research article assessment of global voltage stability ... · with dynamic operating scenario, the...
TRANSCRIPT
Research ArticleAssessment of Global Voltage Stability Margin throughRadial Basis Function Neural Network
Akash Saxena and Ankit Kumar Sharma
Department of Electrical Engineering Swami Keshvanand Institute of Technology Jaipur India
Correspondence should be addressed to Akash Saxena aakashsaxenahotmailcom
Received 30 June 2016 Accepted 30 August 2016
Academic Editor Gorazd Stumberger
Copyright copy 2016 A Saxena and A K SharmaThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Dynamic operating conditions along with contingencies often present formidable challenges to the power engineers Decisionspertaining to the control strategies taken by the system operators at energy management centre are based on the information aboutthe systemrsquos behavior The application of ANN as a tool for voltage stability assessment is empirical because of its ability to doparallel data processing with high accuracy fast response and capability to model dynamic nonlinear and noisy data This paperpresents an effective methodology based on Radial Basis Function Neural Network (RBFN) to predict Global Voltage StabilityMargin (GVSM) for any unseen loading condition of the system GVSM is used to assess the overall voltage stability status of thepower system A comparative analysis of different topologies of ANN namely Feedforward Backprop (FFBP) Cascade ForwardBackprop (CFB) Generalized Regression (GR) Layer Recurrent (LR) Nonlinear Autoregressive Exogenous (NARX) ELMANBackprop and Feedforward Distributed Time Delay Network (FFDTDN) is carried out on the basis of capability of the predictionof GVSM The efficacy of RBFN is better than other networks which is validated by taking the predictions of GVSM at differentlevels of Additive White Gaussian Noise (AWGN) in input features The results obtained from ANNs are validated through theoffline Newton Raphson (N-R) methodThe proposed methodology is tested over IEEE 14-bus IEEE 30-bus and IEEE 118-bus testsystems
1 Introduction
In recent years the power system stability issues are promi-nent and possess more relevance due to a competitivebusiness environmentThe existing generation and transmis-sion utilities are working on their operating limits due toexponential increase in the load demand [1]
The burning issue with the modern power networks isto accommodate an escalating demand without an expansionof the transmission utilities With this constraint the voltagestability has emerged as a potential area of research Twocritical findings are carried out through stability studies Thefirst one is the assessment of the critical point of voltagestability and second area suggests preventive control throughload shedding and generator rescheduling IEEE CIGRE taskforce committee defined voltage stability as an ability of thepower system to maintain acceptable and constant voltagelevel at all buses in the system under normal conditions andafter being subjected to the disturbance [1 2] Therefore
voltage stability analysis is necessary to identify the criticalbuses in a power system
Voltage collapse is a phenomenon characterized by thegradual decrement in system voltage along with the systemoperating point Often the assessment of the voltage stabilityis carried out through the calculation of the stability indices[3ndash15] Table 1 shows the comparative study of differentindicators along with the limitations
On the basis of literature survey it can be concluded thatthe indicator for voltage stability assessment should possessthe following qualities
(a) There should be a simple correlation between theindicator and the systemrsquos controllable parameter
(b) Corrective measures can be derived from the valuesof indicator
With dynamic operating scenario the prediction of voltagecollapse through indicators is a slave of the performance
Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2016 Article ID 4858431 11 pageshttpdxdoiorg10115520164858431
2 Advances in Electrical Engineering
Table 1 Comparative analysis of different voltage stability indicators
Name ofindicator Properties of indicator Summary
GVSM [3]L-index [4]VCPI [5]ENVCI [6]VSI [7]LCPI [8]
Based on Newton Raphson (NR)load flow solution
(1) The line indices are suitable for constant power load Theresults may be pessimistic for changed load profile(2) The convergence of NR algorithm is affected by the vicinityof voltage collapse point
References[9 15]
Index based on sensitivityanalysis
(1) Sensitive based methods are computationally intensive andrequired computation of derivatives(2) Dynamic operating conditions introduce errors in thecalculations of the derivatives Moreover the correlationbetween the voltage and reactive power should be continuous innature
References[10ndash14]
These indices are based on staticand dynamic bifurcations
(1) Bifurcations can be detected through the calculation ofeigenvalues Eigenvalues are system specific and vulnerable totopological changes(2) These indicators cannot be useful for loadability marginevaluation and relative voltage stability determination
of load flow routine and accuracy of the calculation ofderivatives and Jacobian matrices It is to be noted herethat the prediction of the voltage stability under unseenoperating condition or contingency cannot be efficientlyexecuted within a safe time limit Hence the initiation of anypreventive or emergency control strategy is not possibleWiththe development of the smart grid efficient technologiesare invited by the operators to participate in the processof operation and control of the power system With thismotivation the paper presents an application of RBFNapproach for online monitoring of voltage stability
A network equivalence framework to predict the globalscenario of voltage stability is developed by reducing theactual system into an equivalent two-bus system The GVSM[3] is used for indicating the state of the actual system Allthe parameters of the equivalent system are obtained fromthe load flow solution of the original system This equivalentsystem is nothing but a power line having series equivalentimpedance with a load at the receiving end but the sendingend voltage is kept at the reference voltage The concept ofsingle line equivalent is further used to determine the voltagecollapse proximity
In this paper RBFN network of the ANN family isemployed to predict the GVSM for various system operatingloading The proposed online scheme has the ability to getit adapted when subjected to any new and unseen operatingcondition
This scheme is validated on standard IEEE 14-bus 30-busand 118-bus power systems The following are the researchobjectives of this manuscript
(a) To present the mathematical framework of GVSMand calculate the GVSM for IEEE 14-bus 30-bus and118-bus test system
(b) To develop a supervised learning prediction enginewith the help of offline simulation results to identifythe GVSM for different system operating loading
(c) To present a meaningful comparison between differ-ent ANN topologies as a predictor and analyze theefficacy of the supervised learning prediction enginein the presence of AWGN
The remaining part of the paper is presented as follows inSections 2 and 3mathematical work of GVSM is presented InSection 4 brief details of proposed RBFN are incorporatedIn Section 5 simulation results are presented and finally inSection 6 the conclusion and future scope of the work arepresented
2 Equivalent Two-Bus Pi-Network
The equivalent two-bus pi-network model is developed asfollows Let us assume a two-bus equivalent network inwhicha generator bus is assumed as a sending end bus and a load busis assumed as a receiving end bus as shown in Figure 1 Thebehavior and properties of the proposed two-bus equivalentmodel should be the same as the multibus network Hencethis makes the evaluation of voltage stability possible [3]Therefore the power equations for the two-bus equivalentnetwork can be written as
119878g = 119875g + 119895119876g = slowast
s = (119878se + 119878sh) + 119878load
119878se = (s minus r) lowast
se
119878sh = slowast
shs + rlowast
shr
(1)
Applying KCL at node119898 we get
lowast
se =119878g
sminus 119878sh(
lowast
s10038161003816100381610038161003816s10038161003816100381610038161003816
2
+10038161003816100381610038161003816r10038161003816100381610038161003816
2
) (2)
Advances in Electrical Engineering 3
Generator m nIrIseIs
Ishr VrVs Ishs
Zse_eq
Ysh_eqY = Ysh2sh_eq
Sg = Pg + jQg
Sload = Pr + jQr
Figure 1 Two-bus pi-equivalent network
Similarly at node 119899
lowast
se = 119878sh(lowast
s10038161003816100381610038161003816s10038161003816100381610038161003816
2
+10038161003816100381610038161003816r10038161003816100381610038161003816
2
) +119878load
r (3)
where 119881s 119881r and 119868s 119868r are the sending and receivingend voltages and currents 119868se is the current through seriesequivalent impedance 119868shs 119868shr are the shunt branch currentsat sending and receiving end respectively
After the calculations we get the equivalent seriesimpedance and equivalent shunt admittance
119885se eq =(s minus r)
se
119884sh eq =shr
r=shs
s
(4)
This equivalent two-bus pi-network is used for obtaining theGVSM
3 Global Voltage Stability Analysis ofMultibus Power System
When the two-bus network equivalent to a multibus powersystem is obtained the global voltage stability index can beformulated in a straightforwardmanner from the parametersof the global network as follows
Here the voltage-current relation in terms of ABCDparameters for pi-equivalent two-bus circuit of the transmis-sion line is given by
[
119881s
119868s] = [
119860 119861
119862 119863
][
119881r
119868r] (5)
where
119860 = 119863 = 1 +119884119885
2
119861 = 119885
119862 = 119884(1 +119884119885
4)
(6)
Assume
[119885 = 119885se eq119884
2= 119884sh eq] (7)
Let us assume
119860 = |119860| ang120572
119861 = |119861| ang120573
s =10038161003816100381610038161003816s10038161003816100381610038161003816ang120579
r =10038161003816100381610038161003816r10038161003816100381610038161003816ang120575 120575 lt 120579
(8)
Solving for the receiving end current
119868r =
10038161003816100381610038161003816s10038161003816100381610038161003816
|119861|ang120579 minus 120573 minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|ang120572 minus 120573 + 120575 (9)
Complex power of receiving end is given by
119878r = rlowast
r
=10038161003816100381610038161003816r10038161003816100381610038161003816ang120575 [
10038161003816100381610038161003816s10038161003816100381610038161003816
|119861|ang minus 120579 + 120573 minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|ang minus 120572 + 120573 minus 120575]
(10)
Sending end voltage is constant then the active and reactivepower at the receiving end is given by
119875r =
10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|cos (120573 + 120575) minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
|119861|cos (120573 minus 120572)
119876r =
10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|sin (120573 + 120575) minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
|119861|sin (120573 minus 120572)
(11)
The Jacobian matrix is given by
119869 =
[[[[
[
120597119875r120597120575
120597119875r120597119881r
120597119876r120597120575
120597119876r120597119881r
]]]]
]
=1
|119861|
sdot [
[
minus10038161003816100381610038161003816r10038161003816100381610038161003816sin (120573 + 120575) cos (120573 + 120575) minus 2 |119860| 10038161003816100381610038161003816r
10038161003816100381610038161003816cos (120573 minus 120572)
10038161003816100381610038161003816r10038161003816100381610038161003816cos (120573 + 120575) sin (120573 + 120575) minus 2 |119860| 10038161003816100381610038161003816r
10038161003816100381610038161003816sin (120573 minus 120572)
]
]
(12)
4 Advances in Electrical Engineering
Start
Increase active amp reactive loadin small steps with keeping
power factor constant
Runload flow
Converge
Calculatetotal load amp losses
Find the equivalent impedance and admittancefor pi-equivalent circuit
Find A B C D parameterfor pi-equivalent circuit
Stop
Yes
No Stop
Go to step 1Calculate GVSM Vcr and global Vr
Figure 2 Algorithm to compute GVSM 119881cr and global 119881r
The determinant of Jacobian matrix is given in
Δ [119869] =1
|119861|2
[2 |119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
cos (120575 + 120572) minus 10038161003816100381610038161003816r10038161003816100381610038161003816] (13)
At the critical point of voltage stability critical voltage index(CVI) is given in
10038161003816100381610038161003816r10038161003816100381610038161003816= 119881cr =
1
2 |119860| cos (120575 + 120572) (14)
Here 119881cr is the critical value of the receiving end voltage atvoltage stability Low value of 119881cr indicates the system willhave better voltage profile along with higher load handlingcapacity To maintain global voltage stability this conditionshould be satisfied Δ[119869] = 0 Therefore to secure globalvoltage stability the GVSM can be defined as GVSM =
Δ[119869] given in (13) It indicates how far the present operatingcondition is from global system voltage collapse [3 16]Figure 2 shows the flow of algorithm for computation ofGVSM and CVI
4 RBF Neural Network Architecture
The RBFN is a feedforward neural network which consistsof an input layer one hidden layer and one output layer
The value of neurons of the input layer feeds in the hiddenlayer a hidden layer which holds each neuron with radialbasis activation function and an output layer which holdseach neuron with a linear activation function The initiatingcentre width for RBF units and computing weights forconnecters are combined to make a learning process forRBF neural network [17] The idea about RBFN comes outfrom the theory of function approximation According to thistheory there are two layers of feedforward network and aset of radial basis functions implemented by hidden nodeThe Gaussian function is normally used in it The linearsummation function as in a Multilayer Perceptron (MLP)is implemented by the output nodes The network trainingis divided into two stages in the first stage weights aredetermined from input to hidden layer and in the secondstage weights are determined from hidden layer to outputlayer This makes interpolation very effective
For the training of ANN the input data sets are generatedfrom offline N-R load flow analysis by varying both realand reactive loads at all the buses randomly of their basecase value In data collection the input data are divided intothree categories namely train data validation data and testdata NR load flow analysis is conducted at all steps andcorresponding GVSM is calculated The real and reactivepower loads on buses are considered as input features forbuilding up the supervised learning models Total 236 inputsfor IEEE 118-bus system 60 inputs for IEEE 30-bus systemand 28 inputs for IEEE 14-bus system are taken By NRmethod GVSM of each line is obtained and the minimumvalues out of 358 for IEEE 118-bus system out of 82 forIEEE 30-bus systems and out of 40 for IEEE 14-bus systemsare taken as output Total 1000 samples are also generatedby offline N-R load flow analysis method 70 data of thesamples are used for training 20 data for validation and10 data for testing
5 Simulation Results
A computer software programme has been developed in theMATLAB 2015b [18] environment to perform the simulationsand run on a Pentium IV CPU 269GHz and 184GB RAMcomputer To demonstrate the effectiveness of the proposedtechnique IEEE 14-bus test system IEEE 30-bus test systemand IEEE 118-bus test system have been used IEEE 14-bussystem represents a portion of the American Electric PowerSystemwhich is located in theMidwesternUS since February1962 Basically this 14-bus system has 14 buses 5 generatorsand 9 load buses IEEE 30-bus system represents a portion ofthe American Electric Power System (in the Midwestern US)since December 1961 This system has 30 buses 6 generatorsand 24 load buses IEEE 118-bus system has 118 buses 51generators and 67 load buses [19]
51 Case Study of IEEE 14-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Advances in Electrical Engineering 5
1 2 3 4
332 33088 32975 3213333157 32978 32856 32087
N-RRBFNFFBP 33018 32847 32576 32197
314316318
32322324326328
33332334
GVS
M
Figure 3 GVSM for operating Case 1
23527 23367 23206 23043
23488 23234 23167 22978
23558 23401 23242 2308
226227228229
23231232233234235236237
GVS
M
32 41
GVSM
GVSM
GVSM
(Case 2) N-R
(Case 2) RBFN
(Case 2) FFBP
Figure 4 GVSM for operating Case 2
1 2 3 4
13779 1359 13431 1330313778 13467 13149 12888
FFBPRBFNN-R 13812 13541 13264 12982
005
115
225
335
445
GVS
M
Figure 5 GVSM for operating Case 3
1 2 3 4Different operating scenario (Case 1)
RBFNFFBP
05132051340513605138
051405142051440514605148
Criti
cal v
olta
ge in
dex
Figure 6 CVI for Case 1
1 2 34
Different operating scenarios (Case 2)
N-RRBFNFFBP
1 2
Criti
cal v
olta
ge in
dex 0551
054905480547054605450544
055
Figure 7 CVI for Case 2
064
065
066
067
1 2 34
Criti
cal v
olta
ge in
dex
Different operating scenario (Case 3)
N-RRBFNFFBP
Figure 8 CVI for Case 3
Case 2 It is medium load (increase of the system operatingload by 224 pu from base case)
Case 3 It is high system operating load (increase of thesystem operating load by 32 pu from base case)
Figures 3ndash8 show theGVSM andCVI for IEEE 14-bus testsystem for three operating scenarios Different intermediateloading conditions are incorporated to show the predictionefficacy of the supervised learning models For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 3ndash8 Due to spacelimitations detailed values of Jacobian matrices and CVIs areprovided in the form of Supplementary Material availableonline at httpdxdoiorg10115520164858431
For Case 1 it is observed that the values of GVSM andCVI predicted by FFBP and RBFN fall in a secure range Inother words near the base case the value of GVSM is higherfor the system and the values of CVIs are lower These valuesare validated by the offline N-R method It is also observedthat the prediction accuracy of RBFN is higher than FFBPIn Case 2 a considerable amount of decrease in numericalvalues of GVSM and increase in the CVI are observed by allprediction methods For Case 3 after a continuous increasein the system operating load the system has approached thepoint of collapse The values of GVSM possess a decreasing
6 Advances in Electrical Engineering
1 2 3 4
2026 2016 2005812 19954492018 201 2005 1994841976 1967 1957523 1945864
N-RRBFNFFBP
19192194196198
2202204
GVS
M
Figure 9 GVSM for operating Case 1
1 2 3 40302908 0272346 0239781 02045960302878 0272276 0239691 02044860321769 030624 0292178 0279588
N-RRBFNFFBP
0
005
01
015
02
025
03
035
GVS
M
Figure 10 GVSM for operating Case 2
tend and CVI becomes higher The prediction capability ofall the networks is verified by the offline N-R method Fromthis analysis it is concluded that RBFN is a suitable topologyto identify the critical buses in IEEE 14-bus test system
Table 2 gives the GVSM at unknown system operatingloading incorporating AWGN of different signal-to-noiseratios (SNRs) Presence of AWGN is a close replica ofmeasurement errors and presence of harmonic loads andelectronic measurement devices At SNR 05 and 005 forunknown loading the efficacy of RBFN is better than othernetworks as can be observed in Table 2 RBFN and CFBtopologies of ANN give the better results Apart from thisthe results from the FFBP networks GR networks LRnetworks NARX networks ELMAN and FFDTD networksare satisfactory
52 Case Study of IEEE 30-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium operating load (increase of the systemoperating load by 189 pu)
0509
05095
051
05105
0511
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
Figure 11 CVI for Case 1
0532
0533
0534
0535
0536
0537
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 12 CVI for Case 2
Case 3 It is high system operating load (increase of thesystem operating load by 27 pu from base case)
Figures 9ndash13 give the GVSM and CVI for the above-mentioned operating cases
It is observed that the values of GVSM and CVI predictedby FFBP and RBFN fall in a close range and the predictionsare validated by the offline N-R method It is observed thatthe prediction accuracy of RBFN is higher For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 9ndash13 In Case 3 after acontinuous increase in the system operating load the systemhas reached near the point of collapse The values of GVSMtend to zero NR (0204596) FFBP (0279588) and RBFN(0204486) From this analysis it is concluded that RBFN isa suitable topology to identify the critical buses in IEEE 30-bus test system
In Table 3 different unseen operating loading conditionsare simulated and it is observed that the value of GVSM givenby RBFN is the nearest value of GVSM given by the N-Roffline method as compared to other networks
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 Advances in Electrical Engineering
Table 1 Comparative analysis of different voltage stability indicators
Name ofindicator Properties of indicator Summary
GVSM [3]L-index [4]VCPI [5]ENVCI [6]VSI [7]LCPI [8]
Based on Newton Raphson (NR)load flow solution
(1) The line indices are suitable for constant power load Theresults may be pessimistic for changed load profile(2) The convergence of NR algorithm is affected by the vicinityof voltage collapse point
References[9 15]
Index based on sensitivityanalysis
(1) Sensitive based methods are computationally intensive andrequired computation of derivatives(2) Dynamic operating conditions introduce errors in thecalculations of the derivatives Moreover the correlationbetween the voltage and reactive power should be continuous innature
References[10ndash14]
These indices are based on staticand dynamic bifurcations
(1) Bifurcations can be detected through the calculation ofeigenvalues Eigenvalues are system specific and vulnerable totopological changes(2) These indicators cannot be useful for loadability marginevaluation and relative voltage stability determination
of load flow routine and accuracy of the calculation ofderivatives and Jacobian matrices It is to be noted herethat the prediction of the voltage stability under unseenoperating condition or contingency cannot be efficientlyexecuted within a safe time limit Hence the initiation of anypreventive or emergency control strategy is not possibleWiththe development of the smart grid efficient technologiesare invited by the operators to participate in the processof operation and control of the power system With thismotivation the paper presents an application of RBFNapproach for online monitoring of voltage stability
A network equivalence framework to predict the globalscenario of voltage stability is developed by reducing theactual system into an equivalent two-bus system The GVSM[3] is used for indicating the state of the actual system Allthe parameters of the equivalent system are obtained fromthe load flow solution of the original system This equivalentsystem is nothing but a power line having series equivalentimpedance with a load at the receiving end but the sendingend voltage is kept at the reference voltage The concept ofsingle line equivalent is further used to determine the voltagecollapse proximity
In this paper RBFN network of the ANN family isemployed to predict the GVSM for various system operatingloading The proposed online scheme has the ability to getit adapted when subjected to any new and unseen operatingcondition
This scheme is validated on standard IEEE 14-bus 30-busand 118-bus power systems The following are the researchobjectives of this manuscript
(a) To present the mathematical framework of GVSMand calculate the GVSM for IEEE 14-bus 30-bus and118-bus test system
(b) To develop a supervised learning prediction enginewith the help of offline simulation results to identifythe GVSM for different system operating loading
(c) To present a meaningful comparison between differ-ent ANN topologies as a predictor and analyze theefficacy of the supervised learning prediction enginein the presence of AWGN
The remaining part of the paper is presented as follows inSections 2 and 3mathematical work of GVSM is presented InSection 4 brief details of proposed RBFN are incorporatedIn Section 5 simulation results are presented and finally inSection 6 the conclusion and future scope of the work arepresented
2 Equivalent Two-Bus Pi-Network
The equivalent two-bus pi-network model is developed asfollows Let us assume a two-bus equivalent network inwhicha generator bus is assumed as a sending end bus and a load busis assumed as a receiving end bus as shown in Figure 1 Thebehavior and properties of the proposed two-bus equivalentmodel should be the same as the multibus network Hencethis makes the evaluation of voltage stability possible [3]Therefore the power equations for the two-bus equivalentnetwork can be written as
119878g = 119875g + 119895119876g = slowast
s = (119878se + 119878sh) + 119878load
119878se = (s minus r) lowast
se
119878sh = slowast
shs + rlowast
shr
(1)
Applying KCL at node119898 we get
lowast
se =119878g
sminus 119878sh(
lowast
s10038161003816100381610038161003816s10038161003816100381610038161003816
2
+10038161003816100381610038161003816r10038161003816100381610038161003816
2
) (2)
Advances in Electrical Engineering 3
Generator m nIrIseIs
Ishr VrVs Ishs
Zse_eq
Ysh_eqY = Ysh2sh_eq
Sg = Pg + jQg
Sload = Pr + jQr
Figure 1 Two-bus pi-equivalent network
Similarly at node 119899
lowast
se = 119878sh(lowast
s10038161003816100381610038161003816s10038161003816100381610038161003816
2
+10038161003816100381610038161003816r10038161003816100381610038161003816
2
) +119878load
r (3)
where 119881s 119881r and 119868s 119868r are the sending and receivingend voltages and currents 119868se is the current through seriesequivalent impedance 119868shs 119868shr are the shunt branch currentsat sending and receiving end respectively
After the calculations we get the equivalent seriesimpedance and equivalent shunt admittance
119885se eq =(s minus r)
se
119884sh eq =shr
r=shs
s
(4)
This equivalent two-bus pi-network is used for obtaining theGVSM
3 Global Voltage Stability Analysis ofMultibus Power System
When the two-bus network equivalent to a multibus powersystem is obtained the global voltage stability index can beformulated in a straightforwardmanner from the parametersof the global network as follows
Here the voltage-current relation in terms of ABCDparameters for pi-equivalent two-bus circuit of the transmis-sion line is given by
[
119881s
119868s] = [
119860 119861
119862 119863
][
119881r
119868r] (5)
where
119860 = 119863 = 1 +119884119885
2
119861 = 119885
119862 = 119884(1 +119884119885
4)
(6)
Assume
[119885 = 119885se eq119884
2= 119884sh eq] (7)
Let us assume
119860 = |119860| ang120572
119861 = |119861| ang120573
s =10038161003816100381610038161003816s10038161003816100381610038161003816ang120579
r =10038161003816100381610038161003816r10038161003816100381610038161003816ang120575 120575 lt 120579
(8)
Solving for the receiving end current
119868r =
10038161003816100381610038161003816s10038161003816100381610038161003816
|119861|ang120579 minus 120573 minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|ang120572 minus 120573 + 120575 (9)
Complex power of receiving end is given by
119878r = rlowast
r
=10038161003816100381610038161003816r10038161003816100381610038161003816ang120575 [
10038161003816100381610038161003816s10038161003816100381610038161003816
|119861|ang minus 120579 + 120573 minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|ang minus 120572 + 120573 minus 120575]
(10)
Sending end voltage is constant then the active and reactivepower at the receiving end is given by
119875r =
10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|cos (120573 + 120575) minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
|119861|cos (120573 minus 120572)
119876r =
10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|sin (120573 + 120575) minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
|119861|sin (120573 minus 120572)
(11)
The Jacobian matrix is given by
119869 =
[[[[
[
120597119875r120597120575
120597119875r120597119881r
120597119876r120597120575
120597119876r120597119881r
]]]]
]
=1
|119861|
sdot [
[
minus10038161003816100381610038161003816r10038161003816100381610038161003816sin (120573 + 120575) cos (120573 + 120575) minus 2 |119860| 10038161003816100381610038161003816r
10038161003816100381610038161003816cos (120573 minus 120572)
10038161003816100381610038161003816r10038161003816100381610038161003816cos (120573 + 120575) sin (120573 + 120575) minus 2 |119860| 10038161003816100381610038161003816r
10038161003816100381610038161003816sin (120573 minus 120572)
]
]
(12)
4 Advances in Electrical Engineering
Start
Increase active amp reactive loadin small steps with keeping
power factor constant
Runload flow
Converge
Calculatetotal load amp losses
Find the equivalent impedance and admittancefor pi-equivalent circuit
Find A B C D parameterfor pi-equivalent circuit
Stop
Yes
No Stop
Go to step 1Calculate GVSM Vcr and global Vr
Figure 2 Algorithm to compute GVSM 119881cr and global 119881r
The determinant of Jacobian matrix is given in
Δ [119869] =1
|119861|2
[2 |119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
cos (120575 + 120572) minus 10038161003816100381610038161003816r10038161003816100381610038161003816] (13)
At the critical point of voltage stability critical voltage index(CVI) is given in
10038161003816100381610038161003816r10038161003816100381610038161003816= 119881cr =
1
2 |119860| cos (120575 + 120572) (14)
Here 119881cr is the critical value of the receiving end voltage atvoltage stability Low value of 119881cr indicates the system willhave better voltage profile along with higher load handlingcapacity To maintain global voltage stability this conditionshould be satisfied Δ[119869] = 0 Therefore to secure globalvoltage stability the GVSM can be defined as GVSM =
Δ[119869] given in (13) It indicates how far the present operatingcondition is from global system voltage collapse [3 16]Figure 2 shows the flow of algorithm for computation ofGVSM and CVI
4 RBF Neural Network Architecture
The RBFN is a feedforward neural network which consistsof an input layer one hidden layer and one output layer
The value of neurons of the input layer feeds in the hiddenlayer a hidden layer which holds each neuron with radialbasis activation function and an output layer which holdseach neuron with a linear activation function The initiatingcentre width for RBF units and computing weights forconnecters are combined to make a learning process forRBF neural network [17] The idea about RBFN comes outfrom the theory of function approximation According to thistheory there are two layers of feedforward network and aset of radial basis functions implemented by hidden nodeThe Gaussian function is normally used in it The linearsummation function as in a Multilayer Perceptron (MLP)is implemented by the output nodes The network trainingis divided into two stages in the first stage weights aredetermined from input to hidden layer and in the secondstage weights are determined from hidden layer to outputlayer This makes interpolation very effective
For the training of ANN the input data sets are generatedfrom offline N-R load flow analysis by varying both realand reactive loads at all the buses randomly of their basecase value In data collection the input data are divided intothree categories namely train data validation data and testdata NR load flow analysis is conducted at all steps andcorresponding GVSM is calculated The real and reactivepower loads on buses are considered as input features forbuilding up the supervised learning models Total 236 inputsfor IEEE 118-bus system 60 inputs for IEEE 30-bus systemand 28 inputs for IEEE 14-bus system are taken By NRmethod GVSM of each line is obtained and the minimumvalues out of 358 for IEEE 118-bus system out of 82 forIEEE 30-bus systems and out of 40 for IEEE 14-bus systemsare taken as output Total 1000 samples are also generatedby offline N-R load flow analysis method 70 data of thesamples are used for training 20 data for validation and10 data for testing
5 Simulation Results
A computer software programme has been developed in theMATLAB 2015b [18] environment to perform the simulationsand run on a Pentium IV CPU 269GHz and 184GB RAMcomputer To demonstrate the effectiveness of the proposedtechnique IEEE 14-bus test system IEEE 30-bus test systemand IEEE 118-bus test system have been used IEEE 14-bussystem represents a portion of the American Electric PowerSystemwhich is located in theMidwesternUS since February1962 Basically this 14-bus system has 14 buses 5 generatorsand 9 load buses IEEE 30-bus system represents a portion ofthe American Electric Power System (in the Midwestern US)since December 1961 This system has 30 buses 6 generatorsand 24 load buses IEEE 118-bus system has 118 buses 51generators and 67 load buses [19]
51 Case Study of IEEE 14-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Advances in Electrical Engineering 5
1 2 3 4
332 33088 32975 3213333157 32978 32856 32087
N-RRBFNFFBP 33018 32847 32576 32197
314316318
32322324326328
33332334
GVS
M
Figure 3 GVSM for operating Case 1
23527 23367 23206 23043
23488 23234 23167 22978
23558 23401 23242 2308
226227228229
23231232233234235236237
GVS
M
32 41
GVSM
GVSM
GVSM
(Case 2) N-R
(Case 2) RBFN
(Case 2) FFBP
Figure 4 GVSM for operating Case 2
1 2 3 4
13779 1359 13431 1330313778 13467 13149 12888
FFBPRBFNN-R 13812 13541 13264 12982
005
115
225
335
445
GVS
M
Figure 5 GVSM for operating Case 3
1 2 3 4Different operating scenario (Case 1)
RBFNFFBP
05132051340513605138
051405142051440514605148
Criti
cal v
olta
ge in
dex
Figure 6 CVI for Case 1
1 2 34
Different operating scenarios (Case 2)
N-RRBFNFFBP
1 2
Criti
cal v
olta
ge in
dex 0551
054905480547054605450544
055
Figure 7 CVI for Case 2
064
065
066
067
1 2 34
Criti
cal v
olta
ge in
dex
Different operating scenario (Case 3)
N-RRBFNFFBP
Figure 8 CVI for Case 3
Case 2 It is medium load (increase of the system operatingload by 224 pu from base case)
Case 3 It is high system operating load (increase of thesystem operating load by 32 pu from base case)
Figures 3ndash8 show theGVSM andCVI for IEEE 14-bus testsystem for three operating scenarios Different intermediateloading conditions are incorporated to show the predictionefficacy of the supervised learning models For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 3ndash8 Due to spacelimitations detailed values of Jacobian matrices and CVIs areprovided in the form of Supplementary Material availableonline at httpdxdoiorg10115520164858431
For Case 1 it is observed that the values of GVSM andCVI predicted by FFBP and RBFN fall in a secure range Inother words near the base case the value of GVSM is higherfor the system and the values of CVIs are lower These valuesare validated by the offline N-R method It is also observedthat the prediction accuracy of RBFN is higher than FFBPIn Case 2 a considerable amount of decrease in numericalvalues of GVSM and increase in the CVI are observed by allprediction methods For Case 3 after a continuous increasein the system operating load the system has approached thepoint of collapse The values of GVSM possess a decreasing
6 Advances in Electrical Engineering
1 2 3 4
2026 2016 2005812 19954492018 201 2005 1994841976 1967 1957523 1945864
N-RRBFNFFBP
19192194196198
2202204
GVS
M
Figure 9 GVSM for operating Case 1
1 2 3 40302908 0272346 0239781 02045960302878 0272276 0239691 02044860321769 030624 0292178 0279588
N-RRBFNFFBP
0
005
01
015
02
025
03
035
GVS
M
Figure 10 GVSM for operating Case 2
tend and CVI becomes higher The prediction capability ofall the networks is verified by the offline N-R method Fromthis analysis it is concluded that RBFN is a suitable topologyto identify the critical buses in IEEE 14-bus test system
Table 2 gives the GVSM at unknown system operatingloading incorporating AWGN of different signal-to-noiseratios (SNRs) Presence of AWGN is a close replica ofmeasurement errors and presence of harmonic loads andelectronic measurement devices At SNR 05 and 005 forunknown loading the efficacy of RBFN is better than othernetworks as can be observed in Table 2 RBFN and CFBtopologies of ANN give the better results Apart from thisthe results from the FFBP networks GR networks LRnetworks NARX networks ELMAN and FFDTD networksare satisfactory
52 Case Study of IEEE 30-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium operating load (increase of the systemoperating load by 189 pu)
0509
05095
051
05105
0511
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
Figure 11 CVI for Case 1
0532
0533
0534
0535
0536
0537
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 12 CVI for Case 2
Case 3 It is high system operating load (increase of thesystem operating load by 27 pu from base case)
Figures 9ndash13 give the GVSM and CVI for the above-mentioned operating cases
It is observed that the values of GVSM and CVI predictedby FFBP and RBFN fall in a close range and the predictionsare validated by the offline N-R method It is observed thatthe prediction accuracy of RBFN is higher For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 9ndash13 In Case 3 after acontinuous increase in the system operating load the systemhas reached near the point of collapse The values of GVSMtend to zero NR (0204596) FFBP (0279588) and RBFN(0204486) From this analysis it is concluded that RBFN isa suitable topology to identify the critical buses in IEEE 30-bus test system
In Table 3 different unseen operating loading conditionsare simulated and it is observed that the value of GVSM givenby RBFN is the nearest value of GVSM given by the N-Roffline method as compared to other networks
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 3
Generator m nIrIseIs
Ishr VrVs Ishs
Zse_eq
Ysh_eqY = Ysh2sh_eq
Sg = Pg + jQg
Sload = Pr + jQr
Figure 1 Two-bus pi-equivalent network
Similarly at node 119899
lowast
se = 119878sh(lowast
s10038161003816100381610038161003816s10038161003816100381610038161003816
2
+10038161003816100381610038161003816r10038161003816100381610038161003816
2
) +119878load
r (3)
where 119881s 119881r and 119868s 119868r are the sending and receivingend voltages and currents 119868se is the current through seriesequivalent impedance 119868shs 119868shr are the shunt branch currentsat sending and receiving end respectively
After the calculations we get the equivalent seriesimpedance and equivalent shunt admittance
119885se eq =(s minus r)
se
119884sh eq =shr
r=shs
s
(4)
This equivalent two-bus pi-network is used for obtaining theGVSM
3 Global Voltage Stability Analysis ofMultibus Power System
When the two-bus network equivalent to a multibus powersystem is obtained the global voltage stability index can beformulated in a straightforwardmanner from the parametersof the global network as follows
Here the voltage-current relation in terms of ABCDparameters for pi-equivalent two-bus circuit of the transmis-sion line is given by
[
119881s
119868s] = [
119860 119861
119862 119863
][
119881r
119868r] (5)
where
119860 = 119863 = 1 +119884119885
2
119861 = 119885
119862 = 119884(1 +119884119885
4)
(6)
Assume
[119885 = 119885se eq119884
2= 119884sh eq] (7)
Let us assume
119860 = |119860| ang120572
119861 = |119861| ang120573
s =10038161003816100381610038161003816s10038161003816100381610038161003816ang120579
r =10038161003816100381610038161003816r10038161003816100381610038161003816ang120575 120575 lt 120579
(8)
Solving for the receiving end current
119868r =
10038161003816100381610038161003816s10038161003816100381610038161003816
|119861|ang120579 minus 120573 minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|ang120572 minus 120573 + 120575 (9)
Complex power of receiving end is given by
119878r = rlowast
r
=10038161003816100381610038161003816r10038161003816100381610038161003816ang120575 [
10038161003816100381610038161003816s10038161003816100381610038161003816
|119861|ang minus 120579 + 120573 minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|ang minus 120572 + 120573 minus 120575]
(10)
Sending end voltage is constant then the active and reactivepower at the receiving end is given by
119875r =
10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|cos (120573 + 120575) minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
|119861|cos (120573 minus 120572)
119876r =
10038161003816100381610038161003816r10038161003816100381610038161003816
|119861|sin (120573 + 120575) minus
|119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
|119861|sin (120573 minus 120572)
(11)
The Jacobian matrix is given by
119869 =
[[[[
[
120597119875r120597120575
120597119875r120597119881r
120597119876r120597120575
120597119876r120597119881r
]]]]
]
=1
|119861|
sdot [
[
minus10038161003816100381610038161003816r10038161003816100381610038161003816sin (120573 + 120575) cos (120573 + 120575) minus 2 |119860| 10038161003816100381610038161003816r
10038161003816100381610038161003816cos (120573 minus 120572)
10038161003816100381610038161003816r10038161003816100381610038161003816cos (120573 + 120575) sin (120573 + 120575) minus 2 |119860| 10038161003816100381610038161003816r
10038161003816100381610038161003816sin (120573 minus 120572)
]
]
(12)
4 Advances in Electrical Engineering
Start
Increase active amp reactive loadin small steps with keeping
power factor constant
Runload flow
Converge
Calculatetotal load amp losses
Find the equivalent impedance and admittancefor pi-equivalent circuit
Find A B C D parameterfor pi-equivalent circuit
Stop
Yes
No Stop
Go to step 1Calculate GVSM Vcr and global Vr
Figure 2 Algorithm to compute GVSM 119881cr and global 119881r
The determinant of Jacobian matrix is given in
Δ [119869] =1
|119861|2
[2 |119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
cos (120575 + 120572) minus 10038161003816100381610038161003816r10038161003816100381610038161003816] (13)
At the critical point of voltage stability critical voltage index(CVI) is given in
10038161003816100381610038161003816r10038161003816100381610038161003816= 119881cr =
1
2 |119860| cos (120575 + 120572) (14)
Here 119881cr is the critical value of the receiving end voltage atvoltage stability Low value of 119881cr indicates the system willhave better voltage profile along with higher load handlingcapacity To maintain global voltage stability this conditionshould be satisfied Δ[119869] = 0 Therefore to secure globalvoltage stability the GVSM can be defined as GVSM =
Δ[119869] given in (13) It indicates how far the present operatingcondition is from global system voltage collapse [3 16]Figure 2 shows the flow of algorithm for computation ofGVSM and CVI
4 RBF Neural Network Architecture
The RBFN is a feedforward neural network which consistsof an input layer one hidden layer and one output layer
The value of neurons of the input layer feeds in the hiddenlayer a hidden layer which holds each neuron with radialbasis activation function and an output layer which holdseach neuron with a linear activation function The initiatingcentre width for RBF units and computing weights forconnecters are combined to make a learning process forRBF neural network [17] The idea about RBFN comes outfrom the theory of function approximation According to thistheory there are two layers of feedforward network and aset of radial basis functions implemented by hidden nodeThe Gaussian function is normally used in it The linearsummation function as in a Multilayer Perceptron (MLP)is implemented by the output nodes The network trainingis divided into two stages in the first stage weights aredetermined from input to hidden layer and in the secondstage weights are determined from hidden layer to outputlayer This makes interpolation very effective
For the training of ANN the input data sets are generatedfrom offline N-R load flow analysis by varying both realand reactive loads at all the buses randomly of their basecase value In data collection the input data are divided intothree categories namely train data validation data and testdata NR load flow analysis is conducted at all steps andcorresponding GVSM is calculated The real and reactivepower loads on buses are considered as input features forbuilding up the supervised learning models Total 236 inputsfor IEEE 118-bus system 60 inputs for IEEE 30-bus systemand 28 inputs for IEEE 14-bus system are taken By NRmethod GVSM of each line is obtained and the minimumvalues out of 358 for IEEE 118-bus system out of 82 forIEEE 30-bus systems and out of 40 for IEEE 14-bus systemsare taken as output Total 1000 samples are also generatedby offline N-R load flow analysis method 70 data of thesamples are used for training 20 data for validation and10 data for testing
5 Simulation Results
A computer software programme has been developed in theMATLAB 2015b [18] environment to perform the simulationsand run on a Pentium IV CPU 269GHz and 184GB RAMcomputer To demonstrate the effectiveness of the proposedtechnique IEEE 14-bus test system IEEE 30-bus test systemand IEEE 118-bus test system have been used IEEE 14-bussystem represents a portion of the American Electric PowerSystemwhich is located in theMidwesternUS since February1962 Basically this 14-bus system has 14 buses 5 generatorsand 9 load buses IEEE 30-bus system represents a portion ofthe American Electric Power System (in the Midwestern US)since December 1961 This system has 30 buses 6 generatorsand 24 load buses IEEE 118-bus system has 118 buses 51generators and 67 load buses [19]
51 Case Study of IEEE 14-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Advances in Electrical Engineering 5
1 2 3 4
332 33088 32975 3213333157 32978 32856 32087
N-RRBFNFFBP 33018 32847 32576 32197
314316318
32322324326328
33332334
GVS
M
Figure 3 GVSM for operating Case 1
23527 23367 23206 23043
23488 23234 23167 22978
23558 23401 23242 2308
226227228229
23231232233234235236237
GVS
M
32 41
GVSM
GVSM
GVSM
(Case 2) N-R
(Case 2) RBFN
(Case 2) FFBP
Figure 4 GVSM for operating Case 2
1 2 3 4
13779 1359 13431 1330313778 13467 13149 12888
FFBPRBFNN-R 13812 13541 13264 12982
005
115
225
335
445
GVS
M
Figure 5 GVSM for operating Case 3
1 2 3 4Different operating scenario (Case 1)
RBFNFFBP
05132051340513605138
051405142051440514605148
Criti
cal v
olta
ge in
dex
Figure 6 CVI for Case 1
1 2 34
Different operating scenarios (Case 2)
N-RRBFNFFBP
1 2
Criti
cal v
olta
ge in
dex 0551
054905480547054605450544
055
Figure 7 CVI for Case 2
064
065
066
067
1 2 34
Criti
cal v
olta
ge in
dex
Different operating scenario (Case 3)
N-RRBFNFFBP
Figure 8 CVI for Case 3
Case 2 It is medium load (increase of the system operatingload by 224 pu from base case)
Case 3 It is high system operating load (increase of thesystem operating load by 32 pu from base case)
Figures 3ndash8 show theGVSM andCVI for IEEE 14-bus testsystem for three operating scenarios Different intermediateloading conditions are incorporated to show the predictionefficacy of the supervised learning models For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 3ndash8 Due to spacelimitations detailed values of Jacobian matrices and CVIs areprovided in the form of Supplementary Material availableonline at httpdxdoiorg10115520164858431
For Case 1 it is observed that the values of GVSM andCVI predicted by FFBP and RBFN fall in a secure range Inother words near the base case the value of GVSM is higherfor the system and the values of CVIs are lower These valuesare validated by the offline N-R method It is also observedthat the prediction accuracy of RBFN is higher than FFBPIn Case 2 a considerable amount of decrease in numericalvalues of GVSM and increase in the CVI are observed by allprediction methods For Case 3 after a continuous increasein the system operating load the system has approached thepoint of collapse The values of GVSM possess a decreasing
6 Advances in Electrical Engineering
1 2 3 4
2026 2016 2005812 19954492018 201 2005 1994841976 1967 1957523 1945864
N-RRBFNFFBP
19192194196198
2202204
GVS
M
Figure 9 GVSM for operating Case 1
1 2 3 40302908 0272346 0239781 02045960302878 0272276 0239691 02044860321769 030624 0292178 0279588
N-RRBFNFFBP
0
005
01
015
02
025
03
035
GVS
M
Figure 10 GVSM for operating Case 2
tend and CVI becomes higher The prediction capability ofall the networks is verified by the offline N-R method Fromthis analysis it is concluded that RBFN is a suitable topologyto identify the critical buses in IEEE 14-bus test system
Table 2 gives the GVSM at unknown system operatingloading incorporating AWGN of different signal-to-noiseratios (SNRs) Presence of AWGN is a close replica ofmeasurement errors and presence of harmonic loads andelectronic measurement devices At SNR 05 and 005 forunknown loading the efficacy of RBFN is better than othernetworks as can be observed in Table 2 RBFN and CFBtopologies of ANN give the better results Apart from thisthe results from the FFBP networks GR networks LRnetworks NARX networks ELMAN and FFDTD networksare satisfactory
52 Case Study of IEEE 30-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium operating load (increase of the systemoperating load by 189 pu)
0509
05095
051
05105
0511
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
Figure 11 CVI for Case 1
0532
0533
0534
0535
0536
0537
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 12 CVI for Case 2
Case 3 It is high system operating load (increase of thesystem operating load by 27 pu from base case)
Figures 9ndash13 give the GVSM and CVI for the above-mentioned operating cases
It is observed that the values of GVSM and CVI predictedby FFBP and RBFN fall in a close range and the predictionsare validated by the offline N-R method It is observed thatthe prediction accuracy of RBFN is higher For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 9ndash13 In Case 3 after acontinuous increase in the system operating load the systemhas reached near the point of collapse The values of GVSMtend to zero NR (0204596) FFBP (0279588) and RBFN(0204486) From this analysis it is concluded that RBFN isa suitable topology to identify the critical buses in IEEE 30-bus test system
In Table 3 different unseen operating loading conditionsare simulated and it is observed that the value of GVSM givenby RBFN is the nearest value of GVSM given by the N-Roffline method as compared to other networks
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Advances in Electrical Engineering
Start
Increase active amp reactive loadin small steps with keeping
power factor constant
Runload flow
Converge
Calculatetotal load amp losses
Find the equivalent impedance and admittancefor pi-equivalent circuit
Find A B C D parameterfor pi-equivalent circuit
Stop
Yes
No Stop
Go to step 1Calculate GVSM Vcr and global Vr
Figure 2 Algorithm to compute GVSM 119881cr and global 119881r
The determinant of Jacobian matrix is given in
Δ [119869] =1
|119861|2
[2 |119860|10038161003816100381610038161003816r10038161003816100381610038161003816
2
cos (120575 + 120572) minus 10038161003816100381610038161003816r10038161003816100381610038161003816] (13)
At the critical point of voltage stability critical voltage index(CVI) is given in
10038161003816100381610038161003816r10038161003816100381610038161003816= 119881cr =
1
2 |119860| cos (120575 + 120572) (14)
Here 119881cr is the critical value of the receiving end voltage atvoltage stability Low value of 119881cr indicates the system willhave better voltage profile along with higher load handlingcapacity To maintain global voltage stability this conditionshould be satisfied Δ[119869] = 0 Therefore to secure globalvoltage stability the GVSM can be defined as GVSM =
Δ[119869] given in (13) It indicates how far the present operatingcondition is from global system voltage collapse [3 16]Figure 2 shows the flow of algorithm for computation ofGVSM and CVI
4 RBF Neural Network Architecture
The RBFN is a feedforward neural network which consistsof an input layer one hidden layer and one output layer
The value of neurons of the input layer feeds in the hiddenlayer a hidden layer which holds each neuron with radialbasis activation function and an output layer which holdseach neuron with a linear activation function The initiatingcentre width for RBF units and computing weights forconnecters are combined to make a learning process forRBF neural network [17] The idea about RBFN comes outfrom the theory of function approximation According to thistheory there are two layers of feedforward network and aset of radial basis functions implemented by hidden nodeThe Gaussian function is normally used in it The linearsummation function as in a Multilayer Perceptron (MLP)is implemented by the output nodes The network trainingis divided into two stages in the first stage weights aredetermined from input to hidden layer and in the secondstage weights are determined from hidden layer to outputlayer This makes interpolation very effective
For the training of ANN the input data sets are generatedfrom offline N-R load flow analysis by varying both realand reactive loads at all the buses randomly of their basecase value In data collection the input data are divided intothree categories namely train data validation data and testdata NR load flow analysis is conducted at all steps andcorresponding GVSM is calculated The real and reactivepower loads on buses are considered as input features forbuilding up the supervised learning models Total 236 inputsfor IEEE 118-bus system 60 inputs for IEEE 30-bus systemand 28 inputs for IEEE 14-bus system are taken By NRmethod GVSM of each line is obtained and the minimumvalues out of 358 for IEEE 118-bus system out of 82 forIEEE 30-bus systems and out of 40 for IEEE 14-bus systemsare taken as output Total 1000 samples are also generatedby offline N-R load flow analysis method 70 data of thesamples are used for training 20 data for validation and10 data for testing
5 Simulation Results
A computer software programme has been developed in theMATLAB 2015b [18] environment to perform the simulationsand run on a Pentium IV CPU 269GHz and 184GB RAMcomputer To demonstrate the effectiveness of the proposedtechnique IEEE 14-bus test system IEEE 30-bus test systemand IEEE 118-bus test system have been used IEEE 14-bussystem represents a portion of the American Electric PowerSystemwhich is located in theMidwesternUS since February1962 Basically this 14-bus system has 14 buses 5 generatorsand 9 load buses IEEE 30-bus system represents a portion ofthe American Electric Power System (in the Midwestern US)since December 1961 This system has 30 buses 6 generatorsand 24 load buses IEEE 118-bus system has 118 buses 51generators and 67 load buses [19]
51 Case Study of IEEE 14-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Advances in Electrical Engineering 5
1 2 3 4
332 33088 32975 3213333157 32978 32856 32087
N-RRBFNFFBP 33018 32847 32576 32197
314316318
32322324326328
33332334
GVS
M
Figure 3 GVSM for operating Case 1
23527 23367 23206 23043
23488 23234 23167 22978
23558 23401 23242 2308
226227228229
23231232233234235236237
GVS
M
32 41
GVSM
GVSM
GVSM
(Case 2) N-R
(Case 2) RBFN
(Case 2) FFBP
Figure 4 GVSM for operating Case 2
1 2 3 4
13779 1359 13431 1330313778 13467 13149 12888
FFBPRBFNN-R 13812 13541 13264 12982
005
115
225
335
445
GVS
M
Figure 5 GVSM for operating Case 3
1 2 3 4Different operating scenario (Case 1)
RBFNFFBP
05132051340513605138
051405142051440514605148
Criti
cal v
olta
ge in
dex
Figure 6 CVI for Case 1
1 2 34
Different operating scenarios (Case 2)
N-RRBFNFFBP
1 2
Criti
cal v
olta
ge in
dex 0551
054905480547054605450544
055
Figure 7 CVI for Case 2
064
065
066
067
1 2 34
Criti
cal v
olta
ge in
dex
Different operating scenario (Case 3)
N-RRBFNFFBP
Figure 8 CVI for Case 3
Case 2 It is medium load (increase of the system operatingload by 224 pu from base case)
Case 3 It is high system operating load (increase of thesystem operating load by 32 pu from base case)
Figures 3ndash8 show theGVSM andCVI for IEEE 14-bus testsystem for three operating scenarios Different intermediateloading conditions are incorporated to show the predictionefficacy of the supervised learning models For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 3ndash8 Due to spacelimitations detailed values of Jacobian matrices and CVIs areprovided in the form of Supplementary Material availableonline at httpdxdoiorg10115520164858431
For Case 1 it is observed that the values of GVSM andCVI predicted by FFBP and RBFN fall in a secure range Inother words near the base case the value of GVSM is higherfor the system and the values of CVIs are lower These valuesare validated by the offline N-R method It is also observedthat the prediction accuracy of RBFN is higher than FFBPIn Case 2 a considerable amount of decrease in numericalvalues of GVSM and increase in the CVI are observed by allprediction methods For Case 3 after a continuous increasein the system operating load the system has approached thepoint of collapse The values of GVSM possess a decreasing
6 Advances in Electrical Engineering
1 2 3 4
2026 2016 2005812 19954492018 201 2005 1994841976 1967 1957523 1945864
N-RRBFNFFBP
19192194196198
2202204
GVS
M
Figure 9 GVSM for operating Case 1
1 2 3 40302908 0272346 0239781 02045960302878 0272276 0239691 02044860321769 030624 0292178 0279588
N-RRBFNFFBP
0
005
01
015
02
025
03
035
GVS
M
Figure 10 GVSM for operating Case 2
tend and CVI becomes higher The prediction capability ofall the networks is verified by the offline N-R method Fromthis analysis it is concluded that RBFN is a suitable topologyto identify the critical buses in IEEE 14-bus test system
Table 2 gives the GVSM at unknown system operatingloading incorporating AWGN of different signal-to-noiseratios (SNRs) Presence of AWGN is a close replica ofmeasurement errors and presence of harmonic loads andelectronic measurement devices At SNR 05 and 005 forunknown loading the efficacy of RBFN is better than othernetworks as can be observed in Table 2 RBFN and CFBtopologies of ANN give the better results Apart from thisthe results from the FFBP networks GR networks LRnetworks NARX networks ELMAN and FFDTD networksare satisfactory
52 Case Study of IEEE 30-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium operating load (increase of the systemoperating load by 189 pu)
0509
05095
051
05105
0511
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
Figure 11 CVI for Case 1
0532
0533
0534
0535
0536
0537
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 12 CVI for Case 2
Case 3 It is high system operating load (increase of thesystem operating load by 27 pu from base case)
Figures 9ndash13 give the GVSM and CVI for the above-mentioned operating cases
It is observed that the values of GVSM and CVI predictedby FFBP and RBFN fall in a close range and the predictionsare validated by the offline N-R method It is observed thatthe prediction accuracy of RBFN is higher For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 9ndash13 In Case 3 after acontinuous increase in the system operating load the systemhas reached near the point of collapse The values of GVSMtend to zero NR (0204596) FFBP (0279588) and RBFN(0204486) From this analysis it is concluded that RBFN isa suitable topology to identify the critical buses in IEEE 30-bus test system
In Table 3 different unseen operating loading conditionsare simulated and it is observed that the value of GVSM givenby RBFN is the nearest value of GVSM given by the N-Roffline method as compared to other networks
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 5
1 2 3 4
332 33088 32975 3213333157 32978 32856 32087
N-RRBFNFFBP 33018 32847 32576 32197
314316318
32322324326328
33332334
GVS
M
Figure 3 GVSM for operating Case 1
23527 23367 23206 23043
23488 23234 23167 22978
23558 23401 23242 2308
226227228229
23231232233234235236237
GVS
M
32 41
GVSM
GVSM
GVSM
(Case 2) N-R
(Case 2) RBFN
(Case 2) FFBP
Figure 4 GVSM for operating Case 2
1 2 3 4
13779 1359 13431 1330313778 13467 13149 12888
FFBPRBFNN-R 13812 13541 13264 12982
005
115
225
335
445
GVS
M
Figure 5 GVSM for operating Case 3
1 2 3 4Different operating scenario (Case 1)
RBFNFFBP
05132051340513605138
051405142051440514605148
Criti
cal v
olta
ge in
dex
Figure 6 CVI for Case 1
1 2 34
Different operating scenarios (Case 2)
N-RRBFNFFBP
1 2
Criti
cal v
olta
ge in
dex 0551
054905480547054605450544
055
Figure 7 CVI for Case 2
064
065
066
067
1 2 34
Criti
cal v
olta
ge in
dex
Different operating scenario (Case 3)
N-RRBFNFFBP
Figure 8 CVI for Case 3
Case 2 It is medium load (increase of the system operatingload by 224 pu from base case)
Case 3 It is high system operating load (increase of thesystem operating load by 32 pu from base case)
Figures 3ndash8 show theGVSM andCVI for IEEE 14-bus testsystem for three operating scenarios Different intermediateloading conditions are incorporated to show the predictionefficacy of the supervised learning models For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 3ndash8 Due to spacelimitations detailed values of Jacobian matrices and CVIs areprovided in the form of Supplementary Material availableonline at httpdxdoiorg10115520164858431
For Case 1 it is observed that the values of GVSM andCVI predicted by FFBP and RBFN fall in a secure range Inother words near the base case the value of GVSM is higherfor the system and the values of CVIs are lower These valuesare validated by the offline N-R method It is also observedthat the prediction accuracy of RBFN is higher than FFBPIn Case 2 a considerable amount of decrease in numericalvalues of GVSM and increase in the CVI are observed by allprediction methods For Case 3 after a continuous increasein the system operating load the system has approached thepoint of collapse The values of GVSM possess a decreasing
6 Advances in Electrical Engineering
1 2 3 4
2026 2016 2005812 19954492018 201 2005 1994841976 1967 1957523 1945864
N-RRBFNFFBP
19192194196198
2202204
GVS
M
Figure 9 GVSM for operating Case 1
1 2 3 40302908 0272346 0239781 02045960302878 0272276 0239691 02044860321769 030624 0292178 0279588
N-RRBFNFFBP
0
005
01
015
02
025
03
035
GVS
M
Figure 10 GVSM for operating Case 2
tend and CVI becomes higher The prediction capability ofall the networks is verified by the offline N-R method Fromthis analysis it is concluded that RBFN is a suitable topologyto identify the critical buses in IEEE 14-bus test system
Table 2 gives the GVSM at unknown system operatingloading incorporating AWGN of different signal-to-noiseratios (SNRs) Presence of AWGN is a close replica ofmeasurement errors and presence of harmonic loads andelectronic measurement devices At SNR 05 and 005 forunknown loading the efficacy of RBFN is better than othernetworks as can be observed in Table 2 RBFN and CFBtopologies of ANN give the better results Apart from thisthe results from the FFBP networks GR networks LRnetworks NARX networks ELMAN and FFDTD networksare satisfactory
52 Case Study of IEEE 30-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium operating load (increase of the systemoperating load by 189 pu)
0509
05095
051
05105
0511
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
Figure 11 CVI for Case 1
0532
0533
0534
0535
0536
0537
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 12 CVI for Case 2
Case 3 It is high system operating load (increase of thesystem operating load by 27 pu from base case)
Figures 9ndash13 give the GVSM and CVI for the above-mentioned operating cases
It is observed that the values of GVSM and CVI predictedby FFBP and RBFN fall in a close range and the predictionsare validated by the offline N-R method It is observed thatthe prediction accuracy of RBFN is higher For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 9ndash13 In Case 3 after acontinuous increase in the system operating load the systemhas reached near the point of collapse The values of GVSMtend to zero NR (0204596) FFBP (0279588) and RBFN(0204486) From this analysis it is concluded that RBFN isa suitable topology to identify the critical buses in IEEE 30-bus test system
In Table 3 different unseen operating loading conditionsare simulated and it is observed that the value of GVSM givenby RBFN is the nearest value of GVSM given by the N-Roffline method as compared to other networks
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Advances in Electrical Engineering
1 2 3 4
2026 2016 2005812 19954492018 201 2005 1994841976 1967 1957523 1945864
N-RRBFNFFBP
19192194196198
2202204
GVS
M
Figure 9 GVSM for operating Case 1
1 2 3 40302908 0272346 0239781 02045960302878 0272276 0239691 02044860321769 030624 0292178 0279588
N-RRBFNFFBP
0
005
01
015
02
025
03
035
GVS
M
Figure 10 GVSM for operating Case 2
tend and CVI becomes higher The prediction capability ofall the networks is verified by the offline N-R method Fromthis analysis it is concluded that RBFN is a suitable topologyto identify the critical buses in IEEE 14-bus test system
Table 2 gives the GVSM at unknown system operatingloading incorporating AWGN of different signal-to-noiseratios (SNRs) Presence of AWGN is a close replica ofmeasurement errors and presence of harmonic loads andelectronic measurement devices At SNR 05 and 005 forunknown loading the efficacy of RBFN is better than othernetworks as can be observed in Table 2 RBFN and CFBtopologies of ANN give the better results Apart from thisthe results from the FFBP networks GR networks LRnetworks NARX networks ELMAN and FFDTD networksare satisfactory
52 Case Study of IEEE 30-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium operating load (increase of the systemoperating load by 189 pu)
0509
05095
051
05105
0511
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
Figure 11 CVI for Case 1
0532
0533
0534
0535
0536
0537
1 2 34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 12 CVI for Case 2
Case 3 It is high system operating load (increase of thesystem operating load by 27 pu from base case)
Figures 9ndash13 give the GVSM and CVI for the above-mentioned operating cases
It is observed that the values of GVSM and CVI predictedby FFBP and RBFN fall in a close range and the predictionsare validated by the offline N-R method It is observed thatthe prediction accuracy of RBFN is higher For each scenariofour intermediate random loading conditions are consideredand shown on the 119909-axis of Figures 9ndash13 In Case 3 after acontinuous increase in the system operating load the systemhas reached near the point of collapse The values of GVSMtend to zero NR (0204596) FFBP (0279588) and RBFN(0204486) From this analysis it is concluded that RBFN isa suitable topology to identify the critical buses in IEEE 30-bus test system
In Table 3 different unseen operating loading conditionsare simulated and it is observed that the value of GVSM givenby RBFN is the nearest value of GVSM given by the N-Roffline method as compared to other networks
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 7
0580582058405860588
059
1 23
4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 3)
N-RRBFNFFBP
Figure 13 CVI for Case 3
1 2 3 4
538448 537649 535862 516202538452 537659 535855 516187535461 535461 535461 535461
N-RRBFNFFBP
50551
51552
52553
53554
545
GVS
M
Figure 14 GVSM for operating Case 1
53 Case Study of IEEE 118-Bus Test System To validate theproposed approach three operating scenarios are considered
Case 1 It is near the base case (where the load busesare having the nominal values of real and reactive powerloading)
Case 2 It is medium load (increase of the system operatingload by 090 pu from base case) that is it is half of the highsystem operating loading
Case 3 It is high system operating load (increase of thesystem operating load by 177 pu from base case)
Figures 14ndash18 give the GVSM and CVI for the above-mentioned three cases
As previously observed the prediction capability ofRBFN is better in comparison to FFBP The same can beverified through the offline NR simulation results For eachscenario four intermediate random loading conditions areconsidered and shown on the 119909-axis of Figures 14ndash18 It isinteresting to observe that with the increment of the load thatis for Case 3 the system has collapsed Hence it is giving thenegative values of GVSM
Table 4 gives the GVSM at unknown system operatingload incorporating normal as well as noisy operating condi-tions At SNR 05 and 005 for unknown loading the efficacyof RBFN is better than other networks The increment of the
1 2 3 4
4257021 4132501 3835049 3674014
4257199 413251 3835067 3674021
4355856 4363853 437998 337998
005
115
225
335
445
5
GVS
M
GVSM
GVSM
GVSM
(Case 2)
(Case 2)
(Case 2)
N-R
RBFN
FFBP
Figure 15 GVSM for operating Case 2
1 2 3 4
minus391756 minus1029483 minus2764583 minus6519778
minus391776 minus102941 minus2764499 minus651977
minus60985295 minus60985293 minus60985293 minus609852931
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0G
VSM
GVSM
GVSM
GVSM
(Case 3) N-R
(Case 3) RBFN
(Case 3) FFBP
Figure 16 GVSM for operating Case 3
1 2 3 4
Criti
cal v
olta
ge in
dex
Operating scenario (Case 1)
N-RRBFNFFBP
0555
0545
0535
0525
055
054
053
Figure 17 CVI for operating Case 1
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Advances in Electrical Engineering
Table 2 GVSM for IEEE 14-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load(1149 pu) 32212 16369 30838 3236 2478 20081 32252 25647 25566
Operating load1339 28993 16275 30199 29019 27372 20781 28991 25638 25299
Operating load22315 22657 26635 21953 22657 32673 22493 22659 24685 23028
Operating load24951 20474 30925 17732 20474 32687 22684 20476 24184 22467
Operating load31591 13522 3203 13951 13521 31603 21735 13527 23458 21562
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 16077 31262 32315 23469 19414 3211 25832 24903
Operating load1339 28993 16576 29979 2898 27348 2206 28989 23297 26217
Operating load22315 22657 25865 2158 22754 32403 21797 2265 2917 22091
Operating load24951 20474 31172 18 20411 32453 22748 2047 24592 22027
Operating load31591 13522 31185 13689 1342 31761 21339 13492 20972 2159
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1149 32212 14873 30989 323 2644 18481 31988 26125 26644
Operating load1339 28993 17963 31318 29034 2686 26166 28879 19629 26516
Operating load22315 22657 24508 26347 22567 32917 22748 22552 23478 23044
Operating load24951 20474 30864 18855 20361 32661 23683 2011 1819 23264
Operating load31591 13522 31284 13836 1379 31559 19611 13349 23348 21616
054
056
058
06
062
064
12
34
Criti
cal v
olta
ge in
dex
Operating scenario (Case 2)
N-RRBFNFFBP
Figure 18 CVI for operating Case 2
system load in small and equidistant steps makes the valueof GVSM lower with every step Negative values of GVSMare the indication of collapse The simulation results of thevoltage stability analysis using proposed technique give betteraccuracy and reliability
Discussions
(1) It is empirical to judge that the value of CVI isminimum when 120575 + 120572 asymp 0 The value of 119860 is one fortransmission networks From this it can be concludedthat the minimum value of CVI is equal to 05 Withthe increment in system load increment in the valuesof CVI will be observed For IEEE 14-bus system theCVI observed for Case 3 scenario 4 reached up to066 The high values of CVI are an indication ofstress on lines Similar patterns are observed for IEEE30-bus and IEEE 118-bus system Gradual increase in
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 9
Table 3 GVSM for IEEE 30-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19593 20219 19742 07938 14445 19849 04502 17873
Operating load12118 18617 19432 20203 18625 08713 14237 18611 04794 17916
Operating load15122 14953 17104 19924 14959 15513 12909 14951 0587 18094
Operating load18112 12351 13424 18337 12351 19035 11392 1235 16006 18063
Operating load25122 05201 06106 02401 052 19453 06958 05203 19932 17387
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 19654 20249 19539 06874 14587 1981 03749 17461
Operating load12118 18617 1834 20215 18579 03972 16843 18588 04354 17279
Operating load15122 14953 14669 19649 14901 17619 10475 14886 06187 17252
Operating load18112 12351 17676 19424 12286 13787 06792 12257 13806 18446
Operating load25122 05201 07542 02733 05224 19419 07901 05137 19909 18075
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load11628 19862 20004 20038 19539 06863 17635 19788 04354 16397
Operating load12118 18617 20035 20231 18682 03445 15947 18557 05822 18311
Operating load15122 14953 18995 19757 14905 08095 10957 14789 16404 18227
Operating load18112 12351 07702 17993 12248 18876 08717 12122 10576 18252
Operating load25122 05201 08744 02835 05355 17867 04752 0512 19095 16913
the values of CVI is observed with the increase inoperating load
(2) In case of pi-equivalent models the variation ofCVI is mainly dependent on the values of parameter119860 The values of ldquo119860rdquo parameter are system specificand depend on systemrsquos parameters (susceptance andimpedance) With the change in operating conditionsthese values are changed For IEEE 14-bus system thevalues of CVIs are around 0513 near base load 054near medium load and 065 at heavy loading Simi-larly for IEEE 30-bus system the values of CVIs are051 for nominal loading 053 for medium loadingand 0584 for heavy loading For IEEE 118-bus testsystem the values of CVIs are comparatively high ascompared to the small test systems For high loading(Case 3) system has reached to the point of collapseValues of CVIs are nearly equal to 1 in this case After
observing these values it can be concluded that pointof collapse is dependent on system parameters
6 Conclusion
The paper proposes an online assessment of voltage stabilityfor multimachine networks with the application of RBFNAn equivalent two-bus pi-network model is developed forassessment of voltage stability for multibus power systemswhere series and shunt parameters of transmission lines arelumped separately in the formof series and shunt equivalentsGVSM is used to assess the voltage instability or in otherwords to assess the proximity of the existing system state fromvoltage collapse the following are the major highlights of thiswork
(a) GVSM for the given power system networks arecalculated to judge the health of the power system
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Advances in Electrical Engineering
Table 4 GVSM for IEEE 118-bus test system at unknown different operating loading incorporating noise
Value of GVSM for unknown loading at normal caseNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50112 minus1762 5133 minus64245 minus19971 5118 02715 minus86754
Operating load1478 40734 4541 minus31588 41287 minus84655 minus73933 40729 01209 minus85274
Operating load15671 35154 36765 minus5303 35939 minus87183 minus82413 3515 10685 minus84834
Operating load16591 29059 18745 minus72875 305 minus75549 minus86219 29041 13871 minus84485
Operating load18751 minus46571 minus38761 minus101198 minus34453 minus32183 minus88976 minus46565 14657 minus83791
Value of GVSM for unknown loading at signal-to-noise ratio 05 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 50187 02122 51415 minus108296 minus502 5105 minus13566 minus8456
Operating load1478 40734 52299 minus94885 41286 minus80771 minus87955 40235 minus61076 minus81775
Operating load15671 35154 50041 minus85883 36213 minus52684 minus77728 3505 28002 minus79819
Operating load16591 29059 37894 minus74151 305 minus18476 minus7779 28051 07538 minus82134
Operating load18751 minus46571 minus50553 minus85239 minus34719 minus60628 minus89145 minus4637 minus01573 minus85286
Value of GVSM for unknown loading at signal-to-noise ratio 005 dbNR FFBP CFB GR LR NARX RBFN ELMAN FFDTD
Operating load1249 51187 52705 minus03736 5133 minus10638 minus59343 50081 12566 minus82898
Operating load1478 40734 52223 minus27077 41287 minus25001 minus55532 40101 minus13369 minus7759
Operating load15671 35154 3436 14016 35934 minus6853 minus8663 15013 35063 minus84106
Operating load16591 29059 19993 minus79334 30947 minus41361 minus8425 22041 2643 minus80591
Operating load18751 minus46571 minus39595 minus107149 minus34397 minus72241 minus88729 minus43073 14507 minus84814
Prediction of GVSM and CVI by different neuralnetwork topologies is validated through offline NRmethod
(b) The main advantage of the proposed method is thatit indicates a good agreement between target data (N-R) and RBFN output Prediction accuracy of RBFN isbest as compared with other topologies of ANN
(c) The proposed approach provides fast computationof GVSM Operator can analyze severity of anyunknown load pattern by using this supervised learn-ing approach
(d) Prediction accuracy of RBFN is best It is validated byintroduction of AWGN in the system with differentlevels of SNR
Application of Support Vector Machine (SVM) in voltagestability assessment lies in the scope of the future work
Nomenclature
R Resistance of the lineX Reactance of the lineZ Impedance of the line119885se eq Equivalent series impedance119885sh eq Equivalent shunt impedance119881s 119881r Voltage at sending bus and receiving
bus respectively119868s 119868r Current at sending bus and receiving
bus respectively119868se Current through series equivalent
impedance119868shs 119868shr Shunt branch currents at sending and
receiving end sides respectively119881cr Critical voltage index119878g Apparent power of generator119878load Apparent power at load end side
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 11
119875g 119876g Active and reactive power of generatorrespectively
119875r 119876r Active and reactive power at load end siderespectively
119898 119899 Nodes1198762 Reactive power at receiving bus
ldquosrdquo and ldquorrdquo Symbols for sending and receiving siderespectively
[119860 119861
119862 119863] Transmission (A B C D) parameters
120575 Voltage angle120572 Angle of magnitude of parameter 119860120573 Angle of magnitude of parameter 119861120579 Angle of receiving end voltage 119881sΔ[119869] Determinant of Jacobian matrixSNR Signal-to-noise ratioN-R Newton RaphsonGVSM Global Voltage Stability MarginMLP Multilayer PerceptronLM Levenberg-MarquardtAWGN Additive White Gaussian NoiseRBFN Radial Basis Function NetworkFFBP Feedforward BackpropCFB Cascade Forward BackpropGR Generalized RegressionLR Layer RecurrentFFDTDN Feedforward Distributed Time Delay
Network
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
The authors acknowledge the support and encouragementof Swami Keshvanand Institute of Technology Managementamp Gramothan Jaipur Rajasthan India The authors alsowould like to thank Ms Neha Purohit Reader Departmentof English Swami Keshvanand Institute of Technology Man-agement amp Gramothan for valuable suggestions
References
[1] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004
[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994
[3] P Nagendra S H nee Dey and S Paul ldquoAn innovative tech-nique to evaluate network equivalent for voltage stability assess-ment in a widespread sub-grid systemrdquo International Journalof Electrical Power amp Energy Systems vol 33 no 3 pp 737ndash7442011
[4] P Kessel and H Glavitsch ldquoEstimating the voltage stability of apower systemrdquo IEEE Transactions on Power Delivery vol 1 no3 pp 346ndash354 1986
[5] F Gubina and B Strmcnik ldquoVoltage collapse proximity indexdetermination using voltage phasors approachrdquo IEEE Transac-tions on Power Systems vol 10 no 2 pp 788ndash794 1995
[6] Y Wang W Li and J Lu ldquoA new node voltage stability indexbased on local voltage phasorsrdquo Electric Power Systems Researchvol 79 no 1 pp 265ndash271 2009
[7] S Dey C K Chanda and A Chakrabarti ldquoDevelopment of aglobal voltage security indicator (VSI) and role of SVC on it inlongitudinal power supply (LPS) systemrdquoElectric Power SystemsResearch vol 68 no 1 pp 1ndash9 2004
[8] R Tiwari K R Niazi and V Gupta ldquoLine collapse proximityindex for prediction of voltage collapse in power systemsrdquoInternational Journal of Electrical Power andEnergy Systems vol41 no 1 pp 105ndash111 2012
[9] G B Jasmon L H Callistus and C Lee ldquoPrediction of voltagecollapse in power systems using a reduced system modelrdquo inProceedings of the IEE International Conference on Control pp32ndash36 London UK March 1991
[10] H G Kwatny A K Pasrija and L Y Bahar ldquoStatic bifurcationsin electric power networks loss of steady-state stability andvoltage collapserdquo IEEETransactions onCircuits and Systems vol33 no 10 pp 981ndash991 1986
[11] V Ajjarapu and B Lee ldquoBifurcation theory and its application tononlinear dynamical phenomena in an electrical power systemrdquoIEEE Transactions on Power Systems vol 7 no 1 pp 424ndash4311992
[12] B Gao G K Morison and P Kundur ldquoVoltage stability evalua-tion usingmodal analysisrdquo IEEE Transactions on Power Systemsvol 7 no 4 pp 1529ndash1542 1992
[13] C A Canizares A C Z De Souza and V H Quintana ldquoCom-parison of performance indices for detection of proximity tovoltage collapserdquo IEEE Transactions on Power Systems vol 11no 3 pp 1441ndash1450 1996
[14] C A Canizares ldquoOn bifurcations voltage collapse and loadmodelingrdquo IEEE Transactions on Power Systems vol 10 no 1pp 512ndash522 1995
[15] S Konar D Chatterjee and S Patra ldquoVndashQ sensitivity-basedindex for assessment of dynamic voltage stability of powersystemsrdquo IET Generation Transmission amp Distribution vol 9no 7 pp 677ndash685 2015
[16] AK SharmaA Saxena andR Tiwari ldquoVoltage stability assess-ment using GVSM and preventive control using SVCrdquo Inter-national Journal of Computer Applications vol 142 no 11 pp23ndash31 2016
[17] M T Hagan H B Demuth and M H Beale Neural NetworkDesign 2nd edition 1995
[18] httpwwwmathworkscom[19] Power System Test Archive-UWEE University of Washington
httpswwweewashingtoneduresearchpstca
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of