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Reliability Evaluation of Bangladesh Power System Using Cumulant Method

Department of Electrical and Electronic Engineering
Bangladesh University of Engineering and Technology
Dhaka, Bangladesh

S. Ahmad1, G.A.K.Biswas, A.U.Elahi, N.M.Zakaria
Department of Electrical and Electronic Engineering
BRAC University
Dhaka, Bangladesh
E-mail: 1

Abstract—In recent years, power systems are frequently operating under highly stressed and unpredictable conditions because of not only the market-oriented reform of power systems but also the integration of various renewable energy sources. The uncertain factors resulting from the market constraints and the inherent randomness of renewable energy set higher requirements on the reliability of electric networks. This paper presents the reliability assessment of Bangladesh Power System (BPS). Reliability index, loss of load probability (LOLP) of BPS is evaluated using cumulant method. The rationale for using the cumulant method is to take advantage of its computational efficiency. BPS has sixty one generators and a total installed capacity of 5275 MW. The maximum demand of BPS is about 5000 MW. The relevant data of the generators and hourly load profiles are collected from the National Load Dispatch Center (NLDC) of Bangladesh and reliability index ‘LOLP’ is assessed. The simulation results show that the LOLP of BPS is 2.06%.

Keywords—cumulant method, convolution, LOLP, forced outage rate, Gram-Charlier series


An electric utility’s main concern is to plan, design, operate and maintain its power supply to provide an acceptable level of reliability to its users. This clearly requires that standards of reliability be specified and used in all three sectors of the power system, i.e., generation, transmission and distribution. Reliability indices have been defined for the three sectors separately as well as for the bulk power system. Reliability criteria may be determined at the selected load points in the system for different combination of generators and transmission line failures.

A survey of literature reveals the fact that there has been a considerable activity in the development and application of reliability techniques in electric power systems. In power system reliability evaluation, usually component failures are assumed independent and reliability indices are calculated using methods based on the multiplication rule of probabilities [1]. But in some cases, for instance when the effects of fluctuating weather are considered, the previous assumption is invalid. Generally, two kinds of methodologies are adopted to solve this problem, analytical methods based on Markov processes [2], [3] and Monte Carlo simulation [4], [5]. A DC-OPF based Markov cut-set method (DCOPF-MCSM) to evaluate composite power system reliability considering weather effects is presented in [6] where the DC-OPF approach is used to determine minimal cut sets (MCS) up to a preset order and then MCSM is used to calculate reliability indices.

The appropriate incorporation and presentation of the implications of uncertainty are widely recognized as fundamental components in the analyses of complex systems [7].There are two fundamentally different forms of uncertainty in power system reliability assessment [7, 8]. Aleatory and epistemic uncertainties are considered in power system reliability evaluation in [9] where aleatory uncertainty arises because the study system can potentially behave in many different ways. A method for incorporating the failures due to aging in power system reliability evaluation is presented in [10]. It includes the development of a calculation approach with two possible probability distribution models for unavailability of aging failures and implementation in reliability evaluation. Adverse weather such as hurricanes can have significant impact on power system reliability [11, 12]. One of the challenges of incorporating weather effects in power system reliability evaluation is to assess how adverse weather affects the reliability parameters of system components. A fuzzy inference system (FIS) built by using fuzzy clustering method is combined with the regional weather model to solve the preceding problem is illustrated in [13]. A new computationally efficient methodology for calculating the reliability indices of a bulk power system using the state enumeration approach is depicted in [14]. The approach utilizes topological analysis to determine the contribution of each system state to the frequency and duration indices at both the system and the bus level. Common cause outage is also considered in power system reliability evaluation [15]. Power system reliability evaluation and quality assessment using fuzzy logic and genetic algorithm are depicted in [16] and [17], respectively. Cumulant method, a very fast computational technique is used to evaluate the reliability of BPS in this paper. Reliability index ‘LOLP’ is assessed for this intention. LOLP gives the probability that the available generation capacity will be insufficient to meet the daily peak loads. The simulation results show that the LOLP of BPS is 2.06%.

Generator and Load Model of BPS

1 Generator Model

The simplest model for a generating unit for continuous operation is a Run-Fail-Repair-Run cycle that states that every generator has two states. They are— i) Unit availability and ii) Unit unavailability or forced outage rate (FOR). The unit availability means the long term probability that the generating unit will reside in on state and unit unavailability or FOR means the long term probability that the generating unit will reside in off state. Mathematically FOR can be defined as,

[pic] (1)


FOH = Forced outage hours

SH = Service hours or operating hours at full availability

Unit availability of a generating unit can be defined as,

[pic] (2)

For a generating unit with capacity = C MW and FOR = q and unit availability = p, the probability density function (PDF) of forced outage capacity is shown in Fig.1.

PDF of forced outage capacity of a generating unit

BPS has sixty one generators and a total installed capacity of 5275 MW. The individual capacity and FOR of the generators are shown in Table I.

Capacity and FOR of the Generators of BPS

| |(MW) | | |(MW) | |
|1 |40 |1.4 X 10-6 |32 |15 |0.15 |
|2 |40 |1.4 X 10-6 |33 |15 |0.15 |
|3 |50 |1.4 X 10-6 |34 |15 |0.15 |
|4 |50 |1.4 X 10-6 |35 |15 |0.15 |
|5 |50 |1.4 X 10-6 |36 |35 |0.10 |
|6 |210 |0.16 |37 |35 |0.10 |
|7 |50 |0.113 |38 |21 |0.122 |
|8 |109 |0.07 |39 |120 |0.04 |
|9 |55 |0.185 |40 |77 |0.101 |
|10 |55 |0.185 |41 |100 |0.04 |
|11 |210 |0.095 |42 |125 |0.10 |
|12 |210 |0.019 |43 |125 |0.10 |
|13 |210 |0.08 |44 |110 |0.301 |
|14 |210 |0.08 |45 |60 |0.402 |
|15 |64 |0.116 |46 |28 |0.50 |
|16 |64 |0.116 |47 |28 |0.50 |
|17 |150 |0.013 |48 |20 |0.045 |
|18 |150 |0.014 |49 |20 |0.20 |
|19 |150 |0.014 |50 |20 |0.20 |
|20 |56 |0.321 |51 |20 |0.119 |
|21 |56 |0.321 |52 |60 |0.50 |
|22 |30 |0.15 |53 |8 |0.30 |
|23 |100 |0.30 |54 |450 |0.07 |
|24 |210 |0.197 |55 |235 |0.07 |
|25 |210 |0.197 |56 |125 |0.07 |
|26 |60 |0.117 |57 |142 |0.07 |
|27 |28 |0.60 |58 |45 |0.07 |
|28 |28 |0.60 |59 |45 |0.07 |
|29 |12 |0.15 |60 |110 |0.11 |
|30 |12 |0.15 |61 |110 |0.07 |
|31 |12 |0.15 |

2 Load Model

In order to develop the load model of BPS, hourly loads of last five years (2006-2010) are collected from NLDC of Bangladesh. Hourly loads are divided in seven groups having a group size of 500 MW. The occurrence of each group is then counted. The probability of occurrence of each group is calculated as,

[pic] (3)


Pg = Probability of occurrence of a group

Ng = No. of occurring days of that group in observation period of 5 years

Nt = Total no. of days in observation period of 5 years

Finally the average value of each group is taken and the corresponding probabilities reside for that average value of the load. Table II shows the load model of BPS.

Load Model of BPS

|1750 |0.0124 |
|2250 |0.0728 |
|2750 |0.1834 |
|3250 |0.3331 |
|3750 |0.2816 |
|4250 |0.1120 |
|4750 |0.0048 |


The cumulant method also known as the method of moment is an approximation technique which approximates the discrete distribution of load through Gram– Charlier series expansion as a continuous function. In this method, convolution of generating unit outage with the distribution of load is performed through a very fast algorithm. The steps of calculating LOLP using cumulant method are described in what follows.

(i) The moments about the origin for each generating unit is determined at first. For any i-th machine, the moments about the origin can be calculated using the following relations.

[pic] (4)

[pic] (5)

[pic] (6)

… … …

[pic] (7)


mn (i) = n-th moment about the origin of the i-th machine

Ci = Capacity of the i-th machine

qi = FOR of the i-th machine

(ii) In the second step, the central moments or moments about the mean of each generating unit is calculated. For any i-th machine, the central moments can be calculated as,

[pic] (8)

[pic] (9)

[pic] (10)


… … …

[pic] (12)


Mn (i) = n-th central moment of the i-th machine

pi = Availability of the i-th machine

(iii) In the third step, cumulant of each machine is calculated. For i-th machine, the cumulants can be determined as follows,

[pic] (13)

[pic] (14)

[pic][pic] (15)

[pic] (16)

[pic] (17)

(iv) In the fourth step, the cululants of the load is obtained. For this, at first, the moments about the origin and the central moments of the load are calculated. Using these moments, cumulants of the load are obtained using (13) to (17).

(v) In this step, total system cumulant is obtained by summing the machine cumulants and load cumulants. It can be represented as,

[pic] (18)

(vi) Now standardized random variable, z is calculated using the relation,

[pic] (19)


IC = Installed capacity of the power system

k1, k2 = System cumulants

(vii) LOLP can be calculated using the relationship given by,

[pic] (20)

Where, Q (z) can be calculated as,

[pic] (21)


[pic] (22)

[pic] (23)

And r, b1, b2 and b3 are constants

(viii) F (z) is calculated using Gram- Charlier series which is given by,

[pic] … (24)

Where, the expansion factors G1, G2, G3 are calculated using the following relationship.

[pic] (25)

And the derivatives of the normal PDF N (z) may be obtained using the following recursive relations.

[pic] (26)

m = 3, 4, 5, ...


[pic] (27)

[pic] (28)

(ix) The value of constants are set as, r = 0.232, b1 = 0.319, b2 = -0.356, b3 = 1.781. Finally LOLP is evaluated using (20).


BPS has sixty one generators and a total installed capacity of 5275 MW. Using the generator and load [pic]model of BPS shown in Table I and Table II, respectively and employing (4) to (17), the cumulants of generating units and load are calculated. Table II represents the cumulants.

Cumulants of Generators and Load

|K1 |626.76 |3326.83 |
|K2 |69973.58 |333749.80 |
|k3 |1.75x107 |-5.2 x 10 7 |
|k4 |2.18 x 109 |-7.81 x 10 10 |
|k5 |1.53 x 1011 |4.12 x 10 13 |

Now system cumulants are calculated combining the cumulants of the generating units and the load. Table IV presents the system cumulants.

System Cumulants

|K1 |3953.59 |
|K2 |403723.38 |
|K3 |-3.45 X 10 7 |
|K4 |-7.59 X 10 10 |
|K5 |4.14 X 10 13 |

Using (19), standardized random variable z is

[pic] (29)

Using (22) and (23), normal PDF, N (z) and t are calculated as,

[pic] (30)

[pic] (31)

Now using (30) and (31) in (21),

[pic] (32)

Expansion factors G1, G2 and G3 are calculated using (25) as,

[pic] (33)

[pic] (34)

[pic] (35)

Derivatives of normal PDF are calculated using (26) to (28) as,

[pic] (36)

[pic][pic] (37)

[pic] (38)

[pic] (39)

Now using (33) to (35) and (37) to (39) in (24) F (z) is determined as,

[pic] (40)

Finally using (32) and (40) in (20), LOLP is evaluated as,


Thus the reliability index ‘LOLP’ of BPS is 2.06%.


The basic function of a power system is to supply electrical energy to both large and small consumers as economically as possible with an acceptable degree of reliability and quality. Reliability is the ability of a power system to provide service to consumers while maintaining the quality and price of electricity at an acceptable level. This paper evaluates the reliability of BPS using cumulant method which is a very fast computational technique. The simulation results reveal that the LOLP of BPS is 2.06%.

Lower reliability level imperils energy supply continuity and increases the possibility of additional maintenance and the restoration costs due to the higher rate of system outages. The costs associated with low reliability or poor system qualities are enormous and can be largely avoided by enhancing the level of reliability. Thus the reliability assessment of BPS will help estimating the service quality of the system. It will also create awareness among the utility and the consumers of the system and will assist in planning and operation process of BPS.


1] C. Singh, and R. Billinton, System Reliability Modeling and Evaluation, London, U.K., Hutchinson Educational, 1977.
2] R. Billinton , and G. Singh, “Application of Adverse and Extreme Adverse Weather: Modeling in Transmission and Distribution System Reliability Evaluation,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 153, no. 1, pp. 115–120, Jan. 2006.
3] C. Dichirico, and C. Singh, “Reliability Analysis of Transmission Lines with Common Mode Failures When Repair Times are Arbitrarily Distributed,” IEEE Trans. Power Syst., vol. 3, no. 3, pp. 1012–1019, Aug. 1988.
4] R. Billinton, and W. Li, “A Novel Method for Incorporating Weather Effects in Composite System Adequacy Evaluation,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 1154–1160, Aug. 1991.
5] M. R. Bhuiyan and R. N. Allan, “Inclusion of Weather Effects in Composite System Reliability Evaluation Using Sequential Simulation,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 141, no. 6, pp. 575–584, Nov. 1994.
6] Yong Liu, and Singh, C., “Reliability Evaluation of Composite Power Systems Using Markov Cut-Set Method”, IEEE Trans. Power Syst., vol. 25, no. 2, pp. 777-785, May 2010.
7] Guest Editorial, “Alternative Representations of Epistemic Uncertainty”, Reliability Engineering and System Safety 85 (2004), pp. 1-10.
8] F. Owen Hoffman, and Jana S. Hammonds, “Propagation of Uncertainty in Risk Assessments: The Need to Distinguish Between Uncertainty Due to Lack of Knowledge and Uncertainty Due to Variability”, Risk Analysis, vol. 14, no. 5, pp. 707-712, 1994.
9] Roy Billinton, and Dange Huang, “Aleatory and Epistemic Uncertainty Considerations in Power System Reliability Evaluation”, Proc. Inst. Probabilistic Methods Applied to Power Systems, pp. 1-8, May 2008.
10] Li, W. “Incorporating Aging Failures in Power System Reliability Evaluation” , IEEE Power Engineering Review, vol. 22, no. 7, pp. 59, July 2002.
11] R. A. Davidson, H. Liu, I. K. Sarpong, P. Sparks, and D. V. Rosowsky, “Electric Power Distribution System Performance in Carolina Hurricanes,” Nat. Haz. Rev., vol. 4, no. 1, pp. 36–45, Feb. 2003.
12] H. Liu, R. A. Davidson, D. V. Rosowsky, and J. R. Stedinger, “Negative Binomial Regression of Electric Power Outages in Hurricanes,” J. Infrastruct. Syst., vol. 11, no. 4, pp. 258–267, Dec. 2005.
13] Liu, Y., and Singh, C.A, “Methodology for Evaluation of Hurricane Impact on Composite Power System Reliability”, IEEE Trans. Power Syst., vol. 9, no. 9, pp. 1-8, 2007.
14] Jonnavithula, S., and Billinton, R., “Topological Analysis in Bulk Power System Reliability Evaluation”, IEEE Trans. Power Syst., vol. 12, no. 1, pp. 456-463, Feb. 1997.
15] Wenyuan Li , and Billinton, R., “Common Cause Outage Models in Power System Reliability Evaluation”, IEEE Trans. Power Syst., vol. 18, no. 2, pp. 966-968, May 2003.
16] Farahat, M.A, and Al-Shammari, B.M., “Power System Reliability Evaluation and Quality Assessment by Fuzzy Logic Technique”, Proc. Inst. Universities Power Engineering Conference, vol. 1, pp. 478-483, Sept. 2004.
17] Green, R.C., Lingfeng Wang, and Singh, C., “ State Space Pruning for Power System Reliability Evaluation Using Genetic Algorithms”, IEEE Power and Energy Society General Meeting, pp. 1-6, July 2010.






Forced outage capacity

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...Case Study Analysis: A Tale of Two Coaches, Part 3 Elida Roshi Grand Canyon University Leadership Styles and Development LDR-600 Shawna Wentlandt January 29, 2014 Case Study Analysis: A Tale of Two Coaches, Part 3 It is identified that the focus point of leadership is to achieve goals by encouraging people and directing them in the proper direction. Organizations consist of various departments and groups that are required to meet both short term and long term goals of the department and organization. For this reason an exceptional leader is needed to direct and inspire them throughout this process. The most important part of leadership is the quality of a leader to encourage others and offer enhancement and creativity in their submission of ideas and methods and that is exactly where the distinction between management and leadership falls. This author will focus accurately on this distinction illustrating Blanchard’s situational model and Fiedler’s contingency model on the two famous coaches’ leadership style. Furthermore, this writer will discuss which of the leadership styles utilized from them are more effective and what is impacting on their effectiveness. Based on Fiedler’s contingency model it is apparent that Coach Knight and Coach K are matched to their situations. As the Fiedler’s contingency model suggests leaders are assigned to a situation for their specified qualities and abilities and is more like a leader-match structure (Northouse, 2013). It is......

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...Self Assessment Log 2 Jerry Chauvin Thomas Edison State University What have you learned from completing this exercise? Fiedler’s contingency model of leadership, along with his LPC score board revealed to me that my leadership abilities mimic a High LPC, leader. “High LPC leaders are most effective when situational variables are neither extremely favorable nor unfavorable” (Hackman, 2009, p.80). I have also learned that the best way to understand someone else is to imagine myself as them (Hackman, 2009, p.78). How has your self-understanding increased? It has also made me realize how crucial it is to acknowledge the imperfections of some of my followers, which is equally as important to acknowledge the perfections of my most desirable followers in the workplace. It is my understanding that I am more relationship oriented, thus making me less focused on the task. My environment needs to be restructured where my position power is strong, tasks are structured, and relationships between me and my followers are good. According to Hackman (2009), “the most favorable conditions exist when the relationship between the leader and followers is good, the task is highly structured task, and the leader’s position power is strong" (p.80). What does this assessment reveal about your potential strengths and weaknesses as a leader (or follower)? Strengths: I work best when the conditions are favorable. My favorable conditions include pleasant, friendly, supportive, open,......

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Leadership the ability to influence a group toward the achievement of a vision or set of goals. (Robbins and Judge 2005). Leaders have the task of planning, organizing, solving problems, motivating, supporting, delegating, rewarding, networking and rewarding their employees or group members, among numerous other tasks. Consequently it can be said that the performance and effectiveness of an organization is directly related and depends upon the leadership that it has. With reference to Fiedler’s Contingency Model/ Theory, Hersey and Blanchard Situational Leadership Theory, Path-Goal theory and Participative Leadership Model, the validity of this statement shall be proven. The very first complete contingency model for leadership was developed by Fred Fiedler. The Fiedler contingency model proposes that successful group performance depends on the proper coherence between the leader’s style and the extent to which the situation gives the leader control. (Robbins and Judge 2005) In Fielder’s contingency model, first the leadership style of the individual needs to be identified through a special questionnaire referred to as the “least preferred co-worker questionnaire”. This evaluates whether the person is relationship oriented or task oriented. The next step is matching the leader to the situation using leader-member relations, task structure or position power variables. Finally we evaluate. Fiedler concluded that task-oriented leaders perform better in situations very favorable...

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Leadership Theory Analysis

...Leadership Theory Analysis This week in our assignment we take a look at leadership. Within this topic we look at theory analysis of two different types of model’s. First we have the Vroom’s model of leadership which deal with how leaders make decisions. Then you also have the Fiedler’s Contingency model which deals with choosing the right type of leader for the situation. In life or in any work place there is always someone who is in charge. That person who is in charge, is viewed as a leader, in the eyes of subordinates. A video that we watched in class defined a leader as a brave person who is not afraid to stand out among the rest. There are many traits and characteristics of leadership. Many people are born with leading abilities, while others pick up the traits by facing situations in their life’s, within the situations they had to come up with a solution. When we take a look at the Vroom’s model of leadership we see an analysis of what makes a leader make decisions. It opens an analysis to questions on how to make the right decision. Vrooms theory is closely resembles the models of Tannenbaum and Schmidt. Well in Vrooms model there is a series of questions that follow any decision that is to be made by a leader. The importance of the questions is based on the importance of the situation. The process can make it easier to make the decision and help facilitate to what might be the best......

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