Voltage stability studies and formulation of efficient stability indices have been the subject of research over the past few decades. Voltage stability indices play a very important role in the planning and operation of modern power system, when considering stability constraint. Due to the continuous changing nature of power system, lack of continuous monitoring of critical portion of the power system can lead the system to a worse condition of voltage instability. Hence, In very recent time, derivation of voltage stability indices with better performance especially in terms of computational requirements has been a major area of research focus especially for applications such as voltage stability prediction, voltage collapse margin prediction, estimation of transmission line power capability and power outage prediction      . Power system stability is usually classified into rotor angle, voltage and frequency stability, but these events are usually not independent or isolated from one another    . A voltage collapse at a bus can lead to large excursions in rotor angle and frequency. Similarly, large frequency deviations can lead to large changes in voltage magnitude. Hence, dealing with voltage stability issues and its associate problems in power system operation affect the whole condition of the power system   .
Online and offline voltage stability analysis of power system is attracting increasing attention due to continuous rise in electricity demand and deregulation policy of the electricity markets which has forced power systems to operate close to their stability limits. Approaches for monitoring the voltage stability condition of power systems are classified into either node or line stability indices   . This classification is based on the system component of interest; whether we are trying to identify the buses or lines that position the power system closer to the point of voltage collapse. Some of the node voltage stability indices are P-Q-V curves, L-Index, P-index, V/V0 index and Modal analysis     . Some of the line voltage stability are Lmn, FVSI, VCPI, NLSI, LQP  -  . The continuation power flow (CPF) has been extensively employed for estimating the voltage stability margin of a power system     and a new approach for line voltage stability margin estimation which can double as voltage stability index was developed in our previous work, this tool is known as the critical boundary index, CBI  .
Several other approaches for voltage collapse point identification and prevention has been widely studied in available literature  -  . The network component sensitivity model for identifying critical components of a large network is recently being applied for static voltage stability assessment. Critical node identification is an important tool for understanding network vulnerability; a node is considered to be critical if its role in maintaining credible system performances is vital such that when it is removed the system deteriorates greatly   . This approach has been widely used in several area of studies such as communication network, transportation network, smart grid network, financial risk-management, chemical and physical structural analysis etc.  -  . However, the use of this approach for voltage stability assessment of power system is limited and it is just gaining attention in recent time.
In  , a voltage stability index based on the sensitivity of local network and neighbouring network to the local voltage phasors was presented. This index is known as the equivalent node voltage collapse index (ENVCI) and it is derived based on equivalent system model (ESM). Voltage stability assessment and important nodes identification in transmission network using graph theory and network restructuring is been considered in recent research studies.  introduced a network response structural characteristics theory participation factor (NRSCTPF) sensitivity approach for critical nodes determination based on network response structural characteristics indices (NRSCIs) and eigenvalue decomposition. This approach was found to be less computationally intensive.
Most of the existing indices are based on approximation and these approximations are premised mainly on the relationship between the power system’s reactive power limit and bus voltage magnitude. More so, Most of the existing indices are based on minimum singular value of eigen values and Jacobian matrices  . However, the relationship between power system loading (real and reactive demand) and voltage stability can not be overlooked, especially by considering it from the structure of the power system network; and this can be inferred from the relationship between the bus voltage profile and the power transfer capacity of the connecting lines. Hence, the impact of active and reactive power variation on the point at which voltage collapse occurs is not well-elaborated in most of the existing reports. Also, the structural correlation between the buses and the connecting lines from neighbouring buses are not clearly emphasized. In this work, we have considered the importance of active and reactive power changes on the voltage stability of power system using the relationship between nodes (buses) and optimal stability margin of the connecting links (lines). The critical boundary index, CBI is a measure of the effect of active and reactive load variation on the stability margin of the transmission lines. The effectiveness of the proposed approach is tested on IEEE 14, 30, 57 and 118 bus test systems. The results are compared with other approaches in existing literature in order to validate the proposed method in terms of accuracy and computational requirement. The rest of the paper is organized as follows: concept of voltage stability margin estimation using CBI is discussed in Section 2. The concept of bus criticality determination using graph theory was discussed in Section 3. Simulation results are discussed in Section 4 and the study was concluded in Section 5.
2. Evaluation of Lines Voltage Stability Margin (CBI)
Relationship between voltage stability and power flow along each branch of a power system is analyzed using a simple two bus system shown in Figure 1.
The transmitting end terminal voltage is , receiving terminal voltage is , apparent power sent to the receiving bus is , and the line
Figure 1. Transmission line model.
impedance is . represent the complex conjugate.
separating the real and imaginary parts of the above equation, the following equations are obtained;
Solving for from Equations (3) and (4) with yields the following equation.
With the condition that , the receiving end voltage is obtained as
There is a point at which there is a limit to the amount of power that can be transmitted along the transmission line prior to system collapse. These stability condition is obtained by setting A to zero as shown in the following expression:
Hence, the locus of a point that corresponds to the voltage stability limit is obtained as:
(P,Q)-V characteristic which shows the combination of P-Q characteristics for different V is shown on Figure 2. The P-Q characteristic depicts the voltage stability limit as obtained from Equation (5).
From Figure 3, we obtain the voltage stability limit point with respect to the current operating point using Lagrangian undetermined multiplier method as shown.
The shortest distance between current operating point B and voltage
Figure 2. (P,Q)-V characteristics showing the voltage stability limit  .
Figure 3. (P,Q) characteristics showing the voltage stability limit.
instability point C is :
The minimum point on C that corresponds to the shortest distance from point B is thus obtained:
Partial differentiation of the above equation with respect to yields the following equation.
We obtain the values of by simultaneously solving Equations (12) to (14) using fsolve function in matlab. The optimal voltage stability margin known as critical boundary index, CBI of a line is evaluated as  :
The CBI is the measure of the available apparent power transfer limit of the transmission line with regards to voltage stability and as it approaches zero for any transmission line, the voltage stability level of the line deteriorates.
3. Estimation of Bus Criticality Index
The detail analysis of power systems as complex networks using graph theory is presented in  . As shown in Figure 4, each link (line) between any pair of nodes (buses) is described by two parameters; the electrical distance between
Figure 4. A 5 node system showing the link parameters under consideration.
nodes (impedance value) and the optimal voltage stability margin (CBI value). In this study, the voltage stability criticality index of a bus is determined based on the approach of weighted degree of node centrality    using the line optimal voltage stability margin as the line weight:
3.1. Electrical Degree of Bus Centrality
Degree of a node (bus) is the most basic indicator of the importance of the node to the performance of the network. Freeman in  describes the degree of a reference node of a network as the number of other nodes connected to that node. In order to measure this index with respect to a power system network, the electrical distance of connected links are used as the weight of each link and this is described as shown in the given equation:
where is the weighted degree of node centrality for node l, is the impedance value of each line connected to the reference node l and n is the total number of system nodes. The most critical node will have the highest value. The traditional electrical degree of centrality has the same value as . The traditional electrical degree of centrality is described below  :
where is the admittance matrix value at the diagonal corresponding to the node l. The higher the value of , the more critical is node l.
3.2. Proposed Bus Voltage Stability Criticality Indices
The approach for identifying critical buses is based on the voltage stability limit of transmission lines connected to the buses as defined below:
n is the total number of buses and the criticality of a bus l increases as the BVCI value reduces. Hence, the bus with the lowest BVSI value is most susceptible to voltage collapse phenomenon in the power system network.
4. Simulation Results
The derived voltage stability-based critical bus identification approaches were tested on IEEE 14, 30, 57 and 118 buses power systems. A comparative assessment with some other approaches found in literature is presented. To verify the accuracy of the proposed approaches, voltage collapse point was identified by monitoring the voltage magnitude at identified critical buses with heavy loading at all buses at constant power factor, accordingly:
where, i is the bus number and is the loading factor. Three simulation conditions are considered as given on Table 1 and the point at which the power flow solution diverges is the system collapse point under each case.
For the IEEE 14 bus system, buses 14, 12, 11, 8 and 3 are identified as the weakest buses as shown in Table 2. The electrical degree of centrality, Cd identified bus 8 as the weakest; BVSI identified bus 14 as the weakest bus. Figure 5 shows that bus 14 is most sensitive to continuous load increase as its voltage at the point of system collapse was reduced to 0.6858 pu at under heavy real loading, 0.6877 pu at under heavy reactive loading and 0.5774 pu at under simultaneous real and reactive power loading. Table 3 shows the values and ranking of the 5 most critical buses at the identified system collapse point under heavy real, reactive and both real and reactive loading
Table 1. Simulation conditions.
Table 2. Critical bus analysis of IEEE 14 bus at base load.
Table 3. Critical bus analysis of IEEE 14 bus at heavy load.
Figure 5. Voltage collapse point on IEEE 14 system. (a) Only active power increase; (b) Only reactive power increase; (c) Active and reactive power increase.
respectively. Contrary to the result of the continuous power flow that confirms bus 14 as the most critical under the three loading condition based on the voltage magnitude, the proposed index indicates bus 8 as the most critical under case 1 while bus 14 was identified as the most critical under both case 2 and case 3. The proposed index failure to accurately identify the critical bus under case 1 shows the importance of reactive power to voltage stability analysis.
For the IEEE 30 bus system, buses 26, 29 and 30 are the clear candidates for the most critical bus as presented in Table 4. Cd, BVSI identified bus 26 as the most critical closely followed by bus 30. From the result for IEEE 57 as shown in Table 5, buses 18, 19 and 31 are identified as the critical buses susceptible to voltage collapse. In Table 6, buses 87, 117 and 43 are clearly indicated as the critical buses from the result obtained from IEEE 118.
In Table 7, the results obtained from the proposed algorithms are compared with more established indices from the literature. They are mostly derived,
Table 4. Critical bus analysis of IEEE 30 bus at base load.
Table 5. Critical bus analysis of IEEE 57 bus at base load.
Table 6. Critical bus analysis of IEEE 118 bus at base load.
Table 7. Comparison of critical bus identification approaches at base load.
approximately, based on the relationship between the power system’s reactive power limit and bus voltage magnitude. Also, except for Cd and NRSCTPF, all the other indices are not exclusively based on the structural relationship of the power system components; hence the reasons for the difference in the results as the system size increases. For IEEE 14, the proposed algorithm agree with the existing algorithms as bus 14 is shown to be the most critical. For IEEE 30 bus system, bus 26, 30, 29, 27 are clear candidates for the bus responsible for voltage collapse under continuous loading. Although, different set of buses are identified by the proposed indices and the existing indices as the bus number increases, yet we can infer that bus 31 and bus 43 for IEEE 57 and IEEE 118 bus system, respectively are within the range of critical buses, as indicated by the aggregation of all the considered indices.
A formulation of power system voltage stability analysis model based on network structure is presented. The proposed index can perform the dual function of line voltage stability assessment and weakest bus determination. The proposed indices have shown the ability to yield accurate information relating to the closeness of a power system to the voltage collapse point. The computational approach is quite simple and straightforward and the computational requirement is less. The accuracy of the proposed index is tested by comparison with other existing approach and the proposed index agrees with existing indices, especially with the electrical degree of centrality.
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