Research ArticleAn Innovatory Method Based on Continuation Power Flow toAnalyze Power System Voltage Stability with DistributedGeneration Penetration
Le Van Dai1 Ngo Minh Khoa2 and Le Cao Quyen 34
1Faculty of Electrical Engineering Technology Industrial University of Ho Chi Minh City Ho Chi Minh City Vietnam2Faculty of Engineering and Technology Quy Nhon University Quy Nhon City Binh Dinh Vietnam3Department for Management of Science and Technology Development Ton Duc (ang University Ho Chi Minh City Vietnam4Faculty of Electrical and Electronics Engineering Ton Duc (ang University Ho Chi Minh City Vietnam
Correspondence should be addressed to Le Cao Quyen lecaoquyentdtueduvn
Received 4 June 2020 Revised 6 August 2020 Accepted 25 August 2020 Published 7 September 2020
Academic Editor Michele Scarpiniti
Copyright copy 2020 Le Van Dai et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
With the penetration of distributed generation (DG) units the power systems will face insecurity problems and voltage stabilityissues is paper proposes an innovatory method by modifying the conventional continuation power flow (CCPF) method eproposed method is realized on two prediction and correction steps to find successive load flow solutions according to a specificload scenario Firstly the tangent predictor is proposed to estimate the next predicted solution from two previous correctedsolutions And then the corrector step is proposed to determine the next corrected solution on the exact solution is correctedsolution is constrained to lie in the hyperplane running through the predicted solution orthogonal to the line from the twoprevious corrected solutions Besides once the convergence criterion is reached the procedure for cutting the step length controldown to a smaller one is proposed to be implemented e effectiveness of the proposed method is verified via numericalsimulations on three standard test systems namely IEEE 14-bus 57-bus and 118-bus and compared to the CCPF method
1 Introduction
11 Motivation In recent years distributed generations(DGs) as renewable energy sources connected to powersystem networks have been growing rapidly Generallythe units of DG have a small size compared to existingcentral power plants they have little effect if their pen-etration level is low at about 1 to 5 However if thepenetration level is about 20 to 30 the impact of DGunits will be deep [1] With a rapid increase in the DGspenetration level the electric power system stability canchange and it is affecting the voltage profile reliabilityprotection stability power quality and power flow [2 3]among them the voltage stability problem is an impor-tant aspect
e voltage stability is an important problem that hasbeen reported for several years and a lot of papers and
reports of high quality on this subject have been publishedby well-known publishers According to the CIGRE StudyCommittee and IEEE Power System Dynamic Perfor-mance Committee that have set up a joint task force thetopic of voltage stability is classified into two types that arethe large and small disturbances considering the short andlong terms [4] e impact of DGs on the voltage stabilityin the long and short terms is examined in [5ndash9] re-spectively In addition the DGsrsquo optimal allocationconsidering uncertainty in load demand and growth isstudied in [10ndash12]
e voltage stability is the efficiency of an electric systemthat voltage at all buses can maintain steady acceptablevalues under normal operating conditions for all the powersystem depending on a disturbance [13] e voltage in-stability occurs when load dynamics struggle to restorepower consumption beyond the possibilities of the
HindawiComplexityVolume 2020 Article ID 8037837 15 pageshttpsdoiorg10115520208037837
generation system So it could lead to the trip of loads andortransmission lines and the loss of synchronism of somegenerators [14]
During the last periods numerous blackouts of thepower system in the world relating to the voltage insta-bility problems have been occurring in various countriessuch as Japan India the USA France Sweden andGermany It is worth noting that the blackouts have oc-curred in the Greek island of Kefalonia on January 242006 [15] the aftereffect is the cut of electricity energy tobe approximately 3000 MWh e 61800MW capacitylost lasted around 4 to 7 days due to occurring blackoutsthroughout Ohio Michigan Pennsylvania New YorkVermont Massachusetts Connecticut New Jersey andOntario Canada [16] affecting more than 50 millionpeople A blackout affecting the normal life and work of670 million people happened on the power system ofnorthern India on July 30 and 31 2012 [17] Anotherblackout affecting more than 10 million customershappened on the Brazilian power system leading to heavyeconomic losses [18 19]
12 Literature Review e blackouts in the power systemcan cause great damage to socioeconomic and broughtinconvenience to people erefore the need for a trust-worthy method to assess voltage stability in a power systemis developed To achieve this a number of methods havebeen used in the literature for voltage stability analysis Forexample the active power-voltage (P-V) curve developed in[20] and the reactive power-voltage (Q-V) curve introducedin [13] e shortcomings of these two methods are noinformation about key contributing factors for voltagestability problems and high computational time for a largepower system network To overcome these shortcomingsthe continuation power flow (CPF) method has been pro-posed in [21] Actually CPF is also time-consuming for alarge power system network even though it has a lot ofstrong points based on a prediction-correction technique tofind the continuous power flow solutions that start from thebase-load condition to the limit of steady-state voltagestability [22]
e aforesaid methods based on the power flow as themodal analysis have been introduced in [23] is pro-posed method depends on the power flow Jacobianmatrix in which the active power is kept constant and thereactive power margin and voltage instability contributeas factors resulting in the reduced Jacobian matrix of thesystem for computation So that it is more suitable for thelarge power system e methods have used voltagestability indices including the fast voltage stability index[24] line stability index [25] voltage reactive powerindex [26] and L index [27] ese methods are helpful todetermine the proximity of a given operating point to avoltage collapse point but the computational steps andtime are heavy erefore their main disadvantage istime-consuming to calculate for a large power systemnetwork and they can only predict the voltage collapsepoint on the power network [28 29]
13 Contribution and Study Organization is paper pro-poses a feasible method namely the modified continuationpower flow (MCPF) method to analyze the voltage stabilityproblem to the electric power system considering thepenetration of DG units is proposed method is realizedon the base of the conventional continuation power flow(CCPF) method considering the prediction and correctionsteps to find successive load flow solutions according to theload scenarios A next real solution is determined based ontwo previously defined adjacent solutions using the pre-dictor technique based on the arc-length parameterizationand the correction technique based on the local parame-terization e process of next solutions on the whole P-Vcurve will continue until determining the critical solutioncorresponding to the load parameter equal to the criticalBesides in exceptional cases if the real solution is notidentified by the correction step the previous prediction stepshould be adjusted until the real solution is determined
To verify the effectiveness of the proposed method threestandard test systems namely IEEE 14-bus 57-bus and118-bus are proposed to perform the different test casesBased on the obtained results this paper has the maincontributions as follows
(i) Can exactly determine the voltage collapse point foranalyzing voltage stability for large power systemsand especially for the large power systems with DGspenetration
(ii) Cut the number of predictor-corrector steps andreduce computational time
e remaining parts of the paper can be divided into foursections as follows e voltage stability problem is for-mulated in Section 2 to provide inputs to develop theanalysis method Section 3 recalls the CCPF method anddevelops a novel method for analyzing the voltage stabilityissue e effectiveness of the proposed method is verified inSection 4 Finally the conclusions are given in Section 5
2 Voltage Stability Problem
21 Problem Formulation e power flow is an importantproblem in the field of power system engineering where thevoltage magnitudes and angles at buses are desired and thevoltage magnitudes and power levels at other buses areknown considering the available mode of the networkconfiguratione power flow calculation aims to determinevoltage magnitudes and angles for a given generation loadand grid condition e starting step of resolving the powerflow problem is to determine the unknown and knownvariables in the power system Based on these variables asknown in the power systems there exist three bus types thatare the load bus (PQ bus) in which the P andQ variables aredetailed andU and δ variables are necessary to be solved thegeneration bus (PV bus) in which the P and U variables aredetailed and theU and δ variables are necessary to be solvedand the slack bus (swing bus) in which theU and δ variablesare detailed and the P and Q variables are necessary to besolved It can be observed that for the power system havingn buses and g generators (including DG units) the number
2 Complexity
of unknowns which is 2(n minus 1) minus (g minus 1) can be calculatedby using the reactive and active power balance equationserefore the active and reactive power balance equationsinjected into the network at bus j can be defined as follows[30]
Pj PGenerationj minus PLoadj
Qj QGenerationj minus QLoadj
⎧⎨
⎩ (1)
where PGenerationj and QGenerationj are the generatorrsquos activeand reactive powers at bus j respectively PLoadj and QLoadj
are the loadrsquos active and reactive powers at bus j respectivelye complex power at bus j can be given by
Sj Pj + jQj UjIlowastj (2)
where Uj is the voltage at bus j and Ij is the current injectedinto bus j which can be written as follows
Ij 1113944n
k1YjkUk (3)
where Uk is the voltage at bus k and equation (3) can berewritten under the following relationship between the busvoltage and current as follows
I1
I2
⋮
In
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
Y11 Y12 middot middot middot Y1n
Y21 Y22 middot middot middot Y2n
⋮ ⋮ ⋱ ⋮
Yn1 Yn2 middot middot middot Ynn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
U1
U2
⋮
Un
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(4)
Substituting equation (3) into equation (2) one gets
Sj Pj + jQj Uj 1113944
n
k1YjkUk
⎡⎣ ⎤⎦lowast
(5)
where Yjk is mutual admittance between buses j and k eactive and reactive power balance equation at bus j can bewritten as follows
Pj 1113944n
k1Uj
11138681113868111386811138681113868
11138681113868111386811138681113868 Uk
11138681113868111386811138681113868111386811138681113868 Gjk cos δjk + Bjk sin δjk1113872 1113873
Qj 1113944n
k1Uj
11138681113868111386811138681113868
11138681113868111386811138681113868 Uk
11138681113868111386811138681113868111386811138681113868 Gjk sin δjk minus Bjk cos δjk1113872 1113873
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(6)
where Gjk and Bjk are respectively the real andimaginary part of the element in the bus admittance matrixcorresponding to the jth row and kth column e voltagephase angle between buses j and k is δjk δj minus δk in whichδj and δk are the voltage phase angles at buses j and krespectively
22 Analyses e CPF method is a numerical techniqueused to overcome the problems of the Newton-Raphsontechnique such that it is more reliable for obtaining thesolution curve specifically to the ill-conditioned power flowequations and faster via the effective predictor and correctorscheme step length control and parameterization
e CPF determines the consecutive load flow solutionsunder a load scenario using the prediction and correctionsteps e prediction-correction procedure is illustrated inFigure 1 It can be summarized as from a previous knownsolution the tangent predictor step is used to determine anext predictor solution under considering the increase of theloading factor and then the predictor step is applied to findthe exact solution And after that based on the new tangentvector a new prediction for a load increase is built isprocess will be finished when the critical point is reached inother words the critical point is one where tangent vector iszero
e CPF is performed based on the predictor-correctorprocedure via the following system of nonlinear equation[22]
F(x) 0 (7)
where
x [δ U]T (8)
e nonlinear power flow equation in equation (8) isaugmented by the loading factor λ as follows
F(x λ) 0 0le λle λcritical (9)
where U is the vector of bus voltage magnitudes and δ is thevector of bus voltage angles e dimension of F is 2n + g asmentioned n denotes the number of PQ buses and g de-notes the number of PV buses
e power flow equation in equation (1) is rewrittencomprising a loading factor λ as follows
where PLoad oj and QLoad oj are respectively the originalload demands at bus j PGeneration oj is the original generationdemands at bus j kGenerationi is the generation factor at bus jand c is the generator participation coefficient
From equation (9) the base solution corresponding tothe loading factor equal to zero started to determine the nextsolutions corresponding to different load levels and thecritical solution corresponding to the load parameter equalto the critical is can be shown from the tracing of the PVcurve based on the predictor-corrector scheme as shown inFigure 1 e predictor-corrector continuation processprocedure can be realized in two steps
221 (e First Step is step realizes the predictor processto calculate the tangent vector and can be expressed in thefollowing form
t [dx dλ]T (11)
e tangent vector t with a corresponding dimension2n + g + 1 is determined from the following form
Complexity 3
Fx Fλ1113858 1113859t 0 (12)
where Fx and Fλ are the partial derivative of equation (9)with respect to x and λ respectively and T denotes thetranspose operation
From equation (12) the left side is a partial derivativematrix that this matrix is nothing but the original Jacobianaugmented by one column [Fλ] of an unknown variable ofload λ and a vector of differentials t is the differentialsrepresenting the tangent vector Since an unknown variableof the load is added to the nonlinear system (9) this will leadthe augmented Jacobian to be singular at the point ofmaximum possible system load In order to solve thisproblem the tangent vector should be determined based onthe satisfied augment Jacobian as follows
FxFλ
ei
1113890 1113891t 0
ti
1113890 1113891 (13)
where ei is a row vector having an appropriate dimension inwhich the ith element is the only nonzero element
From equation (13) ti is a nonzero element of thetangent vector this parameter can be either +1 or minus 1depending on how the ith state variable is changing as asolution that is being traced An estimate of the next solutioncan be determined by solving system (13) as follows
x(e)
λ(e)⎡⎣ ⎤⎦
x(i)
λ(i)⎡⎣ ⎤⎦ + σ
dx
dλ1113890 1113891 (14)
where the subscripts ldquoirdquo and ldquoerdquo denote the current andestimated solutions of the next step respectively and σ is ascalar defining the predictor step length control
222 (e Second Step After the prediction is finished if thepredictor solution may not be exactly on a desired solutioncurve this step is realized to correct the predicted solutionthat is determined in equation (14) using a technique
namely the local parameterization Now for the correctorstep system (9) is introduced by one equation unit with thepurpose of determining the value of one of the bus voltagemagnitude bus voltage angle and loading factor Hence thenew system is expressed as follows
F(x λ)
xi minus μ1113890 1113891
0
01113890 1113891 (15)
where xi is the ith state variable that is selected as thecontinuation parameter and μ is the predicted value of xi
Equation (15) shows that it is involved due to one ad-ditional equation and a state variable In order to solve thisdifficulty this paper reproposes Newtonrsquos method for thissecond step as follows
FxFλ
ei
1113890 1113891Δx
Δλ1113890 1113891
F(x λ)
xi minus η1113890 1113891 (16)
3 Proposed Method
As analyzed in Section 2 when using the CCPF firstly thetangential predictor step with the P-V curve at the previoussolution with a constant step length control σ as seen inequation (14) is applied en the corrector step based onthe parameterization method is used by solving equation(16) Obviously using the tangential predictive method witha constant step length control value may increase thenumber of predictor and corrector steps and lead to non-convergence when solving equation (16)
In order to overcome these disadvantages this paperproposed a method by using the scant method to performthe predictive solution e executorial predictor-correctorprocedure is illustrated in Figure 2 in which it will help theprocess of finding solutions to the critical point to be fasteras shown in Figure 2(a) If in case that has not found the nextsolution from the previous solution the step length controlwill be cut down and resumed using the corrector step to thenext corrected solution as shown in Figure 2(b) e processof finding solutions to the proposed method is presented inthe following
31 Predictor e secant method and arc-length parame-terization are used to realize this step e function of thepredictor is to find an approximate point for the next so-lution Figure 2(a) shows that if the continuation process atith step is the current corrector solution (x(i)
c λ(i)c ) the
predictor solution will be found in an approximate point(x(i+1)
p λ(i+1)p ) for the next corrector solution (x(iminus 1)
c λ(iminus 1)c )
and can be obtained as follows
x(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ + σ(i)x
(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(iminus 1)c
λ(iminus 1)c
⎡⎢⎢⎣ ⎤⎥⎥⎦⎛⎝ ⎞⎠ (17)
From equation (17) the number of iterations is used toperform the prediction process for finding solutions on theP-V curve depending on the distance between the two so-lutions (x(i)
c λ(i)c ) and (x(iminus 1)
c λ(iminus 1)c ) However the process of
Criticalpoint
Predictor
Corrector
λ
U
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
λcritical0
Figure 1 e predictor-corrector procedure of continuationpower flow
4 Complexity
finding solutions can quickly reach the voltage collapse pointor not depending on the step length control σ(i) e processof corrector may be ineffective if the step length control σ(i)
is too large the process of the previous solution will be keptSo that the step length control needs to be adjusted in orderto find a new solution around the critical point λcritical
32 Corrector From Figure 2(a) in order to find the nextcorrect solution (x(i+1)
c λ(i+1)c ) the corrector step is per-
formed by adding a line equation that is perpendicular toline connected between the two previous correct solutions(x(iminus 1)
c λ(iminus 1)c ) and (x(i)
c λ(i)c ) at the prediction solution (x(i+1)
p λ(i+1)
p ) e added line equation is expressed as follows
ρ(x λ) x minus x(i+1)
p
λ minus λ(i+1)p
⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦
T
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ 0 (18)
e actual correct solution (x(i+1)c λ(i+1)
c ) on the bifur-cation manifold can be calculated on the following set ofequations (9) and (18) for x and λ as follows
F(x λ) 0 0le λle λcritical
ρ(x λ) 01113896 (19)
where ρ() is the additional equation that is expressed inequation (18)
Equation (19) can also be solved by using a slightlymodified Newton-Raphson power flow method Howeverthis corrector step cannot converge if the step length controlis too large To overcome this limit this paper proposed aprocedure for cutting the step length control down to asmaller one until the convergence criteria are reached asshown in Figure 2(b)
e flowchart of the continuation power flow is shown inFigure 3 and can be explained in detail step by step asfollows
Step 1 Read all input data including line data and busdata of the power systemStep 2 Set the initial parameters including the initialsolution [δ U λ](0) [0 1 0] and the initial steplength control σ 1 note that x [δ U]TStep 3 Calculate the initial power flow with equation(3) using the Newton-Raphson method and update theinitial solution [δ U λ](0) note that x [δ U]TStep 4 Predict find the (i+1)th predicted solution withequation (17)Step 5 Correct find the (i+1)th corrected solution bysolving equation (19)Step 6 If the convergence criterion is reached the steplength control will be cut using equation σ σ2 and goto Step 4 If the convergence criterion is not reached itwill go to Step 7Step 7 If the critical point is not reached it will go toStep 4Step 8 If the critical point is reached it stops and putsthe P-V curves
33VoltageStability Index e voltage stability condition ina power system can be represented by using the voltagestability index is index can be calculated based on thetangent vector of the P-V curve at each operating state evoltage stability degree at buses experiencing when thesystem reached the state of the voltage collapse is consid-ered us we have to find out the weakest bus with respectto the voltage stability so that we can find suitable solutionsto improve voltage stability e weak bus is one that has thegreatest ratio of differential change in voltage to differentialchange in active load for the whole system Using thereformulated power flow equations the differential changein active system load is given as follows [19]
Critical point
x
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xc(i) λc(i))
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(a)
Critical point
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xp(i+1) λp(i+1))(xc(i) λc(i))
Successful corrector
Unsuccessful corrector
x
Predictorstep cut
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(b)
Figure 2 e secant predictor-corrector procedure of the proposed method (a) smooth function and (b) flat corner
Complexity 5
dPtotal 1113944n
j1dPLoadj SΔbase 1113944
n
j1kLoadj cos φj
⎡⎢⎢⎣ ⎤⎥⎥⎦dλ Cdλ
(20)
where dPLoadj is the differential change in load jth kLoadj isthe multiplier to designate the rate of load change at bus jthas λ changes φj is the power factor angle of load change atbus jth and SΔbase is the apparent power selected to provideappropriate scaling of λ e weakest bus hth is determinedas follows
When the weakest bus hth reaches its steady-statevoltage stability limit the differential of changes dλ closes tozero and the ratio of dVhCdλ will become infinite isratio is defined as the voltage stability index at the bus hthe ratio of dVhCdλ in equation (21) shows the magnitudecomparison of the elements dVjCdλ of the tangent vector ofthe P-V curve at each operating state If the ratio of
differential change in voltage at a bus to differential changein the connected load is the largest one this bus indicates theweakest one
4 Study and Simulation Results
To evaluate the effectiveness of the proposed method thenumerical case studies were examined through using threeIEEE 14-bus 57-bus and 118-bus systems which are con-sidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively e detailed data of these systems have beenintroduced in [31ndash33] All dynamic models such as gener-ators excitation systems transmission lines and load aremodeled based on [13] For each system the case studies areexamined considering conditions as with or without thepenetration of DGs in which the DG is used to be the doublyfed induction generator-based variable-speed wind turbines(DFIG-VSWT) with total installed capacity being 20MW ateach bus is bus is chosen based on the condition of theplaced location to be far in comparison with the existinggeneration sources Note also that in this paper the DG issuggested to be a pure distributed source with the constantpower model and its model and structure are introduced in[34] e placed locations of DGs at buses of each testedsystem are listed in Table 1e numerical simulation resultsare realized in the following case studies
41 Test on IEEE 14-Bus System e IEEE 14-bus systemwhich is considered as the small-scale network case has twogeneration buses three synchronous compensators at buses3 6 and 8 used only for supporting reactive powers elevenload buses three static compensators and three trans-formers e total load demand is 259MW and 735 MVArMore details on this system can be also found in [33]
In this case study we choose the load bus 5 to install the20MW DG e P-V curve of load bus 14 is shown inFigure 4 using CCPF and MCPF methods to analyze voltagestability considering either penetrating the installed the20MW DG at bus 5 or not and the numerical results aresummarized in Table 2
From the P-V curve in Figure 4 for the case of beingwithout DG using the CCPF method it takes too manypredictor-corrector steps to pass the bifurcation pointfrom the numerical results in Table 2 it takes 41 pre-dictor-corrector steps totally and spends 081162 secondsfinishing the critical point calculation Meanwhile usingMCPF it takes 32 predictor-corrector steps and spends073061 seconds e maximum loading factor (λcritical) atthe critical point when using CCPF is 27437 which issmall compared to that using MCPF which is 28178 Incase of being with DG it takes 37 predictor-correctorsteps totally and spends 086893 seconds finishing thecritical point calculation Meanwhile using MCPF ittakes 25 predictor-corrector steps and spends 075092seconds e maximum loading factor (λcritical) at thecritical point when using CCPF is 36397 which is smallcompared to that using MCPF which is 37412 In
Start
Calculate initial power flow with equation (3)using Newton-Raphson method and update initial
solution [δ U λ](0)
Check ifconvergence criterion
is reached
Cut step length control σi = σi2
No
Set initial parametersInitial solution [δ U λ](0) = [0 1 0]Step length control σ = 1
Note x = [δ U]T
PredictorAt (i + 1) using equation (17)
Corrector At (i + 1) using equation (19)
Check ifcritical point
is reached
Yes
End
No
Yes
Read line and bus data of power system
(i)(ii)
Figure 3e flowchart of voltage stability analyses of the proposedmethod
6 Complexity
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
generation system So it could lead to the trip of loads andortransmission lines and the loss of synchronism of somegenerators [14]
During the last periods numerous blackouts of thepower system in the world relating to the voltage insta-bility problems have been occurring in various countriessuch as Japan India the USA France Sweden andGermany It is worth noting that the blackouts have oc-curred in the Greek island of Kefalonia on January 242006 [15] the aftereffect is the cut of electricity energy tobe approximately 3000 MWh e 61800MW capacitylost lasted around 4 to 7 days due to occurring blackoutsthroughout Ohio Michigan Pennsylvania New YorkVermont Massachusetts Connecticut New Jersey andOntario Canada [16] affecting more than 50 millionpeople A blackout affecting the normal life and work of670 million people happened on the power system ofnorthern India on July 30 and 31 2012 [17] Anotherblackout affecting more than 10 million customershappened on the Brazilian power system leading to heavyeconomic losses [18 19]
12 Literature Review e blackouts in the power systemcan cause great damage to socioeconomic and broughtinconvenience to people erefore the need for a trust-worthy method to assess voltage stability in a power systemis developed To achieve this a number of methods havebeen used in the literature for voltage stability analysis Forexample the active power-voltage (P-V) curve developed in[20] and the reactive power-voltage (Q-V) curve introducedin [13] e shortcomings of these two methods are noinformation about key contributing factors for voltagestability problems and high computational time for a largepower system network To overcome these shortcomingsthe continuation power flow (CPF) method has been pro-posed in [21] Actually CPF is also time-consuming for alarge power system network even though it has a lot ofstrong points based on a prediction-correction technique tofind the continuous power flow solutions that start from thebase-load condition to the limit of steady-state voltagestability [22]
e aforesaid methods based on the power flow as themodal analysis have been introduced in [23] is pro-posed method depends on the power flow Jacobianmatrix in which the active power is kept constant and thereactive power margin and voltage instability contributeas factors resulting in the reduced Jacobian matrix of thesystem for computation So that it is more suitable for thelarge power system e methods have used voltagestability indices including the fast voltage stability index[24] line stability index [25] voltage reactive powerindex [26] and L index [27] ese methods are helpful todetermine the proximity of a given operating point to avoltage collapse point but the computational steps andtime are heavy erefore their main disadvantage istime-consuming to calculate for a large power systemnetwork and they can only predict the voltage collapsepoint on the power network [28 29]
13 Contribution and Study Organization is paper pro-poses a feasible method namely the modified continuationpower flow (MCPF) method to analyze the voltage stabilityproblem to the electric power system considering thepenetration of DG units is proposed method is realizedon the base of the conventional continuation power flow(CCPF) method considering the prediction and correctionsteps to find successive load flow solutions according to theload scenarios A next real solution is determined based ontwo previously defined adjacent solutions using the pre-dictor technique based on the arc-length parameterizationand the correction technique based on the local parame-terization e process of next solutions on the whole P-Vcurve will continue until determining the critical solutioncorresponding to the load parameter equal to the criticalBesides in exceptional cases if the real solution is notidentified by the correction step the previous prediction stepshould be adjusted until the real solution is determined
To verify the effectiveness of the proposed method threestandard test systems namely IEEE 14-bus 57-bus and118-bus are proposed to perform the different test casesBased on the obtained results this paper has the maincontributions as follows
(i) Can exactly determine the voltage collapse point foranalyzing voltage stability for large power systemsand especially for the large power systems with DGspenetration
(ii) Cut the number of predictor-corrector steps andreduce computational time
e remaining parts of the paper can be divided into foursections as follows e voltage stability problem is for-mulated in Section 2 to provide inputs to develop theanalysis method Section 3 recalls the CCPF method anddevelops a novel method for analyzing the voltage stabilityissue e effectiveness of the proposed method is verified inSection 4 Finally the conclusions are given in Section 5
2 Voltage Stability Problem
21 Problem Formulation e power flow is an importantproblem in the field of power system engineering where thevoltage magnitudes and angles at buses are desired and thevoltage magnitudes and power levels at other buses areknown considering the available mode of the networkconfiguratione power flow calculation aims to determinevoltage magnitudes and angles for a given generation loadand grid condition e starting step of resolving the powerflow problem is to determine the unknown and knownvariables in the power system Based on these variables asknown in the power systems there exist three bus types thatare the load bus (PQ bus) in which the P andQ variables aredetailed andU and δ variables are necessary to be solved thegeneration bus (PV bus) in which the P and U variables aredetailed and theU and δ variables are necessary to be solvedand the slack bus (swing bus) in which theU and δ variablesare detailed and the P and Q variables are necessary to besolved It can be observed that for the power system havingn buses and g generators (including DG units) the number
2 Complexity
of unknowns which is 2(n minus 1) minus (g minus 1) can be calculatedby using the reactive and active power balance equationserefore the active and reactive power balance equationsinjected into the network at bus j can be defined as follows[30]
Pj PGenerationj minus PLoadj
Qj QGenerationj minus QLoadj
⎧⎨
⎩ (1)
where PGenerationj and QGenerationj are the generatorrsquos activeand reactive powers at bus j respectively PLoadj and QLoadj
are the loadrsquos active and reactive powers at bus j respectivelye complex power at bus j can be given by
Sj Pj + jQj UjIlowastj (2)
where Uj is the voltage at bus j and Ij is the current injectedinto bus j which can be written as follows
Ij 1113944n
k1YjkUk (3)
where Uk is the voltage at bus k and equation (3) can berewritten under the following relationship between the busvoltage and current as follows
I1
I2
⋮
In
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
Y11 Y12 middot middot middot Y1n
Y21 Y22 middot middot middot Y2n
⋮ ⋮ ⋱ ⋮
Yn1 Yn2 middot middot middot Ynn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
U1
U2
⋮
Un
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(4)
Substituting equation (3) into equation (2) one gets
Sj Pj + jQj Uj 1113944
n
k1YjkUk
⎡⎣ ⎤⎦lowast
(5)
where Yjk is mutual admittance between buses j and k eactive and reactive power balance equation at bus j can bewritten as follows
Pj 1113944n
k1Uj
11138681113868111386811138681113868
11138681113868111386811138681113868 Uk
11138681113868111386811138681113868111386811138681113868 Gjk cos δjk + Bjk sin δjk1113872 1113873
Qj 1113944n
k1Uj
11138681113868111386811138681113868
11138681113868111386811138681113868 Uk
11138681113868111386811138681113868111386811138681113868 Gjk sin δjk minus Bjk cos δjk1113872 1113873
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(6)
where Gjk and Bjk are respectively the real andimaginary part of the element in the bus admittance matrixcorresponding to the jth row and kth column e voltagephase angle between buses j and k is δjk δj minus δk in whichδj and δk are the voltage phase angles at buses j and krespectively
22 Analyses e CPF method is a numerical techniqueused to overcome the problems of the Newton-Raphsontechnique such that it is more reliable for obtaining thesolution curve specifically to the ill-conditioned power flowequations and faster via the effective predictor and correctorscheme step length control and parameterization
e CPF determines the consecutive load flow solutionsunder a load scenario using the prediction and correctionsteps e prediction-correction procedure is illustrated inFigure 1 It can be summarized as from a previous knownsolution the tangent predictor step is used to determine anext predictor solution under considering the increase of theloading factor and then the predictor step is applied to findthe exact solution And after that based on the new tangentvector a new prediction for a load increase is built isprocess will be finished when the critical point is reached inother words the critical point is one where tangent vector iszero
e CPF is performed based on the predictor-correctorprocedure via the following system of nonlinear equation[22]
F(x) 0 (7)
where
x [δ U]T (8)
e nonlinear power flow equation in equation (8) isaugmented by the loading factor λ as follows
F(x λ) 0 0le λle λcritical (9)
where U is the vector of bus voltage magnitudes and δ is thevector of bus voltage angles e dimension of F is 2n + g asmentioned n denotes the number of PQ buses and g de-notes the number of PV buses
e power flow equation in equation (1) is rewrittencomprising a loading factor λ as follows
where PLoad oj and QLoad oj are respectively the originalload demands at bus j PGeneration oj is the original generationdemands at bus j kGenerationi is the generation factor at bus jand c is the generator participation coefficient
From equation (9) the base solution corresponding tothe loading factor equal to zero started to determine the nextsolutions corresponding to different load levels and thecritical solution corresponding to the load parameter equalto the critical is can be shown from the tracing of the PVcurve based on the predictor-corrector scheme as shown inFigure 1 e predictor-corrector continuation processprocedure can be realized in two steps
221 (e First Step is step realizes the predictor processto calculate the tangent vector and can be expressed in thefollowing form
t [dx dλ]T (11)
e tangent vector t with a corresponding dimension2n + g + 1 is determined from the following form
Complexity 3
Fx Fλ1113858 1113859t 0 (12)
where Fx and Fλ are the partial derivative of equation (9)with respect to x and λ respectively and T denotes thetranspose operation
From equation (12) the left side is a partial derivativematrix that this matrix is nothing but the original Jacobianaugmented by one column [Fλ] of an unknown variable ofload λ and a vector of differentials t is the differentialsrepresenting the tangent vector Since an unknown variableof the load is added to the nonlinear system (9) this will leadthe augmented Jacobian to be singular at the point ofmaximum possible system load In order to solve thisproblem the tangent vector should be determined based onthe satisfied augment Jacobian as follows
FxFλ
ei
1113890 1113891t 0
ti
1113890 1113891 (13)
where ei is a row vector having an appropriate dimension inwhich the ith element is the only nonzero element
From equation (13) ti is a nonzero element of thetangent vector this parameter can be either +1 or minus 1depending on how the ith state variable is changing as asolution that is being traced An estimate of the next solutioncan be determined by solving system (13) as follows
x(e)
λ(e)⎡⎣ ⎤⎦
x(i)
λ(i)⎡⎣ ⎤⎦ + σ
dx
dλ1113890 1113891 (14)
where the subscripts ldquoirdquo and ldquoerdquo denote the current andestimated solutions of the next step respectively and σ is ascalar defining the predictor step length control
222 (e Second Step After the prediction is finished if thepredictor solution may not be exactly on a desired solutioncurve this step is realized to correct the predicted solutionthat is determined in equation (14) using a technique
namely the local parameterization Now for the correctorstep system (9) is introduced by one equation unit with thepurpose of determining the value of one of the bus voltagemagnitude bus voltage angle and loading factor Hence thenew system is expressed as follows
F(x λ)
xi minus μ1113890 1113891
0
01113890 1113891 (15)
where xi is the ith state variable that is selected as thecontinuation parameter and μ is the predicted value of xi
Equation (15) shows that it is involved due to one ad-ditional equation and a state variable In order to solve thisdifficulty this paper reproposes Newtonrsquos method for thissecond step as follows
FxFλ
ei
1113890 1113891Δx
Δλ1113890 1113891
F(x λ)
xi minus η1113890 1113891 (16)
3 Proposed Method
As analyzed in Section 2 when using the CCPF firstly thetangential predictor step with the P-V curve at the previoussolution with a constant step length control σ as seen inequation (14) is applied en the corrector step based onthe parameterization method is used by solving equation(16) Obviously using the tangential predictive method witha constant step length control value may increase thenumber of predictor and corrector steps and lead to non-convergence when solving equation (16)
In order to overcome these disadvantages this paperproposed a method by using the scant method to performthe predictive solution e executorial predictor-correctorprocedure is illustrated in Figure 2 in which it will help theprocess of finding solutions to the critical point to be fasteras shown in Figure 2(a) If in case that has not found the nextsolution from the previous solution the step length controlwill be cut down and resumed using the corrector step to thenext corrected solution as shown in Figure 2(b) e processof finding solutions to the proposed method is presented inthe following
31 Predictor e secant method and arc-length parame-terization are used to realize this step e function of thepredictor is to find an approximate point for the next so-lution Figure 2(a) shows that if the continuation process atith step is the current corrector solution (x(i)
c λ(i)c ) the
predictor solution will be found in an approximate point(x(i+1)
p λ(i+1)p ) for the next corrector solution (x(iminus 1)
c λ(iminus 1)c )
and can be obtained as follows
x(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ + σ(i)x
(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(iminus 1)c
λ(iminus 1)c
⎡⎢⎢⎣ ⎤⎥⎥⎦⎛⎝ ⎞⎠ (17)
From equation (17) the number of iterations is used toperform the prediction process for finding solutions on theP-V curve depending on the distance between the two so-lutions (x(i)
c λ(i)c ) and (x(iminus 1)
c λ(iminus 1)c ) However the process of
Criticalpoint
Predictor
Corrector
λ
U
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
λcritical0
Figure 1 e predictor-corrector procedure of continuationpower flow
4 Complexity
finding solutions can quickly reach the voltage collapse pointor not depending on the step length control σ(i) e processof corrector may be ineffective if the step length control σ(i)
is too large the process of the previous solution will be keptSo that the step length control needs to be adjusted in orderto find a new solution around the critical point λcritical
32 Corrector From Figure 2(a) in order to find the nextcorrect solution (x(i+1)
c λ(i+1)c ) the corrector step is per-
formed by adding a line equation that is perpendicular toline connected between the two previous correct solutions(x(iminus 1)
c λ(iminus 1)c ) and (x(i)
c λ(i)c ) at the prediction solution (x(i+1)
p λ(i+1)
p ) e added line equation is expressed as follows
ρ(x λ) x minus x(i+1)
p
λ minus λ(i+1)p
⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦
T
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ 0 (18)
e actual correct solution (x(i+1)c λ(i+1)
c ) on the bifur-cation manifold can be calculated on the following set ofequations (9) and (18) for x and λ as follows
F(x λ) 0 0le λle λcritical
ρ(x λ) 01113896 (19)
where ρ() is the additional equation that is expressed inequation (18)
Equation (19) can also be solved by using a slightlymodified Newton-Raphson power flow method Howeverthis corrector step cannot converge if the step length controlis too large To overcome this limit this paper proposed aprocedure for cutting the step length control down to asmaller one until the convergence criteria are reached asshown in Figure 2(b)
e flowchart of the continuation power flow is shown inFigure 3 and can be explained in detail step by step asfollows
Step 1 Read all input data including line data and busdata of the power systemStep 2 Set the initial parameters including the initialsolution [δ U λ](0) [0 1 0] and the initial steplength control σ 1 note that x [δ U]TStep 3 Calculate the initial power flow with equation(3) using the Newton-Raphson method and update theinitial solution [δ U λ](0) note that x [δ U]TStep 4 Predict find the (i+1)th predicted solution withequation (17)Step 5 Correct find the (i+1)th corrected solution bysolving equation (19)Step 6 If the convergence criterion is reached the steplength control will be cut using equation σ σ2 and goto Step 4 If the convergence criterion is not reached itwill go to Step 7Step 7 If the critical point is not reached it will go toStep 4Step 8 If the critical point is reached it stops and putsthe P-V curves
33VoltageStability Index e voltage stability condition ina power system can be represented by using the voltagestability index is index can be calculated based on thetangent vector of the P-V curve at each operating state evoltage stability degree at buses experiencing when thesystem reached the state of the voltage collapse is consid-ered us we have to find out the weakest bus with respectto the voltage stability so that we can find suitable solutionsto improve voltage stability e weak bus is one that has thegreatest ratio of differential change in voltage to differentialchange in active load for the whole system Using thereformulated power flow equations the differential changein active system load is given as follows [19]
Critical point
x
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xc(i) λc(i))
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(a)
Critical point
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xp(i+1) λp(i+1))(xc(i) λc(i))
Successful corrector
Unsuccessful corrector
x
Predictorstep cut
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(b)
Figure 2 e secant predictor-corrector procedure of the proposed method (a) smooth function and (b) flat corner
Complexity 5
dPtotal 1113944n
j1dPLoadj SΔbase 1113944
n
j1kLoadj cos φj
⎡⎢⎢⎣ ⎤⎥⎥⎦dλ Cdλ
(20)
where dPLoadj is the differential change in load jth kLoadj isthe multiplier to designate the rate of load change at bus jthas λ changes φj is the power factor angle of load change atbus jth and SΔbase is the apparent power selected to provideappropriate scaling of λ e weakest bus hth is determinedas follows
When the weakest bus hth reaches its steady-statevoltage stability limit the differential of changes dλ closes tozero and the ratio of dVhCdλ will become infinite isratio is defined as the voltage stability index at the bus hthe ratio of dVhCdλ in equation (21) shows the magnitudecomparison of the elements dVjCdλ of the tangent vector ofthe P-V curve at each operating state If the ratio of
differential change in voltage at a bus to differential changein the connected load is the largest one this bus indicates theweakest one
4 Study and Simulation Results
To evaluate the effectiveness of the proposed method thenumerical case studies were examined through using threeIEEE 14-bus 57-bus and 118-bus systems which are con-sidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively e detailed data of these systems have beenintroduced in [31ndash33] All dynamic models such as gener-ators excitation systems transmission lines and load aremodeled based on [13] For each system the case studies areexamined considering conditions as with or without thepenetration of DGs in which the DG is used to be the doublyfed induction generator-based variable-speed wind turbines(DFIG-VSWT) with total installed capacity being 20MW ateach bus is bus is chosen based on the condition of theplaced location to be far in comparison with the existinggeneration sources Note also that in this paper the DG issuggested to be a pure distributed source with the constantpower model and its model and structure are introduced in[34] e placed locations of DGs at buses of each testedsystem are listed in Table 1e numerical simulation resultsare realized in the following case studies
41 Test on IEEE 14-Bus System e IEEE 14-bus systemwhich is considered as the small-scale network case has twogeneration buses three synchronous compensators at buses3 6 and 8 used only for supporting reactive powers elevenload buses three static compensators and three trans-formers e total load demand is 259MW and 735 MVArMore details on this system can be also found in [33]
In this case study we choose the load bus 5 to install the20MW DG e P-V curve of load bus 14 is shown inFigure 4 using CCPF and MCPF methods to analyze voltagestability considering either penetrating the installed the20MW DG at bus 5 or not and the numerical results aresummarized in Table 2
From the P-V curve in Figure 4 for the case of beingwithout DG using the CCPF method it takes too manypredictor-corrector steps to pass the bifurcation pointfrom the numerical results in Table 2 it takes 41 pre-dictor-corrector steps totally and spends 081162 secondsfinishing the critical point calculation Meanwhile usingMCPF it takes 32 predictor-corrector steps and spends073061 seconds e maximum loading factor (λcritical) atthe critical point when using CCPF is 27437 which issmall compared to that using MCPF which is 28178 Incase of being with DG it takes 37 predictor-correctorsteps totally and spends 086893 seconds finishing thecritical point calculation Meanwhile using MCPF ittakes 25 predictor-corrector steps and spends 075092seconds e maximum loading factor (λcritical) at thecritical point when using CCPF is 36397 which is smallcompared to that using MCPF which is 37412 In
Start
Calculate initial power flow with equation (3)using Newton-Raphson method and update initial
solution [δ U λ](0)
Check ifconvergence criterion
is reached
Cut step length control σi = σi2
No
Set initial parametersInitial solution [δ U λ](0) = [0 1 0]Step length control σ = 1
Note x = [δ U]T
PredictorAt (i + 1) using equation (17)
Corrector At (i + 1) using equation (19)
Check ifcritical point
is reached
Yes
End
No
Yes
Read line and bus data of power system
(i)(ii)
Figure 3e flowchart of voltage stability analyses of the proposedmethod
6 Complexity
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
of unknowns which is 2(n minus 1) minus (g minus 1) can be calculatedby using the reactive and active power balance equationserefore the active and reactive power balance equationsinjected into the network at bus j can be defined as follows[30]
Pj PGenerationj minus PLoadj
Qj QGenerationj minus QLoadj
⎧⎨
⎩ (1)
where PGenerationj and QGenerationj are the generatorrsquos activeand reactive powers at bus j respectively PLoadj and QLoadj
are the loadrsquos active and reactive powers at bus j respectivelye complex power at bus j can be given by
Sj Pj + jQj UjIlowastj (2)
where Uj is the voltage at bus j and Ij is the current injectedinto bus j which can be written as follows
Ij 1113944n
k1YjkUk (3)
where Uk is the voltage at bus k and equation (3) can berewritten under the following relationship between the busvoltage and current as follows
I1
I2
⋮
In
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
Y11 Y12 middot middot middot Y1n
Y21 Y22 middot middot middot Y2n
⋮ ⋮ ⋱ ⋮
Yn1 Yn2 middot middot middot Ynn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
U1
U2
⋮
Un
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(4)
Substituting equation (3) into equation (2) one gets
Sj Pj + jQj Uj 1113944
n
k1YjkUk
⎡⎣ ⎤⎦lowast
(5)
where Yjk is mutual admittance between buses j and k eactive and reactive power balance equation at bus j can bewritten as follows
Pj 1113944n
k1Uj
11138681113868111386811138681113868
11138681113868111386811138681113868 Uk
11138681113868111386811138681113868111386811138681113868 Gjk cos δjk + Bjk sin δjk1113872 1113873
Qj 1113944n
k1Uj
11138681113868111386811138681113868
11138681113868111386811138681113868 Uk
11138681113868111386811138681113868111386811138681113868 Gjk sin δjk minus Bjk cos δjk1113872 1113873
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(6)
where Gjk and Bjk are respectively the real andimaginary part of the element in the bus admittance matrixcorresponding to the jth row and kth column e voltagephase angle between buses j and k is δjk δj minus δk in whichδj and δk are the voltage phase angles at buses j and krespectively
22 Analyses e CPF method is a numerical techniqueused to overcome the problems of the Newton-Raphsontechnique such that it is more reliable for obtaining thesolution curve specifically to the ill-conditioned power flowequations and faster via the effective predictor and correctorscheme step length control and parameterization
e CPF determines the consecutive load flow solutionsunder a load scenario using the prediction and correctionsteps e prediction-correction procedure is illustrated inFigure 1 It can be summarized as from a previous knownsolution the tangent predictor step is used to determine anext predictor solution under considering the increase of theloading factor and then the predictor step is applied to findthe exact solution And after that based on the new tangentvector a new prediction for a load increase is built isprocess will be finished when the critical point is reached inother words the critical point is one where tangent vector iszero
e CPF is performed based on the predictor-correctorprocedure via the following system of nonlinear equation[22]
F(x) 0 (7)
where
x [δ U]T (8)
e nonlinear power flow equation in equation (8) isaugmented by the loading factor λ as follows
F(x λ) 0 0le λle λcritical (9)
where U is the vector of bus voltage magnitudes and δ is thevector of bus voltage angles e dimension of F is 2n + g asmentioned n denotes the number of PQ buses and g de-notes the number of PV buses
e power flow equation in equation (1) is rewrittencomprising a loading factor λ as follows
where PLoad oj and QLoad oj are respectively the originalload demands at bus j PGeneration oj is the original generationdemands at bus j kGenerationi is the generation factor at bus jand c is the generator participation coefficient
From equation (9) the base solution corresponding tothe loading factor equal to zero started to determine the nextsolutions corresponding to different load levels and thecritical solution corresponding to the load parameter equalto the critical is can be shown from the tracing of the PVcurve based on the predictor-corrector scheme as shown inFigure 1 e predictor-corrector continuation processprocedure can be realized in two steps
221 (e First Step is step realizes the predictor processto calculate the tangent vector and can be expressed in thefollowing form
t [dx dλ]T (11)
e tangent vector t with a corresponding dimension2n + g + 1 is determined from the following form
Complexity 3
Fx Fλ1113858 1113859t 0 (12)
where Fx and Fλ are the partial derivative of equation (9)with respect to x and λ respectively and T denotes thetranspose operation
From equation (12) the left side is a partial derivativematrix that this matrix is nothing but the original Jacobianaugmented by one column [Fλ] of an unknown variable ofload λ and a vector of differentials t is the differentialsrepresenting the tangent vector Since an unknown variableof the load is added to the nonlinear system (9) this will leadthe augmented Jacobian to be singular at the point ofmaximum possible system load In order to solve thisproblem the tangent vector should be determined based onthe satisfied augment Jacobian as follows
FxFλ
ei
1113890 1113891t 0
ti
1113890 1113891 (13)
where ei is a row vector having an appropriate dimension inwhich the ith element is the only nonzero element
From equation (13) ti is a nonzero element of thetangent vector this parameter can be either +1 or minus 1depending on how the ith state variable is changing as asolution that is being traced An estimate of the next solutioncan be determined by solving system (13) as follows
x(e)
λ(e)⎡⎣ ⎤⎦
x(i)
λ(i)⎡⎣ ⎤⎦ + σ
dx
dλ1113890 1113891 (14)
where the subscripts ldquoirdquo and ldquoerdquo denote the current andestimated solutions of the next step respectively and σ is ascalar defining the predictor step length control
222 (e Second Step After the prediction is finished if thepredictor solution may not be exactly on a desired solutioncurve this step is realized to correct the predicted solutionthat is determined in equation (14) using a technique
namely the local parameterization Now for the correctorstep system (9) is introduced by one equation unit with thepurpose of determining the value of one of the bus voltagemagnitude bus voltage angle and loading factor Hence thenew system is expressed as follows
F(x λ)
xi minus μ1113890 1113891
0
01113890 1113891 (15)
where xi is the ith state variable that is selected as thecontinuation parameter and μ is the predicted value of xi
Equation (15) shows that it is involved due to one ad-ditional equation and a state variable In order to solve thisdifficulty this paper reproposes Newtonrsquos method for thissecond step as follows
FxFλ
ei
1113890 1113891Δx
Δλ1113890 1113891
F(x λ)
xi minus η1113890 1113891 (16)
3 Proposed Method
As analyzed in Section 2 when using the CCPF firstly thetangential predictor step with the P-V curve at the previoussolution with a constant step length control σ as seen inequation (14) is applied en the corrector step based onthe parameterization method is used by solving equation(16) Obviously using the tangential predictive method witha constant step length control value may increase thenumber of predictor and corrector steps and lead to non-convergence when solving equation (16)
In order to overcome these disadvantages this paperproposed a method by using the scant method to performthe predictive solution e executorial predictor-correctorprocedure is illustrated in Figure 2 in which it will help theprocess of finding solutions to the critical point to be fasteras shown in Figure 2(a) If in case that has not found the nextsolution from the previous solution the step length controlwill be cut down and resumed using the corrector step to thenext corrected solution as shown in Figure 2(b) e processof finding solutions to the proposed method is presented inthe following
31 Predictor e secant method and arc-length parame-terization are used to realize this step e function of thepredictor is to find an approximate point for the next so-lution Figure 2(a) shows that if the continuation process atith step is the current corrector solution (x(i)
c λ(i)c ) the
predictor solution will be found in an approximate point(x(i+1)
p λ(i+1)p ) for the next corrector solution (x(iminus 1)
c λ(iminus 1)c )
and can be obtained as follows
x(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ + σ(i)x
(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(iminus 1)c
λ(iminus 1)c
⎡⎢⎢⎣ ⎤⎥⎥⎦⎛⎝ ⎞⎠ (17)
From equation (17) the number of iterations is used toperform the prediction process for finding solutions on theP-V curve depending on the distance between the two so-lutions (x(i)
c λ(i)c ) and (x(iminus 1)
c λ(iminus 1)c ) However the process of
Criticalpoint
Predictor
Corrector
λ
U
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
λcritical0
Figure 1 e predictor-corrector procedure of continuationpower flow
4 Complexity
finding solutions can quickly reach the voltage collapse pointor not depending on the step length control σ(i) e processof corrector may be ineffective if the step length control σ(i)
is too large the process of the previous solution will be keptSo that the step length control needs to be adjusted in orderto find a new solution around the critical point λcritical
32 Corrector From Figure 2(a) in order to find the nextcorrect solution (x(i+1)
c λ(i+1)c ) the corrector step is per-
formed by adding a line equation that is perpendicular toline connected between the two previous correct solutions(x(iminus 1)
c λ(iminus 1)c ) and (x(i)
c λ(i)c ) at the prediction solution (x(i+1)
p λ(i+1)
p ) e added line equation is expressed as follows
ρ(x λ) x minus x(i+1)
p
λ minus λ(i+1)p
⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦
T
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ 0 (18)
e actual correct solution (x(i+1)c λ(i+1)
c ) on the bifur-cation manifold can be calculated on the following set ofequations (9) and (18) for x and λ as follows
F(x λ) 0 0le λle λcritical
ρ(x λ) 01113896 (19)
where ρ() is the additional equation that is expressed inequation (18)
Equation (19) can also be solved by using a slightlymodified Newton-Raphson power flow method Howeverthis corrector step cannot converge if the step length controlis too large To overcome this limit this paper proposed aprocedure for cutting the step length control down to asmaller one until the convergence criteria are reached asshown in Figure 2(b)
e flowchart of the continuation power flow is shown inFigure 3 and can be explained in detail step by step asfollows
Step 1 Read all input data including line data and busdata of the power systemStep 2 Set the initial parameters including the initialsolution [δ U λ](0) [0 1 0] and the initial steplength control σ 1 note that x [δ U]TStep 3 Calculate the initial power flow with equation(3) using the Newton-Raphson method and update theinitial solution [δ U λ](0) note that x [δ U]TStep 4 Predict find the (i+1)th predicted solution withequation (17)Step 5 Correct find the (i+1)th corrected solution bysolving equation (19)Step 6 If the convergence criterion is reached the steplength control will be cut using equation σ σ2 and goto Step 4 If the convergence criterion is not reached itwill go to Step 7Step 7 If the critical point is not reached it will go toStep 4Step 8 If the critical point is reached it stops and putsthe P-V curves
33VoltageStability Index e voltage stability condition ina power system can be represented by using the voltagestability index is index can be calculated based on thetangent vector of the P-V curve at each operating state evoltage stability degree at buses experiencing when thesystem reached the state of the voltage collapse is consid-ered us we have to find out the weakest bus with respectto the voltage stability so that we can find suitable solutionsto improve voltage stability e weak bus is one that has thegreatest ratio of differential change in voltage to differentialchange in active load for the whole system Using thereformulated power flow equations the differential changein active system load is given as follows [19]
Critical point
x
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xc(i) λc(i))
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(a)
Critical point
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xp(i+1) λp(i+1))(xc(i) λc(i))
Successful corrector
Unsuccessful corrector
x
Predictorstep cut
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(b)
Figure 2 e secant predictor-corrector procedure of the proposed method (a) smooth function and (b) flat corner
Complexity 5
dPtotal 1113944n
j1dPLoadj SΔbase 1113944
n
j1kLoadj cos φj
⎡⎢⎢⎣ ⎤⎥⎥⎦dλ Cdλ
(20)
where dPLoadj is the differential change in load jth kLoadj isthe multiplier to designate the rate of load change at bus jthas λ changes φj is the power factor angle of load change atbus jth and SΔbase is the apparent power selected to provideappropriate scaling of λ e weakest bus hth is determinedas follows
When the weakest bus hth reaches its steady-statevoltage stability limit the differential of changes dλ closes tozero and the ratio of dVhCdλ will become infinite isratio is defined as the voltage stability index at the bus hthe ratio of dVhCdλ in equation (21) shows the magnitudecomparison of the elements dVjCdλ of the tangent vector ofthe P-V curve at each operating state If the ratio of
differential change in voltage at a bus to differential changein the connected load is the largest one this bus indicates theweakest one
4 Study and Simulation Results
To evaluate the effectiveness of the proposed method thenumerical case studies were examined through using threeIEEE 14-bus 57-bus and 118-bus systems which are con-sidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively e detailed data of these systems have beenintroduced in [31ndash33] All dynamic models such as gener-ators excitation systems transmission lines and load aremodeled based on [13] For each system the case studies areexamined considering conditions as with or without thepenetration of DGs in which the DG is used to be the doublyfed induction generator-based variable-speed wind turbines(DFIG-VSWT) with total installed capacity being 20MW ateach bus is bus is chosen based on the condition of theplaced location to be far in comparison with the existinggeneration sources Note also that in this paper the DG issuggested to be a pure distributed source with the constantpower model and its model and structure are introduced in[34] e placed locations of DGs at buses of each testedsystem are listed in Table 1e numerical simulation resultsare realized in the following case studies
41 Test on IEEE 14-Bus System e IEEE 14-bus systemwhich is considered as the small-scale network case has twogeneration buses three synchronous compensators at buses3 6 and 8 used only for supporting reactive powers elevenload buses three static compensators and three trans-formers e total load demand is 259MW and 735 MVArMore details on this system can be also found in [33]
In this case study we choose the load bus 5 to install the20MW DG e P-V curve of load bus 14 is shown inFigure 4 using CCPF and MCPF methods to analyze voltagestability considering either penetrating the installed the20MW DG at bus 5 or not and the numerical results aresummarized in Table 2
From the P-V curve in Figure 4 for the case of beingwithout DG using the CCPF method it takes too manypredictor-corrector steps to pass the bifurcation pointfrom the numerical results in Table 2 it takes 41 pre-dictor-corrector steps totally and spends 081162 secondsfinishing the critical point calculation Meanwhile usingMCPF it takes 32 predictor-corrector steps and spends073061 seconds e maximum loading factor (λcritical) atthe critical point when using CCPF is 27437 which issmall compared to that using MCPF which is 28178 Incase of being with DG it takes 37 predictor-correctorsteps totally and spends 086893 seconds finishing thecritical point calculation Meanwhile using MCPF ittakes 25 predictor-corrector steps and spends 075092seconds e maximum loading factor (λcritical) at thecritical point when using CCPF is 36397 which is smallcompared to that using MCPF which is 37412 In
Start
Calculate initial power flow with equation (3)using Newton-Raphson method and update initial
solution [δ U λ](0)
Check ifconvergence criterion
is reached
Cut step length control σi = σi2
No
Set initial parametersInitial solution [δ U λ](0) = [0 1 0]Step length control σ = 1
Note x = [δ U]T
PredictorAt (i + 1) using equation (17)
Corrector At (i + 1) using equation (19)
Check ifcritical point
is reached
Yes
End
No
Yes
Read line and bus data of power system
(i)(ii)
Figure 3e flowchart of voltage stability analyses of the proposedmethod
6 Complexity
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
Fx Fλ1113858 1113859t 0 (12)
where Fx and Fλ are the partial derivative of equation (9)with respect to x and λ respectively and T denotes thetranspose operation
From equation (12) the left side is a partial derivativematrix that this matrix is nothing but the original Jacobianaugmented by one column [Fλ] of an unknown variable ofload λ and a vector of differentials t is the differentialsrepresenting the tangent vector Since an unknown variableof the load is added to the nonlinear system (9) this will leadthe augmented Jacobian to be singular at the point ofmaximum possible system load In order to solve thisproblem the tangent vector should be determined based onthe satisfied augment Jacobian as follows
FxFλ
ei
1113890 1113891t 0
ti
1113890 1113891 (13)
where ei is a row vector having an appropriate dimension inwhich the ith element is the only nonzero element
From equation (13) ti is a nonzero element of thetangent vector this parameter can be either +1 or minus 1depending on how the ith state variable is changing as asolution that is being traced An estimate of the next solutioncan be determined by solving system (13) as follows
x(e)
λ(e)⎡⎣ ⎤⎦
x(i)
λ(i)⎡⎣ ⎤⎦ + σ
dx
dλ1113890 1113891 (14)
where the subscripts ldquoirdquo and ldquoerdquo denote the current andestimated solutions of the next step respectively and σ is ascalar defining the predictor step length control
222 (e Second Step After the prediction is finished if thepredictor solution may not be exactly on a desired solutioncurve this step is realized to correct the predicted solutionthat is determined in equation (14) using a technique
namely the local parameterization Now for the correctorstep system (9) is introduced by one equation unit with thepurpose of determining the value of one of the bus voltagemagnitude bus voltage angle and loading factor Hence thenew system is expressed as follows
F(x λ)
xi minus μ1113890 1113891
0
01113890 1113891 (15)
where xi is the ith state variable that is selected as thecontinuation parameter and μ is the predicted value of xi
Equation (15) shows that it is involved due to one ad-ditional equation and a state variable In order to solve thisdifficulty this paper reproposes Newtonrsquos method for thissecond step as follows
FxFλ
ei
1113890 1113891Δx
Δλ1113890 1113891
F(x λ)
xi minus η1113890 1113891 (16)
3 Proposed Method
As analyzed in Section 2 when using the CCPF firstly thetangential predictor step with the P-V curve at the previoussolution with a constant step length control σ as seen inequation (14) is applied en the corrector step based onthe parameterization method is used by solving equation(16) Obviously using the tangential predictive method witha constant step length control value may increase thenumber of predictor and corrector steps and lead to non-convergence when solving equation (16)
In order to overcome these disadvantages this paperproposed a method by using the scant method to performthe predictive solution e executorial predictor-correctorprocedure is illustrated in Figure 2 in which it will help theprocess of finding solutions to the critical point to be fasteras shown in Figure 2(a) If in case that has not found the nextsolution from the previous solution the step length controlwill be cut down and resumed using the corrector step to thenext corrected solution as shown in Figure 2(b) e processof finding solutions to the proposed method is presented inthe following
31 Predictor e secant method and arc-length parame-terization are used to realize this step e function of thepredictor is to find an approximate point for the next so-lution Figure 2(a) shows that if the continuation process atith step is the current corrector solution (x(i)
c λ(i)c ) the
predictor solution will be found in an approximate point(x(i+1)
p λ(i+1)p ) for the next corrector solution (x(iminus 1)
c λ(iminus 1)c )
and can be obtained as follows
x(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ + σ(i)x
(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(iminus 1)c
λ(iminus 1)c
⎡⎢⎢⎣ ⎤⎥⎥⎦⎛⎝ ⎞⎠ (17)
From equation (17) the number of iterations is used toperform the prediction process for finding solutions on theP-V curve depending on the distance between the two so-lutions (x(i)
c λ(i)c ) and (x(iminus 1)
c λ(iminus 1)c ) However the process of
Criticalpoint
Predictor
Corrector
λ
U
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
λcritical0
Figure 1 e predictor-corrector procedure of continuationpower flow
4 Complexity
finding solutions can quickly reach the voltage collapse pointor not depending on the step length control σ(i) e processof corrector may be ineffective if the step length control σ(i)
is too large the process of the previous solution will be keptSo that the step length control needs to be adjusted in orderto find a new solution around the critical point λcritical
32 Corrector From Figure 2(a) in order to find the nextcorrect solution (x(i+1)
c λ(i+1)c ) the corrector step is per-
formed by adding a line equation that is perpendicular toline connected between the two previous correct solutions(x(iminus 1)
c λ(iminus 1)c ) and (x(i)
c λ(i)c ) at the prediction solution (x(i+1)
p λ(i+1)
p ) e added line equation is expressed as follows
ρ(x λ) x minus x(i+1)
p
λ minus λ(i+1)p
⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦
T
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ 0 (18)
e actual correct solution (x(i+1)c λ(i+1)
c ) on the bifur-cation manifold can be calculated on the following set ofequations (9) and (18) for x and λ as follows
F(x λ) 0 0le λle λcritical
ρ(x λ) 01113896 (19)
where ρ() is the additional equation that is expressed inequation (18)
Equation (19) can also be solved by using a slightlymodified Newton-Raphson power flow method Howeverthis corrector step cannot converge if the step length controlis too large To overcome this limit this paper proposed aprocedure for cutting the step length control down to asmaller one until the convergence criteria are reached asshown in Figure 2(b)
e flowchart of the continuation power flow is shown inFigure 3 and can be explained in detail step by step asfollows
Step 1 Read all input data including line data and busdata of the power systemStep 2 Set the initial parameters including the initialsolution [δ U λ](0) [0 1 0] and the initial steplength control σ 1 note that x [δ U]TStep 3 Calculate the initial power flow with equation(3) using the Newton-Raphson method and update theinitial solution [δ U λ](0) note that x [δ U]TStep 4 Predict find the (i+1)th predicted solution withequation (17)Step 5 Correct find the (i+1)th corrected solution bysolving equation (19)Step 6 If the convergence criterion is reached the steplength control will be cut using equation σ σ2 and goto Step 4 If the convergence criterion is not reached itwill go to Step 7Step 7 If the critical point is not reached it will go toStep 4Step 8 If the critical point is reached it stops and putsthe P-V curves
33VoltageStability Index e voltage stability condition ina power system can be represented by using the voltagestability index is index can be calculated based on thetangent vector of the P-V curve at each operating state evoltage stability degree at buses experiencing when thesystem reached the state of the voltage collapse is consid-ered us we have to find out the weakest bus with respectto the voltage stability so that we can find suitable solutionsto improve voltage stability e weak bus is one that has thegreatest ratio of differential change in voltage to differentialchange in active load for the whole system Using thereformulated power flow equations the differential changein active system load is given as follows [19]
Critical point
x
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xc(i) λc(i))
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(a)
Critical point
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xp(i+1) λp(i+1))(xc(i) λc(i))
Successful corrector
Unsuccessful corrector
x
Predictorstep cut
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(b)
Figure 2 e secant predictor-corrector procedure of the proposed method (a) smooth function and (b) flat corner
Complexity 5
dPtotal 1113944n
j1dPLoadj SΔbase 1113944
n
j1kLoadj cos φj
⎡⎢⎢⎣ ⎤⎥⎥⎦dλ Cdλ
(20)
where dPLoadj is the differential change in load jth kLoadj isthe multiplier to designate the rate of load change at bus jthas λ changes φj is the power factor angle of load change atbus jth and SΔbase is the apparent power selected to provideappropriate scaling of λ e weakest bus hth is determinedas follows
When the weakest bus hth reaches its steady-statevoltage stability limit the differential of changes dλ closes tozero and the ratio of dVhCdλ will become infinite isratio is defined as the voltage stability index at the bus hthe ratio of dVhCdλ in equation (21) shows the magnitudecomparison of the elements dVjCdλ of the tangent vector ofthe P-V curve at each operating state If the ratio of
differential change in voltage at a bus to differential changein the connected load is the largest one this bus indicates theweakest one
4 Study and Simulation Results
To evaluate the effectiveness of the proposed method thenumerical case studies were examined through using threeIEEE 14-bus 57-bus and 118-bus systems which are con-sidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively e detailed data of these systems have beenintroduced in [31ndash33] All dynamic models such as gener-ators excitation systems transmission lines and load aremodeled based on [13] For each system the case studies areexamined considering conditions as with or without thepenetration of DGs in which the DG is used to be the doublyfed induction generator-based variable-speed wind turbines(DFIG-VSWT) with total installed capacity being 20MW ateach bus is bus is chosen based on the condition of theplaced location to be far in comparison with the existinggeneration sources Note also that in this paper the DG issuggested to be a pure distributed source with the constantpower model and its model and structure are introduced in[34] e placed locations of DGs at buses of each testedsystem are listed in Table 1e numerical simulation resultsare realized in the following case studies
41 Test on IEEE 14-Bus System e IEEE 14-bus systemwhich is considered as the small-scale network case has twogeneration buses three synchronous compensators at buses3 6 and 8 used only for supporting reactive powers elevenload buses three static compensators and three trans-formers e total load demand is 259MW and 735 MVArMore details on this system can be also found in [33]
In this case study we choose the load bus 5 to install the20MW DG e P-V curve of load bus 14 is shown inFigure 4 using CCPF and MCPF methods to analyze voltagestability considering either penetrating the installed the20MW DG at bus 5 or not and the numerical results aresummarized in Table 2
From the P-V curve in Figure 4 for the case of beingwithout DG using the CCPF method it takes too manypredictor-corrector steps to pass the bifurcation pointfrom the numerical results in Table 2 it takes 41 pre-dictor-corrector steps totally and spends 081162 secondsfinishing the critical point calculation Meanwhile usingMCPF it takes 32 predictor-corrector steps and spends073061 seconds e maximum loading factor (λcritical) atthe critical point when using CCPF is 27437 which issmall compared to that using MCPF which is 28178 Incase of being with DG it takes 37 predictor-correctorsteps totally and spends 086893 seconds finishing thecritical point calculation Meanwhile using MCPF ittakes 25 predictor-corrector steps and spends 075092seconds e maximum loading factor (λcritical) at thecritical point when using CCPF is 36397 which is smallcompared to that using MCPF which is 37412 In
Start
Calculate initial power flow with equation (3)using Newton-Raphson method and update initial
solution [δ U λ](0)
Check ifconvergence criterion
is reached
Cut step length control σi = σi2
No
Set initial parametersInitial solution [δ U λ](0) = [0 1 0]Step length control σ = 1
Note x = [δ U]T
PredictorAt (i + 1) using equation (17)
Corrector At (i + 1) using equation (19)
Check ifcritical point
is reached
Yes
End
No
Yes
Read line and bus data of power system
(i)(ii)
Figure 3e flowchart of voltage stability analyses of the proposedmethod
6 Complexity
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
finding solutions can quickly reach the voltage collapse pointor not depending on the step length control σ(i) e processof corrector may be ineffective if the step length control σ(i)
is too large the process of the previous solution will be keptSo that the step length control needs to be adjusted in orderto find a new solution around the critical point λcritical
32 Corrector From Figure 2(a) in order to find the nextcorrect solution (x(i+1)
c λ(i+1)c ) the corrector step is per-
formed by adding a line equation that is perpendicular toline connected between the two previous correct solutions(x(iminus 1)
c λ(iminus 1)c ) and (x(i)
c λ(i)c ) at the prediction solution (x(i+1)
p λ(i+1)
p ) e added line equation is expressed as follows
ρ(x λ) x minus x(i+1)
p
λ minus λ(i+1)p
⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦
T
x(i)c
λ(i)c
⎡⎢⎢⎣ ⎤⎥⎥⎦ minusx
(i+1)p
λ(i+1)p
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ 0 (18)
e actual correct solution (x(i+1)c λ(i+1)
c ) on the bifur-cation manifold can be calculated on the following set ofequations (9) and (18) for x and λ as follows
F(x λ) 0 0le λle λcritical
ρ(x λ) 01113896 (19)
where ρ() is the additional equation that is expressed inequation (18)
Equation (19) can also be solved by using a slightlymodified Newton-Raphson power flow method Howeverthis corrector step cannot converge if the step length controlis too large To overcome this limit this paper proposed aprocedure for cutting the step length control down to asmaller one until the convergence criteria are reached asshown in Figure 2(b)
e flowchart of the continuation power flow is shown inFigure 3 and can be explained in detail step by step asfollows
Step 1 Read all input data including line data and busdata of the power systemStep 2 Set the initial parameters including the initialsolution [δ U λ](0) [0 1 0] and the initial steplength control σ 1 note that x [δ U]TStep 3 Calculate the initial power flow with equation(3) using the Newton-Raphson method and update theinitial solution [δ U λ](0) note that x [δ U]TStep 4 Predict find the (i+1)th predicted solution withequation (17)Step 5 Correct find the (i+1)th corrected solution bysolving equation (19)Step 6 If the convergence criterion is reached the steplength control will be cut using equation σ σ2 and goto Step 4 If the convergence criterion is not reached itwill go to Step 7Step 7 If the critical point is not reached it will go toStep 4Step 8 If the critical point is reached it stops and putsthe P-V curves
33VoltageStability Index e voltage stability condition ina power system can be represented by using the voltagestability index is index can be calculated based on thetangent vector of the P-V curve at each operating state evoltage stability degree at buses experiencing when thesystem reached the state of the voltage collapse is consid-ered us we have to find out the weakest bus with respectto the voltage stability so that we can find suitable solutionsto improve voltage stability e weak bus is one that has thegreatest ratio of differential change in voltage to differentialchange in active load for the whole system Using thereformulated power flow equations the differential changein active system load is given as follows [19]
Critical point
x
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xc(i) λc(i))
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(a)
Critical point
λCritical
F(x λ) = 0
(xc(indash1) λc
(indash1))
(xc(i+1) λc
(i+1))
(xp(i+1) λp(i+1))
(xp(i+1) λp(i+1))(xc(i) λc(i))
Successful corrector
Unsuccessful corrector
x
Predictorstep cut
λ
Corrected solutionPredicted solutionCritical point
PredictorCorrectorExact solution
(b)
Figure 2 e secant predictor-corrector procedure of the proposed method (a) smooth function and (b) flat corner
Complexity 5
dPtotal 1113944n
j1dPLoadj SΔbase 1113944
n
j1kLoadj cos φj
⎡⎢⎢⎣ ⎤⎥⎥⎦dλ Cdλ
(20)
where dPLoadj is the differential change in load jth kLoadj isthe multiplier to designate the rate of load change at bus jthas λ changes φj is the power factor angle of load change atbus jth and SΔbase is the apparent power selected to provideappropriate scaling of λ e weakest bus hth is determinedas follows
When the weakest bus hth reaches its steady-statevoltage stability limit the differential of changes dλ closes tozero and the ratio of dVhCdλ will become infinite isratio is defined as the voltage stability index at the bus hthe ratio of dVhCdλ in equation (21) shows the magnitudecomparison of the elements dVjCdλ of the tangent vector ofthe P-V curve at each operating state If the ratio of
differential change in voltage at a bus to differential changein the connected load is the largest one this bus indicates theweakest one
4 Study and Simulation Results
To evaluate the effectiveness of the proposed method thenumerical case studies were examined through using threeIEEE 14-bus 57-bus and 118-bus systems which are con-sidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively e detailed data of these systems have beenintroduced in [31ndash33] All dynamic models such as gener-ators excitation systems transmission lines and load aremodeled based on [13] For each system the case studies areexamined considering conditions as with or without thepenetration of DGs in which the DG is used to be the doublyfed induction generator-based variable-speed wind turbines(DFIG-VSWT) with total installed capacity being 20MW ateach bus is bus is chosen based on the condition of theplaced location to be far in comparison with the existinggeneration sources Note also that in this paper the DG issuggested to be a pure distributed source with the constantpower model and its model and structure are introduced in[34] e placed locations of DGs at buses of each testedsystem are listed in Table 1e numerical simulation resultsare realized in the following case studies
41 Test on IEEE 14-Bus System e IEEE 14-bus systemwhich is considered as the small-scale network case has twogeneration buses three synchronous compensators at buses3 6 and 8 used only for supporting reactive powers elevenload buses three static compensators and three trans-formers e total load demand is 259MW and 735 MVArMore details on this system can be also found in [33]
In this case study we choose the load bus 5 to install the20MW DG e P-V curve of load bus 14 is shown inFigure 4 using CCPF and MCPF methods to analyze voltagestability considering either penetrating the installed the20MW DG at bus 5 or not and the numerical results aresummarized in Table 2
From the P-V curve in Figure 4 for the case of beingwithout DG using the CCPF method it takes too manypredictor-corrector steps to pass the bifurcation pointfrom the numerical results in Table 2 it takes 41 pre-dictor-corrector steps totally and spends 081162 secondsfinishing the critical point calculation Meanwhile usingMCPF it takes 32 predictor-corrector steps and spends073061 seconds e maximum loading factor (λcritical) atthe critical point when using CCPF is 27437 which issmall compared to that using MCPF which is 28178 Incase of being with DG it takes 37 predictor-correctorsteps totally and spends 086893 seconds finishing thecritical point calculation Meanwhile using MCPF ittakes 25 predictor-corrector steps and spends 075092seconds e maximum loading factor (λcritical) at thecritical point when using CCPF is 36397 which is smallcompared to that using MCPF which is 37412 In
Start
Calculate initial power flow with equation (3)using Newton-Raphson method and update initial
solution [δ U λ](0)
Check ifconvergence criterion
is reached
Cut step length control σi = σi2
No
Set initial parametersInitial solution [δ U λ](0) = [0 1 0]Step length control σ = 1
Note x = [δ U]T
PredictorAt (i + 1) using equation (17)
Corrector At (i + 1) using equation (19)
Check ifcritical point
is reached
Yes
End
No
Yes
Read line and bus data of power system
(i)(ii)
Figure 3e flowchart of voltage stability analyses of the proposedmethod
6 Complexity
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
dPtotal 1113944n
j1dPLoadj SΔbase 1113944
n
j1kLoadj cos φj
⎡⎢⎢⎣ ⎤⎥⎥⎦dλ Cdλ
(20)
where dPLoadj is the differential change in load jth kLoadj isthe multiplier to designate the rate of load change at bus jthas λ changes φj is the power factor angle of load change atbus jth and SΔbase is the apparent power selected to provideappropriate scaling of λ e weakest bus hth is determinedas follows
When the weakest bus hth reaches its steady-statevoltage stability limit the differential of changes dλ closes tozero and the ratio of dVhCdλ will become infinite isratio is defined as the voltage stability index at the bus hthe ratio of dVhCdλ in equation (21) shows the magnitudecomparison of the elements dVjCdλ of the tangent vector ofthe P-V curve at each operating state If the ratio of
differential change in voltage at a bus to differential changein the connected load is the largest one this bus indicates theweakest one
4 Study and Simulation Results
To evaluate the effectiveness of the proposed method thenumerical case studies were examined through using threeIEEE 14-bus 57-bus and 118-bus systems which are con-sidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively e detailed data of these systems have beenintroduced in [31ndash33] All dynamic models such as gener-ators excitation systems transmission lines and load aremodeled based on [13] For each system the case studies areexamined considering conditions as with or without thepenetration of DGs in which the DG is used to be the doublyfed induction generator-based variable-speed wind turbines(DFIG-VSWT) with total installed capacity being 20MW ateach bus is bus is chosen based on the condition of theplaced location to be far in comparison with the existinggeneration sources Note also that in this paper the DG issuggested to be a pure distributed source with the constantpower model and its model and structure are introduced in[34] e placed locations of DGs at buses of each testedsystem are listed in Table 1e numerical simulation resultsare realized in the following case studies
41 Test on IEEE 14-Bus System e IEEE 14-bus systemwhich is considered as the small-scale network case has twogeneration buses three synchronous compensators at buses3 6 and 8 used only for supporting reactive powers elevenload buses three static compensators and three trans-formers e total load demand is 259MW and 735 MVArMore details on this system can be also found in [33]
In this case study we choose the load bus 5 to install the20MW DG e P-V curve of load bus 14 is shown inFigure 4 using CCPF and MCPF methods to analyze voltagestability considering either penetrating the installed the20MW DG at bus 5 or not and the numerical results aresummarized in Table 2
From the P-V curve in Figure 4 for the case of beingwithout DG using the CCPF method it takes too manypredictor-corrector steps to pass the bifurcation pointfrom the numerical results in Table 2 it takes 41 pre-dictor-corrector steps totally and spends 081162 secondsfinishing the critical point calculation Meanwhile usingMCPF it takes 32 predictor-corrector steps and spends073061 seconds e maximum loading factor (λcritical) atthe critical point when using CCPF is 27437 which issmall compared to that using MCPF which is 28178 Incase of being with DG it takes 37 predictor-correctorsteps totally and spends 086893 seconds finishing thecritical point calculation Meanwhile using MCPF ittakes 25 predictor-corrector steps and spends 075092seconds e maximum loading factor (λcritical) at thecritical point when using CCPF is 36397 which is smallcompared to that using MCPF which is 37412 In
Start
Calculate initial power flow with equation (3)using Newton-Raphson method and update initial
solution [δ U λ](0)
Check ifconvergence criterion
is reached
Cut step length control σi = σi2
No
Set initial parametersInitial solution [δ U λ](0) = [0 1 0]Step length control σ = 1
Note x = [δ U]T
PredictorAt (i + 1) using equation (17)
Corrector At (i + 1) using equation (19)
Check ifcritical point
is reached
Yes
End
No
Yes
Read line and bus data of power system
(i)(ii)
Figure 3e flowchart of voltage stability analyses of the proposedmethod
6 Complexity
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
addition the stability margin is expanded with two caseswhen using MCPF
Figure 5(a) plots the voltage magnitude profile of the 14-bus system without DG when the loading point has reacheda maximum value which is obtained using the CCPF andMCPF methods From this figure when using MCPF it candetect the unstable voltage corresponding to the maximumloading point at a voltage collapse point to be more sensitivewhen using CCPF In case of being with DG the simulationresults are shown in Figures 4(b) and 5(b) and the rest of thenumerical results are shown in Table 2 using the CCPFmethod it takes too many continuation steps to pass thebifurcation point the stability margin is limited and thedetection of the voltage stability properties is low comparedto the proposed method
In the analysis of voltage stability the relation betweenthe power transfer to the load and voltage of the load bus is
not weak As known the variation in power transfer fromone bus to the other ones will affect the bus voltages at isthe reason why the proposed method is applied to considerits effectiveness based on the P-V curve e voltage at buses4 5 7 9 10 and 14 is plotted concerning the loading factoras shown in Figure 6 As the loading factor is increased thevoltage at load buses decreases e most reduction in busvoltages occurs on bus 14
e buses have a large voltage stability index to thechange in the total load active power in the system to bebuses numbers 4 5 7 9 10 and 14 in which buses 5 and 14are the weakest in voltage stability aspect under cases ofbeing with and without DG respectively as shown inFigure 7 It can be concluded that buses 5 and 14 areidentified as a critical bus for cases of being without and withDG in the system respectively After installing DG at bus 5bus 5 has become the strongest as shown in Figure 7
Table 1 e modification of the test systems
Test system DG placed at busesIEEE 14-bus 5IEEE 57-bus 26 39 54IEEE 118-bus 2 21 41 57 101
05 1 15 2 25 30Loading factor
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(a)
1 2 3 40Loading factor
05
06
07
08
09
1
11
Volta
ge (p
u)
CCPFMCPF
(b)
Figure 4 e P-V curve at bus 14 during the load-increasing process (a) without DG (b) with DG
Table 2 Comparisons between CCPF and MCPF of the IEEE 14-bus test system
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
naturally it is an expected result since there are hugegenerating units
From the simulated results in Figures 5ndash7 it can beconcluded that bus 5 is the weakest for the case of beingwithout DG and meanwhile the bus 14 is the weakest for twocases Buses 1 2 3 6 and 8 are applied for the voltagecontrol in which bus 1 is the slack bus so the voltage profileis constant despite increasing the loading factor and it ispretty obvious bus 1 is the strongest e voltage of theremaining buses decreases since they are load buses Besideswhen DG is connected at bus 5 the stability margin and
loading factor are expanded as shown in the fourth row ofTable 2
42 Test on IEEE 57-Bus System In this section in order toverify the proposed method the IEEE 57-bus system can bechosen as a medium-scale system is system has 7 syn-chronous machines with IEEE type-1 exciters three of whichare synchronous compensators 64 buses 65 transmissionlines 22 transformers and 42 constant impedance loadsetotal load demand is 12508MW and 3364 MVAr More
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(a)
05
06
07
08
09
1
11
Vol
tage
(pu
)
2 4 6 8 10 12 14Bus number
CCPFMCPF
(b)
Figure 5 Voltage profile of 14-bus system (a) without DG (b) with DG
11
1
09
08
07
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 05 1 15 2 25 3Loading factor
(a)
11
1
09
08
07
06
05
Volta
ge (p
u)
Bus 4Bus 5Bus 7
Bus 9Bus 10Bus 14
0 1 2 3 4Loading factor
(b)
Figure 6 e P-V curve of buses 4 5 7 9 10 and 14 during the load-increasing process when using MCPF (a) without DG (b) with DG
8 Complexity
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
details on the IEEE 57-bus test system can be also found in[32]
For this test case we propose three 20MW DG unitswhich are installed at buses 26 39 and 54 respectively andthe load bus 31 is selected to verify the proposed method Inorder to compare easily the P-V curve of load bus 31 isplotted out in Figure 8 considering either installing three20MW DGs at buses 26 39 and 54 or not e numericalresults are summarized in Table 3
e obtained results from the case of being without theintegration of DGs show that using the CCPF method ittakes too many predictor-corrector steps to pass the bi-furcation point from the numerical results in Table 3 ittakes 50 predictor-corrector steps totally and spends 093464seconds finishing the critical point calculation Meanwhileusing MCPF it takes 38 predictor-corrector steps andspends 085021 seconds e maximum loading factor(λcritical) at the critical point when using CCPF is 1774 whichis small compared to that using MCPF which is 18434 Incase of being with DGs it takes 27 predictor-corrector stepstotally and spends 074566 seconds finishing the criticalpoint calculation Meanwhile using MCPF it takes 20predictor-corrector steps and spends 071025 seconds emaximum loading factor (λcritical) at the critical point whenusing CCPF is 27936 which is small compared to that usingMCPF which is 28915 In addition the stability margin isexpanded with two cases when using MCPF as shown in thefourth row of Table 3
e zoom-in for this case is shown in Figure 8 Fromthis figure the effect of step length control reduction isclearly shown and exactly in this case the maximumloading factors reach 19241 without DGs and 29152with DGs corresponding to the step length control inequation (17) which is cut down to values 12 and 14respectively
Figure 9 plots the voltage magnitude profile of the systemwithout and with the integration of DGs when the loading
point has reached a maximum value which is obtained usingthe CCPF and MCPFmethods From this figure when usingMCPF it can detect the unstable voltage corresponding tothe maximum loading point at a voltage collapse point to bemore sensitive when using CCPF
Figure 10 shows the voltage profiles of buses 25 and30ndash34 when using the proposed method It is seen from thisfigure that after the load point with maximum loadingfactors λcritical of 18434 and 28915 for the cases of beingwithout and with the integration of DGs respectively thebus voltages start to decrease because of the deficient powergeneration At these points the system is defined as theoperating conditions and after these points the systementers into an unstable condition which can cause thephenomena of voltage collapse When comparing thevoltage profiles of buses bus 34 seems to be the strongestand bus 31 seems to be the weakest bus under voltagestability facet
Figure 11 plots the voltage stability index of the wholesystem From the obtained result buses 25 and 31 have thelarge voltage stability index to the change in the total loadactive power where bus 31 is the weakest in the voltagestability aspect under cases of being with and without theintegration of DGs in the system From the obtained resultsin Figures 9ndash11 it can be concluded that bus 31 is identifiedas a critical bus
43 Test on IEEE 118-Bus System Lastly to add practicalityand to ensure the efficiency of the proposed method this testcase is considered on the IEEE 118-bus system presumed as acomplicated system having 19 generators 35 synchronouscondensers 177 lines 9 transformers and 91 loadse totalload demand is 4242MW and 1438 MVar Many detailscan be also found in [31] Besides the differences in resultsdiscussed in the two previous test cases for this test case thepaper proposes five 20MWDG units installed at buses 2 2141 57 and 101 respectively e load bus 44 is selected to
Volta
ge st
abili
ty in
dex
08
07
06
05
04
03
02
01
2 4 8 10 12 146Bus number
Weakestbus 14
Weakestbus 5
Without DGsWith DGs
Figure 7 e voltage stability index of the 14-bus system when using MCPF without and with DG at bus 5
Complexity 9
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
verify and ensure the efficiencies of the proposed methode results of the P-V curve of load bus 44 using CCPF andMCPF methods considering either penetrating the installed
five 20MWDG units at buses 2 21 41 57 and 101 or not areshown in Figure 12 and the numerical results are summa-rized in Table 4
Table 3 Comparisons between CCPF and MCPF of the IEEE 57-bus test system
Figure 9 Voltage profile of the 57-bus system buses (a) without DG at buses 26 39 and 54 (b) with DG at buses 26 39 and 54
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 2
Loading factor
07
06
05
15 16 17 18
CCPFMCPF
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 05 1 15 3252
Loading factor
07
06
0526 28 3
CCPFMCPF
(b)
Figure 8 e P-V curve at bus 31 during the load-increasing process (a) without DG (b) with DG
10 Complexity
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
From the obtained results the use of the proposedmethod to analyze the voltage stability has the advantagecompared to the use of the CCPF method For example thepredictor-corrector steps are less the processing time isquicker and the effect of adding the new generation units tothe system is also observed clearly Without the integrationof DGs it takes 177 predictor-corrector steps totally andspends 32385 seconds finishing the critical point calcu-lation when using the CCPF method Meanwhile usingMCPF it takes 162 predictor-corrector steps and spends31846 secondsemaximum loading factor (λcritical) at the
critical point when using CCPF is 105820 which is smallcompared to that using MCPF which is 109091 It is thesame For the integration of DGs it takes 215 predictor-corrector steps totally and spends 38142 seconds finishingthe critical point calculation when using the CCPF methodMeanwhile using MCPF it takes 198 predictor-correctorsteps and spends 37251 seconds e maximum loadingfactor (λcritical) at the critical point when using CCPF is202583 which is small compared to that usingMCPF whichis 211519 e stability margin is expanded when usingMCPFe obtained results are also summarized in Table 4
Volo
tage
(pu
)
12
11
1
09
08
07
06
050 05 1 15 2
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(a)Vo
lota
ge (p
u)
11
1
09
08
07
06
050 05 1 15 32 25
Loading factor
Bus 25Bus 30Bus 31
Bus 32Bus 33Bus 34
(b)
Figure 10 Voltage profile of buses from 25 to 30ndash34 during the load-increasing process when using MCPF (a) without DG at buses 26 39and 54 (b) with DG at buses 26 39 and 54
0
1
2
3
4
5
6
7
8
Volta
ge st
abili
ty in
dex
10 20 30 40 50Bus number
Weakestbus 31
Weakestbus 31
Without DGsWith DGs
Figure 11 e voltage stability index of 57-bus system when using MCPF without and with DG at buses 26 39 and 54
Complexity 11
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
e zoom-in for this case is shown in Figure 12 From thisfigure the effect of step length control reduction is clearlyshown and exactly in this case the maximum loading
factors reach 109091 without DGs and 211519 with DGscorresponding to the step length control in equation (17)which is cut down to value 12 e stability margin is
105
1
095
09
085
08
075
07
065
Volta
ge (p
u)
0 2 4 6 8 10 12Loading factor
CCPFMCPF
075
07
065
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
040 5 10 15 20 25
Loading factor
CCPFMCPF
07
065
06
055
05
(b)
Figure 12 e P-V curve at the load bus 44 during the load-increasing process (a) without DGs and (b) with DGs
Table 4 Comparisons between CCPF and MCPF of the IEEE 118-bus test system
Figure 13 Voltage profile of 118 buses (a) without the integration of DGs and (b) with the integration of DGs
12 Complexity
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
expanded when using MCPF and can be shown in thefourth row of Table 4
Figure 13 plots the voltage magnitude profile of thesystem without and with the integration of DGs when theloading point has reached a maximum value which isobtained using the CCPF and MCPF methods From thisfigure when using MCPF it can detect the unstablevoltage corresponding to the maximum loading point at avoltage collapse point to be more sensitive when usingCCPF
Figure 14 shows the voltage profiles of buses 33 and43ndash47 when using the proposed method It is seen fromthis figure that after the load point with maximumloading factors λcritical of 109091 and 211519 for the casesof being without and with the integration of DGs re-spectively the bus voltages start to decrease because ofthe deficient power generation At these points thesystem is defined as the operating conditions and afterthese points the system enters into an unstable conditionwhich can cause the phenomena of voltage collapseWhen comparing the voltage profiles of buses bus 47seems to be the strongest and buses 33 and 44 seem to bethe weakest ones under the voltage stability facet for thecases of being without and with the integration of DGsrespectively
Figure 15 plotted the voltage stability index of thewhole system From this figure buses 33 and 44 have alarge voltage stability index to the change in the total loadactive power in the system For this case buses 33 and 44are the weakest buses in the voltage stability aspect undercases of being with and without the integration of DGsrespectively erefore the simulated results inFigures 13ndash15 could conclude that buses 33 and 44 areidentified as critical ones
5 Conclusions
is paper proposes an innovatory method for analyzing thevoltage stability and specifically monitoring the bus voltagewhen the system is operating at a load point near the criticalone is proposed method is developed based on thecontinuation power flow (CPF)methode voltage stabilityproblem of the power system has been analyzed to establishthe proposed method the CPF method based on the tangentand local parameterization methods is recalled to be com-pared with the proposed method e proposed method isrealized on the predictor and corrector procedures to draw
Volta
ge (p
u)
105
1
095
09
085
08
075
07
0650 2 4 6 8 10 12
Loading factor
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(a)
Volta
ge (p
u)
11
1
09
08
07
06
05
04
Loading factor0 5 10 15 20 25
Bus 33Bus 43Bus 44
Bus 45Bus 46Bus 47
(b)
Figure 14 Voltage profile of buses during the load-increasing process when using MCPF (a) without the integration of DGs and (b) withthe integration of DGs
25
20
15
10
5
ndash5
0
Volta
ge st
abili
ty in
dex
20 40 60 80 100Bus number
Weakestbus 44
Weakestbus 33
1
05
0
40 6050
Without DGsWith DGs
Figure 15 e voltage stability index of 118 buses when usingMCPF without and with the integration of DGs
Complexity 13
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
the P-V curves at buses according to a specified generation-load scenario
ree IEEE 14-bus 57-bus and 118-bus test systems areconsidered as a quite simple and small-scale network case amedium-scale one and a complex and large-scale systemrespectively and the distributed generation (DG) is used tobe the doubly fed induction generator-based variable-speedwind turbines (DFIG-VSWT) with the constant power toverify the efficiency of the proposed method e numericalresults are simulated for all study cases based on the load-increasing process and voltage stability index by usingMATLAB software on a PC with Intel(R) Core processor(TM) i7 32GHz e obtained results show that using theproposed method to analyze the voltage stability has theadvantages compared to the CCPF method namely thepredictor-corrector steps are less the processing time isquicker the effect of adding the new generation units intothe system is also observed clearly during the load-increasingprocess and the stability margin and loading factor areexpanded In addition the proposed method was shown tobe effective on a large power system with the integration ofmany DG units and the voltage stability index indicatedcloser proximity to voltage collapse when the system isoperating at a load point near the critical one
Data Availability
e data used to support the study are presented in [31ndash33]
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
e authors sincerely acknowledge the financial supportprovided by Industrial University of Ho Chi Minh City TonDuc ang University and Quy Nhon University Vietnamfor carrying out this work
References
[1] Y M Atwa and E F El-Saadany ldquoOptimal allocation of ESSin distribution systems with a high penetration of wind en-ergyrdquo IEEE Transactions on Power Systems vol 25 no 4pp 1815ndash1822 2010
[2] S Eftekharnejad V Vittal G T Heydt B Keel and J LoehrldquoImpact of increased penetration of photovoltaic generationon power systemsrdquo IEEE Transactions on Power Systemsvol 28 no 2 pp 893ndash901 2013
[3] J M Sexauer and S Mohagheghi ldquoVoltage quality assessmentin a distribution system with distributed generation a prob-abilistic load flow approachrdquo IEEE Trans Power Delvol 28no 3 pp 1653ndash1662 2013
[4] P Kundur J Paserba and V Ajjarapu ldquoDefinition andclassification of power system stability IEEECIGRE joint taskforce on stability terms and definitionsrdquo IEEE Trans PowerSystvol 19 no 3 pp 1387ndash1401 2004
[5] W Freitas J C M Vieira A Morelato L C P daSilvaV F da Costa and F A B Lemos ldquoComparative analysis
between synchronous and induction machines for distributedgeneration applicationsrdquo IEEE Transactions on Power Sys-tems vol 21 no 1 pp 301ndash311 2006
[6] W Freitas L C P DaSilva and A Morelato ldquoSmall-dis-turbance voltage stability of distribution systems with in-duction generatorsrdquo IEEE Transactions on Power Systemsvol 20 no 3 pp 1653-1654 2005
[7] I Xyngi A Ishchenko M Popov and L van der SluisldquoTransient stability analysis of a distribution network withdistributed generatorsrdquo IEEE Transactions on Power Systemsvol 24 no 2 pp 1102ndash1104 2009
[8] J Slootweg and W Kling ldquoImpacts of distributed generationon power system transient stabilityrdquo in Proceedings of the 2002IEEE Power Engineering Society SummerMeeting Chicago ILUSA July 2002
[9] E G Potamianakis and C D Vournas ldquoShort-term voltageinstability effects on synchronous and induction machinesrdquoIEEE Transactions on Power Systems vol 21 no 2 pp 791ndash798 2006
[10] P Kundur ldquoPower System Stability and Controlrdquo McGraw-Hill New York NY USA 1994
[11] H Abdel-mawgoud S Kamel M Ebeed and A R YoussefldquoOptimal allocation of renewable DG sources in distributionnetworks considering load growthrdquo in Proceedings of the 2017Nineteenth International Middle East Power Systems Con-ference (MEPCON) Cairo Egypt December 2017
[12] S Kamel A Ramadan M Ebeed J Yu K Xie and T WuldquoAssessment integration of wind-based DG and DSTATCOMin Egyptian distribution grid considering load demand un-certaintyrdquo in Proceedings of the 2019 IEEE Innovative SmartGrid Technologies-Asia (ISGT Asia) Chengdu China May2019
[13] A Ramadan M Ebeed S Kamel and L Nasrat ldquoOptimalallocation of renewable energy resources considering un-certainty in load demand and generationrdquo in Proceedings ofthe 2019 IEEE Conference on Power Electronics and RenewableEnergy (CPERE) Aswan City Egypt October 2019
[14] T V Cutsem and C Vournas ldquoVoltage Stability of ElectricPower Systemsrdquo Springer Berlin Germany 1998
[15] S D Anagnostatos C D Halevidis A D PolykratiP D Bourkas and C G Karagiannopoulos ldquoExamination ofthe 2006 blackout in Kefallonia island Greecerdquo InternationalJournal of Electrical Power amp Energy Systems vol 49pp 122ndash127 2013
[16] J E Chadwick ldquoHow a smarter grid could have prevented the2003 US cascading blackoutrdquo in Proceedings of the IEEEPower and Energy Conference at Illinois (PECI) ChampaignIL USA February 2013
[17] S Sarkar G Saha G Pal and T Karmakar ldquoIndian expe-rience on smart grid application in blackout controlrdquo inProceedings of the 2015 National Systems Conference (NSC)Noida India December 2015
[18] H Haes Alhelou M Hamedani-Golshan T Njenda andP Siano ldquoA survey on power system blackout and cascadingevents research motivations and challengesrdquo Energies vol 12no 4 p 682 2019
[19] V A P Miranda and A V M Oliveira ldquoAirport slots and theinternalization of congestion by airlines an empirical modelof integrated flight disruption management in BrazilrdquoTransportation Research Part A Policy and Practice vol 116pp 201ndash219 2018
[20] T Van Cutsem and C Vournas ldquoVoltage Stability of Elec-trical Power Systemsrdquo New York Springer Science NewYork NY USA 1998
14 Complexity
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
Complexity 15
[21] V Ajjarapu and C Christy ldquoe continuation power flow atool for steady state voltage stability analysisrdquo IEEE Trans-actions on Power Systems vol 7 no 1 pp 416ndash423 1992
[22] V Ajjarapu ldquoComputational Techniques for Voltage StabilityAssessment and Controlrdquo Springer Berlin Germany 2007
[23] B Gao G K Morison and P Kundur ldquoVoltage stabilityevaluation using modal analysisrdquo IEEE Transactions on PowerSystems vol 7 no 4 pp 1529ndash1542 1992
[24] I Musirin and T K A Rahman ldquoEstimating maximumloadability for weak bus identification using FVSIrdquo IEEEPower Engineering Review vol 22 no 11 pp 50ndash52 2002
[25] M Moghavemmi and F M Omar ldquoTechnique for contin-gency monitoring and voltage collapse predictionrdquo IEEEProceedings on Generation Transmission and Distributionvol 145 no 6 pp 634ndash640 1998
[26] F A Althowibi and M W Mustafa ldquoVoltage stability cal-culations in power transmission lines indication and allo-cationsrdquo in Proceedings of the IEEE International Conferenceon Power and Energy Kuala Lumpur Malaysia November2010
[27] P Kessel and H Glavitsch ldquoEstimating the voltage stability ofa power systemrdquo IEEE Transactions on Power Delivery vol 1no 3 pp 346ndash354 1986
[28] M Javad G Eskandar and A Khodabakhshian ldquoA com-prehensive review of the voltage stability indicesrdquo RenewSustain Energy Revvol 63 pp 1ndash12 2016
[29] WM Villa-Acevedo J M Lopez-Lezama and D G ColomeldquostabilitVoltage y margin index estimation using a hybridkernel extreme learning machine approachrdquo Energies vol 13no 4 p 857 2020
[30] A R Bergen ldquoPower System Analysisrdquo Prentice-Hall UpperSaddle River NJ USA 2000
[31] ldquoIEEE 118-bus test systemrdquo 2020 httplabseceuwedupstcapf118pg_tca118bushtm
[32] ldquoIEEE 57-bus test systemrdquo 2020 httplabseceuwedupstcapf57pg_tca57bushtm
[33] ldquoIEEE 14-bus test systemrdquo 2020 httplabseceuwedupstcapf14pg_tca14bushtm
[34] V Le X Li Y Li T L T Dong and C Le ldquoAn innovativecontrol strategy to improve the fault ride-through capabilityof DFIGs based on wind energy conversion systemsrdquo Ener-gies vol 9 no 2 69 pages 2016
