4. Composite power system reliability assessment

The research work presented in this book concentrates on HL-II, i.e., composite power system reliability assessment. These reliability studies will assess the ability of the composite generation and transmission system (composite power system) to not only satisfy the consumer requirements but also tolerate the random failures and execute preventive maintenance of electrical components. The reliability performance of any system is generally evaluated by the reliability parameters or indices. The composite power system reliability is judged in this research work by considering "average power availability" and "loss of load expected" as reliability indices at the bulk consumer buses. There are many publications that are dealing with composite power system reliability assessment [1–14]. In the evaluation of reliability in composite power system, many technical issues are involved such as load uncertainty, generation adequacy, integration of nonconventional sources, multiple outages, etc. [4, 13]. By keeping these technical issues in view, this research work addresses some new and improved methods for the assessment of composite power system reliability.

The composite power system reliability assessment presented in this book considers all equipment in the system such as circuit breakers, transformers, generators, buses, lines, etc.

5.1. Series-parallel approach

ity was evaluated.

5.2. Star-delta approach

Figure 3. Components in series.

Figure 4. Components in parallel.

the branch are given in the Eqs. (3) and (4).

failure and repair rates are given in Eqs. (5) and (6).

If two components are connected in series in a branch of the network and each component has its failure rate and repair rate as shown in Figure 3. The equivalent failure and repair rates for

> <sup>μ</sup>eq <sup>¼</sup> <sup>μ</sup>1μ<sup>2</sup> μ<sup>1</sup> þ μ<sup>2</sup>

Similarly if two components are connected in parallel as shown in Figure 4, then the equivalent

<sup>λ</sup>eq <sup>¼</sup> <sup>λ</sup>1λ<sup>2</sup> λ<sup>1</sup> þ λ<sup>2</sup>

Initially using this series-parallel approach, most of the simple power system network reliabil-

In the complex interconnected power systems, there exist a number of star and delta configurations, and series-parallel approach alone is not enough to reduce the network. During the evaluation of the availability, there will be a need for star-delta transformation for network

λeq ¼ λ<sup>1</sup> þ λ<sup>2</sup> (3)

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μeq ¼ μ<sup>1</sup> þ μ<sup>2</sup> (6)

(4)

105

(5)

#### 5. Reliability assessment techniques

The electric power systems are good examples for reliability assessment. In many power systems, the average duration of interruptions faced by a customer is just a few hours per year, which indicates that high availability of power supply to consumers is ensured considering scheduled and unscheduled outages (random failures). The high power availability can be achieved by proper maintenance and monitoring of the equipment. There are several methodologies developed over the years for the reliability measurement. The early methods used were all deterministic and are not convenient to apply for large interconnected power system. Also the deterministic methods cannot consider the stochastic nature of the system and load [14].

Later probabilistic methods are developed to obtain meaningful information regarding the system reliability. The probability of random failure and repair durations during the operating life of the component is assumed to be exponentially distributed. Based on this, the mean time to failure (MTTF = 1/λ) and the mean time to repair (MTTR = 1/μ) can be evaluated [15]. These indices for each component are used to obtain the overall system reliability. In this chapter, the average power availability at the load bus is used as a measure for the reliability assessment of the system. The reliability study in the interconnected power system is complex due to the large number of components and network topology. So far the reliability assessment in interconnected power system is achieved through tracing of the power flow paths [10–14]. But tracing of power flow paths in a large power system network becomes difficult and takes time. Simple and more convenient method based on electrical circuit approach is presented here.

The probability of power availability and unavailability of a component having failure and repair rates (λ and μ) is given in Eqs. (1) and (2). The failure and repair rates of each component in the power system are assumed to be constant throughout the operation. The probabilities of failure and repair rates are exponentially distributed. The component failure and repair rates are independent of other components, and their future states are not dependent on their past history. So the probability of present state changes is governed by the exponential distribution and not dependent on past history of the component.

$$Availability = \frac{\mu}{\lambda + \mu} \tag{1}$$

$$\text{Un} \text{available} = (1 - \text{Availability}) = \frac{\lambda}{\mu + \lambda} \tag{2}$$

The existing methods for the reliability assessment of composite power system are explained in the following chapters. The limitations and difficulties of those methods are also discussed.

#### 5.1. Series-parallel approach

The composite power system reliability assessment presented in this book considers all equip-

The electric power systems are good examples for reliability assessment. In many power systems, the average duration of interruptions faced by a customer is just a few hours per year, which indicates that high availability of power supply to consumers is ensured considering scheduled and unscheduled outages (random failures). The high power availability can be achieved by proper maintenance and monitoring of the equipment. There are several methodologies developed over the years for the reliability measurement. The early methods used were all deterministic and are not convenient to apply for large interconnected power system. Also the deterministic methods cannot consider the stochastic nature of the system and load [14]. Later probabilistic methods are developed to obtain meaningful information regarding the system reliability. The probability of random failure and repair durations during the operating life of the component is assumed to be exponentially distributed. Based on this, the mean time to failure (MTTF = 1/λ) and the mean time to repair (MTTR = 1/μ) can be evaluated [15]. These indices for each component are used to obtain the overall system reliability. In this chapter, the average power availability at the load bus is used as a measure for the reliability assessment of the system. The reliability study in the interconnected power system is complex due to the large number of components and network topology. So far the reliability assessment in interconnected power system is achieved through tracing of the power flow paths [10–14]. But tracing of power flow paths in a large power system network becomes difficult and takes time. Simple and more convenient method based on electrical circuit approach is presented

The probability of power availability and unavailability of a component having failure and repair rates (λ and μ) is given in Eqs. (1) and (2). The failure and repair rates of each component in the power system are assumed to be constant throughout the operation. The probabilities of failure and repair rates are exponentially distributed. The component failure and repair rates are independent of other components, and their future states are not dependent on their past history. So the probability of present state changes is governed by the exponential distribution

Availability <sup>¼</sup> <sup>μ</sup>

Unavailability <sup>¼</sup> <sup>ð</sup><sup>1</sup> � AvailabilityÞ ¼ <sup>λ</sup>

The existing methods for the reliability assessment of composite power system are explained in the following chapters. The limitations and difficulties of those methods are also discussed.

<sup>λ</sup> <sup>þ</sup> <sup>μ</sup> (1)

<sup>μ</sup> <sup>þ</sup> <sup>λ</sup> (2)

ment in the system such as circuit breakers, transformers, generators, buses, lines, etc.

5. Reliability assessment techniques

and not dependent on past history of the component.

here.

104 Smart Microgrids

If two components are connected in series in a branch of the network and each component has its failure rate and repair rate as shown in Figure 3. The equivalent failure and repair rates for the branch are given in the Eqs. (3) and (4).

$$
\lambda\_{eq} = \lambda\_1 + \lambda\_2 \tag{3}
$$

$$
\mu\_{eq} = \frac{\mu\_1 \mu\_2}{\mu\_1 + \mu\_2} \tag{4}
$$

Similarly if two components are connected in parallel as shown in Figure 4, then the equivalent failure and repair rates are given in Eqs. (5) and (6).

$$
\lambda\_{eq} = \frac{\lambda\_1 \lambda\_2}{\lambda\_1 + \lambda\_2} \tag{5}
$$

$$
\mu\_{eq} = \mu\_1 + \mu\_2 \tag{6}
$$

Initially using this series-parallel approach, most of the simple power system network reliability was evaluated.

#### 5.2. Star-delta approach

In the complex interconnected power systems, there exist a number of star and delta configurations, and series-parallel approach alone is not enough to reduce the network. During the evaluation of the availability, there will be a need for star-delta transformation for network

Figure 3. Components in series.

Figure 4. Components in parallel.

reduction. The equivalent failure and repair rate transformations from star to delta or vice versa are given in the following equations from Eqs. (7) to (12). The equivalents are based on the condition that the equivalent failure and repair rates for both the configuration should be same across any two terminals. The equivalent star-delta reliability models are shown in Figures 5 and 6.

The equivalent failure rates are given by

$$
\lambda\_{ab} = \frac{\lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_3 \lambda\_1}{\lambda\_3} \tag{7}
$$

The interconnected power system network (IEEE 6 bus reliability test system) used here consists of a number of circuit breakers, two generating units and four load points as shown in Figure 7 [15]. In this IEEE 6 bus reliability test system, the failure and repair rates (λ and μ) of each component are given in [15]. The average probability of power availability at load bus is calculated by reducing the network by series-parallel and star-delta or delta-star conversion methods between the source node and the sink or load node. The equivalent reliability model

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In the interconnected power system shown in Figure 7, the IEEE 6 bus reliability test system is reduced to simple delta connection, where it has two nodes of generating units and one node for load. The reduced reliability network is shown in Figure 9. Using the methodology explained above, equivalent λ and μ are obtained between generator nodes 1 and 2 and load node. From this, the probability of average power availability at the load is obtained using Eq. (1). The same procedure is used to find the probability of availability at all load points one by one. The probabilities of average power availability at each load are calculated using series-

of the IEEE 6 bus reliability test system is shown in Figure 8.

parallel and star-delta approach and are given in Table 3.

Figure 5. Equivalent star connected reliability model.

Figure 6. Equivalent delta connected reliability model.

$$
\lambda\_{bc} = \frac{\lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_3 \lambda\_1}{\lambda\_1} \tag{8}
$$

$$
\lambda\_{\rm ac} = \frac{\lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_3 \lambda\_1}{\lambda\_2} \tag{9}
$$

Equivalent repair rates are given in the equations from Eqs. (10) to (12) as follows

$$
\mu\_{\rm ab} = \frac{\mu\_1 \mu\_2}{\mu\_1 + \mu\_2 + \mu\_3} \tag{10}
$$

$$
\mu\_{\rm bc} = \frac{\mu\_2 \mu\_3}{\mu\_1 + \mu\_2 + \mu\_3} \tag{11}
$$

$$
\mu\_{\rm ac} = \frac{\mu\_1 \mu\_3}{\mu\_1 + \mu\_2 + \mu\_3} \tag{12}
$$

#### 5.3. Delta-star approach

Similarly the conversion from star to delta is as follows. The equivalent failure rates are given by equations from Eqs. (13) to (18),

$$
\lambda\_1 = \frac{\lambda\_{ab}\lambda\_{ac}}{\lambda\_{ab} + \lambda\_{bc} + \lambda\_{ca}} \tag{13}
$$

$$
\lambda\_2 = \frac{\lambda\_{ab}\lambda\_{bc}}{\lambda\_{ab} + \lambda\_{bc} + \lambda\_{ca}} \tag{14}
$$

$$
\lambda\_3 = \frac{\lambda\_{ac}\lambda\_{bc}}{\lambda\_{ab} + \lambda\_{bc} + \lambda\_{ca}} \tag{15}
$$

Equivalent repair rates are given by

$$
\mu\_1 = \frac{\mu\_{ab}\mu\_{bc} + \mu\_{bc}\mu\_{ac} + \mu\_{ab}\mu\_{ac}}{\mu\_{bc}} \tag{16}
$$

$$
\mu\_2 = \frac{\mu\_{ab}\mu\_{bc} + \mu\_{bc}\mu\_{ac} + \mu\_{ab}\mu\_{ac}}{\mu\_{ac}} \tag{17}
$$

$$
\mu\_3 = \frac{\mu\_{ab}\mu\_{bc} + \mu\_{bc}\mu\_{ac} + \mu\_{ab}\mu\_{ac}}{\mu\_{ab}} \tag{18}
$$

The interconnected power system network (IEEE 6 bus reliability test system) used here consists of a number of circuit breakers, two generating units and four load points as shown in Figure 7 [15]. In this IEEE 6 bus reliability test system, the failure and repair rates (λ and μ) of each component are given in [15]. The average probability of power availability at load bus is calculated by reducing the network by series-parallel and star-delta or delta-star conversion methods between the source node and the sink or load node. The equivalent reliability model of the IEEE 6 bus reliability test system is shown in Figure 8.

In the interconnected power system shown in Figure 7, the IEEE 6 bus reliability test system is reduced to simple delta connection, where it has two nodes of generating units and one node for load. The reduced reliability network is shown in Figure 9. Using the methodology explained above, equivalent λ and μ are obtained between generator nodes 1 and 2 and load node. From this, the probability of average power availability at the load is obtained using Eq. (1). The same procedure is used to find the probability of availability at all load points one by one. The probabilities of average power availability at each load are calculated using seriesparallel and star-delta approach and are given in Table 3.

Figure 5. Equivalent star connected reliability model.

reduction. The equivalent failure and repair rate transformations from star to delta or vice versa are given in the following equations from Eqs. (7) to (12). The equivalents are based on the condition that the equivalent failure and repair rates for both the configuration should be same across any two terminals. The equivalent star-delta reliability models are shown in

> <sup>λ</sup>ab <sup>¼</sup> <sup>λ</sup>1λ<sup>2</sup> <sup>þ</sup> <sup>λ</sup>2λ<sup>3</sup> <sup>þ</sup> <sup>λ</sup>3λ<sup>1</sup> λ3

> <sup>λ</sup>bc <sup>¼</sup> <sup>λ</sup>1λ<sup>2</sup> <sup>þ</sup> <sup>λ</sup>2λ<sup>3</sup> <sup>þ</sup> <sup>λ</sup>3λ<sup>1</sup> λ1

> <sup>λ</sup>ac <sup>¼</sup> <sup>λ</sup>1λ<sup>2</sup> <sup>þ</sup> <sup>λ</sup>2λ<sup>3</sup> <sup>þ</sup> <sup>λ</sup>3λ<sup>1</sup> λ2

> > <sup>μ</sup>ab <sup>¼</sup> <sup>μ</sup>1μ<sup>2</sup>

<sup>μ</sup>bc <sup>¼</sup> <sup>μ</sup>2μ<sup>3</sup>

<sup>μ</sup>ac <sup>¼</sup> <sup>μ</sup>1μ<sup>3</sup>

Similarly the conversion from star to delta is as follows. The equivalent failure rates are given

λab þ λbc þ λca

λab þ λbc þ λca

λab þ λbc þ λca

<sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>λ</sup>abλac

<sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>λ</sup>abλbc

<sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>λ</sup>acλbc

<sup>μ</sup><sup>1</sup> <sup>¼</sup> <sup>μ</sup>abμbc <sup>þ</sup> <sup>μ</sup>bcμac <sup>þ</sup> <sup>μ</sup>abμac μbc

<sup>μ</sup><sup>2</sup> <sup>¼</sup> <sup>μ</sup>abμbc <sup>þ</sup> <sup>μ</sup>bcμac <sup>þ</sup> <sup>μ</sup>abμac μac

<sup>μ</sup><sup>3</sup> <sup>¼</sup> <sup>μ</sup>abμbc <sup>þ</sup> <sup>μ</sup>bcμac <sup>þ</sup> <sup>μ</sup>abμac μab

μ<sup>1</sup> þ μ<sup>2</sup> þ μ<sup>3</sup>

μ<sup>1</sup> þ μ<sup>2</sup> þ μ<sup>3</sup>

μ<sup>1</sup> þ μ<sup>2</sup> þ μ<sup>3</sup>

Equivalent repair rates are given in the equations from Eqs. (10) to (12) as follows

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Figures 5 and 6.

106 Smart Microgrids

5.3. Delta-star approach

by equations from Eqs. (13) to (18),

Equivalent repair rates are given by

The equivalent failure rates are given by

Figure 6. Equivalent delta connected reliability model.

The evaluation of reliability in the interconnected power system is complex due to the large number of components connected and growing network topology. So far the reliability assessment in interconnected power system is obtained through tracing of the power flow paths [14]. Tracing of paths is time consuming in the case of large networks. Simple and more convenient

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109

The interconnected power system (IEEE 6 bus reliability test system) consists of a number of components, and each component has its own failure and repair rates (λ and μ). From Eqs. (3), (4), (5) and (6), it can be observed the failure rate (λ) is similar to the resistance (R) and the repair rate (μ) is similar to the capacitance (C) in an equivalent electrical network. Hence the reliability model of interconnected power network shown in Figure 8 can be replaced by an equivalent R-C network for reliability assessment. The classical node elimination method is a known technique. The classical node elimination method is used for power system analysis and has not been used so far for reliability studies. This is the first time the classical node elimination method for reliability assessment in interconnected power system is adapted. It is used to reduce the equivalent electrical network to calculate power availability at load bus.

The equivalent reliability model between generator nodes 1 and 2 and the load bus 4 is shown

method based on electrical circuit approach is presented in the following section.

5.4. Node elimination method

Figure 10. Equivalent reliability model for load 4.

Figure 9. Reduced reliability network.

in Figure 10.

Figure 7. IEEE 6 bus reliability test system.

Figure 8. Equivalent reliability model of the IEEE 6 bus reliability test system.

Figure 9. Reduced reliability network.

The evaluation of reliability in the interconnected power system is complex due to the large number of components connected and growing network topology. So far the reliability assessment in interconnected power system is obtained through tracing of the power flow paths [14]. Tracing of paths is time consuming in the case of large networks. Simple and more convenient method based on electrical circuit approach is presented in the following section.

#### 5.4. Node elimination method

Figure 7. IEEE 6 bus reliability test system.

108 Smart Microgrids

Figure 8. Equivalent reliability model of the IEEE 6 bus reliability test system.

The interconnected power system (IEEE 6 bus reliability test system) consists of a number of components, and each component has its own failure and repair rates (λ and μ). From Eqs. (3), (4), (5) and (6), it can be observed the failure rate (λ) is similar to the resistance (R) and the repair rate (μ) is similar to the capacitance (C) in an equivalent electrical network. Hence the reliability model of interconnected power network shown in Figure 8 can be replaced by an equivalent R-C network for reliability assessment. The classical node elimination method is a known technique. The classical node elimination method is used for power system analysis and has not been used so far for reliability studies. This is the first time the classical node elimination method for reliability assessment in interconnected power system is adapted. It is used to reduce the equivalent electrical network to calculate power availability at load bus.

The equivalent reliability model between generator nodes 1 and 2 and the load bus 4 is shown in Figure 10.

Figure 10. Equivalent reliability model for load 4.

In the analogous electrical model, this network is replaced by two networks where in the first one, all failure rates (λ) in each branch are represented by a resistance (equal to λ) and in the second one each branch is represented by a capacitance equal to μ. For reliability assessment, each of these equivalent electrical networks is reduced to a simplified network connecting the sources to the load nodes where the average power availability is required to be calculated. For simplification of the network, node elimination method is used as explained in the following paragraph.

The power system network consists of eight nodes. The power supply node is considered as a current injection node, and the load node where the availability is to be computed is treated as current sink. This reliability model is used to obtain the power availability at load bus 4 only. The other load nodes do not have any current injection. To reduce the network, the nodes in which the current does not enter or leave are eliminated. The equivalent electrical network is described by the nodal equation.

$$
\begin{bmatrix} I\_1 \\ I\_2 \\ \vdots \\ \vdots \\ \vdots \\ I\_8 \end{bmatrix} = \begin{bmatrix} Y\_{11} & Y\_{12} & \dots & \dots & Y\_{18} \\ Y\_{21} & Y\_{22} & \dots & \dots & Y\_{28} \\ \vdots & \vdots & \dots & \dots & \vdots \\ \vdots & \vdots & \dots & \dots & \vdots \\ Y\_{81} & Y\_{82} & \dots & \dots & Y\_{88} \end{bmatrix} \begin{bmatrix} V\_1 \\ V\_2 \\ \vdots \\ \vdots \\ \vdots \\ V\_8 \end{bmatrix} \tag{19}
$$

<sup>P</sup><sup>1</sup> <sup>¼</sup> <sup>μ</sup>G<sup>1</sup>

The incident paths for load L1 are a and g in addition to path from generator G2. The branch b is not incident on bus 2. The sending end probability of power availability of path a is P1 and

> μlb λlb þ μlb � � <sup>1</sup> � <sup>P</sup><sup>2</sup>

> > <sup>P</sup><sup>4</sup> <sup>¼</sup> <sup>μ</sup>G<sup>3</sup>

P<sup>5</sup> ¼ P<sup>4</sup>

P<sup>6</sup> ¼ P<sup>5</sup>

μlf <sup>λ</sup>lf <sup>þ</sup> <sup>μ</sup>lf ! <sup>1</sup> � <sup>P</sup><sup>1</sup>

where P1, P2, P3, P5, P6 and P7 are the probability of power available at respective buses; λla, λlb, λlc, λld, λle, λlf, λlg and λlh are the failure rates of the respective branches; and μla, μlb, μlc, μld,

λG<sup>3</sup> þ μG<sup>3</sup>

μld λld þ μld

μle λle þ μle

similarly for path g is P6 and for generator branch is <sup>μ</sup>G<sup>2</sup>

Similarly the power availability at other buses is given by

P<sup>3</sup> ¼ 1 � 1 � P<sup>4</sup>

P<sup>7</sup> ¼ 1 � 1 � P<sup>6</sup>

λG<sup>2</sup> þ μG<sup>2</sup> � � <sup>1</sup> � <sup>P</sup><sup>1</sup>

So the power availability at bus 2 is given by

Figure 11. Interconnected power system.

<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup> � <sup>μ</sup>G<sup>2</sup>

λG<sup>1</sup> þ μG<sup>1</sup>

λG2þμG<sup>2</sup> .

μla λla þ μla � � <sup>1</sup> � <sup>P</sup><sup>6</sup>

<sup>λ</sup>lg <sup>þ</sup> <sup>μ</sup>lg " # ! (22)

μlc λlc þ μlc � � � � (23)

Assessment of Reliability of Composite Power System Including Smart Grids

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μlh <sup>λ</sup>lh <sup>þ</sup> <sup>μ</sup>lh " # � � (27)

μlg

(21)

111

(24)

(25)

(26)

#### 5.5. Concept of conditional probability

The approach is used to evaluate the power availability in the composite power system, and it is based on conditional probability. A system/component is said to be connected if there exists a path between the source and the sink. The availability of power at the receiving end of a branch not only depends on the failure and repair rates of the components in that branch but also depends on the state of associated components of the branches. These branches can form a power flow path for the particular branch. In the literature, most of the methods are based on the tracing of power flow paths. For example, if a load bus is supplied by three paths a, b and c with power availability at the sending end of each path assumed to P1, P2 and P3 and the probabilities of availability of paths a, b and c are Pa, Pb and Pc, then the probabilities of power unavailable at the ends of paths a, b and c are 1ð Þ � P1Pa , 1ð Þ � P2Pb and 1ð Þ � P3Pc . Then net probability of average power available (PL) at the receiving end load bus is given by

$$P\_L = 1 - (1 - P\_1 P\_a)(1 - P\_2 P\_b)(1 - P\_3 P\_c) \tag{20}$$

The sending end probabilities of each path are termed as conditional probabilities. The concept of conditional probability is explained with the example given in Figure 11. In this directed graph, the generators are connected at the buses 1, 2 and 4. The load buses are 2, 3, 5 and 7. The average power availabilities at the different buses are calculated using the concept of conditional probability as follows.

The availability of power injected into the system by generator G1 at bus 1 is given by

Assessment of Reliability of Composite Power System Including Smart Grids http://dx.doi.org/10.5772/intechopen.75268 111

Figure 11. Interconnected power system.

In the analogous electrical model, this network is replaced by two networks where in the first one, all failure rates (λ) in each branch are represented by a resistance (equal to λ) and in the second one each branch is represented by a capacitance equal to μ. For reliability assessment, each of these equivalent electrical networks is reduced to a simplified network connecting the sources to the load nodes where the average power availability is required to be calculated. For simplification of the network, node elimination method is used as explained in the following

The power system network consists of eight nodes. The power supply node is considered as a current injection node, and the load node where the availability is to be computed is treated as current sink. This reliability model is used to obtain the power availability at load bus 4 only. The other load nodes do not have any current injection. To reduce the network, the nodes in which the current does not enter or leave are eliminated. The equivalent electrical network is

> Y<sup>11</sup> Y<sup>12</sup> … … Y<sup>18</sup> Y<sup>21</sup> Y<sup>22</sup> … … Y<sup>28</sup> ⋮ ⋮ … … ⋮ ⋮ ⋮ … … ⋮ Y<sup>81</sup> Y<sup>82</sup> … … Y<sup>88</sup>

The approach is used to evaluate the power availability in the composite power system, and it is based on conditional probability. A system/component is said to be connected if there exists a path between the source and the sink. The availability of power at the receiving end of a branch not only depends on the failure and repair rates of the components in that branch but also depends on the state of associated components of the branches. These branches can form a power flow path for the particular branch. In the literature, most of the methods are based on the tracing of power flow paths. For example, if a load bus is supplied by three paths a, b and c with power availability at the sending end of each path assumed to P1, P2 and P3 and the probabilities of availability of paths a, b and c are Pa, Pb and Pc, then the probabilities of power unavailable at the ends of paths a, b and c are 1ð Þ � P1Pa , 1ð Þ � P2Pb and 1ð Þ � P3Pc . Then net

probability of average power available (PL) at the receiving end load bus is given by

The availability of power injected into the system by generator G1 at bus 1 is given by

The sending end probabilities of each path are termed as conditional probabilities. The concept of conditional probability is explained with the example given in Figure 11. In this directed graph, the generators are connected at the buses 1, 2 and 4. The load buses are 2, 3, 5 and 7. The average power availabilities at the different buses are calculated using the concept of condi-

PL ¼ 1 � ð Þ 1 � P1Pa ð Þ 1 � P2Pb ð Þ 1 � P3Pc (20)

V1 V2

(19)

⋮

⋮ V8

paragraph.

110 Smart Microgrids

described by the nodal equation.

5.5. Concept of conditional probability

tional probability as follows.

I1 I2

¼

⋮

⋮ I8

$$P\_1 = \frac{\mu\_{G1}}{\lambda\_{G1} + \mu\_{G1}} \tag{21}$$

The incident paths for load L1 are a and g in addition to path from generator G2. The branch b is not incident on bus 2. The sending end probability of power availability of path a is P1 and similarly for path g is P6 and for generator branch is <sup>μ</sup>G<sup>2</sup> λG2þμG<sup>2</sup> .

So the power availability at bus 2 is given by

$$P\_2 = 1 - \left[ \left( 1 - \frac{\mu\_{G2}}{\lambda\_{G2} + \mu\_{G2}} \right) \left( 1 - P\_1 \frac{\mu\_{\rm{la}}}{\lambda\_{\rm{la}} + \mu\_{\rm{la}}} \right) \left( 1 - P\_6 \frac{\mu\_{\rm{lg}}}{\lambda\_{\rm{lg}} + \mu\_{\rm{lg}}} \right) \right] \tag{22}$$

Similarly the power availability at other buses is given by

$$P\_3 = 1 - \left[ \left( 1 - P\_4 \frac{\mu\_{lb}}{\lambda\_{lb} + \mu\_{lb}} \right) \left( 1 - P\_2 \frac{\mu\_{lc}}{\lambda\_{lc} + \mu\_{lc}} \right) \right] \tag{23}$$

$$P\_4 = \frac{\mu\_{G3}}{\lambda\_{G3} + \mu\_{G3}} \tag{24}$$

$$P\_5 = P\_4 \frac{\mu\_{ld}}{\lambda\_{ld} + \mu\_{ld}} \tag{25}$$

$$P\_{\text{6}} = P\_{\text{5}} \frac{\mu\_{\text{le}}}{\lambda\_{\text{le}} + \mu\_{\text{le}}} \tag{26}$$

$$P\_7 = 1 - \left[ \left( 1 - P\_6 \frac{\mu\_{l\bar{f}}}{\lambda\_{l\bar{f}} + \mu\_{l\bar{f}}} \right) \left( 1 - P\_1 \frac{\mu\_{l\bar{h}}}{\lambda\_{l\bar{h}} + \mu\_{l\bar{h}}} \right) \right] \tag{27}$$

where P1, P2, P3, P5, P6 and P7 are the probability of power available at respective buses; λla, λlb, λlc, λld, λle, λlf, λlg and λlh are the failure rates of the respective branches; and μla, μlb, μlc, μld, μle, μlf, μlg and μlh are the repair rates of the respective branches. Based on the generation availability, the direction of power can change in the network. Similarly another way to calculate the average power availability at the bus 2 is calculated by breaking the branch "b" at bus 2, and new node 2I is created. The power availability at this node is P<sup>ɪ</sup> <sup>2</sup> equal to P<sup>3</sup> μb μbþλ<sup>b</sup> h i. The new power availability at bus 2, when the branch "b" terminates on bus 2, is given by

$$P\_2^{\rm II} = 1 - (1 - P\_2^{\rm \circ})(1 - P\_2) \tag{28}$$

in Figure 9. The equivalent failure and repair rates are obtained from the reduced YBus one at a time by assuming λ as resistance R and μ as capacitance C. Since the generator failure and repair rates are already considered in the YBus formation, the nodes 1 and 2 of generators in the equivalent reliability model of the network shown in Figure 9 have 1.0 availability and so can be combined together to evaluate the average availability of power at the load node. So the corresponding network elements between generator 1, generator 2 and load will be in parallel, and overall equivalent λ and μ are calculated. The same procedure is used if there are more than two generators in the power system network. The average power availability at the remaining load points is calculated by adapting the same procedure. The results obtained from

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As discussed in previous sections, the composite power system reliability assessment becomes difficult in complex network because of the large number of equipment, components and integration of renewable power generation. Hence the calculation average power availability becomes complex in the modern power system. For power system reliability assessment, usually component failures are assumed to be independent, and reliability indices are calculated using traditional methods like series-parallel and star-delta equivalents of network connections. This section discusses one new evaluation algorithm for the estimation of average power availability based on modified minimal cut set method using conditional probability. Due to the rapid growth in the power demand, environmental constraints and the competitive power market scenario, the transmission and distribution systems are frequently being operated under heavily loaded conditions, which tend to make the system less stable. The recent literature indicates that most of the blackouts took place due to overloaded transmission system. Further failure of components in the power system causes supply interruptions to connected loads. Statistically, the majority of the service interruptions are happening due to lack of proper planning and operation of power system [15–18]. Therefore complete reliability assessment in transmission and distribution systems (composite power system) is needed in planning of power system. For the above-stated reasons, there is a need for the reliability assessment of composite power systems. One of the objectives used for the evaluation of

S. no. Cut set Components in cut

1. 1 C 2. 2 a, b 3. 3 a, c 4. 4 a, b, c 5. 5 a, c

this method are given in Table 1.

Table 1. Available cut sets.

5.6. Modified minimal cut set algorithm

Knowing the probability of power availability at generators using their respective failure and repair rates, the probability of power availability at all load buses can be computed.

The matrix YBus in the above Eq. (19) is the nodal admittance matrix using the concept of conditional probability, and I and V are the fictitious nodal injected current vector and voltage vector of the equivalent R-C network. To evaluate the equivalent failure rate, the nodal YBus is made up of only the resistive component (λ) for each element, and for equivalent repair rate, the capacitance component (μ) is used for each element. From the equivalent reliability model shown in Figure 10, it is clear that currents I1, I3 and I8 are injected currents and remaining currents are made zero for eliminating the corresponding nodes in the reduced network. Hence the name of this method is called node elimination method. Then Eq. (19) becomes as

$$
\begin{bmatrix} I\_A \\ I\_B \end{bmatrix} = \begin{bmatrix} X & Y \\ Y^T & Z \end{bmatrix} \begin{bmatrix} V\_A \\ V\_B \end{bmatrix} \tag{29}
$$

In Eq. (29), IA is a vector containing the currents that are injected (I1, I3, I8), IB vector is a null vector (I2, I4, I5, I6, I7), VA is a vector containing the voltages at the injected currents (V1, V3, V8), VB is a vector of null vector (V2, V4, V5, V6, V7) and YBus is formed by the combination of matrices X, Y and Z.

From Eq. (29) the following variables are derived as.

$$\mathbf{I}\_{\mathbf{A}} = \mathbf{X} \mathbf{V}\_{\mathbf{A}} + \mathbf{Y} \mathbf{V}\_{\mathbf{B}} \tag{30}$$

$$\mathbf{0} = \mathbf{I}\_{\mathbf{B}} = \mathbf{Y}^{\mathrm{T}} \mathbf{V}\_{\mathbf{A}} + \mathbf{Z} \mathbf{V}\_{\mathbf{B}}$$

$$\mathbf{V}\_{\mathbf{B}} = -\mathbf{Z}^{-1} \mathbf{Y}^{\mathrm{T}} \mathbf{V}\_{\mathbf{A}}$$

$$I\_{A} = \left(\mathbf{X} - \mathbf{Z}^{-1} \mathbf{Y}^{T} V\_{A}\right) V\_{A} \tag{31}$$

The reduced YBus is given in Eq. (32), and with the help of this reduced YBus matrix, we can draw the simple equivalent delta network as shown in Figure 9.

$$Y\_{Bus}^{Reduced} = \left(X - Z^{-1}Y^T V\_A\right) \tag{32}$$

From the above Eq. (32), the equivalent λ and μ between the source node and the load node are obtained. The reduced YBus indicates the nodal equation of the simplified delta network shown in Figure 9. The equivalent failure and repair rates are obtained from the reduced YBus one at a time by assuming λ as resistance R and μ as capacitance C. Since the generator failure and repair rates are already considered in the YBus formation, the nodes 1 and 2 of generators in the equivalent reliability model of the network shown in Figure 9 have 1.0 availability and so can be combined together to evaluate the average availability of power at the load node. So the corresponding network elements between generator 1, generator 2 and load will be in parallel, and overall equivalent λ and μ are calculated. The same procedure is used if there are more than two generators in the power system network. The average power availability at the remaining load points is calculated by adapting the same procedure. The results obtained from this method are given in Table 1.

#### 5.6. Modified minimal cut set algorithm

μle, μlf, μlg and μlh are the repair rates of the respective branches. Based on the generation availability, the direction of power can change in the network. Similarly another way to calculate the average power availability at the bus 2 is calculated by breaking the branch "b"

The new power availability at bus 2, when the branch "b" terminates on bus 2, is given by

Knowing the probability of power availability at generators using their respective failure and

The matrix YBus in the above Eq. (19) is the nodal admittance matrix using the concept of conditional probability, and I and V are the fictitious nodal injected current vector and voltage vector of the equivalent R-C network. To evaluate the equivalent failure rate, the nodal YBus is made up of only the resistive component (λ) for each element, and for equivalent repair rate, the capacitance component (μ) is used for each element. From the equivalent reliability model shown in Figure 10, it is clear that currents I1, I3 and I8 are injected currents and remaining currents are made zero for eliminating the corresponding nodes in the reduced network. Hence the name of this method is called node elimination method. Then Eq. (19) becomes as

> <sup>¼</sup> X Y Y<sup>T</sup> Z � � VA

In Eq. (29), IA is a vector containing the currents that are injected (I1, I3, I8), IB vector is a null vector (I2, I4, I5, I6, I7), VA is a vector containing the voltages at the injected currents (V1, V3, V8), VB is a vector of null vector (V2, V4, V5, V6, V7) and YBus is formed by the combination of

<sup>0</sup> <sup>¼</sup> IB <sup>¼</sup> YTVA <sup>þ</sup> ZVB

The reduced YBus is given in Eq. (32), and with the help of this reduced YBus matrix, we can

From the above Eq. (32), the equivalent λ and μ between the source node and the load node are obtained. The reduced YBus indicates the nodal equation of the simplified delta network shown

YTVA

YTVA

YTVA

VB ¼ �Z�<sup>1</sup>

Bus <sup>¼</sup> <sup>X</sup> � <sup>Z</sup>�<sup>1</sup>

IA <sup>¼</sup> <sup>X</sup> � <sup>Z</sup>�<sup>1</sup>

VB � �

IA ¼ XVA þ YVB (30)

� �VA (31)

� � (32)

<sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup> � <sup>P</sup><sup>ɪ</sup>

repair rates, the probability of power availability at all load buses can be computed.

PII

IA IB � �

From Eq. (29) the following variables are derived as.

draw the simple equivalent delta network as shown in Figure 9.

YReduced

is created. The power availability at this node is P<sup>ɪ</sup>

2

<sup>2</sup> equal to P<sup>3</sup>

� �ð Þ <sup>1</sup> � <sup>P</sup><sup>2</sup> (28)

μb μbþλ<sup>b</sup> h i .

(29)

at bus 2, and new node 2I

112 Smart Microgrids

matrices X, Y and Z.

As discussed in previous sections, the composite power system reliability assessment becomes difficult in complex network because of the large number of equipment, components and integration of renewable power generation. Hence the calculation average power availability becomes complex in the modern power system. For power system reliability assessment, usually component failures are assumed to be independent, and reliability indices are calculated using traditional methods like series-parallel and star-delta equivalents of network connections. This section discusses one new evaluation algorithm for the estimation of average power availability based on modified minimal cut set method using conditional probability.

Due to the rapid growth in the power demand, environmental constraints and the competitive power market scenario, the transmission and distribution systems are frequently being operated under heavily loaded conditions, which tend to make the system less stable. The recent literature indicates that most of the blackouts took place due to overloaded transmission system. Further failure of components in the power system causes supply interruptions to connected loads. Statistically, the majority of the service interruptions are happening due to lack of proper planning and operation of power system [15–18]. Therefore complete reliability assessment in transmission and distribution systems (composite power system) is needed in planning of power system. For the above-stated reasons, there is a need for the reliability assessment of composite power systems. One of the objectives used for the evaluation of


Table 1. Available cut sets.

composite power system reliability is power availability at load buses. Some assumptions made in the proposed algorithms are given below.

procedure for a modified minimal cut set method is explained in the following sections using

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A system is said to be connected if there exists a path between the source (generator) and the sink (load). The removal of the cut set results in the separation of the system into two independent subsystems. One contains all generator nodes and the other system contains all load nodes. A cut set is a set of components or equipments whose failure will cause system failure [55–60]. The general cut set method is given in the following simple example where the cut sets

The definition of a minimal cut set as a cut set in which there is no other subset of components or equipments whose failure alone will cause the system to fail, implies that a normal cut set corresponds to more component failures than are required to cause system failure. The available minimal cut sets for the load in the given example are shown in Table 2. The order of the

The concept of conditional probability is explained with the example given in Figure 12. In this system the generator is connected at the left side. The load bus is 3. The second-order cut set is supplied by two paths and has sending end power availability of P1. The equivalent system is shown in Figure 14. λ and μ are the overall equivalent failure and repair rates of branches a

S. no. Cut set Components in cut

1. 1 C 2. 2 a, b

IEEE 6 bus, 14 bus and single-area IEEE RTS-96 system.

for load in Figure 12 are given in Table 1.

and b in parallel and in series with branch c.

Figure 12. Simple power system to illustrate the concept of cut set.

cut set is shown in Figure 13.

Table 2. Minimal cut sets.

Figure 13. Minimal cut sets for example system.


In literature there are several methods available for the calculation of network reliability. Monte Carlo simulation method has been used by many authors for the estimation of reliability indices including power availability. This method is very popular but takes large computational time. However, it is widely used for the testing of the new methods.

The proposed algorithm discussed in this chapter has the following advantages:


The results obtained by the proposed step-by-step methods are validated by Monte Carlo simulation and also by classical node elimination method discussed in this chapter [64]. The steps for the methodology used in the proposed method are discussed in the following sections and are applied on practical example in [16]. Some of the relevant work regarding the reliability assessment of complex networks is available in [17, 18].

#### 5.7. Introduction to minimal cut set method

The composite power system reliability assessment is generally based on minimal path or cut enumeration, tracing of power flow paths from which the related reliability indices are calculated. The minimal cut set is a popular method in the reliability assessment for simple and complex configurations. There are several methods available for the calculation of average power availability, which is one of the important reliability indices. Some of the popular methods used are minimal cut set, series-parallel, star-delta, tracing of power flow paths, node elimination method and step-by-step algorithm using conditional probability. Yong Liu et al. [13] have assumed that all the branches included in each cut set of order 1 are assumed to be in parallel, with the sending end of each branch in the cut set having the same probability of power availability which is not correct. This assumption is not used in the proposed method. The procedure adapted is explained in the following sections. The initial step in the cut set method is to figure out the minimal cut sets of the system. The identification of minimal cuts becomes more difficult in large complex systems. Some algorithms like node elimination method presented in [15] can be used to reduce this effort for identification. A step-by-step procedure for a modified minimal cut set method is explained in the following sections using IEEE 6 bus, 14 bus and single-area IEEE RTS-96 system.

A system is said to be connected if there exists a path between the source (generator) and the sink (load). The removal of the cut set results in the separation of the system into two independent subsystems. One contains all generator nodes and the other system contains all load nodes. A cut set is a set of components or equipments whose failure will cause system failure [55–60]. The general cut set method is given in the following simple example where the cut sets for load in Figure 12 are given in Table 1.

The definition of a minimal cut set as a cut set in which there is no other subset of components or equipments whose failure alone will cause the system to fail, implies that a normal cut set corresponds to more component failures than are required to cause system failure. The available minimal cut sets for the load in the given example are shown in Table 2. The order of the cut set is shown in Figure 13.

The concept of conditional probability is explained with the example given in Figure 12. In this system the generator is connected at the left side. The load bus is 3. The second-order cut set is supplied by two paths and has sending end power availability of P1. The equivalent system is shown in Figure 14. λ and μ are the overall equivalent failure and repair rates of branches a and b in parallel and in series with branch c.

Figure 12. Simple power system to illustrate the concept of cut set.


Table 2. Minimal cut sets.

composite power system reliability is power availability at load buses. Some assumptions

1. The failure and repair rates during the operating life of the component are assumed to be constants, and the probability distribution of the failure and repair states of the component

2. Each component repair and failure rate is independent of the states of other components. In literature there are several methods available for the calculation of network reliability. Monte Carlo simulation method has been used by many authors for the estimation of reliability indices including power availability. This method is very popular but takes large computa-

2. The power availability at each bus can be computed easily without reducing actual net-

The results obtained by the proposed step-by-step methods are validated by Monte Carlo simulation and also by classical node elimination method discussed in this chapter [64]. The steps for the methodology used in the proposed method are discussed in the following sections and are applied on practical example in [16]. Some of the relevant work regarding the reliabil-

The composite power system reliability assessment is generally based on minimal path or cut enumeration, tracing of power flow paths from which the related reliability indices are calculated. The minimal cut set is a popular method in the reliability assessment for simple and complex configurations. There are several methods available for the calculation of average power availability, which is one of the important reliability indices. Some of the popular methods used are minimal cut set, series-parallel, star-delta, tracing of power flow paths, node elimination method and step-by-step algorithm using conditional probability. Yong Liu et al. [13] have assumed that all the branches included in each cut set of order 1 are assumed to be in parallel, with the sending end of each branch in the cut set having the same probability of power availability which is not correct. This assumption is not used in the proposed method. The procedure adapted is explained in the following sections. The initial step in the cut set method is to figure out the minimal cut sets of the system. The identification of minimal cuts becomes more difficult in large complex systems. Some algorithms like node elimination method presented in [15] can be used to reduce this effort for identification. A step-by-step

tional time. However, it is widely used for the testing of the new methods.

3. The proposed algorithm is applicable to any number of bus systems.

4. It takes less computation time compared to other methods.

ity assessment of complex networks is available in [17, 18].

5.7. Introduction to minimal cut set method

The proposed algorithm discussed in this chapter has the following advantages:

made in the proposed algorithms are given below.

is exponentially distributed.

1. It is very efficient and easy to program.

work.

114 Smart Microgrids

Figure 13. Minimal cut sets for example system.

Figure 14. Cut set branch.

Considering the probability of power availability at the source end, the equivalent failure, repair rates between source and load are given by

$$
\lambda' = \frac{\lambda}{P\_1} \tag{33}
$$

$$
\frac{1}{\mu'} = (1 - P\_1)\frac{1}{\lambda} + \frac{P\_1}{\mu}
$$

$$
\mu' = \frac{\lambda \times \mu}{(1 - P\_1)\mu + P\_1\lambda} \tag{34}
$$

6. Obtain the cut set which isolates this node.

the chosen load node.

ity at the chosen load node.

14. Repeat this exercise for all the load nodes.

procedure steps 1 to 15 described above.

6. Results and discussions

nodes.

ity of power at the node at the other end of the branch.

ities at the other end of the branches in the cut set.

9. Find the cut set which isolates all these above nodes identified in step 7.

15. Obtain the system overall average power availability from step 14.

7. For those cut branches which are incident in this node, assume the probability of availabil-

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8. Based on these probabilities (P), compute the probability of average power availability at

10. Repeat steps 7 and 8 to find the power availabilities at these nodes assuming the probabil-

11. Using these probabilities, evaluate the probability of power availabilities at these cut

13. Using these probabilities works backwards to compute the probability of power availabil-

The proposed algorithm is tested with the practical example taken from the Roy Billinton paper. The configuration of the practical example is shown in Figure 15. The system is connected to generators at the buses 1, 7 and 8 through interconnecting transformers. The failure and repair rates are assumed to be identical for all components throughout the system. This is only for convenience. If different failure and repair rates are specified for each component like generator, transformer, line, etc., the same can be used. There will be no change in the

The algorithms/methods presented in this chapter have been applied to practical example. In this practical example, all components are assumed to have identical reliability data (λ = 0.1; μ = 10). The results are shown in Table 3. The proposed methodology is validated by the Monte Carlo simulation method, node elimination method [15] and step-by-step algorithm using conditional probability [16]. The algorithm developed in this chapter is also applied on IEEE suggested power system network to validate the results. The IEEE 6 bus reliability test system is shown in [15]. The reliability data of the system is given in [15]. The average power availability at the load buses is given in Table 4. To show the efficiency of the proposed method for reliability assessment of large systems, the IEEE 14 bus system and IEEE RTS-96 system are used. The IEEE 14 bus system is shown in [64]. The reliability data of the IEEE 14

12. Repeat this exercise until all the nodes are covered including all generator nodes.

The net average power availability at the receiving end is given by

$$\text{Average power availableity} = \frac{\mu'}{\mu' + \lambda'} \tag{35}$$

The proposed technique is used to find the average power availability at the consumer end in a composite power system and is based on the minimal cut sets.

The steps involved in the proposed algorithm are:


Figure 15. Practical example.

6. Obtain the cut set which isolates this node.

Considering the probability of power availability at the source end, the equivalent failure,

<sup>λ</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup> P1

> 1 λ þ P1 μ

ð Þ <sup>1</sup> � <sup>P</sup><sup>1</sup> <sup>μ</sup> <sup>þ</sup> <sup>P</sup>1<sup>λ</sup> (34)

<sup>μ</sup><sup>0</sup> <sup>þ</sup> <sup>λ</sup><sup>0</sup> (35)

<sup>μ</sup><sup>0</sup> <sup>¼</sup> ð Þ <sup>1</sup> � P1

<sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup> � <sup>μ</sup>

Average power availability <sup>¼</sup> <sup>μ</sup><sup>0</sup>

The proposed technique is used to find the average power availability at the consumer end in a

4. All branches are represented by the reliability parameters failure and repair rates (λ and μ).

2. Generators are connected to the network node through a branch toward that node.

1

The net average power availability at the receiving end is given by

composite power system and is based on the minimal cut sets.

3. Loads are directly connected to the bus called the load node.

The steps involved in the proposed algorithm are:

1. Draw the graph of the network.

5. Choose a particular load node.

Figure 15. Practical example.

(33)

repair rates between source and load are given by

Figure 14. Cut set branch.

116 Smart Microgrids


The proposed algorithm is tested with the practical example taken from the Roy Billinton paper. The configuration of the practical example is shown in Figure 15. The system is connected to generators at the buses 1, 7 and 8 through interconnecting transformers. The failure and repair rates are assumed to be identical for all components throughout the system. This is only for convenience. If different failure and repair rates are specified for each component like generator, transformer, line, etc., the same can be used. There will be no change in the procedure steps 1 to 15 described above.

#### 6. Results and discussions

The algorithms/methods presented in this chapter have been applied to practical example. In this practical example, all components are assumed to have identical reliability data (λ = 0.1; μ = 10). The results are shown in Table 3. The proposed methodology is validated by the Monte Carlo simulation method, node elimination method [15] and step-by-step algorithm using conditional probability [16]. The algorithm developed in this chapter is also applied on IEEE suggested power system network to validate the results. The IEEE 6 bus reliability test system is shown in [15]. The reliability data of the system is given in [15]. The average power availability at the load buses is given in Table 4. To show the efficiency of the proposed method for reliability assessment of large systems, the IEEE 14 bus system and IEEE RTS-96 system are used. The IEEE 14 bus system is shown in [64]. The reliability data of the IEEE 14


Table 3. Average power availability in practical example.


Table 4. Average power availability in IEEE 6 bus system.


7. Conclusions

S. no. Load no. Modified minimal cut

set method

1 Load 1 0.881 0.885 0.885 2 Load 2 0.822 0.819 0.812 3 Load 3 0.555 0.558 0.555 4 Load 4 0.852 0.846 0.852 5 Load 5 0.812 0.812 0.812 6 Load 6 0.813 0.812 0.813 7 Load 7 0.815 0.812 0.813 8 Load 8 0.833 0.836 0.833 9 Load 9 0.855 0.859 0.859 10 Load 10 0.854 0.857 0.854 11 Load 11 0.811 0.818 0.811 12 Load 12 0.836 0.832 0.836 13 Load 13 0.844 0.845 0.844 14 Load 14 0.786 0.788 0.788 15 Load 15 0.764 0.763 0.764 16 Load 16 0.862 0.868 0.864 17 Load 17 0.800 0.808 0.800

Table 6. Average power availability at different loads in IEEE single-area RTS-96 system.

Conflict of interest

There is no potential conflict of interest.

In this chapter reliability modeling of power system components is analyzed by the node elimination method and modified minimal cut set method. The IEEE 6 bus system, IEEE 14 bus systems and single-area IEEE RTS-96 system are used to evaluate the reliability. The two methods gave similar results on average power availability at load bus. The series-parallel and star-delta method is quite difficult for the reduction of complex networks, whereas the node elimination method is easy even for large systems. The new methodologies proposed in this chapter are very useful for power system planners and utility consumers. The electrical circuit approach method is further useful to the power system operators to make decision on the future average power availability. The proposed method on minimal cut set is useful for the

Node elimination method

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http://dx.doi.org/10.5772/intechopen.75268

reliability assessment in the planning and operation of larger power system network.

Table 5. Average power availability at different loads in IEEE 14 bus system.

bus system is given in [15]. The results obtained for the IEEE 14 bus system are shown in Table 5.

The proposed modified minimal cut set algorithm is also applied and tested on IEEE singlearea RTS-96 system shown in [15]. The reliability data for IEEE single-area RTS-96 system is taken from [15]. The average power availability at the load buses for the system is shown in Table 6.


Table 6. Average power availability at different loads in IEEE single-area RTS-96 system.

## 7. Conclusions

In this chapter reliability modeling of power system components is analyzed by the node elimination method and modified minimal cut set method. The IEEE 6 bus system, IEEE 14 bus systems and single-area IEEE RTS-96 system are used to evaluate the reliability. The two methods gave similar results on average power availability at load bus. The series-parallel and star-delta method is quite difficult for the reduction of complex networks, whereas the node elimination method is easy even for large systems. The new methodologies proposed in this chapter are very useful for power system planners and utility consumers. The electrical circuit approach method is further useful to the power system operators to make decision on the future average power availability. The proposed method on minimal cut set is useful for the reliability assessment in the planning and operation of larger power system network.

#### Conflict of interest

bus system is given in [15]. The results obtained for the IEEE 14 bus system are shown in

Node elimination method

Node elimination method

1 Load 1 0.994 0.994 0.92737 0.990 2 Load 2 0.964 0.967 0.89847 0.968 3 Load 3 0.935 0.939 0.90790 0.936 4 Load 4 0.883 0.884 0.85934 0.887

1 Load 1 0.956 0.967 0.967 2 Load 2 0.965 0.967 0.966 3 Load 3 0.967 0.967 0.966 4 Load 4 0.933 0.938 0.933 5 Load 5 0.911 0.914 0.914 6 Load 6 0.911 0.917 0.917 7 Load 7 0.942 0.951 0.950 8 Load 8 0.933 0.939 0.939

1 Load 1 0.994 0.999 0.989 0.999 2 Load 2 0.984 0.985 0.974 0.992 3 Load 3 0.985 0.995 0.956 0.995 4 Load 4 0.991 0.998 0.985 0.998 5 Load 5 0.988 0.998 0.965 0.998

> Monte Carlo method

Node elimination method

Monte Carlo method

> Step-by-step algorithm using conditional probability

> Step-by-step algorithm using conditional probability

Step-by-step algorithm using conditional probability

The proposed modified minimal cut set algorithm is also applied and tested on IEEE singlearea RTS-96 system shown in [15]. The reliability data for IEEE single-area RTS-96 system is taken from [15]. The average power availability at the load buses for the system is shown in

Table 5.

S. no. Load no. Modified minimal

Load no. Modified minimal

Table 3. Average power availability in practical example.

cut set method

S. no.

118 Smart Microgrids

cut set method

Table 4. Average power availability in IEEE 6 bus system.

S. no. Load no. Modified minimal cut

set method

Table 5. Average power availability at different loads in IEEE 14 bus system.

Table 6.

There is no potential conflict of interest.
