membership values: Fuzzy sets for the combined contingency ranking
inputs are the performance indices to the contingency ranking.
Here the membership functions for all the linguistic terms are taken as triangular function. The values are shown in the following Table 1.
The developed fuzzy interference matrix fro the IEEE14 bus system consists of seven rules and is shown in Table 1. Figure 6 shows the interface fuzzy membership value for the IEEE14 bus system.
Fuzzy IFTHEN rules: The output and input membership functions to evaluate the severity of a postcontingent quality divided into three categories using Fuzzy set notation: low, moderate and high (Fig. 7).
Based on these rules and corresponding contingency ranking membership value a area is selected in fuzzification. This area is further defuzzified which gives the fuzzy coefficient. The following rules are implemented to obtain the fuzzy coefficient for IEEE14 bus system and a practical 75 bus Indian system (Table 2 and 3).
PROCEDURAL STEPS FOR THE CONTINGENCY RANKING LIST
The multicriteria based contingency rank is prepared using the following flow chart. The additional performance indices of the entire system can be ranked as to generate rank lists.
The additional multicriteria based ranking approach generally gives idea about the contingency planning in the deregulated environment. The flow chart shown in Fig. 8 shows the steps involved in preparing a contingency ranking list.
 Fig. 6:  Membership
functions of the performance indices 

 Fig. 7:  Membership
functions of the performance indices for practical indian 75 bus system 

Table 2:  Fuzzy
membership values for practical Indian 75 bus system 


Table 3:  Fuzzy
rules for practical Indian 75 bus system 


 Fig. 8:  Flow
chart for fuzzy based multi criteria contingency ranking 

RESULTS AND DISCUSSION
The proposed fuzzy inference system is tested on the IEEE14 bus system and
75 bus practical Indian systems. Obtain the postcontingency (VSM) for each
contingency of the list by using some known method as for example the continuation
method or by successive load flow computations for gradually increasing load
until load flow solutions are no longer found.
Table 4:  Contingency
ranking by loading parameter (λ) for ieee 14 bus system 


Table 5:  Fuzzy
membership values for ieee14 bus system 


The latter is of course very time consuming, however its results are acceptable
from a practical point of view. The results were used as a reference in this
study. Rank contingencies according to their VSM computed using Continuation
power flow obtaining list N. Perform the multi criteria contingency ranking
and obtain five ordered lists corresponding to the five PI’s. Obtain list
P which corresponds to the union of the five ordered lists (five PI’s).
IEEE14 bus system: For the sake of illustration, a detailed description of a simulation for the IEEE 14bus, 20branch system is shown. Table 4 shows the ranking of the ten most severe single contingencies. They were ranked according to the respective value of the loading factor as shown in Table 5. Note that MVAR limits at generation units have been enforced in all simulations shown here. The outage of branch 1 (connecting buses 1 and 2) results in an infeasible operating state and λ_{max}<1 (negative VSM). On the other hand they may also result in small impact on the system’s maximum loading. In fact, this is the usual case for realistic systems.
Table 4 shows the ranking of the twenty single contingencies.
They were ranked according to the respective value of the loading parameter.
The line 1(12) have lowest value of λ_{max }that’s why it
is ranked no.1 that means the outage of line 1 is most severe.
Table 6:  Performance
indices of ieee 14 bus system for single line outage 


Table 7:  Contingency
ranking for IEEE14 bus system based on PI values 


Table 6 shows the performance indices and VSI_{min}
value for each contingency. The outage of line 1 and 14 gives negative value
of VSI which shows infeasible case that’s why its VSI value and respective
PI value are made 0. Table 7 shows the contingency ranking
based on performance indices. The line outage for which the PI value is lowest
(for first 3 PI) is ranked 1 i.e., most severe. Where as for last two the line
which has highest PI value is ranked 1 as most severe. The line 1 has ranked
no.1 in all PI rankings which shows the outage of line 1 is most severe.
Using the fuzzy approach a fuzzy coefficient is generated by combining the five contingency rankings which is shown in Table 8. The line outage whose fuzzy coefficient is lowest is ranked highest. Here also the line no1 is ranked as most severe contingency which shows the effectiveness of this method.
Table 8:  Fuzzy
based contingency ranking for IEEE 14 bus system 


Table 9:  Contingency
ranking by loading parameter (λ) for IEEE 14 bus system 


Table 10:  Contingency
ranking for indian utility 75 bus system based on PI values. Fiures in
parenthesis shows the line connection i.e., from one bus to other bus 


Table 11:  Fuzzy
based contingency ranking for Indian utility 75 bus system 


Practical 75 bus Indian system: Practical Indian 75Bus System having
15 generators including one slack bus, 60 load buses and 24 transformers lines.
The Table 9 shows the ranking of the single contingencies.
They were ranked according to the respective value of the loading parameter.
The line 1 (1920) have lowest value of λ_{max} that’s why
it is ranked no. 1 that means the outage of line 1 is most severe.
Table 10 shows the contingency ranking based on performance indices. The line outage for which the PI value is lowest (for first 3 PI) is ranked 1 i.e., most severe. Where as for last two the line which has highest PI value is ranked 1 as most severe.
Using the fuzzy approach a fuzzy coefficient is generated by combining the five contingency rankings which is shown in Table 11. The line outage whose fuzzy coefficient is lowest is ranked highest. Here line no. 64 has minimum fuzzy coefficient which shows this is the most severe contingency of this system.
CONCLUSION
The contingency ranking method for voltage stability applied in this project shows a great potential to be used as a tool for real time operation. This project demonstrated that various performance indices couldn’t reliably capture all the instable cases individually.
Each index can’t rank the severity of contingency for different system under different conditions but the combination of indices can give an overall evaluation from different aspects of the system. Results on two test systems showed that combination of indices CI with use of fuzzy will provide a better ranking for worst cases.
As evidenced by the results of IEEE 14 bus system and a practical Indian 75 bus system the approach used can provide the user with those outages that may cause immediate loss of load or islanding at a certain bus. This is a kind of information which is very helpful to system operators.
An overall severity index is given for which outage case. These severity indices can be used as a guideline for deciding whether corrective control actions should be taken.
ACKNOWLEDGEMENT
The researchers would like to thank the management and faculty of VIT University for their kind support and encouragement throughout this research.