Identification and Ranking of Weak Buses Using Modified Counterpropagation Neural Network 学术搜索 Pandit, M.;
Srivastava, L.;
Singh, V.Power India Conference, 2006 IEEE318-324April 10-12, 2006学术搜索
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Identification and Ranking of Weak Buses Using Modified Counterpropagation Neural Network Manjaree Pandit, Laxmi Srivastava, Member, IEEE, and Vijay Singh
Abstract--The task of maintaining power system security in the recently deregulated environment is gigantic with uncertain and diverse power transactions and benefit based operational schemes. Monitoring of the system and predicting possible voltage collapses can be accomplished by defining suitable indices that can trigger the preventive or corrective control actions when predefined thresholds are reached to make the system insecure. This paper proposes a modified counterpropagation neural network (MCPN) enhanced by the concept of neo fuzzy neurons for static voltage security assessment. The proposed method works by identifying weak buses on the basis of available reactive margin and ranks them in order of their sensitivity to voltage collapse. A novel feature selection method is used to reduce the dimension and training time of the neural network. The proposed method has been tested on a practical 75-bus Indian system and is found to identify weak buses correctly even for previously unseen operating conditions, instantaneously.
by defining suitable indices that can trigger the preventive or corrective control actions when predefined thresholds are reached to make the system insecure. Timely corrective actions can be taken only if an advance warning can be issued to the system operator at the energy control center. Several indices have been proposed for analyzing voltage stability of power system. Eigen value [1], singular value [2], and sensitivity methods [3], and indices based on worst-case reactive margin [4] are available in literature. The classical eigen/singular value approaches give reliable indications on the areas more affected by voltage problems [1,5] but the dominant eigen/singular value shows very non-linear behavior near the collapse point, especially when the reactive margin of generators vanishes. Higher order indices have also been proposed to increase accuracy and refine the linear estimates. Chowdhury and Taylor [6] have shown that power flow based approaches are suitable for initial screening but decision regarding operating limits should be taken by using time simulation. All the methods discussed above require comparatively large computation time and do not offer quick screening of outages and hence are not suitable for on-line applications. It is well known that the indices based on linearization of the load flow, are suitable for assessing voltage collapse condition under slow variation of load only. However, they offer quick screening and ranking of contingencies. More detailed time domain analysis may be done later to evaluate the selected cases and find voltage stability margins more accurately [6]. Artificial neural networks (ANN) have been proposed for static and dynamic voltage security/stability evaluation [7-10] as they have the ability to produce the value of the severity indices accurately and almost instantaneously under uncertain and changing load and generation conditions. Reference [7] employs multi-layer perceptron networks trained by backpropagation (BP) algorithm. Reference [8] applied Kohonen’s self-organizing feature map, [9] proposed a counter propagation neural (CPNN) network, [10] applied a parallel self-organizing hierarchical neural network (PSHNN) and in reference [11] a hybrid neural network was proposed for quantifying and analyzing voltage security problems. Integrated fuzzy-neural approaches have also been employed for the assessment of voltage security [12,13]. This paper proposes a modified counterpropagation neural
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Index Terms— Coherency based clustering, feature selection, modified counterpropagation neural network, neo fuzzy neuron sensitivity index, voltage security assessment, weak bus identification
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I. INTRODUCTION
T
HE problem of voltage instability and subsequent system collapse has been experienced by growing number of utilities due to the continual increase in demand for electric power. This has forced the utilities to operate their systems closer to the stability limits. Monitoring and predicting possible voltage collapses in the system can be accomplished The authors sincerely acknowledge the financial support provided by Department of Science and Technology (DST), New Delhi, India under research project entitled Integrated fuzzy neural network approach for power system voltage security assessment of Madhya Pradesh State Electricity Board System vide letter no. SR/S3/EECE/14/2003-SERC dated 11/5/2004. and thank the Director M.I.T.S., Gwalior for providing facilities for carrying out this work. M. Pandit (e mail:
[email protected]) and L. Srivastava (e mail:
[email protected]) are with Department of Electical Engineering, MITS, Gwalior, India and Vijay Singh(
[email protected] ) is with Department of Electrical Engineering, RJIT, Tekanpur, India).
0-7803-9525-5/06/$20.00 ©2006 IEEE.
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network (MCPN) enhanced by the concept of neo fuzzy neurons [14,15] for static voltage security assessment. The proposed method works by identifying weak buses on the basis of their sensitivity indices for any given loading condition, or contingency; ranking them in order of their sensitivity to voltage collapse. The MCPN produces the sensitivity indices of the load buses, based on which the buses may be classified into different sensitivity classes, on-line, for any given system condition The performance of the neural network depends on the input features selected for its training. All the available parameters of a large scale network can not be applied as inputs to ANN as they would increase the dimension of the ANN and make its response slow during testing; also large training time would be required for such a case. A number of different methods for feature selection of ANN have been proposed. The application of real-time monitoring and control needs very fast processing of system data. The direct measurement of system state vector (V and δ) is being used as an alternative to the conventional indirect methods which require iterative calculations of the state vector from the power flow results and are hence quite slow. Begovic and Phadke [16] proposed a coherency based measurement clustering technique for reducing the number of measurements of the elements of the state vector, for monitoring the proximity of the system state to voltage instability. An approximate Jacobian was constructed by assuming that all the elements of the state vector from one cluster have the same value, as the one representative measurement taken from that cluster. In this paper, the above approach is extended by including a number of load scenarios in the clustering process. As the MCPN will now require voltages at a few buses only, for the assessment of voltage security, there will be a saving in the cost of hardware (such as phasor measurement units) installed at the measurement sites and communication burden on the central processing computer would also reduce. It has been shown that the MCPN trains very fast, even faster that CP which makes it a very attractive for on-line monitoring and control of power system voltage security.
competitive
layer
by
weight
vector
u k = [u k 1 ,..., u kj ,..., u kN ] . CP can approximate an inputoutput relationship
( y = f ( x)) by adjusting these weight
vectors through the following two stage learning algorithm. The objective of Kohonen training is to bring the weight vector close to the input data so as to assign similar patterns to one neuron. It is a common practice to randomize the weights to small numbers before starting the training of a neural network. But when a random weight vector was selected for CPNN, serious training problems were observed, with most of the Kohonen neurons producing zero output. The most desirable solution is to distribute the weight vectors according to the density of the input vectors, that are to be clustered in a unsupervised manner. In this work, the convex combination method [19] is employed for weight initialization, which makes all the weight vectors coincident, having a value equal to
1
n
where n equals the number of inputs. The training is
continued as per the following steps till the mean squared error becomes very small. When an input vector x is applied to the input layer, the units which has the closest weight vectors to the input vector is defined as winner unit. The weight vector of the winner unit and its neighboring units are updated by:
w new = w old + α ( x − w old j j j ),
(1)
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where, α is a learning rate which decreases with learning iterations. The procedures from i) to ii) are repeated for all the training patterns till stable clusters are formed.
www.libsou.com Y1
Y2 ………….Ym
Grossberg Layer
Ujk competitive layer
Z1
II. METHODOLOGY In this section, the structure and learning algorithm of conventional counterpropagation (CP)[15] and neo fuzzy neuron (NFN) [17] are illustrated. The NFN module replaces the Grossberg layer in the conventional CP and has been found to enhance its capabilities tremendously. A. Counterpropagation Fig. 1 shows the structure of the conventional CPNN. It consists of three layers, an input layer, a competitive layer and an output layer including n, N and m units, respectively. The unit j in the competitive layer is connected to the units in the input layer by weight vector
w j = [ w j1 ,...w ji ,..., w jn ] . The
unit k in the output layer is connected to the units in the
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Estimated SI Values
Wij Input Layer
X1
X2 ........Xn
Selected Bus Voltages and realreactive flows
Target SI of critical buses
Fig. 1. CPNN for Weak bus ranking.
The set of input vectors are classified by this learning. After this training, the weight vector w j is fixed and the weight vectors
u k between competitive and output layers are
adjusted as follows using supervised learning.
1.
The input vector
2.
The output
x is applied to the input layer.
z j of the jth neuron of the competitive
layer is calculated such that for the winner unit c it is z c = 1 and the outputs of other units are
μ11
μ12
μi1
μ ij
μi1+1
μ in
Grade
z j = 0( j ≠ c) . 3.
The outputs of the units in the output layer
y k is
calculated by: N
y k = ∑ u kj z j
(2)
xmin
0 Fuzzy segment
j =1
4.
To reduce the error between the target output
tk
and Yk, weight vectors are updated by gradient descent method as follows:
u where
u
new kj
new kj
= u kjold + η (t k − y k ) z j
and
u
old kj
(3)
are weights after and before
updating, respectively, and
η is the learning rate.
5.
The steps from 1 to 4 are repeated for all training patterns till desired performance is achieved. In the CP, the input vectors are classified to N categories and the output of the network is calculated by weighted sum of vectors of that particular category. But, the output is linear weighted sum so mapping heavy nonlinearity is not possible.
f1
x2
f2
B. Neo Fuzzy Neuron Fig. 2 shows a structure of the conventional NFN with m inputs and one output. A neo fuzzy neuron [17] has non-linear synaptic transfer characteristics where the synapse is realized by a set of fuzzy implication rules of the type If Xi is Aij then the output is wij Aij is a fuzzy set whose membership is
μ ij for the jth rule.
The membership functions are arranged to cover the full range of an input parameter xi. Each membership function in the antecedent is triangular and assigned to be complementary to each other (Fig. 4) such that an input parameter xi activates
μ ij and
μ ij +1 simultaneously. When a nonlinear synapse shown in Fig. 3 is described by L fuzzy if then rules, i -th nonlinear synapse
is represented by: L
f i ( xi ) = ∑ μ il ( xi ) wil
www.libsou.com ∑ y
(4)
i =1
μ il ( xi ) and wil , are membership value and singleton of l -th rule in i -th nonlinear synapse. Where
fi
. .
The output NFN can be represented by:
fm
N
y = ∑ fi ( xi ) ,
.
Membership function
By adjusting singletons wil , NFN can approximate an
Weight
μ i1 wi1
wi 2
wij
μil wil
Fig. 3. Structure of nonlinear synapse fi.
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input-output relationship. The singleton is updated based on a gradient descent method as follows:
Weighted sum
μi 2 μ ij
(5)
i =1
Fig. 2. General structure of a neo- fuzzy neuron.
xi
Fig. 4. Complementary triangular membership functions in the antecedent.
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xi
xm
xi
only two neighboring membership functions,
Nonlinear synapse
x1
xmax
∑
wilnew = wilold + β (t − y ) μ il ( xi ) f i ( xi )
(6)
wilnew and wilold are singletons after and before updating, respectively. t is the desired input and β is the Where
learning rate. In the NFN, an output is calculated by sum of the result of fuzzy inferences for corresponding inputs, thus number of fuzzy if-then rules can be reduced compared to multi-inputs fuzzy inference. An NFN however, cannot be applied to a system, which has correlation between inputs as the output is given by sum of nonlinear functions of inputs.
C. Modified Counterpropagation Neural Network In the ordinary Counterpropagation Neural Network the output is calculated by the weighted sum of outputs of the units in the competitive layer, so the network has poor mapping ability. In the MCPN, the output is obtained by summing the outputs of non-liner functions represented by fuzzy synapses as the output layer consists of neo fuzzy neurons in place of ordinary neurons. A modified CP (MCP) [14] employing NFN in place of the Grossberg layer is found to be very efficient for order to cope with the problems of the conventional CP and NFN described previous section. The modified CP consists of three layers, an input layer, a competitive layer and an output layer including n, N, m units, respectively, same to the conventional CP. The unit j in the competitive layer is connected to all the units in the input layer by weight vector w j . The unit k in the output layer is a neo fuzzy neuron whose inputs are the outputs of the units in the competitive layer. In the learning of the modified CP, the weight vectors between the input and the Kohonen layer are decided by competetive learning law and then the nonlinear synapses in the NFN are trained. The outputs of the unit j in the competitive layer
z j is calculated by
z j = exp(−γ x − w j ) where γ is a parameter for similarity.
(7)
The singletons are updated based on a gradient descent method as follows:
of applying all available information of the sixty buses. The coherency based clustering algorithm was applied to the 60 Fig. 6 Performance of Modified CPNN load buses of the 75bus Indian system. The following algorithm is applied to group coherent buses into one cluster. A. Coherency Criteria A coherency relation on load buses may be defined as
C =
{(i , j ) f (i ,
j,V , δ
Where
(8)
new old u kjl and u kjl are singletons of l -th if then rule of
K
Vector ε ∈ R is a vector of criteria thresholds chosen for a particular coherency relation f. A criterion, which is suitable for monitoring voltage security of power system is given below.The inverse Jacobian matrix [A B]t shown in (3) relates the changes of phase angles and voltage magnitudes to the changes in system loading [ Δ ψ ] which are causing these K
changes.
⎡ ∂δ ∂δ ∂δ ⎤ ⎢ ∂P ∂P ∂Q ⎥ L ⎢ G L ⎥ ⎢ ⎥ ∂ ∂ ∂ φ φ φ ⎡Δδ ⎤ ⎢ Δ φ ⎥ = ⎢ ∂PG ∂PL ∂Q L ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎣⎢ ΔV ⎥⎦ ⎢ − − − − − − − ⎥ ⎢ ∂V ∂V ∂V ⎥ ⎢ ⎥ ⎣⎢ ∂PG ∂PL ∂Q L ⎦⎥
⎡ Δ PG ⎤ ⎢ Δ P ⎥ = ⎡ A ⎤ [ΔΨ ] ⎢ L ⎥ ⎢⎣ B ⎥⎦ ⎢⎣ Δ Q L ⎥⎦
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j -th synapse of k -th output of the NFN after and before updating, and μ kjl is a grade of l -th if then rule of j -th synapse of k -th output of NFN.Parameters β , t k and
yk
represent learning rate, target output and output of k -th output of the NFN. In the proposed model, the synapses represented by the coefficient
u kj in the conventional CP are substituted with the
nonlinear functions
f kj (.) , which results in tremendous
increase in its modeling ability. The input vectors are mapped into the competitive layer output vector space, so the proposed method can be applied to a system, which has a correlation between inputs. Thus limitations of CP and NFN are overcome in the MCPN. III. FEATURE SELECTION The coherency between load buses, with respect to voltage dynamics [16] and graph theory has been used to form clusters of load buses. Then only one representative bus voltage from each cluster along with respective real and reactive powers and total real and reactive powers are used as input features instead
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(9)
where f(.) → R is the criterion function for selection of coherent pairs of load buses and it is symmetrical with respect to bus i and bus j. f (i, j ,V , δ ) = f ( j , i,V , δ ) (10)
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new old u kjl = u kjl + ηMCP(t k − y k ) μ kjl ( z j )
)≤ ε }
(
(11)
∀ i ∈ W K ∀ j ∈ W K ∀ ΔΨ ]) Δ Vi − Δ V j ≤ ε which is equivalent to
(r
i
− r j )ΔΨ
≤ ε
)
(12)
If rows ri and rj are very close then (4) holds good for all load changes and an m x m matrix C can be constructed whose elements satisfy the following criterion.
〈 ri , r j 〉 ⎧ ≥ ε 1 and ri − r j ⎪1, if C ij = ⎨ ri . r j ⎪ ⎩0, otherwise
≤ ε2 (13)
The coherency relation may be defined as
C =
{(i ,
j )C
ij
= 1
}
(14)
IV. WEAK BUS IDENTIFICATION AND RANKING USING MCPN A. Feature selection To find out the weak buses of the test system, 70 patterns were generated by changing the load from 80-120% of the base case for the 75-bus practical system [18]. The coherency based clustering algorithm (equations 9–14) was applied to the 60 load buses of the 75-bus Indian system and it was found that the number of on-line measurements could be reduced considerably to almost one eighth. Best results were obtained with ε1=0.455 and ε2=0.015, for which the load buses were grouped into 8 coherent clusters. Increasing the number of voltage inputs beyond 8 did not improve the performance of the MCPN any further. The result of clustering of 60 load buses is shown in Table I. It can be seen that cluster number 3 groups as many as 31 coherent buses in one cluster. Representative buses are selected from each cluster and real and reactive loads at these buses are employed as inputs to train the MCPN, along with total real and reactive loads. 1
Sensitivity Index
0.8 0.6 0.4
Target Value(Bus 24) ANN value (Bus 24) Target Value (Bus 25) ANN Value (Bus 25) Targer value(Bus27) ANN Value(Bus 27) Target value(Bus 30) ANN Value(Bus 30)
included in the list of weak buses for on-line classification by the MCPN. IV. The values of the normalized sensitivity index for identified buses are classified into five sensitivity classes as given in Table II, and ANN is trained for producing the sensitivity indices, based on which the selected buses are classified into different sensitivity classes. C. Architecture of MCPN The MCPN (26-12-8) was trained for producing the sensitivity index of these buses using 55 patterns. The remaining 15 patterns were kept aside for testing the performance of the trained neural network. The value of α is taken as 0.7 and β is 0.001. The similarity measure has been chosen as 1.0 for the case under study. The neo fuzzy neuron as the output layer has been employed with nine rules and triangular membership functions. D. Results and Discussion Different values of Kohonen layer neurons were tried and it was found that 12 numbers of neurons were giving optimum performance for this problem. Table III presents the comparison of performance of CP and MCPN after 50 iterations with a learning rate of 0.001. Table IV shows that MCPN ranks the buses correctly and the ranking order matches with the rank found from conventional load flow method. TABLE I RESULTS OF COHERENCY BASED FEATURE SELECTION
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0.2
13
10
7
4
1
0 Testing Samples
No of buses
Names of Buses
1 2 3 4 5 6 7 8
13 4 31 1 7 1 1 2
16,17,19,20,35,36,37,40,46,48,49,64,66 18,47,51,52 21-33,38,39,43,53-57,60-63,6567,69,71,73,75 34 41,42,45,50,58,59,74 44 68 70,72
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Fig. 5. Estimation of Sensitivity Index for Bus No. 24, 25, 27, 30.
B. Weak Bus Identification and Classification The real and reactive loads at a bus were increased by 5% of their base values and their Sensitivity Index was determined for all the generated base cases using conventional NewtonRaphson method.. Bus numbers 24, 25, 27, 30, 52, 55, 57 and 60 were found to be most sensitive for voltage collapse and hence were selected for on-line monitoring and ranking. The steps followed to find out the weak buses of the system using MCPN are: I. For all the generated patterns, real and reactive loads at a bus are increased by a small fraction of the base value in steps till the load flow solution is no longer feasible. The procedure is repeated for all the load buses of the system. II. The bus that supports a higher p.u. increase in load, is assumed to be less sensitive to voltage collapse as compared to a bus which can support a lesser value of p.u. increase in its base load. Therefore the bus sensitivity index is defined as, SI = 1 − p.u. increase in base load which can be supported. III. Buses that have a high SI are identified and
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Cluster No
TABLE II CLASSIFICATION OF SENSITIVITY INDEX FOR BUS SENSITIVITY RANKING
Class
I
II
III
IV
V
SI
0.9 - 0.8
0.8 – 0.7
0.7 – 0.55
0.55 – 0.35
0.35-0.1
TABLE III COMPARISON OF CP AND MCPN
S.No
Kohone n Nodes
1 2 3 4
08 10 12 15
Training Error(CP ) 0.01633 0.02130 0.02075 0.02467
Testing Error(CP ) 0.01552 0.0221 0.02136 0.02509
Training Error(MCP ) 0.0019 0.0017 0.0015 0.0013
Testing Error(MCP ) 0.0032 0.0024 0.0017 0.00262
Mean Squared Error
The trained MCPN was found to classify all the selected vulnerable buses quite accurately for all the test patterns. The mean square testing error for 15 test patterns was 0.0017 pu. The result of bus classification by MCPN for the 8 critical buses of the system for all the 15 testing patterns is graphically represented in Fig. 5 and Fig. 6. The training and testing convergence of both approaches can be seen for Fig. 7 and Fig. 8. The performance superiority of the MCPN can be clearly seen.
0.4 0.3 0.2 0.1
Target SI
24 25 27 30 52 55 57 60
0.900 0.100 0.588 0.473 0.153 0.365 0.876 0.645
Target Rank I VIII IV V VII VI II III
Computed SI 0.899 0.096 0.567 0.468 0.153 0.361 0.881 0.641
Computed Rank I VIII IV V VII VI II III
1 Target value(Bus52)
Sensitivity index
0.8
28
25
22
19
16
13
10
ITERATIONS
Training Error Testing Error Fig. 7. Performance of CPNN.
0.06
Mean Squared Error
Bus No
7
TABLE IV WEAK BUS RANKING BASED ON SI FOR A SYSTEM STATE
4
1
0
0.04
0.02
ANN Value(Bus 52)
0
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0.6
ANN Value(Bus 55)
0 5 10 ITERATIONS
15
20
25 Training Error Testing Error
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0.4
Fig. 8. Performance of Modified CPNN.
Target Value(Bus 57)
0.2
ANN Vlaue(Bus57) target value(Bus60)
13
10
7
4
1
0 Testing Samples
ANN Value(Bus 60)
Fig. 6. Estimation of Sensitivity index of Bus Nos. 52,55,57,60.
V. CONCLUSIONS As only voltages at a few buses are required by the MCPN for assessment of voltage security, there will be saving in the cost of hardware installed at the measurement sites and communication burden on the central processing computer would also reduce. The conventional counterpropagation neural network when enhanced with neo fuzzy neurons is found to be a better and more accurate non-linear mapping tool. The proposed approach requires almost negligible training time and hence may be used effectively for on-line security assessment
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VI. REFERENCES
[1] [2] [3]
[4] [5] [6] [7]
[8]
Gao, B.,Morison, G.K., and Kundur, P., “Voltage stability evaluation using modal analysis, IEEE Trans., Power Systems, vol. 7, pp. 1529 – 1542, 1992 Barquin, Julian, Gomez,Tomas and Pagola, F. Luis, “Estimating the loading limit margin taking into account voltage collapse areas”, IEEE Trans. Power Systems, vol. 10, pp. 1952 – 1962, Nov. 1995. Flatabo, N.,Fosso, O.B., Ognedal. R., Carlsen, T., and Heggland, K.R., “A method for calculation of margins to voltage instability applied on the Norwegian System for maintaining required security level”, IEEE Trans. Power Systems, vol. 8, pp. 920 – 936, Aug. 1993. Chang, C. S. and Huang, J.S., “Worst case identification of reactive power margin and local weakness of power systems”, Electric Power System Research, Vol. 44, pp. 77 – 83, 1998. Lof, P.A., Smed, T., Anderson, G. and Hill, D.H., “Fast calculation of a voltage stability index”, IEEE Trans. Power Systems, vol. 11, pp. 1654 – 1659, Aug. 1996. Chowdhary, B.H., and Taylor, C.W., “Voltage stability analysis: V-Q power flow simulation versus dynamic simulation”, IEEE Trans. Power Systems, vol. 15, pp. 1354 – 1359, November 2000. Mansour, Yakout, Vaahedi, Ebrahim and El Sharkawi, Mohammed A., “Large-Scale dynamic security contingency screening and ranking using neural networks”, IEEE Trans. Power Systems, vol. 12, pp. 954 – 960, May. 1997. Song, Y.H., Wan, H. B., and Johns, A.T., “Kohonen’n neural network based approach to voltage weak buses / areas identification”, IEE Proceedings, Gener. Trans. Distrib , vol. 144, pp. 340 – 344, 1997.
[9]
Lo, K.L., Peng, L.J., Macqueen, J.F., Ekwue, A.O. and Cheng, D.T.Y., “Fast real power contingency ran king using a counter-propagation network”, IEEE Trans. Power Systems”, vol 13, pp. 1259 – 1264, Nov. 1998
[10] Pandit, Manjaree, Srivastava L., and Sharma J., “Contingency ranking for voltage collapse using parallel self-organizing hierarchical neural network”, Int. Journal of Electrical Power and Energy Systems, vol. 23(5), pp. 369 – 379, 2001. [11] Wan, H.B. and Song, Y.H., “Hybrid supervised and unsupervised neural network approach to voltage stability analysis”, Electric Power Systems Research, vol. 47, pp. 115 – 122, 1998. [12] Lui Chih-Wen, Chang, Chen-Sung and Su, Mu-Chun “Neuro-Fuzzy networks for voltage security monitoring based on synchronized phasor measurements”, IEEE Trans. Power Systems, vol. 13, pp. 326 – 332, May 1998. [13] Pandit M., Srivastava L. and Sharma, J., “Fast voltage contingency selection using fuzzy Parallel Self-organizing Hierarchical Neural Network”, IEEE Tran. Power System, vol. 18, pp. 657-664, May 2003,. [14] Horio, Keichi, Yamakawa, Takeshi, “Modified counterpropagation employing Neo Fuzzy neurons and its application to system modeling” in proc. 2001 Info-tech and info-net international conference, vol. 4, pp. 50-55. [15] Robert Hecht Nielsen, ‘“Counter propagation networks”, Applied Optics, 1987, vol 26, pp 4979-4984. [16] Begovic, M., and Phadke, A.G., “Voltage stability assessment th rough measurement of a reduced state vector”, IEEE Trans. Power Systems, vol. 5, pp. 198 – 202, Feb. 1990. [17] Eiji Uchino and Takeshi Yamakawa, “Neo -Fuzzy neuron based new approach to system modeling with application to actual system”, proc. 1994 VI International Conference on Tools with AI, pp.564-570.
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[18] Singh, S.N., and Srivastava, S.C., “Corrective action planning to achieve a feasible optimum power flow solution”, IEE Proc. Gener. Trans. Distrib , vol. 142, pp. 576 – 582, 1995. [19] Phillip D. Wasserman, Neural computing, theory and practice, New York: Van Nostrand Reinhold, 1989, p. 69.
VII. BIOGRAPHIES M. Pandit obtained her M. Tech degree in Electrical Engineering from Maulana Azad Collage of Technology, Bhopal, (India) in 1989 and PhD degree from Jiwaji University Gwalior, (India) in 2001. She is currently Professor & Head in the Department of Electrical Engineering, M.I.T.S., Gwalior, (India). Her areas of interest are Power System security analysis, optimization and ANN and Fuzzy neural application to Power Systems. L. Srivastava obtained her M. Tech degree in Electrical Engineering from Indian Institute of technology, Kanpur (India) in 1990 and PhD degree from University of Roorkee, Roorkee, (India), in 1998. She is working as a Professor in the Department of Electrical Engineering, M.I.T.S., Gwalior, (India). She is currently involved in research in the area of Power System optimization and control, security analysis, and ANN and Fuzzy neural application to Power Systems. Vijay Singh obtained his M.E. degree in Electrical Engineering from Madhav Institute of Technology and Science, Gwalior in 2004 and he is currently perusing PhD degree in Electrical Engineering from Rajiv Gandhi Proudyogiki Vishwavidyalaya Bhopal (India) through Electrical Engineering Department, M.I.T.S., Gwalior.
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