A Fast Continuation Load Flow Analysis

International Journal of Energy Engineering (IJEE) Nov. 2012, Vol. 2 Iss. 4, PP. 126-136

- 126 -

A Fast Continuation Load Flow Analysis for an Interconnected Power System

D. Hazarika Assam Engineering College, Gauhati-13, Assam, India

[email protected]

Abstract- The paper provides an algorithm for a fast continuation load flow to determining critical load for a bus with respect to its voltage collapse limit of an inter connected multi-bus power system using the criteria of singularity of load flow Jacobian matrix. For this purpose, load flow Jacobian matrix of an interconnected multi-bus power system is transformed into a two by two elements Jacobian matrix with respect to a target/selected bus by incorporating the effect of all the other buses of the system on the target bus. The validity of the proposed method has been investigated for the IEEE 30 and IEEE 118 bus system.

Keywords- Voltage Stability; Load Margin; Voltage Collapse; Power System

List of symbols

N = Total number of buses in the system.

NG = Total number of generation buses in the system.

npv = Total number of PV buses (including slack bus) in the system.

Pi = Injected active power at ith bus.

Qi= Injected reactive power at ith bus.

PDi = Active power demand at ith bus.

QDi= Reactive power demand at ith bus.

PGi = Active power generated by ith generator.

QGi= Reactive power generated by ith generator.

Vi = Magnitude of voltage at ith bus.

δi = Angle of the bus voltage at ith bus.

Cos Øi = Load power factor of the ith load bus..

Gij + jBij = Element of Y-BUS matrix at ith row and jth column.

I. INTRODUCTION

Information/knowledge of load margin of a bus with respect to its voltage collapse limit constitutes important criteria for load pick-up step during power system restoration planning and operation. In a situation, when a load bus has to be energized through network building process (i.e., connecting the load to energized bus through transmission line(s)) and several energized buses are available for the connection, it becomes essential for power system planner/operator to select a proper energized bus for this purpose. Under such circumstances, the bus having highest load margin appears to be the best choice[1] and magnitude of load to be picked-up must be less than the load margin for the

connected bus. Voltage collapse of a bus is characterized by a slow variation in system operating point, due to increase in the loads, in such a way that the voltage magnitude gradually decreases until a sharp accelerated change occurs. It has been observed that voltage magnitudes in general, do not give a good indication of proximity to voltage stability limit [2]. In recent literature many voltage stability and voltage collapse prediction methods have been presented. Some of the important ones are: voltage collapse index based on a normal load flow solution[3, 4, 5], voltage collapse index based on closely located power-flow solution pairs [6], voltage collapse index based on sensitivity analysis [7], and minimum singular value of Newton - Raphson power flow Jacobian matrix [8, 9].

Continuation power flow analysis is based on locally parametrized continuation technique [10]. The key idea here is to avoid the singularity of the Jacobian by slightly reformulating the power flow equations, and applying the locally parametrized continuation technique [11, 12]. The method uses predictor-corrector approach and some parameter (continuation parameter) as additional condition to overcome ill conditioning of Jacobian matrix. Network partitioning technique has been employed in determination of load margin at a load bus [13]. A two parameter continuation technique has been proposed for evaluating branch outage contingencies introducing new branch parameter [14].

A voltage stability index VMPI (voltage margin proximity index) is proposed considering voltage limits, especially lower voltage limits [15]. A simple, computationally very fast local voltage-stability index has been proposed using Tellegen’s theorem [16]. It is easy to implement in the wide-area monitoring and control center or locally in a numerical relay. A new node voltage stability index called the equivalent node voltage collapse index ENVCI), which is based on ESM and uses only local voltage phasors, is presented [17]. Well-known ZIP model has been used to represent loads having components with different power to voltage sensitivities [18] and also, the choice of voltage stability index in the context of load modeling has been suggested.

These methods assess the closeness to the critical loading by looking into the voltage stability sensitivity indices or the smallest Eigen-value or singular value of load flow Jacobian matrix. The performance indices are used to provide the measure of the severity of static voltage stability problem. However, the disadvantages with the performance indices are that they cannot directly provide the margin of load at a load bus with respect to its voltage collapse limit.


- 127 -

The proximity of voltage collapse in a power system is indicated by the condition of Newton-Raphson power flow Jacobian matrix. At the point of voltage collapse, the determinant of the Jacobian matrix of a Newton-Raphson power flow analysis becomes zero. As such, the elements of load flow Jacobin matrix can be used to indicate voltage instability problem of a power system. This paper proposes a continuation load flow for determining the load margin of a bus with respect to its voltage collapse limit. For this purpose, load flow Jacobian matrix of an interconnected multi-bus power system is transformed into a two by two elements Jacobian matrix with respect to a target/selected bus. The elements of the transformed two by two matrix and the bus voltage of the target bus are used to develop an algorithm to determine load margin of a target bus of the system.

II. ANALYSIS OF VOLTAGE STABILITY PROBLEM FOR A MULTI-BUS INTER CONNECTED POWER SYSTEM

It is established that at the point of voltage collapse in a power system, the determinant of load flow Jacobian matrix becomes zero. To quantify the condition of singularity of load flow Jacobian matrix of a multibus system at the point of voltage collapse, it is transformed into a two by two matrix relating Δδk, ΔVk, ΔPk and ΔQk for a target kth bus. The change in voltage magnitudes and voltage phase angles of a multi-bus inter connected power system are related to change in real and reactive power injection at the buses as follows:

Where H, N, M, and L are the elements of Jacobian matrix

used for the load flow analysis of Newton Raphson method. Equation (1) can be expressed as:

For the convenience of representing the system (load flow)

equation, the following assumptions are made:

(i) bus Number 1 is considered as the slack bus; (ii) the system has npv number of voltage control buses which includes the slack bus also, i.e., the system has (npv−1) number of PV buses.

Now, using the elements of W, X, Y and Z of Equation (2) the change in voltage phase angle and voltage magnitude of a kth target bus can be expressed as:

The right hand side of the Equations (3) and (4) contain

change in real and reactive power terms for all buses including the kth target load bus. To relate Δδk and ΔVk of Equations (3) and (4) to only ΔPk and ΔQk, it is required to transform the

Load Flow Jacobian matrix of an interconnected system into a two by two matrix with respect to a target bus k. The paper presents a sensitivity analysis approach to relating change in real and reactive power of the target bus due to change in real and reactive power in other buses. Recognizing the fact that the change in bus voltage angle is primarily influenced by the change in real power injection at the buses [19], the change in real power injection at busbars can be expressed as [20]:

Similarly, for other buses the relation among voltage phase

angle and real power injections at the buses can be expressed as:

Equation (7) can be rearranged as given below:

where;

From Equation (6) and Equation (8) we have,


- 128 -

The Equation (10) can be rearranged as follow:

is a row matrix, therefore, [ΔP] values are to be [ץ]

calculated using pseudoinverse technique i.e. [20],

Equation (12) is solved for terms ΔPi for i = 2...N for

change in ΔPk and expressed as:

Similarly, recognizing the fact that the change in bus

voltage angle is primarily influenced by the change in real power injection at the buses [19], the change in real power injection at busbars can be expressed as:

The change in slack bus reactive power in terms of change in voltage phase angles can be expressed as:

Similarly, for other buses (excluding PV buses, where voltage magnitudes are assigned as fixed value) the relation among

voltage phase angle and real power injections at the buses can be written as:

Equation (16) can be rearranged as given below:

where,

From Equation (15) and Equation (17) we have,

The Equation (19) can be rearranged as follow:

[α] is a row matrix, therefore, [ΔQ] values are to be calculated using pseudoinverse technique i.e. [20],


- 129 -

Substituting values of ΔPi and ΔQi from Equations (13) and (22) in Equations (3) and (4) we have

where,

Therefore,

At the point of voltage collapse

The element of the transformed two by two matrix for the

kth target bus are function of network state namely bus voltages and their phase angles.

III. DETERMINATION OF LOAD MARGIN

In this section, a procedure for determination of load margin of the target kth bus has been developed using the elements of the modified two by two Jacobian matrix given by the Equation (28) and the bus voltage of the target bus. Equation (28) can be represented as two bus system with the target kth load bus, Y-Bus elements and an equivalent source as depicted in Figure 1.

Fig. 1 Equivalent two bus system with target kth bus, Y-Bus elements and an

equivalent source

Now, for this equivalent two bus system, the expression for real and reactive power injections at the target kth bus are as follows:

Where, Gkk, Bkk, GkG and BkG are the elements of

admittance matrix [Y] for the equivalent two bus system. But, the values of Gkk, Bkk, GkG and BkG depend on the system bus voltages and their phase angles. Therefore, while developing the mathematical model for load margin for the kth bus, parameters Gkk, Bkk, GkG and BkG are represented in terms of the elements of transformed two by to matrix represented by Equation (29).

The elements of Jacobian matrix for the equivalent two bus system can be expressed as

At the point of voltage collapse the determinant of the

Jacobian matrix becomes zero, i.e.,

The change in determinant value of the two bus equivalent

system of the multibus system with respect to change in Vk and δk can be expressed as:

Again, the expression for JD in terms of

The terms are derived utilizing equations from (32) to (35). It is to be ensured that

the terms contain only the elements of the modified two by two Jacobian matrix and the bus voltage of the target bus. Because, they are the known values for the transformed two bus system.


- 130 -

Now, applying values of of Equations (40), (41), (42) and (43) in Equation (39) we have,

Similarly, the term can be expressed as:

Again, utilizing equations from (32) to (35), we have

Processing Equation (33) and Equation (34), Gkk can be expressed as:

Processing Equation (32) and Equation (35), Bkk can be expressed as:

Now, applying values of of Equations (46), (47), (48), (49), (50) and (51) in Equation(45) we have,

The terms of Equations (44) and (52) are the elements of the modified Jacobian matrix of a multibus system represented by Equation (28) and Vk is the bus voltage of the kth target bus. Now, to relate change in real power injection at target kth bus to the change in determinant of the modified Jacobian matrix of a multibus system, variables Δδk and ΔVk of Equation (37) are replaced by ΔPk

and ΔQk as follows:

The system collapse occurs when JD becomes zero. Now, if is the determinant value corresponding to the defined

operating condition of the system, then the required change in ΔJD of Equation (53) to force JD zero is

and corresponding change in injection at kth bus will be

Therefore, load margin at kth load bus, i.e., the additional

load that can be supplied to the kth load bus to push it to the proximity of voltage collapse is:

Therefore, predicted critical load for kth load bus at the

point of voltage collapse is:

But, ΔPDk

margin is determined using linear relation between ΔJD and ΔPk. As such, load margin for the kth load bus (PDk

margin) would be more than the actual load margin of the bus. Further, for wide change in JD, the predicted load value will be considerably high compared to that of actual critical load value at the point of voltage collapse. Slight over prediction of load margin could mislead system planner and operator in the decision making related to allowing more load in the load buses which are somewhat close to voltage stability limit. Therefore, it is required to confirm the actual load margin or critical load of a load bus through an iterative load flow analysis using the load margin determined by the Equation (55). To ensure convergence of the load flow analysis in the proposed iterative procedure, it is required to normalize the predicted critical load value PDk

predt in such a way that the modified predicted critical load value remains below the actual critical load of the bus.

For this purpose, a reduction factor (RF) and a distance factor (DF) are used to normalize the predicted critical load value for the kth load bus as follows:

The distance factor is expressed as


- 131 -

The distance factor (DF) will be higher when prediction is

done for wide change of ΔJD, i.e., far from the proximity of voltage collapse limit. Thus, it will normalize the effect of over prediction due to wide change of ΔJD. Therefore, as the iterative procedure approaches the proximity of voltage collapse limit PDk

margin will become very small; thus, the term

will also approach to 1. When the value of JD becomes negative the iterative procedure has to be terminated.

IV. PROCEDURE FOR DISTRIBUTION OF NORMALIZED PREDICTED LOAD MARGIN TO THE GENERATORS

Equation (11) represents sensitivity relation for all the buses with respect to the target kth bus. It is justified to supply

the normalized predicted load margin ( ) of the kth target bus from the generators having higher sensitivity factor values. Therefore, to distribute the normalized predicted load margin of kth target bus, the sensitivity relation for generator buses with respect to target kth bus are utilized. Equation (11) can be expressed for generator buses only as follows;

Therefore, modified generation values are

Where, PGi

max is the maximum limit on generation for ith generating station.

V. ALGORITHM OF THE ITERATIVE PROCEDURE

Reduction factor (RF) is used to normalize the over prediction of load margin (which is determined using linear relation between ΔPDk

margin and ΔJD governed by Equation (53)) with the objective of ensuring convergence of load flow analysis used in the proposed iterative procedure. As such, load at a bus must be always less than actual critical load of the bus, i.e. to say that PDk

crt determined using Equation (57)

must be less than actual critical load of the bus. In case, PDkcrt

determined using Equation (57) becomes slightly more than actual critical load of the bus (due to improper selection of Reduction Factor (RF), the load flow analysis of the iterative procedure will not converge. To take into account of such a situation, the proposed algorithm is equipped with a step after the load flow analysis. This step reloads the loads and generations of the system of the previous iteration values and reduces the reduction factor (RF) as RF = RF/3.0 and load flow analysis is carried out again before proceeding to the other portion of the algorithm. This step helps in changing the value of reduction factor (RF) to ensure proper normalization (by Reduction Factor (RF)) of critical load governed by Equation (53). The algorithm of the proposed iterative procedure is as follows:

1. Initialize the load flow data for the system and set iteration count K = 0, RF = (0.3 to 0.5).

2. Conduct base case load flow analysis of the system. 3. Determine PDk

crt using Equation (57) and assign PDkK+1

= PDkcrt , QDk

k+1 = tan Øk PDkK+1 and distribute the additional

load (RF/DF) PDkmargin among the generating stations based on

defined criteria subjected to generation limits of the generating stations as described in Section-III. In case of reactive power limit violation, a PV bus has to be changed into a PQ bus by assigning reactive power at its limit.

4. Conduct the load flow analysis and Check for load flow convergence criteria. If load flow has converged, then go to Step-5.

Otherwise, reduce reduction factor RF =RF/1.5 1.5 and reload PDk

K+1 = PDkK, QDk

K+1 = QDkK+1, V [i]K+1 = V [i]K for i

= 1...N and δ[i]K+1 = δ[i]K for i = 1...N, also the generating stations output with their Kth iteration values and go to Step-3.

5. Check for |JD| < 0.0 go to Step-6, otherwise, set K = K + 1 and go to Step-3.

6. Stop.

VI. SIMULATON, RESULTS AND DISCUSSIONS

To verify the validity and applicability of the proposed method, simulations were carried out on IEEE 30 bus and IEEE 118 bus systems. The aim of the simulations was to examine the nature of change of JD for different load bus of IEEE 30 and IEEE 118 bus system with respect to change in load at the target bus. For this purpose, load at a target bus is increased gradually manually using an interactive load flow program and the change in JD values were recorded. It is found that the JD reduces parabolically with increase in load and at the point of voltage collapse it becomes zero. Figures 2 and 3 illustrate the variation of JD value of two bus equivalent system of the IEEE 30 bus system with respect to change in load at load buses 21 and 29 for load power factors 0.8 and 0.9. Figures 4 and 5 illustrate the variation of JD value of two bus equivalent system of the IEEE 118 bus system with respect to change in load at load buses 88 and 118 for load power factors 0.8 and 0.9.


- 132 -

Fig. 2 Variation of JD for load change at load bus 21 of IEEE 30 bus system

for load power factors pf=0.8 & 0.9





Fig. 5 Variation of JD for load change at load bus 118 of IEEE 118 bus system for load power factors pf=0.8 & 0.9

It is observed that JD value of two bus equivalent system of IEEE 30 and 118 bus systems have different initial values for different target buses, as such, JD value for a bus reflects the voltage stability characteristic of the bus.

The proposed algorithm is used to determine critical load of a bus with different reduction factors (RF). It has been found that RF values between 0.1 to 0.5 ensures convergence of load flow analysis in the iterative process for IEEE 30 and IEEE 118 bus system for any target bus of the system. But, with RF = 0.3 to 0.5, the iterative procedure terminates with less number of iterations. During iteration, the additional load assigned on the target (k) bus was distributed among all the generating stations based on their load contribution (subjected to their limits) as described in Section-IV. Continuation power flow analysis technique is also used to determine the critical load at the target buses with step size σ= 0.001 to ensure that it arrives the point of voltage collapse. In the continuation power flow analysis technique the point of voltage collapse is arrived when the predicted continuation parameter (Δλ) becomes negative. However, in this case predicted normalized load is not distributed among the generators. Therefore, the proposed algorithm is also used to determine the critical load at the target buses without distributing predicted normalized load among the generators.

Table 1 and Table 2 represent the simulation results for some of the buses of IEEE 30 bus system with RF = 0.4 with power factor 0.9 and 0.8 respectively. Table 3 and Table 4 represent the simulation results for some of buses of IEEE 118 bus system with RF = 0.4 with power factor 0.9 and 0.8 respectively. Tables represent the target bus selected for determination of critical load with respect to its voltage collapse limit, initial value of the determinant of the two by two Jacobian matrix for the target bus, initial value of predicted continuation parameter (Δλ), initial load at the target bus, final value of the determinant of the two by two Jacobian matrix ,final value of predicted continuation parameter (at the point of collapse), critical load for the target bus and remarks about the methods. The abbreviation used in the remarks column of the tables stand for:

PMWGD = Proposed method with generation distribution.


- 133 -

PMWOGD = Proposed method without generation distribution.

CLFWOGD = Continuation power flow without generation distribution.


- 134 -

It is observed that predicted continuation parameter (Δλ) of

IEEE 30 and 118 bus systems has different initial values for different target buses, as such, initial value of Δλ reflects the voltage stability characteristic of the target bus. Further, the


- 135 -

simulation results for the IEEE systems indicate that the predicted continuation parameter (Δλ) becomes slightly negative when the continuation power flow analysis is

terminated. It signifies that the systems are at the point of voltage collapse. Also, the determinant value of the transformed two by two matrix of the system becomes slightly negative at the point of voltage collapse for the IEEE systems. It is to be noted that sensitivity based distribution of normalized predicted load margin of a target bus among generators ensures higher critical load of a target bus in comparison to that of continuation power flow analysis without distribution of normalized predicted load margin among generators. It is due to the fact that the distribution of normalized predicted load is carried out among generators using sensitivity relation governed by Equation (61). This ensures that the higher portion of the normalized predicted load would be assigned to the generator having higher sensitivity factor i.e., the generator electrically closed to the target load bus.It has been observed that the distance factor

becomes close to 1, when the iterative process terminates. But, at the beginning of the iterative process it appears to be high depending upon change of ΔJD used for the prediction of load margin for the load bus. It normalizes the effect of over prediction of load margin due to wide change of ΔJD and ensures convergence of the load flow analysis of the system during the iterative process.

To examine the effect of limits on real and reactive power supply from generator(s) on the load margin, simulations were carried out on IEEE 30 bus system, since; it has only six generating stations. It is observed that Bus-29 is more sensitive to generator located at Bus-8. Different limits on real and reactive power supply are set this generator to examine the value of load margin at Bus-29.

Case I: The maximum real and reactive power limits on generator located at Bus-8 are set as PG8

max = 0.55 pu QG8max

= 0.34 respectively and the bus is treated as PQ bus. Table-5 represents the load margin of bus-29 of IEEE 30 bus system with limits mention above. TABLE 5: CRITICAL LOAD OF BUS-29 OF IEEE 30 BUS SYSTEM FOR LOAD PF 0.8 (APPLYING LIMITS ON REAL AND REACTIVE POWER

SUPPLY ON GENERATOR LOCATED AT BUS-8)

With the limits on Generator 8, the contribution of real and reactive power supply from the generator located at Bus-1, Bus-2, Bus-5, Bus-11, Bus-13 are PG1 =0.578443 pu QG1 = -0.03426 pu, PG2 =0.478458 pu QG2 = 0.252048 pu, PG5 =0.653096 pu QG5 =0.153756 pu PG11 =0.287925 pu QG11 =0.179893 pu and PG13 =0.287096 pu QG13 =0.255502 pu respectively.

Case II: The maximum real and reactive power limits on generator located at Bus-8 are set as as PG8

max = 0.55 pu QG8

max = 0.24 respectively and the bus is treated as PQ bus. Table-6 represents the load margin of Bus-29 of IEEE 30 bus system with limits mention above.

TABLE 6: CRITICAL LOAD OF BUS-29 OF IEEE 30 BUS SYSTEM FOR LOAD PF 0.8 (APPLYING LIMITS ON REAL AND REACTIVE POWER

SUPPLY ON GENERATOR LOCATED AT BUS-8)

With the limits on Generator 8, the contribution of real and

reactive power supply from the generator located at Bus-1, Bus-2, Bus-5, Bus-11, Bus-13 are PG1 =0.563591 pu QG1 =- 0.023318 , PG2 =0.477695 pu QG2 =0.274476 pu ,PG5 =0.652348 pu QG5 =0.166568 pu PG11 =0.287079 pu QG11 =0.183276 pu and PG13 = 0.286284 pu QG13 =0.257491 pu respectively.

Table 6 indicates that the load margin of Bus-29 of IEEE 30 bus system is found to be 0.312 pu without any violation on real and reactive power supply of all the six generators of the system. Whereas, with limits on real and reactive power supply on generator located at Bus 8, the load margin decreased for the bus as indicated by Table-5 and Table-6. For Case I, the load margin is found to be 0.299 pu with limits on Generator-8 as PG8

max = 0.55 pu QG8max = 0.34 pu. But, the

load margin decreases to 0.296 pu with limits on Generator-8 as PG8

max = 0.55 pu QG8max = 0.24 pu. It indicates that the both

real and reactive supply limits on generating station(s) influence the load margin of a bus. Further, case study Case I and Case II indicate that real and reactive supply limits on generating station(s) influence the contribution from the generating station(s) supplying loads within their limits.

Bifactorization matrix solution technique with optimal ordering of the sparse load flow jacobian matrix has been incorporated in the load flow analysis used for proposed method and that of continuation power flow analysis. The programs were executed on a P C with Pentium-4 processor having processor speed of 1.5 GHz and LINUX operating system. The simulation results show that the proposed method requires considerably less CPU time compared to that of continuation power flow analysis. Continuation power flow analysis requires on an average 2-3 times more CPU time than that of proposed method for IEEE systems.

VII. CONCLUSION

The paper proposed algorithm for a fast continuation load flow analysis to determine load margin of a target/selected load bus using the singularity condition of the load flow Jcaobian matrix. The load flow Jacobian matrix of an interconnected multi-bus power system is transformed into a two by two elements Jacobian matrix with respect to a target/selected load bus by incorporating the effect of all the


- 136 -

other buses of the system to the target bus. To incorporate the effect of other buses to the target bus, sensitivity relations have been formulated to relate change in injection at the 118 bus system indicated that the determinant of two bus equivalent system of a multibus system reduces with increase in load at the target bus and it becomes zero (near to zero) when load flow solution does not converge i.e., voltage collapse takes place. Therefore, it could be concluded that when the modified two by two matrix become singular, actual load flow Jacobian matrix also becomes singular.

But, it is observed that determinant value of two bus equivalent system of IEEE 30 and 118 bus systems have different initial values for different target buses, as such, it shows that the transformed two by two elements Jacobian matrix reflects the property/quality of the target bus.

The algorithm proposed for the determination of critical load of a bus with respect to its voltage collapse limit of a power system works for all buses of IEEE 30 and IEEE 118 bus system, as such, it could be used for any inter connected power system. The use of reduction factor (RF) and distance factor (DF) ensures convergence of the load flow analysis of the system during the proposed iterative procedure. These two factors effectively normalize the prediction of load margin, which is carried out using the linear relation between ΔJD and ΔPk governed by Equation (55). The simulation results show that the proposed method requires considerably less CPU time compared to that of continuation power flow analysis.

REFERENCES

[1] D. Hazarika and A.K. Sinha “Power system restoration: Planning and simulation”. Int. Journal of Electrical Power & Energy Systems, Vol- 25, 2003, pp. 209–218.

[2] Clark. H. K. “New challenges: Voltage stability” IEEE Power ENG. Rev (April 1990), pp. 33-37.

[3] P. Kessel and H. Glavitsch” Estimating the voltage stability of a power system.” IEEE Trans. on Power Delivery, Vol. PWRD-1, No.3 July/1986, pp. 346 - 354.

[4] Gubina F and Strmcnk B “Voltage collapse proximity index determination using voltage phaser approach.” IEEE Trans. Power System, Vol. 10, No. 2 1995, pp. 788 – 793.

[5] A.K. Sinha and D. Hazarika. “Comparative study of voltage stability indices in a power system”. Int. Journal of Electrical Power & Energy Systems, Vol- 22, No. 8, Nov. 2000, pp. 589–596.

[6] Y. Tamura, H. Mori and S. Lwanoto “Relationship between voltage instability and multiple load flow solutions in electrical system”, IEEE Trans. Vol. PAS-102, pp. 115-1125.

[7] Crisan and M Liu “Voltage collapse prediction using an improved sensitivity approach” Electrical Power System Research, 1994, pp. 181-190.

[8] P A. Lof, G. Anderson and D. J. Hill “Voltage stability indices of stressed power system”, IEEE Tran. PWRS, Vol. 8, No.1, 1993, pp, 326-335.

[9] A. Tiranuchit and R. J. Thomas, “A posturing strategy against voltage instability in electrical power systems” IEEE Tran. PWRS, Vol. 3, No.1, 1989, pp. 87-93.

[10] W. C. Rheinboldt and J.V. Burkhardt. “A locally parameterized continuation process”. 28 ACM Transactions on Mathematical Software, Vol. 9(2), June 1983, pp. 215-235.

[11] V. Ajjarapu and C. Christy. “The continuation power flow - a tool for steady state voltage stability analysis”. IEEE Transactions on Power Systems, Vol. 7(1), Feb. 1992, pp. 416-423.

[12] V. Ajjarapu, P. L. Lau, and S. Battula. “An optimal reactive power planning strategy against voltage collapse”. IEEE Transactions on Power Systems, Vol. 9(2) May, 1993, pp. 906-917.

[13] A. C. Souza and V. H. Quintana. “New technique of network partitioning for voltage collapse margin calculation”, IEE Proceedings Gerer. Transm. Distrib., vol. 141(6), Nov, 1994, pp. 630-636.

[14] A. J. Flueck and Qiu Wei “A new technique for evaluating the severity of branch outage contingencies based on two-parameter continuation.” Power Engineering Society General Meeting, Vol. 1, June, 2004, pp. 323-329.

[15] Y. Kataoka, M. Watanabe and S. Iwamoto, “A New Voltage Stability Index Considering Voltage Limits.” Power Systems Conference and Exposition, 2006. PSCE ’06. 2006, IEEE PES Oct./Nov, 29, 2006, pp. 1878 - 1883.

[16] I. Smon, G. Verbic and F. Gubina, “Local voltage-stability index using tellegen’s Theorem.” IEEE Transactions on Power Systems, Vol. 21, no. 3, 2006, pp. 1267-1275.

[17] Wanga Yang, Lib Wenyuan and Lua Jiping, “A new node voltage stability index based on local voltage phasors.” Electric Power Systems Research, Volume 79, Issue 1, January, 2009, pp. 265-271.

[18] Saikat Chakrabarti, “Static load modelling and voltage stability indices.” International Journal of Power and Energy Systems, 29(3). pp. 200-205.

[19] B. Stott and O. Alsac, “Fast decoupled load flow”, IEEE Transactions on Power System, Vol, 93, May/june, 1974, pp. 859-869.

[20] D. Hazarika and A K Sinha, “Standing phase angle reduction for power system restoration”, IEE Proccedings Genr. transm. Distrib., Vol. 145, No. 1, January, 1998, pp. 82-88.

Durlav Hazarika received the degree in Electrical Engineering from Jorhat Engineering College, Jorhat, Assam, India in 1983, the Master degree from Indian Institute of Technology (IIT) Bombay, Mumbai, India in 1986 and the Ph.D degree from Indian Institute of Technology (IIT) Kharagpur, Kharagpur, West Bangle, and India in 2000.

He is currently working as a Professor with the Department of Electrical

Engineering, Assam Engineering College, Guwahati, Assam, India.

Date post:	21-Jul-2016
Category:	Documents
Upload:	qbjohnnydp
View:	12 times
Download:	3 times

A Fast Continuation Load Flow Analysis

Documents