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Voltage security assessment of power system is an important and all-inclusive aspect of power system operation and preventive control actions. Fast and accurate detection of critical components of the power system is one essential approach for preventing the occurrence of voltage collapse phenomenon. Over the years, several approaches for voltage collapse point identification and prevention have been widely studied using the continuous power flow approach, minimum singular value of eigenvalues, Jacobian matrices, and power transfer concept. In this work, critical node (bus) identification based on power system network structure is proposed. In this approach, the power system is treated as a multidimensional graph with several nodes (buses) linked together by the transmission lines. An improved line voltage stability margin estimator which is based on active and reactive power changes in a power system is used as the weight of each transmission line and an adaptation of the degree of centrality approach is used to determine the criticality of the system buses. A comparative analysis with other bus voltage stability indices is presented to test the suitability of the proposed approach using the IEEE 14, 30, 57 and 118 bus test systems.

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 [

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 [

Several other approaches for voltage collapse point identification and prevention has been widely studied in available literature [

In [

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 [

Relationship between voltage stability and power flow along each branch of a power system is analyzed using a simple two bus system shown in

P k − j Q k = ( V k ∠ δ k ) * V l ∠ δ l − V k ∠ δ k r l k + j x l k (1)

The transmitting end terminal voltage is V l , receiving terminal voltage is V k , apparent power sent to the receiving bus is S k = P k + j Q k , and the line

impedance is z ^ l k = r l k + j x l k . * represent the complex conjugate.

P k r l k + x l k Q k + j ( P k x l k − r l k Q k ) = V l V k cos ( δ l − δ k ) − j V l V k sin ( δ l − δ k ) − V k 2 (2)

separating the real and imaginary parts of the above equation, the following equations are obtained;

P k r l k + x l k Q k + V k 2 = V l V k cos ( δ l − δ k ) (3)

P k x l k − r l k Q k = − V l V k sin ( δ l − δ k ) (4)

Solving for V k 2 from Equations (3) and (4) with ( sin 2 δ + cos 2 δ = 1 ) yields the following equation.

V k 2 = − ( r l k P k + x l k Q k − V l 2 2 ) ± ( r l k P k + x l k Q k − V l 2 2 ) 2 − ( r l k 2 + x l k 2 ) ( P k 2 + Q k 2 ) (5)

With the condition that V k > 0 , the receiving end voltage V k is obtained as

V k = − ( r l k P k + x l k Q k − V l 2 2 ) ± A (6)

where

A = ( r l k P k + x l k Q k − V l 2 2 ) 2 − ( r l k 2 + x l k 2 ) ( P k 2 + Q k 2 )

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:

( r l k P k + x l k Q k − V l 2 2 ) 2 − ( r l k 2 + x l k 2 ) ( P k 2 + Q k 2 ) = 0 (7)

Hence, the locus of a point that corresponds to the voltage stability limit is obtained as:

C ( X , Y ) = ( r l k P k + x l k Q k − V l 2 2 ) 2 − ( r l k 2 + x l k 2 ) ( P k 2 + Q k 2 ) (8)

(P,Q)-V characteristic which shows the combination of P-Q characteristics for different V is shown on

From

The shortest distance between current operating point B and voltage

instability point C is d ( X , Y ) :

d ( X , Y ) = ( ( X − P o ) 2 + ( Y − Q o ) 2 ) 1 2 (9)

The minimum point on C that corresponds to the shortest distance from point B is thus obtained:

F ( X , Y , λ ) = d ( X , Y ) − λ C ( X , Y ) (10)

F ( X , Y , λ ) = { ( X − P o ) 2 + ( Y − Q o ) 2 } 1 2 − λ { ( r l k X + x l k Y − V l 2 2 ) 2 − ( r l k 2 + x l k 2 ) ( X 2 + Y 2 ) } (11)

Partial differentiation of the above equation with respect to X , Y , λ yields the following equation.

( X − P o ) { ( X − P o ) 2 + ( Y − Q o ) 2 } − 1 2 − 2 λ { ( r l k X + x l k Y − V l 2 2 ) r l k + ( r l k 2 + x l k 2 ) X } = 0 (12)

( Y − Q o ) { ( X − P o ) 2 + ( Y − Q o ) 2 } − 1 2 − 2 λ { ( r l k X + x l k Y − V l 2 2 ) x l k + ( r l k 2 + x l k 2 ) Y } = 0 (13)

( r l k X + x l k Y − V l 2 2 ) 2 + ( r l k 2 + x l k 2 ) ( X 2 + Y 2 ) = 0 (14)

We obtain the values of X , Y , λ 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 [

Δ P l k = X − P o (15)

Δ Q l k = Y − Q o (16)

C B I l k = Δ P l k 2 + Δ Q l k 2 (17)

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.

The detail analysis of power systems as complex networks using graph theory is presented in [

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 [

Degree of a node (bus) is the most basic indicator of the importance of the node to the performance of the network. Freeman in [

C d ( l ) = ∑ k ‖ y ^ l k ‖ n − 1 ; ( l ≠ k ) (18)

where C d ( l ) is the weighted degree of node centrality for node l, y ^ l k 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 C d value. The traditional electrical degree of centrality has the same value as C d ( l ) . The traditional electrical degree of centrality is described below [

C d Y ( l ) = ∑ k ‖ Y l , l ‖ n − 1 (19)

where Y l , l is the admittance matrix value at the diagonal corresponding to the node l. The higher the value of C d Y ( l ) , the more critical is node l.

The approach for identifying critical buses is based on the voltage stability limit of transmission lines connected to the buses as defined below:

B V S I ( l ) = ∑ k ‖ C B I l k ‖ n − 1 ; ( l ≠ k ) (20)

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.

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:

P ′ l o a d i = P l o a d i × σ (21)

Q ′ l o a d i = Q l o a d i × σ (22)

where, i is the bus number and σ is the loading factor. Three simulation conditions are considered as given on

For the IEEE 14 bus system, buses 14, 12, 11, 8 and 3 are identified as the weakest buses as shown in _{d} identified bus 8 as the weakest; BVSI identified bus 14 as the weakest bus.

Case | Simulation condition |
---|---|

1 | Increase only Active power (Equation (21)) |

2 | Increase only Reactive power (Equation (22)) |

3 | Increase Active and Reactive power (Equations (21), (22)) |

C_{d} | BVSI | |||
---|---|---|---|---|

Rank | Bus | Value | Bus | Value |

1 | 8 | 0.4367 | 14 | 0.0991 |

2 | 14 | 0.4559 | 8 | 0.1028 |

3 | 12 | 0.5292 | 12 | 0.1239 |

4 | 11 | 0.7173 | 11 | 0.1640 |

5 | 3 | 0.7968 | 3 | 0.1815 |

6 | 13 | 0.9831 | 13 | 0.2230 |

7 | 10 | 1.2203 | 10 | 0.2818 |

8 | 6 | 1.4740 | 6 | 0.3359 |

9 | 7 | 1.5120 | 7 | 0.3497 |

10 | 1 | 1.5706 | 1 | 0.3546 |

Case 1 | Case 2 | Case 3 | |||
---|---|---|---|---|---|

Bus | Value | Bus | Value | Bus | Value |

8 | 0.0028 | 14 | 0.0421 | 14 | 0.0258 |

14 | 0.0429 | 8 | 0.0553 | 8 | 0.0301 |

11 | 0.0457 | 11 | 0.0735 | 11 | 0.0597 |

10 | 0.0603 | 12 | 0.0865 | 12 | 0.0780 |

7 | 0.0896 | 13 | 0.1281 | 1 | 0.0906 |

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 _{d}, BVSI identified bus 26 as the most critical closely followed by bus 30. From the result for IEEE 57 as shown in

In

C_{d} | BVSI | |||
---|---|---|---|---|

Rank | Bus | Value | Bus | Value |

1 | 26 | 0.0754 | 26 | 0.0179 |

2 | 30 | 0.1178 | 30 | 0.0275 |

3 | 29 | 0.1406 | 29 | 0.0336 |

4 | 11 | 0.1658 | 11 | 0.0416 |

5 | 14 | 0.2372 | 14 | 0.0621 |

6 | 13 | 0.2463 | 13 | 0.0663 |

7 | 23 | 0.2677 | 23 | 0.0674 |

8 | 25 | 0.3127 | 25 | 0.0779 |

9 | 16 | 0.3215 | 16 | 0.0829 |

10 | 27 | 0.3602 | 27 | 0.0882 |

C_{d} | BVSI | |||
---|---|---|---|---|

Rank | Bus | Value | Bus | Value |

1 | 31 | 0.0497 | 18 | 0.0136 |

2 | 19 | 0.0561 | 19 | 0.0168 |

3 | 20 | 0.0565 | 31 | 0.0192 |

4 | 57 | 0.0705 | 20 | 0.0196 |

5 | 42 | 0.0870 | 57 | 0.0250 |

6 | 18 | 0.0973 | 42 | 0.0283 |

7 | 25 | 0.1031 | 25 | 0.0313 |

8 | 30 | 0.1035 | 54 | 0.0368 |

9 | 54 | 0.1225 | 41 | 0.0381 |

10 | 41 | 0.1350 | 30 | 0.0387 |

C_{d} | BVSI | |||
---|---|---|---|---|

Rank | Bus | Value | Bus | Value |

1 | 87 | 0.0408 | 87 | 0.0094 |

2 | 117 | 0.0594 | 117 | 0.0133 |

3 | 43 | 0.0832 | 43 | 0.0192 |

4 | 72 | 0.0884 | 72 | 0.0192 |

5 | 107 | 0.0897 | 107 | 0.0196 |

6 | 86 | 0.1077 | 111 | 0.0252 |

7 | 111 | 0.1087 | 86 | 0.0256 |

8 | 53 | 0.1192 | 53 | 0.0261 |

9 | 33 | 0.1235 | 33 | 0.0280 |

10 | 98 | 0.1239 | 112 | 0.0292 |

Indices | IEEE 14 | IEEE 30 | IEEE 57 | IEEE 118 |
---|---|---|---|---|

BVSI | 14, 8, 12, 11 | 26, 30, 29, 11 | 18, 19, 31, 20, 57 | 87, 117, 43, 72 |

C_{d} [ | 8, 14, 12, 11 | 26, 30, 29, 11 | 31, 19, 20, 57, 42 | 87, 117, 43, 72 |

ENVSI [ | 14, 4, 9, 13 | 30, 29, 27 | 33, 32 | 44, 23, 43 |

L-index [ | 14, 9, 10, 8, 11 | - | 31, 33, 32, 30 | 44, 45, 43 |

P-Index [ | 14, 9, 10, 5, 4 | - | 31, 33, 32, 30 | 44, 22, 45 |

CMAT [ | - | 30, 27, 29, 28 | 31, 33, 32, 30 | - |

IMAT [ | - | 27, 30, 29, 26 | 31, 33, 32, 30 | - |

VCPI [ | - | 27, 30, 29, 23 | 31, 33, 32, 30 | - |

PVDBI [ | - | 27, 30, 29, 23 | 31, 33, 32, 30 | - |

NRSCTPF [ | 14, 10, 9, 12, 13 | - | 33, 31, 32, 30 | - |

approximately, based on the relationship between the power system’s reactive power limit and bus voltage magnitude. Also, except for C_{d} 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.

The authors declare no conflicts of interest regarding the publication of this paper.

Adewuyi, O.B., Danish, M.S.S., Howlader, A.M., Senjyu, T. and Lotfy, M.E. (2018) Network Structure-Based Critical Bus Identification for Power System Considering Line Voltage Stability Margin. Journal of Power and Energy Engineering, 6, 97-111. https://doi.org/10.4236/jpee.2018.69010