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Abstract--A new method for probabilistic power flow

considering transmission network contingency is proposed for the purpose of removing the unrealistic assumptions of conventional analytical PLF techniques. Because any component failure in a transmission network will change the system configuration, the calculation of power flow distribution needs to consider those contingencies. The method proposed in this paper is able to provide a summation of conditional probability of the load flow solutions assuming all network configurations resulted from contingencies have occurred. The proposed method is based on the combination of Cumulant method for probabilistic power flow and Logarithmic barrier interior point method type solution, which is used to estimate the statistics of load flows. By using this method, the exact level of risk can be expressed easily, which is impossible by deterministic philosophy. This method can bring more economical benefits than that of other PLF methods with respect to the costs and risk levels. The method is validated via a real large transmission system – Queensland network, with forced outages of circuits and generating units. More accurate boundary of variables is calculated with this efficient method.

Index Terms—Probabilistic Load Flow, Cumulants, Probabilistic Distribution function, Gram-Charlier A Series, Normalized Coefficient of Probability.

I. NOMENCLATURE

k = network configuration u = unavailability of component a = availability of component σ = standard deviation µ = the mean of normal distribution

II. INTRODUCTION

N recent years, much effort has been devoted to produce a practical computational procedure for large power system

evaluation. Probabilistic load flow (PLF) has brought more confidence in these power system studies. PLF techniques have been directly applied to both short-term operational planning and long-term expansion planning problems [1], [2]. The PLF assessment can measure the impact of generating unit unavailabilities and load uncertainties on the steady state

Acknolwedgement: The research is supported by an APA scholarship and ACCS Top-Up Scholarship with the first author from the University of Queensland.

Miao Lu, Zhao Yang Dong and Tapan Kumar Saha are with The School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia (e-mail: lumiao, zdong, saha@itee.uq.edu.au).

performance of a given system. It is commonly accepted that PLF can be used to assess adequacy indices, such as: the probability of a line flow being greater than its thermal rating, and probability of a bus bar voltage being outside its operational constraint. These are extremely useful parameters in planning and operation of power systems. Usually, this analysis can be carried out by Monte Carlo Simulation (MCS) techniques, analytical techniques or by a combination of these two techniques.

MCS [3] has been applied for this purpose for many years. This technique evaluates the indices by simulating the actual process and random behavior of the system. MCS treats the problem as a series of experiments. In order to obtain meaningful results, thousands of simulations are needed. Theoretically, there is no constraint for this method. However, the computation is very time consuming. The high computational cost often makes this technique unattractive.

Analytical techniques represent the problem by a mathematical model and assess the indices using direct numerical solutions. A combined Cumulants and Gram-Charlier Series expansion method for DC PLF has been presented in [4]. It enables the operators and planners to obtain the possible ranges of power flow and the probability of occurrence quickly since a simple arithmetic process is used instead of the complex convolution calculation. Moreover, it can obtain results in one run. Therefore, it is able to handle large system with accurate solutions quickly. The Edgeworth Expansion is used by authors of [5] for optimal power flow analysis. It utilized some matrices and computed components as part of the Newton method step. While maintaining a higher level of accuracy, this algorithm has reduced the computational burden compared with the MCS. In [6], First-Order Second-Moment Method and Cumulant method have been compared for their suitability with different assumptions. The major demerits of analytical techniques are that the assumptions are required to simplify the problem. For PLF analysis, the most commonly used assumptions are, [1], [4-5]:

• Total independence between nodal power injections; • Stable system configuration; • Most of the variables are based on normal

distributions. The assumption of stable system configuration somehow

limits the application of those methods. Under this assumption, the probability of losing any network components

A Probabilistic Load Flow Method Considering Transmission Network Contingency

Miao Lu, Student Member, IEEE, Zhao Yang Dong, Senior Member, IEEE, and Tapan Kumar Saha, Senior Member, IEEE

I

1-4244-1298-6/07/$25.00 ©2007 IEEE.

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is neglected. However, for a given system state, the probability of the fully available state occurrence must be less than unity. Consequently, whether for system operational planning or long term expansion planning, the performance of various generation and transmission line configurations must be considered. Leite da Silva et al. [7] concluded two formulations which considered the effects of network outages in the PLF analysis and extended them for more efficient calculation based on MCS. The method in [7] obtains the distributions of unknown variables from a weighted sum of density function. As aforementioned, the computational burden of method based on MCS is very heavy. Recently, authors of [8] proposed a method where the variations of outputs are produced by normally and discretely distributed input variables. Random branch outages are simulated by fictitious power injections with 0-1 distributions at the corresponding nodes.

In order to overcome the above discussed limitations of PLF methods, a new PLF method is proposed in this paper. It is carried out by analytical analysis to produce a practical computational procedure for bulk power system and takes the system contingency into account. The following sections of this paper are organized as follows: Section III presents the formulation of the proposed analytical PLF method. Section IV briefly introduced the cumulant based PLF method that is applied in this paper. Finally, one case study based on the Queensland network from Australia is implemented for illustration purpose.

III. PROBLEM FORMULATION

A. Network Configuration

In a power system, different component failures may lead to different network configurations. Accordingly, the input and output probabilistic distribution function (PDF) for analytical model will be changed as well. Thus, the PLF in this paper considers each network configuration as an individual case. Moreover, each configuration has an associated probability. For example, if a particular transmission line with failure rate P is out of service, then, the probability of occurrence for the network configuration at this state is P. In addition, it is impossible for two or more network configurations to appear at the same time.

According to total probability theorem, given n mutually

exclusive events kAA ,,1 L , whose probabilities sum to unity

( 1)(...)( 1 =++ kAPAP ), then

)()|(...)()|()( 11 kk APABPAPABPBP ++= (1)

Through this section, the Ak is defined as the kth network

configuration; )( kAP is the probability of the kth network

configuration occurrence; B represents the variables of power flow; )(BP is the final PDF of each power flow variable,

such as active power, and )|( kABP is the probability of

power flow variables when the kth network configuration occurred.

Following the above analysis, the PDF of each power flow variable )(BP can be seen as a summation of the product of

the probability of each configuration occurrence )( kAP and

the probability distribution under this network

configuration )|( kABP . )|( kABP can be obtained easily

through conventional PLF calculation. )( kAP can also be

computed by using system components’ failure rate. The crucial problems for this evaluation are: the analysis of

all possible configurations and calculation of the associated

probability )( kAP , especially for a large scale power system.

For an N components system, it has 2N combinations of each component. Thereby, there are 2N network configurations. It is impossible to evaluate all configurations for a practical system. In order to simplify the problem, we reduce the number by considering maximum 3 components fail simultaneously. It’s more than n-2 criteria, and is sufficient for most of power system operations and planning requirement. Accordingly, the computational burden is greatly reduced.

In regard to the probability of a network configuration k, it can be expressed as a function of components availability and unavailability. If the outage of each component is independent, then the probability of the situation (c1,c2…cn, d1,d2,...dm) can be expressed as:

)()()(1 1

∏ ∏= =

⋅=n

i

m

j

kj

kik dacuAP (2)

where components ( kn

kk ccc L,, 21 ) are out of service and

components ( km

kk ddd L,, 21 ) are in service. In large systems,

the probability can be a very small number. For example, a system with 5000 components, the availability of each component is 0.98, and then the probability of any N-1 situation is:

P = 0.02 × (0.98)4999 = 2.755 × 10−46 (3)

In this paper, the probability )( kAP will be replaced with

Normalized Coefficient of Probability (NCP), which is defined as the current state probability )( kAP divided by a

common factor∏Ω∈ic

kica )( , [4], [9], as below.

∏

∏∏

Ω∈

∈∈

⋅

=ki

kj

ki

c

ki

INd

kj

OUTc

ki

k

ca

dacu

NCP)(

)()(

(4)

where Ω represents the set of all connected components of the system, so that OUTIN U=Ω . The merit of using NCP is

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that it does not depend on the system size. Therefore, only the failed components need to be considered.

Mathematically, the common factor can be represented as:

∏∏∏∈∈Ω∈

⋅=INd

kj

OUTc

ki

c

ki

kj

ki

ki

dacaca )()()( (5)

Then, the definition of NCP can be expressed according to the following equation:

∏∈

=OUTc

ki

kik

ki

ca

cuNCP

)(

)( (6)

Therefore, the computation of probabilistic load flow can be seen as a process to calculate the conditional probability

with the replacement of kNCP .

∑=

⋅=

++=k

k

kk

kk

NCPABP

APABPAPABPBP

1

11

))|((

)()|(...)()|()( (7)

Because the maximum number of failure components are

three, in Eqn. (3), kNCP is a big enough value for calculation. Hereby, it can greatly reduce the frequency of computation of small value and improve the calculation accuracy sequentially.

B. Computation of Conditional Probability of PLF Indices

Problem 1: Compute the PLF indices under different system configuration by using Cumulant method based PLF;

Problem 2: Calculate the probability of each system configuration. For large systems, in order to avoid the handling of utterly small numbers and the cumulating of numerical errors, Eqn. (7) is used.

The overall flow chart for calculation is shown in Fig.1.

Distributions of

system

variables

Cumulants

Method Based

PLF

Probability of

configuration

occurrence

Conditional

Probability for

output

Output indices

of PLF

Adjusted

output indices

of PLF

Fig.1. Sequence of PLF studies

IV. CUMULANT METHOD BASED PLF

In the Cumulant Method, the probabilistic description of load and generation are represented by a set of cumulants, which are functions of the moments of the probability distributions. The cumulants for unknown random variables are computed from known random variables and reconstructed PDFs.

Two fundamental concepts of cumulant method – moments and cumulants are briefly introduced as follows.

A. Moments and Cumulants

For a random variable x, which is the linear combination of independent random variables, y1, y2, …, yn.

nn ybybybx +++= ...2211 (8)

The expected value of x is given as:

xdxxfxE x∫∞

∞−= )(][ (9)

where )(xf x is the PDF of x.

Given a random variable x, with a continuous distribution

function )(xFx , the nth-order raw moment is defined as:

∫∞

∞−= )(][ xdFxxE x

nn (10)

Mathematically, the moment generating function can be written as:

][)( sxx eEs =Φ (11)

The nth-order raw moment can be computed by taking successive derivatives with respect to s and evaluating at s = 0.

The cumulant generating function )(sxΨ is usually

expressed in terms of the moment generating function )(sxΦ .

)(ln)( ss xx Φ=Ψ (12)

Therefore, the raw moments can also be expressed in terms of the cumulants by exponentiation of both sides of the series. To calculate the cumulants for variable x in terms of the component variables, the following equation can be utilized.

)()()(

))()()(ln(

)(ln)(

21

21

21

21

sbsbsb

sbsbsb

ss

nyyy

nyyy

xx

n

n

Ψ++Ψ+Ψ=

ΦΦΦ=Φ=Ψ

L

L (13)

B. PLF Method Proposed in [5]

This method uses a combination of logarithmic barrier interior point method type solution and cumulant method for PLF calculation. In this method, the Hessian of the Lagrangian is necessary for the Newton step. Moreover, the inverse of the Hessian can be used as the coefficients for the linear combination of random variables around the current

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operating point. The cumulants for unknown variables are reconstructed

with the use of the Gram-Charlier A Series:

∑∞

=

=

++=

0

221100

)(

)()()()(

jjj xHec

xHecxHecxHecxf

α

ααα L

(14)

where f(x) is the PDF of the random variable x, cj is the jth series coefficient, Hej(x) is the jth Tchebycheff-Hermite polynomial, and α(x) is the standard normal distribution function.

Given the cumulants for a distribution in standard form, the coefficients for the Edgeworth form of the A series can be computed and the following result is obtained. Detailed information for the derivative can be found in [5].

L+++

+++

++=

)(!7

35

)(!610

)(!5

)(!4

)(!3

1)(

7437

6

236

55

44

33

xHeKKK

xHeKK

xHeK

xHeK

xHeK

xf

α

αα

αα

(15)

The nth Tchebycheff-Hermite polynomial can be symbolically written as:

)()()()( xDxxHe nn αα −= (16)

where the operator D is the derivative with respect to x to simplify notation. The recursive relationship is available to determine third-order and higher polynomials

)()1()()( 21 xHenxxHexHe nnn −− −−= (17)

V. CASE STUDY

The proposed method was tested using a real large scale power system - Queensland (QLD) Transmission Network from Australia as shown in Fig.2. [10].

QLD transmission network includes more than 8,400km of high-voltage transmission lines which stretch from far north of Cairns to the border of New South Wales. The whole system has an installed generating capacity about 10.6GW. In this paper, QLD system has been simplified and grouped into 10 regions: Far North (FN), Ross, North, Central West, Gladstone, Wide Bay, South West, Moreton North, Moreton South and Gold Coast. Each region is treated as a HV substation; the generators within one region are considered individually but connected to the same bus bar. The load of each region has also been cumulated as one big demand point.

Fig.2. Queensland 330/275/132/110KV Network [10]

A. Load Modeling

The probability density functions of each HV substation load were collected and processed. Fig. 3 shows a series of daily active peak load over one year in QLD, [11].

0 50 100 150 200 250 300 350 4005000

5500

6000

6500

7000

7500

8000

Act

ive

Load

(M

W)

Days

Fig. 3. Daily Active Peak Load Sample of the QLD System

As a general rule, active and reactive load can be modeled by normal distributions. Therefore, it is possible to evaluate the standard deviations of daily peak values and the expected values, which is shown in Table I.

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TABLE I: LOAD DISTRIBUTION IN THE QLD SYSTEM

Region Mean (MW) Standard deviation Far North 348 44.33 Ross 485 61.78 North 436 55.54 Central West 632 80.51 Gladstone 1207 153.75 Wide Bay 259 32.99 South West 422 53.76 Moreton North 1791 228.14 Moreton South 2580 328.65 Gold Coast 829 105.60

B. Generation Modeling

In this cumulant method, another important input is the generating unit outage capacity. It is based on normal distributions as well. Some of the generation information has been shown in Table II. More details of generators’ type and failure can be found in [11] and [12].

TABLE II: CUMULATED GENERATION CAPACITY OF THE QLD SYSTEM

Region Mean (MW) Far North 141 Ross 575 North 224 Central West 3115 Gladstone 1680 South West 3324 Moreton North 500 Moreton South 913

C. Results and Discussions

Initially one case is solved with considering network contingency. For comparison purpose, the solution of original Cumulants method has also been obtained. The results are illustrated in Fig. 4 and 5. Table III shows the distributions of active power for 11 circuits.

TABLE III: PLF RESULTS OF THE QLD SYSTEM ( µ IS MEAN AND σ IS

THE STANDARD DEVIATION)

From To Active Power ( µ ) MW

Active Power (σ )

FN Ross 326 38.48 Ross North 614 59.31 North CW 902 98.15 CW Gladstone 760 75.22 CW SW 1517 160.50 Gladstone WB 428 47.56 WB MN 862 90.18 SW MN 3846 400.83 SW MS 724 75.84 MN MS 2938 3015.7 MS GC 975 110.54

From Fig. 4, it can be seen that the PDF of active power is

not a smooth line anymore comparing with original cumulants

method. This will give relatively accurate results. For example, after considering the network contingency, the probability of transmission line between Far North and Ross equals to 600MW is 32.29% instead of 29.99%.

100 200 300 400 500 600 700 8000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

(600,0.29998)

(600,0.3229798)

Active Power (MW)

Pro

babi

lity

Fig. 4. Active Power PDF of Transmission Line between Far North and ROSS

100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(600,0.79051)

(600,0.75951)

Active Power (MW)

Pro

babi

lity

(%)

Fig. 5. CDF of Active Power of Transmission Line between Far North and ROSS

Typical results of cumulative probability distribution function of power flow magnitude of circuit between Far North and Ross are shown in Fig. 5. The dotted line represents cumulative probability without considering network contingency. The solid line is the cumulative probability with network contingency.

Traditional methods consider the planning according to the ranking of hazards, which may lead to a dangerous state or a system failure. Sometimes a hazard is of little consequence if it cannot occur or is so unlikely, even if it is extremely undesirable. Planning alternatives based on such hazard analyses will lead to overinvestment. Thereby, the economical saving can be obtained since the introduction of probabilistic techniques. The accurate adjustment between costs and risk levels can even increase the benefit greatly. Let’s assume the

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real power rating of this circuit is 600MW, it can be seen that the risk of overloading is increased from 75.95% to 79.05% due to the consideration of network configuration.

The PLF method proposed in this paper can provide an accurate reinforcement for future system planning as well as the operation planning by compromising the possible technical performance and their effective costs.

TABLE IV: COMPUTATION TIME

Methods Computation Time (seconds)

Original Cumulants Method 9.57 Proposed Method 13.29

Table IV lists the computation time for each method used

in this paper. It can be seen that the original cumulants method is faster than the newly proposed method. However, the new method can remove the undesirable assumption. If we compare with other PLF method, the proposed method is still considered as a faster method, especially it can provide more reasonable results than the original cumulants method.

VI. CONCLUSION

This paper presents a new probabilistic load flow method, which considers the random transmission network contingency and removes the undesirable assumptions as required by many conventional analytical PLF methods. The method is based on the conditional probability approach and uses cumulant based probabilistic load flow method to obtain more accurate PLF indices. The NCP is used to avoid the computation of small probability.

The method has all the advantages of conventional PLF assessment, such as the measurement of generation and load uncertainties, and the likelihood of system unreliability while no remedial actions will be considered. It has been found that the network contingency has distinct influence in the solution when an analytical technique is applied. This method can give operators and planners a more accurate analytic tool for decision making, cost benefit analysis and better resources utilisation.

VII. REFERENCES [1] A. M. Leite da Silva, S. M. P. Ribeiro, V. L. Arienti, R. N. Allan and M.

B. Do Coutto Filho, “Probabilistic Load Flow Techniques Applied to Power System Expansion Planning”, IEEE Transactions on Power Systems, vol. 5, no.4, pp.1047 – 1053, Nov. 1990.

[2] M. B. Do Coutto Filho, A. M. Leite Da Silva, V. L. Arienti and S. M. P. Ribeiro, “Probabilistic Load Modelling For Power System Expansion Planning”, Proc. of the Third International Conference on Probabilistic Methods Applied to Electric Power Systems, 3-5 July, 1991, pp. 203 – 207.

[3] G. K. Stefopoulos, A. P. Meliopoulos and G. J. Cokkinids, “Probabilistic Power Flow with Non-Conforming Electric Loads”, Proc. of the 8th International Conference on Probabilistic Methods Applied to Power Systems, Iowa State University, Ames, Iowa, September 12-16, pp. 525 – 531, 2004

[4] P. Zhang and S. T. Lee, “Probabilistic Load Flow Computation Using the Method of Combined Cumulants and Gram-Charlier Expansion”, IEEE Transactions on Power Systems, vol. 19, no. 1, pp. 676 – 682, February, 2004

[5] A. Schellenberg, W. Rosehart and J. Aguado, “Cumulant-Based Probabilistic Optimal Power Flow (P-OPF) With Gaussian and Gamma Distributions”, IEEE Transactions on Power Systems, vol. 20, no.2, pp.773 – 781, May 2005

[6] A. Schellenberg, W. Rosehart and J. Aguado, “Cumulant Based Probabilistic Optimal Power Flow (P-OPF)”, Proc of the 8th International Conference on Probabilistic Methods Applied to Power Systems, Iowa State University, Ames, Iowa, September 12-16, pp. 506 – 511, 2004

[7] A. M. Leite da Silva, R. N. Allan, S. M. Soares and V. L. Arienti, “Probabilistic Load Flow Considering Network Outages”, IEE Proceedings, vol.132, Pt. C, no.3, pp. 139 – 145, May 1985,

[8] Z. Hu and X. F. Wang, “A Probabilistic Load Flow Method Considering Branch Outages,” IEEE Transactions On Power Systems, vol. 21, no. 2. pp. 507 - 514, 2006.

[9] N. Maruejouls, V. Sermanson, S. T. Lee and P. Zhang, “A Practical Probabilistic Reliability Assessment Using Contingency Simulation”. Proc. IEEE PES Power Systems Conference and Exposition. pp: 1074 – 1080, 2004

[10] Powerlink Queensland, “Annual Planning Report 2006”, [Online]: http://www.powerlink.com.au/asp/index.asp?sid=5056&page=Corporate/Documents&cid=5250&gid=476

[11] National Electricity Market Management Company Limited (NEMMCO) website: http://www.nemmco.com.au

[12] Queensland Government, “Queensland’s Main Power Stations”, [Online]. Available: http://www.energy.qld.gov.au/power_stations.cfm

VIII. BIOGRAPHIES

Miao Lu (S’M05) obtained her Master of Information Technology degree from Griffith University, Australia in 2003 and B.E. in Electrical Engineering from North China Electrical Power University (Beijing) in 1998. She is now pursuing her PhD in Electrical Engineering degree at The School of Information Technology and Electrical Engineering, The University of Queensland, Australia. Her research interests include power system analysis, power system planning, reliability assessment and risk management, and power system computations. Zhao Yang Dong (M'99-SM’06) received his PhD in Electrical and Information Engineering from The University of Sydney, Australia in 1999. He is now a senior lecturer at the School of Information Technology Electrical Engineering, The University of Queensland, Australia. His research interests include power system security assessment and enhancement, electricity market, artificial intelligence and its application in electric power engineering, power system planning and management, and power system stability & control. Tapan Kumar Saha (M’93–SM’97) was born in Bangladesh and immigrated to Australia in 1989. Dr Saha is a Professor of Electrical Engineering in the School of Information Technology and Electrical Engineering, The University of Queensland, Australia. Before joining the University of Queensland he taught at the Bangladesh University of Engineering and Technology, Dhaka, Bangladesh for three and a half years and at James Cook University, Townsville, Australia for two and a half years. He is a Fellow of the Institution of Engineers, Australia. His research interests include power systems, power quality, and condition monitoring of electrical plants.