probabilistic approach to voltage stability analysis with load uncertainty considered

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2009; 19:209–224Published online 18 October 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/etep.207
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Probabilistic approach to voltage stability analysiswith load uncertainty considered

B. Wu*,y, Y. Zhang and M. J. Chen

Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China

SUMMARY

This paper proposes a new probabilistic assessment algorithm of voltage stability based onMonte-Carlo simulationand modal analysis considering uncertainty of the load level and load parameters. By Monte-Carlo sampling, thebus load level is determined according to the forecasted bus load curve of a research period. The coefficients of theload polynomials, which include induction motors, are treated as random variables with normal distribution. Atechnique of normal distribution sampling is utilized to simulate these coefficients uncertainty. Voltage stability isevaluated in the form of indices such as the expected maximum loadability and the statistics of systemparticipations, which are obtained from modal analysis near the point of collapse. A case study of the IEEE118-node system is given to demonstrate the validity of the proposed algorithm, and the effects of load uncertaintyand the proportion of motors on probabilistic assessment of voltage stability are investigated. Copyright # 2007John Wiley & Sons, Ltd.

key words: load uncertainty; voltage stability; modal analysis; probabilistic assessment

1. INTRODUCTION

Voltage stability has often been viewed as a steady-state problem [1]. With the continuing increase in

the demand for electric power and a tremendous change for electric power utility industry toward a

deregulated environment, hidden trouble induced by long transmission, heavy load and the lack of

reactive power emerges, and the voltage stability problem becomes more complex and serious. A

number of methods have been developed for assessing voltage stability, most of which are deterministic

[2–5]. Deterministic techniques analyze the mechanism of voltage instability clearly and require less

raw data. However, some inherent defects exist in them. They are based on the analysis of a

predetermined set of sever situation, which will lead to conservative results and is unable to tell how far

the system is to voltage instability, that is, what the voltage instability probability of the system is. It is

because deterministic methods do not reflect the probabilistic or stochastic nature of the system, for

example, customer demands. The unavoidable uncertainties of the loads are the results of demand

variation, measurement errors, and the modeling errors involved in the representation of loads. In

rrespondence to: B. Wu, Department of Electrical Engineering, Shanghai Jiao Tong University, No. 800, Road Dongchuan,A0403121, Shanghai, China.

mail: [email protected]

pyright # 2007 John Wiley & Sons, Ltd.

210 B. WU, Y. ZHANG AND M. J. CHEN

respect that uncertainties and inaccuracies are the inherent characteristics of the loads and as now well

known that loads with different characteristics have great impact on voltage stability, it is necessary to

evaluate voltage stability using probabilistic methods.

A few efforts have been done to overcome the shortages of deterministic methods’ inability of

considering the uncertainty of the system [6–11]. A contingency enumeration-based approach, taking

into stochastic elements faults and load forecast uncertainty account, is presented in Reference [6] to

evaluate the voltage stability of a power system. Reference [7] deals with the static voltage security

problem under stochastic conditions including uncertain load level, the random-forced outage of

generation, transmission, and protection facilities. In Reference [8], a voltage sensitive load model

considering the uncertainties involved in the coefficients of the load polynomials is developed and

sensitivity criteria of voltage stability in the case of series compensated lines is defined. In Reference

[9], a new nodal loading model, called the ‘hyper-cone’ model, is developed to simulate the uncertainty

of future loading. Reference [10] proposes a Cumulant Method-based solution to solve a maximum

loading problem incorporating a constraint on the maximum variance of the loading parameter. In

Reference [11], optimal power flow solved by an interior point algorithm is introduced into a

probabilistic simulator, allowing the calculation of probabilistic indicators of maximum loadability,

taking into account random-forced outages of generators, lines, and transformers. These methods

present valid voltage stability criterions taking into account the uncertainties of the system, yet few of

them consider the uncertainty of load composition, and the criterions proposed are unable to provide

enough information for identifying the weak buses and weak lines.

In this paper, the consideration of the uncertainties involved in the load level and the coefficients of

the load polynomials is dealt with, and voltage stability is evaluated by using modal analysis technique.

Bus load level, which is random, is determined by Monte-Carlo sampling according to the forecasted

bus load curve of a research period. As motors consume high reactive power at lower voltage level, they

are included in load composition in addition to constant impedance, constant current, and constant

power loads. The coefficients of the load polynomials, which refer to the fractions of load components

on the total power, are treated as random variables with normal distributions. These coefficients

uncertainty are simulated by utilizing normal distribution sampling technique. Once bus load level and

the coefficients of the load polynomials are determined, the system is loaded to near its point of

collapse, and modal analysis is used on this stressed system to assess the voltage stability. Voltage

stability is quantified in the form of indices such as the expected maximum loadability and the statistics

of system participations, which help to obtain the information of voltage instability probability, provide

criteria for contingency selection, and determine the areas close to voltage instability from a point of

view of probability.

2. DESCRIPTION OF LOAD UNCERTAINTY

This section describes the load representation adopted in the proposed algorithm and the sampling

methods used to simulate the bus load level and the load composition.

2.1. Load representation

Voltage stability is highly dependent on system load characteristics as well as on the modeling of the

subtransmission network and associated voltage control equipment. A polynomial model (1) has been

Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:209–224

DOI: 10.1002/etep

VOLTAGE STABILITY ANALYSIS WITH LOAD UNCERTAINTY 211

widely used to represent the voltage dependency of loads.

Pl ¼ Pl0ða2U2 þ a1U þ a0ÞQl ¼ Ql0ðb2U2 þ b1U þ b0Þ

�(1)

This model is commonly referred to as the ZIP model, as it is composed of constant impedance (Z),

constant current (I), and constant power (P) components. In this model a2–a0 and b2–b0 define the

proportion of each component, Pl, Ql are active and reactive components of the load when the bus

voltage magnitude is U, Pl0, and Ql0 are those at rating voltage.

Studies on load composition revealed that motors constitute more than half of the total load [1],

especially of the industrial energy consumption. Induction motors are loads that present high reactive

power consumption during depressed voltage conditions and therefore deserve careful consideration in

voltage stability studies. As neither the polynomial model nor the exponential model could represent

the characteristics of motors with accuracy, an aggregate induction motor model that has a two-node

structure is developed in Reference [12]. One of the nodes represents the motor terminals and the other

represents the transient internal voltage E0 and the mechanical torque. If subtransient effects on the

rotor, iron core losses and magnetic saturation are neglected, and motor parameters are assumed

independent of rotor speed, bus load with induction motors at bus i may be depicted in Figure 1.

In Figure 1, P0 and Q0 denote the loads except motors, which can be modeled as ZIP model. RS, X0,

and X0 are stator resistance, magnetizing reactance, and transient reactance of equivalent motor,

respectively.

Assuming that the active consumption of the motor Pmec is constant and it can be written in the form:

Pmec ¼ a3Pl0 (2)

here Pl0 is active component of the total load at bus i, including non-motor load and motors, when

U¼ 1.0. Thus, neglecting active loss of the motor, from Equations (1) and (2), Pl and Ql at bus i can be

expressed as:

Pl ¼ P0 þ Pmec ¼ Pl0ða2U2 þ a1U þ a0 þ a3ÞQl ¼ Q0 þ Qm ¼ Q0

0ðb2U2 þ b1U þ b0Þ þ Qm

�(3)

where from a3 to a0 define the proportion of motors and each component of non-motor load in Pl0,

respectively. Q00 is the reactive component of the non-motor load at rating voltage, and bi is the

proportion of each component in it. So the sum of the coefficients ai or bi satisfies Equation (4). Qm is

the reactive consumption of the induction motor, which is represented by reactive consumption of the

Figure 1. Load model of a bus load with induction motor loads.


DOI: 10.1002/etep


additional shunt element j(X0�X0) and the additional impedance (Rsþ jX0).

P3i¼0

ai ¼ 1

P2i¼0

bi ¼ 1

8>><>>:

(4)

2.2. Probability sampling

Bus load uncertainty always exists in an actual power system and it has been recognized that different

load characteristics have different impacts on voltage stability [13]. It is therefore necessary to carry out

the voltage stability analysis for various load level and all combinations of possible loads considered

over their range of variability.

Monte-Carlo simulation is a technique based on statistical test and obtains statistical results by

sampling and data analysis. It samples according to sample distribution of the stochastic variable, such

as load level, the status of generators, lines, transformers, and so on. Probabilistic indices can be

estimated as the average of the test function values calculated for each of N samples, if sample size N is

large enough.

In this paper, two steps are performed for probability sampling. The first step is to determine the

system load level, which is consisted of active load at every bus. For a given system, the forecasted bus

active power load curve provides all possible load levels and their occurrence probability over the

research period. The load level Pl0 of each bus may be then determined by sampling according to the

occurrence probability of these load levels. Assuming that power factor of ZIP load, pf, is definite. Then

for the bus without induction motors, the corresponding reactive power load may be calculated by pfand Pl0. However, for the bus with motors, reactive power load of the ZIP load model, Q0, will becalculated after the second step and Qm will be calculated by the power flow program.

The second step is to sampling the coefficients of the load polynomials that denote the fractions of

load components on the total power. For a particular bus, load components vary due to many factors

such as time, temperature, and so on. However, the fractions of load components vary not drastically

but over a limited range, so these coefficients may be treated as random variable with normal

distributions.

Assuming that ai (i¼ 0, 1, 2) is a normally distributed variable with desired value mi and standard

deviation si. According to probability theory, linear combination of normal distributions is still a

normal distribution. So, if as,ms, and ss are the sums of ai,mi, and si (i¼ 0, 1, 2) respectively, aswill be

a normal distribution whose standard deviation is ms and desired value is ss. Then a3, which has to

satisfy Equation (4), that is, (5) can be proved to be a normally distributed variable with desired value

(1�ms) and standard deviation ss. The certification process is given in Appendix.

a3 ¼ 1�X2i¼0

ai ¼ 1� as (5)

In a similar way, once b0 and b1 are supposed to be normally distributed variables, b2 will be proved

to be normally distributed and may be calculated by using Equation (6).

b2 ¼ 1� b0 � b1 (6)


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A technique of normal distribution sampling [14] is utilized to simulate the coefficients uncertainty

of ai (i¼ 0, 1, 2) and bi (i¼ 0, 1). a3 and b2 are then obtained by Equations (5) and (6).

Once coefficients of the load polynomials are determined, for the buses with motors, the active and

reactive components of non-motor load may be expressed by using Equation (7).

P0 ¼ Pl0ða2U2 þ a1U þ a0ÞQ0 ¼ Q0

0ðb2U2 þ b1U þ b0Þ

�(7)

where Q00¼P0

0 ctg (arccos pf), and P00 is P

0 at rating voltage, that is, P00¼Pl0 (a2þ a1þ a0).

3. PROBABILISTIC ASSESSMENT OF VOLTAGE STABILITY

Although, besides load, the forced outages of generators, lines, and transformers affect the voltage

stability of power system, they may be easily introduced in probabilistic assessment by Monte-Carlo

sampling. In order to stress the voltage stability affected by the uncertainty of load composition, these

forced outages are not involved in this paper.

3.1. Probabilistic assessment of voltage stability

On the basis of continuation power flow and modal analysis in the vicinity of the point of collapse, a set

of indices for probabilistic assessment of voltage stability is presented in this paper. The expected

maximum loadability and the probability of voltage instability are basic probabilistic indices. The

maximum loadability is calculated for each load sample and the expected maximum loadability is

given by the following equation:

Elm ¼ 1

N

XNk¼1

Pkmax (8)

where N is the number of load samples, Pkmax is the maximum loadability calculated by continuation

power flow for the k-th load sample.

The probability of voltage instability for a particular load level is then obtained by probability

distribution of maximum loadability. The expected system participation factors, such as bus

participations and branch participations, are defined in a way similar to the expected maximum

loadability as:

Epb;i ¼1

N

XNk¼1

Fk;i (9)

Epl;lj ¼1

N

XNk¼1

Fk;lj (10)

where Fk,i and Fk,lj, obtained by modal analysis in the vicinity of the point of collapse, are the

participations of bus i and branch lj for the k-th sample, and Epb,i and Epl.lj are the desired values of

them, respectively.


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For the k-th sample, let jk and hk stand for the right and left eigenvector of the system Jacobian matrix

corresponding to critical mode, respectively. Fk,i, the participation factor of bus i is defined as:

Fk;i ¼ jk;ihk;i (11)

where jk,i, hk,i are the i-th elements of jk and hk. The participation factor of branch lj to critical mode,

Fk,lj, is defined as:

Fk;lj ¼DQk;lj

maxðDQk;ljÞ(12)

where DQk,lj is the linearized reactive loss variation across transmission branch lj. Branch

participations indicate which branches consume the most reactive power in response to an incremental

change in reactive load.

As a result of statistical calculation, the higher desired values of participations indicate which buses

are more vulnerable to voltage instability and which branches are more heavily loaded when consumer

demands vary. The expected system participations determine the areas close to voltage instability and

help to bring forward effectual remedial actions to alleviate the loading on branches from the viewpoint

of probability.

According to the probabilistic indices, the measures taken for voltage stability enhancement will

have effects on large numbers of operating conditions instead of on a few of predetermined severe

situations occurring with smaller probability that are considered in deterministic method. Thus, the

proposed probabilistic analysis provides basic information for power system planning and is better

suited for the economic trade-off between the cost of installing new equipment and the benefits in terms

of enhanced voltage stability. The proposed probabilistic method can also detect some severe situations

that will occur due to the system randomicity yet may be neglected by deterministic method.

3.2. The procedures of probabilistic voltage stability assessment

As an aggregate induction motor model may be modeled as a two-node structure, an n-buses system, of

which m buses are loaded with ZIP load and induction motors, may be equivalent to an expanded

system with total nþm buses. Of the new expanded system, n buses are loaded with ZIP load and m

buses are only loaded with constant active power load. Then the procedures of probabilistic assessment

can be described as follows:

Step 1: Determine the system active power load level of initial n-buses system according to the

forecasted bus load curve.

Step 2: Sampling the coefficients of the load polynomials, ai (i¼ 0, 1, 2), bi (i¼ 0, 1), and then

calculate a3 and b2 using Equations (5) and (6).

Step 3: For expanded system, the load at each of extra m buses is calculated by Equation (2), and the

load at buses that connect to these extra m buses will be expressed by Equation (7).

Step 4: With the help of continuation power flow, the maximum loadability, each critical bus voltage

of expanded system and thenQm may be obtained. Thus the load at buses with motors in initial n-buses

system can be calculated by using Equation (3).

Step 5: For initial n-buses system, modal analysis in the vicinity of the point of collapse is performed,

and system participations are calculated by using Equations (11) and (12).

Step 6: Repeat step 1–step 5 for N times.


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Step 7: Calculate expected maximum loadability and expected system participations using Equations

(8)–(10).

4. NUMERICAL EXAMPLE

The technique described in the previous sections has been applied to IEEE standard system with

118 nodes [15]. Let it be supposed that 10 of the 118 buses, bus 20–23, 43–45, and 95–97, are loaded

with ZIP load and induction motors, whose equivalent parameters are derived from Reference [1], and

that the others are loaded without motors. In all of the numerical examples reported in this paper, the

desired values of the coefficients of ZIP load are assumed to be equal. That is to say, if induction motors

account for 70% of the total load of the bus at which motors are connected, the desired values of the

coefficients of ZIP load at the bus are all equal to 10%. For simplicity, all of the standard deviation (SD)

of the sampled coefficients ai (i¼ 0, 1, 2) and bi (i¼ 0, 1) is given to be 5%. The forecasted bus load

curve of a research period is referred to Reference [16], from which 8736 load levels of a year are given.

A computer program has been developed to calculate Elm, Epb,i, and Epl.lj as defined in Section III,

and case studies are conducted on a Pentium 4 2.0-GHz PC. In order to demonstrate the validity of the

proposed algorithm and illustrate the effects of different degrees of SD and motor load on probabilistic

voltage stability assessment, the following three cases are considered.

4.1. Case I

The comparison between the proposed method and the deterministic methods, continuation power flow

and modal analysis, is conducted. The expected percentage of motor load of the buses with motors is

supposed to be 70%, and the standard deviation of the sampled coefficients is 5%. Samplings (10 000)

are attempted, and then Elm of the test system, 108.3770 per unit, with standard deviation 4.32% is

obtained. Due to repeated samplings, it takes about 2.31� 104 seconds to have the results. The

minimum value of the maximum loadability, that is, the shortest distance to instability, for possible load

levels is also obtained, which is 92.3238 per unit. Probability density curve of maximum loadability is

illustrated in Figure 2. It shows the possibility of maximum loadability of thewhole systemwith clarity.

Figure 3 gives the information about voltage instability probability when the total system demand

reaches a certain level. For example, if the total system active power demand reaches 100.75 per unit,

from Figure 3, it can be seen that the voltage instability probability of the system is 5%. This illustration

is useful for power system planning in order to determine whether it is necessary to put money into new

electric equipments to improve voltage stability, so that voltage instability probability for a certain

demand level of the system may be reduced.

The expected bus participations for each bus are ranked from the highest to the lowest. Due to space

limitation, only the top 15 buses are listed in Table I, and the top 15 branches that have the highest

expected branch participations are listed in Table II.

Based on expected bus participations, the areas prone to voltage instability are identified. The

heavily loaded branches, such as 19–20, are found, which are useful for identifying remedial measures

to alleviate voltage stability problems and for contingency selection when treated in a probabilistic

way.

For the purpose of comparing with deterministic method, the maximum loadability is calculated by

continuation power flow for the peak load of the year on the supposition that the percentage of motor

load of the buses with motors is 70%, namely a severe condition. The result is 100.3834 per unit, larger

than the minimum value of the maximum loadability, 92.3238 per unit, calculated by proposed


DOI: 10.1002/etep

Figure 2. Probability density curve of maximum loadability.


probabilistic method. It is just that some more severe conditions are left out when predetermining the

severe condition in deterministic method, and yet these neglected conditions are taken into account by

probabilistic method.

Modal analysis for the predetermined severe condition is employed near the point of voltage

collapse. The top 15 buses and branches that have the highest expected branch participations are also

listed in Table I and Table II, respectively. As the great impact of motors on voltage stability, the

weakest buses in the first six identified by deterministic method and probabilistic method are in

the same order in Table I. However, for those buses without motors the marshaling sequences of the

weakest buses identified by the two methods are different. It is because for many conditions induced by

Figure 3. Probability distribution of maximum loadability.


DOI: 10.1002/etep

Table I. Top of buses with the highest expected participations.

Rank Proposed method Deterministic method

Bus Participation Bus Participation

1 21 0.4287 21 0.43002 22 0.3549 22 0.35483 20 0.2014 20 0.20654 23 0.0063 23 0.00635 44 0.0045 44 0.00246 45 0.0021 45 0.00117 43 0.0019 43 0.00118 38 1.449 e-4 30 7.8678 e-59 30 1.416 e-4 38 7.6366 e-510 17 2.489 e-5 17 1.3766 e-511 37 1.111 e-5 37 3.7295 e-612 68 1.027 e-5 33 2.3754 e-613 33 9.265 e-6 16 1.6095 e-614 72 9.234 e-6 47 6.5621 e-715 75 8.742 e-6 13 5.5358 e-7


system random the effectiveness of remedial actions applied at that bus in voltage stability is not

invariable.

4.2. Case II

Two SD of 5% and 10% are considered, respectively, with 70% motors at the ten buses mentioned

above. The results are shown in Tables III and IV. The values in the ‘M’ row are the expected percentage

of motor load at those ten buses.

Table II. Top of branches with the highest expected participations.

Rank Proposed method Deterministic method

Branch Participation Branch Participation

1 19–20 0.7650 19–20 1.00002 38–65 0.7256 22–23 0.62533 26–30 0.5848 38–65 0.35234 23–32 0.4169 26–30 0.30955 25–27 0.3974 23–32 0.23306 22–23 0.3673 25–27 0.19957 30–17 0.2819 20–21 0.18698 23–25 0.2428 30–17 0.14569 20–21 0.1781 23–24 0.133310 45–46 0.1615 23–25 0.096011 38–37 0.1594 45–46 0.079212 45–49 0.1237 38–37 0.076613 42–49 0.1171 45–49 0.060214 17–18 0.1171 42–49 0.058915 23–24 0.1099 17–18 0.0589


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Table III. Maximum loadability indices for different SD.

SD 5% 10%

Elm (p.u.) 108.3770 108.5078Standard deviation of Elm (%) 4.32 8.14Minimum Pkmax (p.u.) 92.3238 79.0552Computation time (seconds) 2.31� 104 2.35� 104


The results in Table III indicate that the expected maximum loadability Elm with different SD are

almost the same, the difference between them is quite small and may be covered by computational

numeric errors. In the case of a relatively large SD of 10%, however, the standard deviation of Elm is

much larger than that of 5%, and the shortest distance to instability is smaller than that of 5%, which

denote that uncertainty of load components are prone to result in voltage instability. Table III also

provides the computation time and shows that the effect of different degree of SD on it is little. The

probability density curves and probability distributions of maximum loadability are shown in Figures 4

and 5.

It may be seen from Figure 5 that the voltage instability probability will be 5% when load demand

reaches 100.75 per unit in the case of a relatively small SD of 5%, while in the case of SD of 10% it is

94.3 per unit.

The top buses and branches with the highest expected participations for different SD listed in

Table IV show that different degrees of SD have little impact on the rank of buses and branches that are

prone to induce to voltage instability.

4.3. Case III

Induction motor reactive power is sensitive to voltage level. When the voltage is well below rating

voltage, the reactive power drawn by induction motors will increase noticeably with decreased voltage

Table IV. Top of buses and branches with the highest expected participations for different SD and differentproportion of motor load.

Rank Bus Branch

M 70% 70% 10% 70% 70% 10%

SD 10% 5% 5% 10% 5% 5%

1 21 21 44 19–20 19–20 38–372 22 22 22 38–65 38–65 38–653 20 20 21 26–30 26–30 44–454 44 23 38 23–32 23–32 26–305 23 44 43 25–27 25–27 25–276 45 45 45 22–23 22–23 69–707 43 43 20 30–17 30–17 23–328 38 38 30 23–25 23–25 42–499 30 30 23 45–46 20–21 42–4910 17 17 37 20–21 45–46 30–38


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Figure 4. Probability density curves of maximum loadability with different SD.


[1]. The percentage of load modeled as motors are therefore plays an important role on voltage stability

of the system.

Suppose that SD is 5%, Tables IVand V show the results with different expected percentage of motor

load of the total load at the ten buses.

The results in Table V indicate that the expected maximum loadability Elm and the shortest distance

to instability obtained considering 70% motor load are much smaller than that obtained considering

10% motor load, while the standard deviation of Elm in the case of 70% motor load is much larger than

that with 10%motor load. It can be seen that different percentage of motor load has very little influence

on computation time. Figures 6 and 7 illustrate the probability density curves and probability

distributions of maximum loadability considering various expected proportion of motor load.

Figure 5. Probability distributions of maximum loadability with different SD.


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Table V. Maximum loadability indices for different proportion of motor load.

M 10% 70%

Elm (p.u.) 140.6226 108.3770Standard deviation of Elm (%) 0.52 4.32Minimum Pkmax(p.u.) 138.4780 92.3238Computation time (seconds) 2.30� 104 2.31� 104

Figure 6. Probability density curves of maximum loadability considering different proportion of motor load.

Figure 7. Probability distributions of maximum loadability considering different proportion of motor load.


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From Figure 7 we can see that the voltage instability probability will be 5% when load demand

reaches 139.5 per unit in the case of a low percentage of motor load of 10%, while in the case of 70%

motor load it is 100.75 per unit, much smaller than that in the case of 10%.

In Table IV, the top buses and branches with the highest expected participations obtained considering

various percentage of motor load are dissimilar. Of the top ten buses with the highest participations, the

buses with motors are more centered on the top when the expected percentage of motor load is

relatively high of 70%. For the top ten branches with the highest participations, it is similar. More

branches with one end belonging to one of the ten buses with motors put into the first ten. The different

percentage of motor load has great effect on the rank of buses or branches that are vulnerable to voltage

instability.

5. CONCLUSIONS

This paper proposes a new probabilistic assessment algorithm of voltage stability for power system

planning based on Monte-Carlo simulation and modal analysis, considering uncertainties of the load

level and the load parameters. Bus load level is determined by load curve of a research period, and the

coefficients of the load polynomials, which refer to the fractions of load components on the total power,

are sampled by normal distribution sampling technique. Since induction motors consume much of

reactive power at low voltage level, they are considered in the load polynomials. Once the load level

and the load parameters have been decided, modal analysis is performed in the vicinity of the point of

collapse. As a result of statistical calculation, a set of probabilistic indices are proposed to evaluate

voltage stability of the system.

The numerical study on IEEE system with 118 nodes proves the suggested approach suitable for

probabilistic assessment of voltage stability. Based on the indices derived from the suggested

algorithm, the areas prone to voltage instability and the strength or weakness of the transmission lines

can be identified. The probability density curve, as well as probability distribution of maximum

loadability provides information about voltage instability probability when the total system demand

rises to a certain level. These results are better suited for the economic trade-off between the cost of

installing new equipment and the benefits in terms of enhanced voltage stability in the planning of

power systems.

The effects of different degrees of SD and different percentage of motor load on probabilistic voltage

stability assessment are investigated. The case studies show that the system is more close to voltage

instability in the case of a relatively large SD or a relatively high percentage that motor load accounts

for. However, these two factors mentioned above have little effect on computation time. Comparing

with deterministic method, the proposed probabilistic approach spends much more time and is suitable

for off-line computation and analysis. Work is in the process to employ efficient sampling method or

computational method of voltage stability indices to reduce the computation time of probabilistic

assessment.

6. LIST OF SYMBOLS AND ABBREVIATIONS

ai th

Copyright #

e proportion of each component on the total active power of the composite load

as th
e sum of ai (i¼ 0, 1, 2)
bi th
e proportion of each component on the total reactive power of the composite load
2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:209–224

DOI: 10.1002/etep


E0 t

Copyright #

ransient internal voltage of equivalent motor

Elm t
he expected maximum loadability
Epb,i t
he expected bus participations of bus i
Epl.lj t
he expected branch participations of branch lj
fX (x) p
robabilistic density function for variable X
Fk.i t
he bus participation of bus i for the k-th sample
Fk.lj t
he bus participation of branch lj for the k-th sample
FX (x) c
umulative distribution function for variable X
N t
he number of load samples
pf p
ower factor
Pl a
ctive component of the load when the bus voltage magnitude is U
Pl0 a
ctive component of the bus load at rating voltage
P0 a
ctive loads except motors
P00 a
ctive loads except motors at rating voltage
Pmec a
ctive consumption of equivalent motor
Pkmax t
he maximum loadability calculated for the k-th load sample
Ql r
eactive component of the load when the bus voltage magnitude is U
Q0 r
eactive loads except motors
Q00 r
eactive loads except motors at rating voltage
RS s
tator resistance of equivalent motor
SD t
he standard deviation of the sampled coefficients
U t
he bus voltage magnitude
X0 m
agnetizing reactance of equivalent motor
X0 t
ransient reactance of equivalent motor
si t
he standard deviation of ai ss t he sum of si (i¼ 0, 1, 2)
mi t
he desired value of ai ms t he sum of mi (i¼ 0, 1, 2)
7. APPENDIX

Assuming that X is a normally distributed variable with desired value m and standard deviation s,

whose probabilistic density function is

fXðxÞ ¼1ffiffiffiffiffiffi2p

pse�ðx�mÞ2

2s2 ; x 2 ð�1;þ1Þ (A1)

Y is defined to be a random variable as

Y ¼ 1� X ¼ gðxÞ (A2)

If y 2 ð�1;þ1Þ, cumulative distribution function of Y is

FYðyÞ ¼ PfY � yg ¼ PfX � hðyÞg ¼Z hðyÞ

�1fXðxÞdx (A3)

2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:209–224

DOI: 10.1002/etep


where P{�} is probability that meets the inequality condition in {}, h(y) is inverse function of g(x)

hðyÞ ¼ 1� y (A4)

Then probabilistic density function of Y can be represented as

fYðyÞ ¼ fXðhðyÞÞh0ðyÞ (A5)

where h0(y) is the derivate of h(y) in relation to y,

h0ðyÞ ¼ �1 (A6)

Substituting Equations (A1), (A4), and (A6) into (A5) yields

fYðyÞ ¼1ffiffiffiffiffiffi2p

pse�½y�ð1�mÞ�2

2s2 (A7)

Thus, Y is proved to be a normally distributed variable with desired value (1�m) and standard

deviation s.

REFERENCES

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DOI: 10.1002/etep


AUTHORS’ BIOGRAPHIES

Bei Wu was born in China in 1978. She obtained the B.Sc. and M.Sc. degrees in ElectricalEngineering from Hefei University of Technology, China, in 2000 and 2004, respectively, andonce had been worked there for 1 year. She is presently pursuing the Ph.D. in the Department ofElectrical Engineering at Shanghai Jiaotong University, where she is Graduate ResearchAssistant. Her main research interests are in the areas of power system voltage stability andcontrol.

Yan Zhang was born in China in 1958. She received B.Sc. degree from Hefei University ofTechnology in 1982, the M.Sc. degree from China Electric Power Research Institute in1987,and the Ph.D. from Shanghai Jiaotong University in 1998. Currently she is Professor and Headof the School of Electronic, Information, and Electrical Engineering at Shanghai JiaotongUniversity. Her research interests include power system planning, reliability, and distributionsystem automation.

Minjiang Chen was born in China in 1975. He obtained the B.Sc. and M.Sc. degrees inElectrical Engineering from Hefei University of Technology, China, in 1997 and 2004,respectively. Since 2004, he has been a Ph.D. candidate in the Department of ElectricalEngineering at Shanghai Jiaotong University, where he is Graduate Research Assistant. Hisareas of research interests are voltage stability and the effects of FACTs on power system.


DOI: 10.1002/etep

probabilistic approach to voltage stability analysis with load uncertainty considered

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