Probabilistic voltage stability assessment considering renewable sources with the help of the PV and QV curves

Published in IET Renewable Power GenerationReceived on 31st January 2012Revised on 9th March 2013Accepted on 26th March 2013doi: 10.1049/iet-rpg.2012.0265

ISSN 1752-1416

Probabilistic voltage stability assessment consideringrenewable sources with the help of the PV and QVcurvesAdriano Batista Almeida1, Eliane Valença De Lorenci1, Rafael Coradi Leme2,

Antonio Carlos Zambroni de Souza1, Benedito Isaias Lima Lopes3, Kwok Lo4

1Electrical and Energy Systems Institute, Federal University of Itajubá, Brazil2Institute of Industrial Engineering and Management, Federal University of Itajubá, Brazil3Systems Engineering and Information Technology Institute, Federal University of Itajubá, Brazil4Department of Electronic & Electrical Engineering, University of Strathclyde, Glasgow, UK

E-mail: [email protected]

Abstract: The use of renewable energy sources has increased year-on-year. Thus, there is an increasing rate of small generatingunits connected directly to distribution networks and micro-grids close to consumers. At the same time, these micro-sources mustprovide stability and reliability of electrical energy to the power network to which they are connected. In the technical literature,several studies have been done to ensure power systems with traditional generating sources to operate in a stable and reliable way,but there is an issue regarding generation uncertainty when a distribution system has many micro-sources. This is because of theuncertainty of primary sources, for example, wind and radiation intensity, and could result in intermittent generation. In this study,stability and reliability of voltage in a power system with distributed generation is analysed using simulation techniques. In theproposed method in this study voltage security analysis is jointly considered with probability laws. Moreover reliability theory isalso considered in the proposed voltage collapse analysis methodology. The responsibility of generator in the voltage collapseprocess, the probabilistic risk of voltage collapse of each operating point and the probability of enlarging the system load as afunction of different operating points are the outcome of the methodology, and it is validated by using the IEEE34 test feeder.

1 Introduction

Distributed energy resources (DER) usage is quicklyincreasing their share in power systems around the world.An advantage of this scenario is that energy is generated insmall scale as close as possible to the load centre, and insome cases, could well be placed on the consumer side ofmetre [1]. This reduces the electrical distance betweengeneration and loads, improving energy efficiency andpostponing new investments [2]. As environmental policiesare being discussed in many countries, DER appears as anattractive alternative to centralised generation [3, 4]. Manyemerging DER technologies present lower emissions, suchas wind power, photovoltaic systems, fuel cells, geothermaland gas micro-turbines. These DER are known asrenewable energy sources (RES).Nevertheless, the use of DER also impacts network power

flow [5], voltage stability [6], protection [7] and quality ofpower supply [8]. Indeed, the intermittent nature of RESmakes the system stability control a very difficult task, andmany researchers have been focusing on control strategyand stability evaluation [9–13]. When the focus is voltagecollapse [14], multiple scenarios and/or contingencies areconsidered, and often the continuation power flow method

is employed to determine load margins, margin to thermallimits [15] and voltage violations. Voltage collapse,normally studied for transmission systems, has also beenaddressed for distribution systems, as shown in [16–18].Williams et al. [19] discusse an interaction between thetransmission and distribution systems by describing a realevent. The study shows that a fault in the transmissionsystem may cause a voltage dip, stalling some airconditioners and driving the system to voltage instability,illustrating how the distribution system may affect thesystem overall voltage stability. Voltage collapse isassociated with the singularity of the power flow Jacobianmatrix [20]. At this singular point, known as saddle-nodebifurcation, the power flow Jacobian matrix presents a zeroeigenvalue and no solution can be obtained beyond thiscritical point, which is the system maximum loadabilitypoint. Induced limits bifurcation may also occur as aconsequence of sudden exhaustion of important reactivepower sources. Calculating the system load margin, that is,the amount of load variation that the system may sustainplays a crucial role. In this paper, the continuation methodis employed for this purpose. In order to provide a morecomplete set of information to operators, QV curve mayalso be used [21]. Such a curve computes the reactive

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IET Renew. Power Gener., pp. 1–10 1doi: 10.1049/iet-rpg.2012.0265 & The Institution of Engineering and Technology 2013

margin of each bus and is an important indication of a bus’robustness. Combining the information obtained by PV andQV curves is also exploited here for stability analysis.Regarding reliability studies, many researches focus on

optimal DER location for improving reliability indices[22–24]. However, when renewable sources are consideredin electrical power supply in a DER layout, a factor ofuncertainty also arises on voltage security monitoring. Forthis sake, simulation techniques [25] and reliability theory[26] may be used to evaluate the system operating conditions.A powerful reliability analysis is that of Monte Carlo

simulation (MCS), which samples power system states, anduses load flow, dispatch algorithms and operating policiesto infer information from the samples. In this work, MCS involtage security analysis is considered. Owing to thedifficulty of obtaining parametric distribution in jointfunctions, empirical distribution function (EDF) approach isused [27]. Reliability measures such as survival and hazardfunctions are used.This paper addresses the probabilistic risk of voltage

collapse by considering an uncertainty in the amount ofgeneration provided by the DER. The focus lies onstudying different generating scenarios and evaluating theload margin with respect to a base case considered. Thelink of these results with the ones provided by the QVcurve is explored, stressing the importance of a goodreactive power control. The main contributions of this papermay be summarised as

† The responsibility of generator in the voltage collapseprocess is determined.† The relation between reactive power margin of a generatorand the system load margin is studied.† The probabilistic risk of voltage collapse of each operatingpoint is assessed.† The probability of enlarging the system load margin iscalculated as a function of different operating points.

This paper is organised as follows: Section 2 brieflydiscusses some aspects about voltage stability. Section 3introduces some features of DER. The simulation of thevoltage stability index and the voltage collapse reliabilityare addressed in Section 4. Finally, Section 5 presents testresults and discussion.

2 Voltage stability assessment

Power systems voltage stability depends on the relationshipsbetween the load bus voltage magnitude (V ), the reactivepower injection or absorption (Q) and the power supplied tothe load (P). Traditionally, the PV curve (Fig. 1) displaysthe relationship between voltage (V ) and transmitted power(P), while the QV curve (Fig. 2) plots the relationshipbetween the voltage level (V ) in a bus (if it is a PQ-type, itis fictitiously considered as a generation unit) and itsreactive power injection (Q) [28].As depicted in Fig. 1, the point A corresponds to the critical

operating point, which is the maximum loadability point.Only the operating points above A represent satisfactoryoperating conditions. On the other hand, as shown inFig. 2, the minimum in the QV curve (for which dQ/dV =0) represents the voltage stability limit. The system is stablein the region where dQ/dV > 0. Hence, an increase in Q isaccompanied by an increase in V. Finally, the system isunstable in the region where is dQ/dV < 0. The information

provided by the QV curve has faced some criticisms in theliterature [29], but recently, it has been shown that tracingthis curve may unfold a variety of reactive power controlcharacteristics [30]. For this reason, this curve is used here,hence the connection between the system load margin andthe reactive power reserve is considered.Voltage stability properties of power systems can be

analysed by computing PV and QV characteristics. This isbecause they provide different pieces of information, sincethe PV curve depicts the system load margin when load andgeneration vary according to a pre-specified pattern;whereas the QV curve measures the robustness of a buswith respect to the reactive support to the system. Thesecurves can be generated by executing a large number ofpower flows using conventional approaches. Conventionalpower flow algorithms may fail to converge at operatingpoints near the loadability limit, because the power flowJacobian matrix becomes ill conditioned. In contrast withthe conventional methods, the PV and QV curves can betraced by using the continuation method [21], which is aniterative process that works with predictor and correctorsteps. The power flow equations are reformulated, assuming

Fig. 1 PV curve

Fig. 2 QV curve

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2 IET Renew. Power Gener., pp. 1–10& The Institution of Engineering and Technology 2013 doi: 10.1049/iet-rpg.2012.0265

the form

f (x, l) = 0 (1)

where x represents the state variables and λ is the systemparameter that moves the system from an equilibrium pointto another. The value for λ is such that 0≤ Δλ≤ Δλcriticaland Δλ = 0 represents the base case. Beginning from aknown operating point (xi, λi), a tangent predictor is used toestimate the solution (xi + Δxi, λi + Δλi), which is correctedby the corrector step. The tangent predictor can be obtainedby manipulating the power flow Jacobian

DuDV

[ ]= [J ]−1 DP

DQ

[ ](2)

where θ is the vector of bus voltage angles and V is the vectorof bus voltage magnitudes. Relating (2) to (1), P and Q areassociated with λ, whereas θ and V are represented by x.Let the active and reactive powers into each bus beexpressed as P = Po(1+ Δλ) and Q=Qo(1+ Δλ),respectively. Po and Qo, respectively, stand for the initialnet active and reactive power at each bus. Using theseassumptions in (2) yields

TV =Du

DlDV

Dl

⎡⎢⎣

⎤⎥⎦ = [J ]−1 Po

Qo

[ ](3)

Since the parameter step size must diminish as the systemapproaches a bifurcation point, the following equation isemployed to control the step size variation

Dl = K

||TV|| (4)

where ||TV|| stands for tangent vector Euclidian norm and K isa parameter adjusting the ‘velocity’ of the process. Thetangent vector determines how the state variables (θ and V )change as the system parameter λ is varied. Equation (4)resolves the problem of the predictor step. As for thecorrector step, it is carried out in this paper by consideringthe predictor step as the initial guess of a power flowprogram. As described in the literature, this convergesrapidly for the actual solution [21].

3 Features of RES

There are commonly accepted benefits of distributedgeneration in power system operation. In distributionsystems, if large blocks of generation are distant electricallyor the transmission system is close to its limits, DER mayplay an important role. If the DER technology used is RES,the energy conversion emits low (or near to zero)greenhouse gases, which is a great advantage in comparisonwith traditional fossil-fuelled generation systems. Manyother advantages with respect to technical, environmental orinvestment aspects may be found in [3]. Nevertheless, theuse of DER may deteriorate the power quality and bringsome technical drawbacks, such as inversion of energyflow, difficult in voltage control and/or problems in systemprotection devices.Another important difference of some RES is the

intermittent characteristic of generation. Unlike

conventional fuel fossil generators, RES relies onavailability of primary (natural) resources to generatepower, which cannot be controlled. Note that RES canprovide energy with no fuel cost and reduced emissions.However, it cannot guarantee full power generation duringpeak hour. For this reason the term capacity credit (CC) iscreated which represents the reduction (in %) ofconventional power generation that can be made withoutaffecting the reliability index, ‘loss of load probability’.Thus, CC represents assured RES power for the system.Altogether, voltage collapse and intermittent generation

should not be analysed deterministically. An alternative andmore realistic way of analysis is the incorporation of pastexperience, associating their likelihood or probability ofoccurrence. Once the past experience is represented bystatistical parameters, the next step, comprising theapplication of special analysis and design techniques, iscarried out within this stochastic environment. Thisprocedure forms the basis of the probabilistic approach.Therefore past performance can be understood and futureperformance can be consistently predicted for the systemoperating planning.In this paper, uncertainty is assumed in RES generation.

Each generator is modelled by normal density functiontruncated between generator limits. Note, however, that anyother stochastic approach can be used.

4 Probabilistic features on simulating RES

As discussed in the previous sections, the intermittentgenerating characteristics of RES may be modelled by astochastic process. Hence, MCS technique may be used totest different system operating conditions.The system of Fig. 3 is used to introduce voltage stability

reliability [The system data and base case are presented inAppendix 1.]. Each generator is modelled by a normaldensity function truncated between generator limits, whichare considered between 0 and 60 MW for generator G1 and0 and 75 MW for generator G2, respectively. The meanvalues are 40 and 60 MW for G1 and G2, respectively.Such a normal density function is usually adopted in thiskind of study, as is reported in [31]. For each dispatchsampling, a load flow is executed, providing active loadmargin λP and reactive power margin λQ. If there exists nofeasible solution (voltage instability), λP is assumednegative [In this case, load shedding would be required inorder to restore a feasible operating point. Such a conditionis treated as a separate case in the simulations.] and λQ doesnot exist.The random variables λP and λQ are modelled using the

EDF approach [27]. If the size sample n is generated by aprobability distribution f, the EDF f is defined as a discreterandom variable whose each outcome has a probability of1/n. The EDF may be represented by the frequency functionof (5) and (6)

f lP . lP0

( )= 1

n

∑ni=1

1 lP . lP0

{ }(5)

g lQ . lQ0

( )= 1

n

∑ni=1

1 lQ . lQ0

{ }(6)

In (5) and (6), 1{A} indicates the occurrence of an event A, forinstance lP . lP0 and lQ . lQ0

. It is straightforward that for

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a given lP0 , 1{lP . lP0} is a Bernoulli random variable with

parameter p, and n f (x) represents a binomial random variablewith mean n f(x) and variance n f(x)[1 − f (x)]. Thus, f (x) is anunbiased estimator of f (x) (see [27] for a fuller explanation).The same results are obtained for g. This result enables oneto use the plug-in principle to estimate parameters fromsamples using (7), as follows

g = t(f ) (7)

where g is the estimated parameter γ and t(·) is a function ofthe EDF. Note that one may also use the EDF to model thejoint distributions. Fig. 4 presents two examples of jointEDF. In Fig. 4a, one can see the joint EDF of RES dispatchand active load margin λP. Likewise, Fig. 4b depicts thejoint EDF of RES dispatch and reactive power margin λQ.From Fig. 4, one can see that RES dispatch and active load

margin λP may be considered as a normal distribution. On theother hand, reactive power margin λQ seems to follow, forinstance, a shifted gamma distribution.

4.1 Voltage collapse reliability

Reliability theory is concerned to measure the risk of anundesirable random event. In this sense, the random natureof system parameters may be modelled with the help of theprobabilistic theory.Usually, reliability analysts are interested in the probability

of a system working in a time interval such as its failure rate,remain life and availability. The results are then considered insystem expansion, maintenance and operation planning.Sometimes, the concept of time is translated to othermeasures, such as distance or volume.Owing to the intermittent nature of DER generation,

discussed in Section 3, it has created uncertainty in theoperating condition of distribution systems. Thus, somequestions may be addressed

† How does the RES generation impact the operatingconditions?† What is the probability or risk of a system voltage collapsein the next time interval?† What are the best points in the system to generate reactivepower, bearing in mind the cost-benefit relation, in order tominimise the risk of voltage collapse for the next, or afuture, time interval?

If the RES generation is modelled by a stochastic process,one may answer the above questions from a probabilisticpoint of view and efficiently plan the distribution systemoperation. In this case, the concept of time is translated toRES dispatch. Thus, this work enables measurements of therisk of voltage collapse in distribution systems with RESpenetration.

Fig. 3 Four-bus test system

Fig. 4 Joint and Marginal EDF for

a RES dispatch and λPb RES dispatch and λQ

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When analysing the reliability of a system, somedistribution representations may be used: probability,cumulative probability, survivor and hazard functions.Probability function gives the probability of a specific

value of occurrence of a random variable. Note thatprobability distribution is the EDF discussed in a previoussection and is obtained by simulation. Moreover,cumulative probability yields information on intervaloccurrence. If the probability of a load margin below lP0 isof interest, one may then use the cumulative probabilityfunction to obtain this query. For a discrete random variablesuch as EDF, cumulative probability function is a sum ofprobability distribution up to the threshold value of interest,as given in (8)

F(x) = p lP , lP0

( )=

∑j|xj,lP0

f xj

( )(8)

Likewise, the probability of an active load margin greater thanthe base case (lP . lP0) for an RES generation below aspecific amount (Pg , Pg0

) can be obtained with the helpof the conditional probability law, as follows

f lP . lP0 |Pg , Pg0

( )=

f lP . lP0 > Pg , Pg0

( )

f Pg , Pg0

( ) (9)

Note, from (9), that (lP . lP0 > Pg , Pg0) partitions the

sample space and f (lP . lP0 > Pg , Pg0) is the

probability of events lP . lP0 and Pg , Pg0to happen

simultaneously. Note also, that f (lP . lP0 |Pg , Pg0) may

be seen as the hazard function of usual reliability analysis.Hazard function, which is the most popular reliabilitymeasure, gives the occurrence rate of an event of interest,that is, the risk associated to the system at a specific valuex. It is an intuitive representation of the risk and itsusefulness in comparing the change of risk over time forseveral populations (operating conditions) its importance is

widely acknowledged. In voltage stability analysis underDER penetration, it may represent the risk associated to asystem operating condition.If one is interested in measuring the risk associated with the

reactive power margin, the hazard function (9) may be writtenas

g lQ . lQ0|Pg , Pg0

( )=

g lQ . lQ0> Pg , Pg0

( )

f Pg , Pg0

( ) (10)

Note that the risk analysis may be extended to any othersystem random variable, such as voltage level, power flowand system loss. For this sake, the next section exploitssome numerical results for voltage collapse analysis underintermittent DER penetration.

5 Test results

To address the points proposed in the foregoing sections, theIEEE34 test feeder shown in Fig. 5 is used [The system dataand base case are presented in Appendix 2.]. Only RES areassumed as random variables. Thus, uncertainty because ofthe equipment unavailability, such as load and transmissionsystem equipment, is not considered in this paper.Furthermore, reactive power limits are considered in theload margin calculation. Simulation results are obtained bythe methodology described in Section 4. The probabilisticapproach is used to analyse the risks on the systemoperating and planning conditions. Both approaches arediscussed in the next subsections.

5.1 System operating condition analysis

In the first approach stressed in this paper, one is interested inextracting some pieces of information about the systemoperating condition with RES penetration. Thus, theprobability approach discussed in Section 4 is applied.Table 1 shows some probability indices obtained through

the simulated data. The sample space is constrained by the

Fig. 5 IEEE34 test feeder

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conditions displayed in the first column of Table 1. It is worthnoting that from the second column of Table 1 the risk ofobtaining no solution for the sampling case is reduced asthe RES generation increases.For those sampling cases with a feasible initial solution, the

active and reactive load margins are obtained and hazardfunctions are computed according to (9) and (10),respectively. Hence, for a given RES aggregated dispatchinterval, one can estimate the risk of the system to present aload margin lower than a pre-specified value. In this paper,this value is given by the load margin associated with thebase case. An important observation of Table 1 is the effecton the system when RES is dispatching < 600 kW, sincethe load margin obtained is smaller than the one calculatedfor the base case, and thus dispatching RES < 600 kWproduces a serious deterioration in system operatingconditions. The voltage collapse hazard function may bedepicted in a Cartesian plot, such as the one shown inFig. 6. In that figure, the curves represent different loadmargins obtained when the power generation at G1 and G2varies.From Fig. 6 it can be observed that when the RES dispatch

increases the system load margin also increases. The approachemployed enables one to assess the risks and costs associatedwith each generation value, since the gain in the load marginincrease is probabilistically determined.Tables 2 and 3 show the same results of Table 1 when using

marginal EDF for G1 and G2, respectively. One can concludethat G1 and G2 have almost the same efficiency when one

aims to reduce the risk of voltage collapse (or when theload margin increase is the objective).The result discussed above is closely related to the reactive

power margin behaviour. Fig. 7 depicts the joint probabilitydistribution for the simulated data of the active load marginλP and the reactive power margin λQ. Note that thecorrelation between λP and λQ is very strong. As aconsequence, a good reactive power margin is associatedwith a larger load margin, rendering reactive power marginas a good tool for effective reactive power controls.The probabilistic approach may also be used for system

operating planning, as will be discussed in the nextsubsection.

5.2 Planning system operation

In planning distribution systems operation, the operator isinterested in setting a benchmark index for securityoperation. For this reason, the results previously obtainedmay be used. For instance, if one is interested in operatingthe system while maintaining a secured distance fromvoltage collapse, the active load margin should be biggerthan a pre-specified value. This specified value of themargin is determined by how much reserve generation is

Table 1 Probability indices – IEEE test feeder

RES Generation(G1 +G2), kW

Risk of voltageinstabilityλP < 0, %

Active load marginprobability lower than the

base caselP , lP0 , %

≤ 100 100.00 100.00≤ 200 81.03 100.00≤ 300 37.97 100.00≤ 400 18.27 100.00≤ 600 6.58 99.76≤ 800 3.69 97.78≤ 1000 2.86 76.10≤ 1200 2.68 71.23

Fig. 6 Hazard functions for voltage collapse assessment– IEEEtest feeder

Table 2 Probability indices for G1 – IEEE test feeder

RES generation(G1), kW


Active load marginprobability lower than thebase case lP , lP0 , %

≤ 100 13.37 100.00≤ 300 6.13 95.27≤ 450 3.66 82.70≤ 600 2.68 71.23

Table 3 Probability indices for G2 – IEEE test feeder

RES generation(G2), kW



≤ 100 21.54 100.00≤ 300 6.17 96.83≤ 450 3.67 84.22≤ 600 2.68 71.23

Fig. 7 Joint and Marginal EDF for λP and λQ – IEEE test feeder

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available and the priority level of the load being supplied.Table 4 presents six generation scenarios and the riskassociated with them. Each scenario is given by thecomposition of generation at units 1 and 2 (G1 + G2).Columns 2–5 are obtained with the help of (9). Note fromTable 4 that the risk of having an unstable operating pointis very low, which suggests there is no need of reactivepower support for the base case. Furthermore, one canmeasure the probability of the system load margin beingbigger than any threshold value, such as 0.50lP0 , lP0 ,1.30lP0 , for example. This table may then be used as areference table for the system operation, where bulkoperating conditions reduce the expected load margin.

5.3 Comparison between the four-bus test systemand the IEEE34 test feeder

Further important conclusions may arise if different operatingscenarios are considered. As discussed above for the IEEE34test feeder, G1 and G2 have similar influence on the systemload margin. Besides, the reactive power margins of bothgenerators present negative values. The reader not familiarwith the meaning of positive and negative reactive powermargins is referred to [30], where a discussion about theQV curve features is carried out. In order to show someimportant information that can be obtained about the systemoperating conditions, the same indices obtained in theprevious section are presented here for the four-bus testsystem.Table 5 shows some probability indices obtained through

the simulated data for the four-bus test system. As expected,the same behaviour is observed when compared withTable 1, that is, the risk of obtaining no solution for thesampling case reduces as the RES generation increases.The voltage collapse hazard function is shown in Fig. 8,

has the same meaning as discussed in Fig. 6.Tables 6 and 7, however, presents some discrepancies with

respect to the results shown in Tables 2 and 3. Unlike

observed for the IEEE34 test feeder, when both generatorsplay similar role in the system load margin, this time G1 ismore efficient than G2 if the risk of voltage collapse (or theload margin increase) is the main issue.Another important aspect to be discussed regards the

reactive power margin. Fig. 9 depicts the joint probabilitydistribution for simulated data of the active load margin λPand the reactive power margin λQ, respectively. Once again,the correlation between λP and λQ is very strong.

Table 4 Reference table for the system operation – IEEE test feeder

RES generation(G1 +G2), kW

VoltageInstabilityλP < 0, %

Load margin probability higherthan half of base case,%

Load margin probabilityhigher than the base case,%

Load margin probability 30%higher than the base case,%

≥ 100 2. 20 73.65 28.91 6.63≥ 300 0.00 78.70 30.89 7.08≥ 500 0.00 96.31 38.50 8.83≥ 700 0.00 100.00 66.73 15.30≥ 900 0.00 100.00 100.00 42.00≥ 1100 0.00 100.00 100.00 100.00

Table 5 Probability indices – four-bus test system

RES generation(G1 +G2), MW



≤ 10 71.89 100.00≤ 20 32.19 100.00≤ 30 19.31 100.00≤ 50 5.81 100.00≤ 55 4.38 99.76≤ 60 3.67 98.77≤ 70 2.19 93.92≤ 90 1.06 79.25≤ 110 0.74 63.52

Fig. 8 Hazard functions for voltage collapse assessment –four-bus test system

Table 6 Probability indices for G1 – four-bus test system

RES generation(G1), MW



≤ 10 1.11 62.03≤ 20 1.02 60.57≤ 30 0.92 59.85≤ 40 0.82 59.16

Table 7 Probability indices for G2 – four-bus test system

RES generation(G2), MW



≤ 10 22.07 100.00≤ 30 3.23 100.00≤ 40 1.78 100.00≤ 50 1.16 98.25

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Recalling that when λQ > 0 stands for the lack of reactivepower margin, flagging imminent voltage collapseproblems. This was not observed in Fig. 7, when nopositive reactive power margin is shown. It can be observedin Fig. 9, that a low value of load margin (λP = 0.05) isobserved when λQ > 0. There exists 2.5% of chance tooperate in this situation.

6 Conclusions

This work discusses the use of distributed and renewableenergy resources in electrical distribution systems regardingtheir voltage stability and reliability. Voltage collapse isstudied by employing the PV and QV curves; hence thesystem load margin and the reactive power margin of eachbus are computed. A discussion on DER when RES ispresent, emphasising the drawbacks of reactive powerconsumption and the intermittent nature of RES generators.The voltage collapse and the intermittent generation are

analysed jointly with the help of MCS and EDF. Theresults obtained stress different operating ranges and theirassociated risk of collapse. In addition to that, the resultsobtained for the reactive power margin flag an importantcharacteristic of reactive power control, since this margin isreflected in the system load margin.

7 Acknowledgments

The authors would like to thank CNPq, CAPES andFAPEMIG for the financial support.

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Fig. 9 Joint and Marginal EDF for λP and λQ – four-bus testsystem

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9 Appendix

9.1 Appendix 1

The four-bus test system data are shown in Tables 8 and 9.

9.2 Appendix 2

The IEEE34 fest feeder data are shown in Tables 10 and 11.

Table 8 Data bus – four-bus test system

Bus Type V, p.u. θ, rad Pg, MW Qg, MVar PLPL, MW QL, MVar Qmin, MVar Qmin, MVar shunt, MVar

1 Swing 1.020 0.0 65.7 9.3 0.0 0.0 – – –2 PV 1.000 − 3.59 40.0 − 7.6 40.0 12.5 − 20.0 30.0 –3 PQ 0.976 − 12.11 0.0 0.0 50.0 16.5 – – 25.04 PV 1.000 − 14.31 50.0 29.9 60.0 12.5 − 20.0 30.0 –Load margin: 0.4362 p.u.

Table 9 Branch data – four-bus test system

Branch R, p.u. X, p.u.

1–2 0.02 0.102–3 0.10 0.204–3 0.10 0.20

Table 10 Data bus – IEEE34 test feeder

Bus Type V, p.u. θ, rad Pg, kW Qg, kVar PL, kW QL, kVar Qmin, kVar Qmax, kVar shunt, kVar

800 Swing 1.050 0.00 2,688.6 1,036.4 0.0 0.0 – – –802 PQ 1.047 − 0.09 0.0 0.0 55.0 29.0 – – –806 PQ 1.044 − 0.14 0.0 0.0 55.0 29.0 – – –808 PQ 1.004 − 1.23 0.0 0.0 16.0 8.0 – – –810 PQ 1.004 − 1.23 0.0 0.0 16.0 8.0 – – –812 PQ 0.959 − 2.60 0.0 0.0 0.0 0.0 – – –814 PQ 0.924 − 3.78 0.0 0.0 0.0 0.0 – – –850 PQ 1.010 − 4.33 0.0 0.0 0.0 0.0 – – –816 PQ 1.010 − 4.34 0.0 0.0 5.0 2.0 – – –818 PQ 1.009 − 4.35 0.0 0.0 34.0 17.0 – – –820 PQ 1.005 − 4.58 0.0 0.0 169.0 87.0 – – –822 PQ 1.004 − 4.61 0.0 0.0 135.0 70.0 – – –824 PQ 0.997 − 4.64 0.0 0.0 49.0 24.0 – – –826 PQ 0.997 − 4.63 0.0 0.0 40.0 20.0 – – –828 PQ 0.996 − 4.66 0.0 0.0 11.0 5.0 – – –830 PQ 0.972 − 5.25 0.0 0.0 97.0 43.0 – – –854 PQ 0.972 − 5.27 0.0 0.0 4.0 2.0 – – –852 PQ 0.932 − 6.37 0.0 0.0 0.0 0.0 – – –832 PQ 1.000 − 6.75 0.0 0.0 15.0 7.0 – – –858 PQ 0.995 − 6.88 0.0 0.0 49.0 25.0 – – –834 PQ 0.990 − 7.02 0.0 0.0 178.0 90.0 – – –842 PQ 0.990 − 7.03 0.0 0.0 9.0 5.0 – – –844 PQ 0.989 − 7.06 0.0 0.0 864.0 658.0 – – 300.0846 PQ 0.991 − 7.11 0.0 0.0 68.0 34.0 – – –848 PV 0.991 − 7.11 400.0 250.0 143.0 107.0 250.0 − 250.0 450.0860 PQ 0.989 − 7.03 0.0 0.0 348.0 212.0 – – –836 PQ 0.988 − 7.03 0.0 0.0 122.0 63.0 – – –840 PQ 0.988 − 7.03 0.0 0.0 94.0 62.0 – – –862 PQ 0.988 − 7.03 0.0 0.0 28.0 14.0 – – –838 PQ 0.988 − 7.03 0.0 0.0 28.0 14.0 – – –864 PQ 0.995 − 6.88 0.0 0.0 2.0 1.0 – – –888 PQ 1.000 − 7.04 0.0 0.0 0.0 0.0 – – –890 PV 0.997 − 7.65 400.0 250.0 450.0 225.0 250.0 − 250.0 –856 PQ 0.972 − 5.26 0.0 0.0 4.0 2.0 – – –Load margin: 0.1688 p.u.

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Table 11 Branch data – IEEE34 test feeder

Branch R, p.u. X, p.u.

800–802 0.00235 0.00244802–806 0.00157 0.00164806–808 0.02934 0.03048808–810 0.00858 − 0.00332808–812 0.03414 0.03547812–814 0.02706 0.02812814–850 0.00001 0.00900816–818 0.00056 0.00146816–824 0.01383 0.01009818–820 0.01590 0.04108820–822 0.00454 0.01172824–826 0.00448 − 0.00173824–828 0.00114 0.00083828–830 0.02769 0.02019830–854 0.00070 0.00051832–858 0.00664 0.00484832–888 0.09500 0.20400834–860 0.00274 0.00200834–842 0.00038 0.00028836–840 0.00117 0.00085836–862 0.00038 0.00028842–844 0.00183 0.00133844–846 0.00493 0.00360846–848 0.00072 0.00052850–816 0.00042 0.00031852–832 0.00001 0.00900854–856 0.03447 − 0.01334854–852 0.04990 0.03638858–864 0.00053 0.00138858–834 0.00790 0.00576860–836 0.00363 0.00265862–838 0.00567 − 0.00127888–890 0.34442 0.35781

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Probabilistic voltage stability assessment considering renewable sources with the help of the PV and QV curves

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