4. Reliability assessment of the coal-fired generating station

3. Graphical representation of the coal-fired generating station

54 Recent Improvements of Power Plants Management and Technology

characterization at system level provides for component interactions mapping.

then is termed as the system structure function (i.e., as discussed in Section2) [2–4].

once reached, the absorbing nodes cannot be left [2, 3].

Ni 0

Lij 0

In general, networks are any systems which admit abstract mathematical representations as graphs. The nodes (vertices) of these networks indicate these systems' components [2, 3]. The occurrence of a relation or intercommunication in the midst of the components in these networks is represented by the set of connecting links (edges). It can be noted that the preceding high level of abstraction can, in general, be applied to wide-ranging systems. Consequently, within that sense, a theoretical framework is provided by networks. This theoretical framework enables convenience of conceptual representation of system relations in which

The thermal power plant system as shown in Figure 1 is represented in the form of a graph G ¼ ðN, LÞ of Figure 2, where N is the set of nodes (or vertices) and L the set of links (or edges) [2, 3]. Let each of the six sub-systems of the generating station be denoted by nodes

sði ¼ 1, 2, …: 6Þ, and the interconnection between the systems ðNi, NjÞ is represented by links

s ði, j ¼ 1, 2, …: 6 and i 6¼ jÞ joining nodes Ni and Nj. The flow of heat and energy, steam, water, pulverized fuel, and pre-heated air connects all the six sub-systems. Nodes and links aid in illustrating this flow in Figure 2. When the thermal power plant is graphically represented, this

When the links (arcs/edges) can be traversed in both directions, the graph is undirected. On the other hand, the graph is directed if the links (arcs/edges) can be traversed only in one direction indicated by an arrow. If an undirected graph has no self-loops the presence of at least one link per node guarantees that all the nodes are connected [2, 3]. Practical structures are in general substantially more connected as compared to this minimal threshold. Consequently, there exist numerous paths between any node pair. For directed graphs, the connectivity property is unwieldy because the nodes can relate to any of the following three of categories: (1) the nodes that are strongly connected (i.e., for this subset the nodes can be arrived at from any other node that is a member of the subset. The access is through following the direction of the links); (2) the transient nodes that only have outgoing links. Therefore, transient nodes cannot be accessed from any other node; and (3) the absorbing nodes that only have ingoing links. Thus,

The main intuitive and illustrative tool to be used, for the coal-fired generating station system structure function, is the directed graph [2, 3]. Intuitively the directed graph can be considered as a set of points (or vertices/nodes) with arrows (or arcs/links/edges) joining some of the points [7]. An arc may be labelled. Conventionally, a digraph consists of a set of vertices (nodes/points), V(N) and a subset of ordered pairs of arrows called the arcs (links). The labelling of the digraph is a function from the arcs (links) to the real numbers. One can visualize a labelled digraph by considering the vertices (nodes) as points with arcs (links) as arrows going from vertex (node) i to vertex (node) j whenever ði, jÞ belongs to the sets of arcs (links). The ith vertex (node) of the arc (link) ði, jÞ is called its initial vertex (node), while the jth vertex (node) is called its terminal vertex (node). The arc (link) is then given a label which is the representation of that arc (link) under the labelling function. When the initial and terminal vertices (nodes) are identical, the arc (link) is called a loop. It can be noted that sometimes an According to Ref. [8] as cited in ([4], pp. 5): "When the words are used sloppily, concepts become fuzzy, thinking is muddled, communication is ambiguous, and decisions and actions are suboptimal, to say the least." Therefore, it follows from this saying that a precise definition of reliability and some associated concepts is needed. There are several definitions of what reliability is. In this study reliability is defined as the ability of an item to perform a required function, under given environmental and operational conditions and for a stated period of time (ISO 8402) ([4], pp. 5). The terms in this definition are explained as follows [4]:


Reliability has two abstract meanings; probabilistic and deterministic. The probabilistic approach is based upon statistical failure modelling, without researching and itemizing causes of failure. On the other hand, the deterministic approach focuses on understanding how and why a component or system has failed, and how it can be designed, repaired and tested to prevent such failure from occurrence or recurrence. In the present analysis, the probabilistic approach in conjunction with the graph theory is applied for the steam power plant.

Let Riði ¼ 1, 2, …: 6Þ denotes the reliability of node Ni and rijði, j ¼ 1, 2, …: 6 and i 6¼ jÞ, the reliability of the link (or interconnection) between the nodes, Ni and Nj [2, 3]. Consequently, associating reliability to the system structure of Figure 2 results in the system reliability graph modelling. The system reliability graph for the coal-fired generating station corresponding to its abridged system structure graph is obtained by associating Ri with Ni and rij with Lij, and this is shown in Figure 3 [2, 3].

Figure 3. System reliability digraph for a coal-fired generating station.

The reliability structure function of the system of Figure 3 is estimated by computing the graph's characteristic polynomial [2, 3]. Each and every finite directed graph has a characteristic polynomial [9]. In the beginning, the characteristic polynomial was believed to be a complete invariant, or unique to a graph and all its isomorphism. Later, it was discovered that there are cases where structurally different graphs share the same characteristic polynomial [9]. Although there are cases where structurally different graphs share the same characteristic polynomial, characteristic polynomials are highly studied because they provide much information about a graph in concise format. Characteristic polynomials are useful in steam power plants, mathematics, chemistry, economics, and physics, among others. For example, a graph's spectrum (i.e., the roots of its characteristic polynomial) has significance in the atomic structure.

There exist several ways of determining the characteristic polynomial of a graph with n vertices. In this research, we utilize three methods, namely: (1) the linearly independent cycles; (2) the formula called the figure equation; and (3) the adjacent matrix method. These three methods are in turn employed to estimate the system reliability structure function. Estimating reliability is always an imperfect endeavour (i.e., the reliability (estimate) ranges from the lower bound to the upper bound) and hence the use of these three methods for comparison purposes [2, 3]. The three methods are discussed in the subsequent sections.

### 4.1. The linearly independent cycles method

In the linearly independent cycles procedure, the structure function is characterized by the presence of a sufficient number of certain cycles which have the property that they are linearly independent [2, 3, 10]. These cycles are denoted in a matrix form (i.e., the cycles are denoted by the links (edges/arcs) present in them). Let A ¼ ðaijÞ of dimension L � L, where L denotes the number of links and an element aij denotes whether link j is present in cycle i or not. When the cycle accumulates the link metrics on its path, then the value observed at m is the sum of the link metrics. Furthermore, the assumption here is that the monitor node is responsible for starting and terminating probes. In the preceding case, m is the monitor node.

One assigns B to denote ðL � 1Þ a column matrix which contains the accumulated metrics that correspond to the linearly independent cycles. Furthermore, one assigns x to denote ðL � 1Þ a column matrix that contains the link variables which one has to identify. One's ultimate goal is to solve Ax ¼ B which represents a system of linear equations. For one to be able to uniquely determine x, A has to be invertible. A is also referred to as identifiable, because it has full rank. Cycles that make up such a matrix (i.e., matrix A) are referred to as linearly independent cycles. All link metrics can be uniquely identified by solving Eq. (6) [2, 3, 10]:

$$\mathbf{x} = \mathbf{A}^{-1}\mathbf{B} \tag{6}$$

### 4.2. The figure equation method

The reliability structure function of the system of Figure 3 is estimated by computing the graph's characteristic polynomial [2, 3]. Each and every finite directed graph has a characteristic polynomial [9]. In the beginning, the characteristic polynomial was believed to be a complete invariant, or unique to a graph and all its isomorphism. Later, it was discovered that there are cases where structurally different graphs share the same characteristic polynomial [9]. Although there are cases where structurally different graphs share the same characteristic polynomial, characteristic polynomials are highly studied because they provide much information about a graph in concise format. Characteristic polynomials are useful in steam power plants, mathematics, chemistry, economics, and physics, among others. For example, a graph's spectrum (i.e.,

Energy r36

R6

R3

r23

R2

r12

Steam

Pulverised fuel + Air

Pre-heated air

r21

56 Recent Improvements of Power Plants Management and Technology

Figure 3. System reliability digraph for a coal-fired generating station.

Input Coal

R1

R4

r54

R5

Condensate

r42

Exhaust steam

Water + Steam

r35

Output Power

There exist several ways of determining the characteristic polynomial of a graph with n vertices. In this research, we utilize three methods, namely: (1) the linearly independent cycles; (2) the formula called the figure equation; and (3) the adjacent matrix method. These three methods are in turn employed to estimate the system reliability structure function. Estimating reliability is always an imperfect endeavour (i.e., the reliability (estimate) ranges from the lower bound to the upper bound) and hence the use of these three methods for comparison

In the linearly independent cycles procedure, the structure function is characterized by the presence of a sufficient number of certain cycles which have the property that they are linearly independent [2, 3, 10]. These cycles are denoted in a matrix form (i.e., the cycles are denoted by the links (edges/arcs) present in them). Let A ¼ ðaijÞ of dimension L � L, where L denotes the

the roots of its characteristic polynomial) has significance in the atomic structure.

purposes [2, 3]. The three methods are discussed in the subsequent sections.

4.1. The linearly independent cycles method

The figure equation procedure provides a direct link between a graph's structure function and the coefficients of its characteristic polynomial. Unlike the linearly independent cycles and the adjacent matrix methodologies, the figure equation method does not use determinants but calculates the characteristic polynomial of any graph by counting the cycles in the graph [2, 3, 9]. In this procedure, coefficients of the graphs' characteristic polynomials are calculated. The calculations are done when one considers the set of linearly-directed sub-graphs of a corresponding length. i nodes and i links constitute a linearly-directed sub-graph of length i in such a way that each node bears in degree and out degree of one (1).

The figure equation states that for any graph G ¼ ðN, LÞ with n vertices, the characteristic polynomial is [2, 3, 9] as follows:

$$X(G) = \mathbf{x}^n + c\_1 \mathbf{x}^{n-1} + \dots + c\_{n-1} \mathbf{x} + c\_n \tag{7}$$

such that for 1 ≤ i ≤ n, the coefficient is as follows:

$$c\_i = \sum\_{L \in L\_i} (-1)^{P(L)} \tag{8}$$

where Li is the set of all linearly directed subgraphs of G and PðLÞ is the number of linearly directed cycles, or the number of pieces in L.

#### 4.3. The adjacent matrix method

An adjacency matrix tells which vertex (node) in the graph is connected to which. The adjacency matrix of the system reliability digraph for a coal-fired generating station is defined to represent the steam power plant. The adjacent matrix should be defined such that it incorporates the structural information of the components and sub-systems (i.e., of the steam power plant) and interconnections between them [2–4].

The system structure matrix of the steam power plant is defined as follows [2, 3]. Here, a general case of a macro level coal-fired generating station with N sub-systems is considered. Thus, leading to a symmetric adjacency, matrix 0f , 1g of order N � N. cijði, j ¼ 1, 2, …:6 and i 6¼ jÞ represents the connectivity between node (vertex) i and j such that cij ¼ 1 if node (vertex) i is connected to node (vertex) j. In the system reliability digraph of Figure 3, this is represented by the link (edge) reliability rij between nodes i and j. cij is equal to zero otherwise. Thus, cij ¼ 0 for all i, as a node (vertex) cannot be connected to itself. In the case, where the node (vertex) is connected to itself, cij ¼ 1. This implies a self-loop at node (vertex) i in the graph.

Each row or column of the system structure matrix corresponds to a node (vertex). The six subsystems of the system reliability digraph of Figure 3 correspond to the six columns or rows of this matrix [2, 3]. The off-diagonal matrix elements, cij, represent a connection between nodes (vertices) i and j. In the adjacent matrix cij 6¼ cji, as only directional connections between nodes (vertices) are considered. The characteristic polynomial of the graph is the characteristic polynomial of its adjacency matrix. The determinant of the characteristic system reliability matrix is called the characteristic system reliability polynomial.

The reliability of the system is estimated by obtaining the determinant of the characteristic system reliability matrix as follows [2–6]:

$$R\_{system} = \det\{RI - A\_{adjacent}\} \tag{9}$$

where R represents the reliability of the nodes constituting the system; I is the node identity matrix; and Aadjacent is the system structure adjacency matrix.

## 5. Illustration of the system reliability methodology assessment

#### 5.1. Illustration of the linearly independent cycles procedure

The linearly independent cycles (LIC) method computes the list of cycles (top) and the corresponding cycle-link matrix (bottom) as shown in Figure 4. The two (2) cycles computed in Figure 4 are linearly independent. Thus, all link metrics may be identified. We use node R1 (see Figure 3) as the monitor that can start and terminate probes. The reliability of the structure of Figure 3 is determined using Eq. (6) as [2–6]:

$$R\_{\text{system}} = \det\{ (\mathbf{R} \times \text{eye}(7)) - A\_{\text{LIC}} \} = \mathbf{R}^7 - 2\mathbf{R}^6 \tag{10}$$

For constant unit failure rate, substituting <sup>R</sup>ðtÞ ¼ <sup>e</sup>�λ<sup>t</sup> into Eq. (10) yields:

$$R\_{\text{system}}(t) = e^{-7\lambda t} - 2e^{-6\lambda t} \tag{11}$$

where RsystemðtÞ is the coal-fired generating station reliability at time t and λ is the unit constant failure rate.

$$\begin{aligned} r\_{12} & \rightarrow r\_{21} \\ r\_{12} & \rightarrow r\_{23} & \rightarrow r\_{35} & \rightarrow r\_{54} & \rightarrow r\_{42} & \rightarrow r\_{21} \\ & \begin{pmatrix} r\_{12}r\_{21}r\_{23}r\_{35}r\_{36}r\_{42}r\_{54} \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \end{aligned}$$

Figure 4. List of linearly independent cycles (top) and the corresponding cycle-link matrix (bottom).

#### 5.2. Illustration of the figure equation procedure

The system structure matrix of the steam power plant is defined as follows [2, 3]. Here, a general case of a macro level coal-fired generating station with N sub-systems is considered. Thus, leading to a symmetric adjacency, matrix 0f , 1g of order N � N. cijði, j ¼ 1, 2, …:6 and i 6¼ jÞ represents the connectivity between node (vertex) i and j such that cij ¼ 1 if node (vertex) i is connected to node (vertex) j. In the system reliability digraph of Figure 3, this is represented by the link (edge) reliability rij between nodes i and j. cij is equal to zero otherwise. Thus, cij ¼ 0 for all i, as a node (vertex) cannot be connected to itself. In the case, where the node (vertex) is connected to itself, cij ¼ 1. This implies a self-loop at node

Each row or column of the system structure matrix corresponds to a node (vertex). The six subsystems of the system reliability digraph of Figure 3 correspond to the six columns or rows of this matrix [2, 3]. The off-diagonal matrix elements, cij, represent a connection between nodes (vertices) i and j. In the adjacent matrix cij 6¼ cji, as only directional connections between nodes (vertices) are considered. The characteristic polynomial of the graph is the characteristic polynomial of its adjacency matrix. The determinant of the characteristic system reliability matrix is

The reliability of the system is estimated by obtaining the determinant of the characteristic

where R represents the reliability of the nodes constituting the system; I is the node identity

The linearly independent cycles (LIC) method computes the list of cycles (top) and the corresponding cycle-link matrix (bottom) as shown in Figure 4. The two (2) cycles computed in Figure 4 are linearly independent. Thus, all link metrics may be identified. We use node R1 (see Figure 3) as the monitor that can start and terminate probes. The reliability of the structure

5. Illustration of the system reliability methodology assessment

For constant unit failure rate, substituting <sup>R</sup>ðtÞ ¼ <sup>e</sup>�λ<sup>t</sup> into Eq. (10) yields:

RsystemðtÞ ¼ e

Rsystem ¼ detfRI � Aadjacentg ð9Þ

Rsystem <sup>¼</sup> detfð<sup>R</sup> � eyeð7ÞÞ � ALICg ¼ <sup>R</sup><sup>7</sup> � <sup>2</sup>R<sup>6</sup> <sup>ð</sup>10<sup>Þ</sup>

�6λ<sup>t</sup> <sup>ð</sup>11<sup>Þ</sup>

�7λ<sup>t</sup> � <sup>2</sup><sup>e</sup>

where RsystemðtÞ is the coal-fired generating station reliability at time t and λ is the unit constant

(vertex) i in the graph.

called the characteristic system reliability polynomial.

58 Recent Improvements of Power Plants Management and Technology

matrix; and Aadjacent is the system structure adjacency matrix.

5.1. Illustration of the linearly independent cycles procedure

of Figure 3 is determined using Eq. (6) as [2–6]:

failure rate.

system reliability matrix as follows [2–6]:

The calculation of characteristic polynomials using the figure procedure is relatively unfamiliar [9]. In the figure equation procedure, coefficients of the graphs' characteristic polynomials are calculated. The calculations are done when one considers the set of linearly-directed subgraphs of a corresponding length. i nodes and i links constitute a linearly-directed sub-graph of length i in such a way that each node bears in degree and out degree of one (1) [9].

From Section4.2, we recall that a factor F of a digraph H is a subgraph containing all the vertices of H in which each vertex (node) has both in degree and out degree equal to one. In other words, it consists of a collection of disjoint cycles that go through each vertex (node) of H. The number of cycles in the factor F is denoted nðFÞ. If the digraph H is labelled, then WðFÞ denotes the weight of the factor [7]. Given the digraph H of order n; F its factor; F its set of all linearly directed subgraphs; nðFÞ the number of cycles in the factor F; WðFÞ the weight of the factor F, then the coefficients, ci of H are determined as follows:

$$\mathfrak{c}\_{i} = \sum\_{F \in F} (-1)^{n(F)} \mathcal{W}(F) \tag{12}$$

where 1 ≤ i ≤ n and in this study we assume WðFÞ ¼ 1.

Using the figure equation formulae (Eqs. (7), (8), and (12)), we get the list of sub-graphs (left) and the corresponding coefficients (right) as shown in Table 1. From the information as shown in Table 1, the characteristic polynomial of the structure in Figure 3 is as follows:


Table 1. Linearly directed cycles and their coefficients.

$$R(G) = \mathbf{x}^6 + \mathbf{x}^5 + \mathbf{x}^4 + \mathbf{x}^3 \tag{13}$$

For constant unit failure rate, substituting <sup>R</sup>ðtÞ ¼ <sup>e</sup>�λ<sup>t</sup> into Eq. (13) yields:

$$R\_{system}(t) = e^{-6\lambda t} + e^{-5\lambda t} + e^{-4\lambda t} + e^{-3\lambda t} \tag{14}$$

where RsystemðtÞ is the coal-fired generating station reliability at time t and λ is the unit constant failure rate.

#### 5.3. Illustration of the adjacent matrix procedure

The digraph for the coal-fired generating station of Figure 3 characterizes the visual representation of the system and its interdependence [2, 3]. The adjacent matrix procedure converts the digraph into a mathematical form, and the structure function is a mathematical model that helps to determine the reliability index [2, 3]. It may be noted here that the development of a structure function is not merely the determinant of the matrix. The structure function is developed in such a manner that no information regarding the system reliability is lost [2, 3]. For this purpose, a step-by-step procedure is proposed in Section 4.3.

Using the adjacent matrix method, the reliability of the structure of Figure 3 is determined using Eq. (9) as [2–6]:

$$\begin{aligned} \mathbf{R}\_{\text{system}} &= \det\left\{ \begin{pmatrix} \begin{bmatrix} \text{R}\_{1} \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \\ \text{0} \ \text{R}\_{2} \ 0 \ 0 \ 0 \ 0 \ 0 \\ \text{0} \ \text{R}\_{3} \ 0 \ 0 \ 0 \ 0 \\ \text{0} \ \text{0} \ \text{R}\_{4} \ 0 \ 0 \ 0 \\ \text{0} \ \text{0} \ \text{0} \ \text{R}\_{5} \ 0 \\ \text{0} \ \text{0} \ 0 \ 0 \ \text{R}\_{5} \ 0 \\ \text{0} \ \text{0} \ \text{0} \ \text{0} \ \text{R}\_{6} \end{bmatrix} \times \begin{bmatrix} 1 \ 0 \ 0 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \ 0 \ 0 \\ 0 \ 0 \ 1 \ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 1 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 0 \ 1 \ 0 \\ 0 \ 0 \ 0 \ 0 \ 0 \ 1 \end{bmatrix} \end{aligned} \right\} - \left( \begin{bmatrix} 0 & \ \mathbf{r}\_{12} \ 0 \ 0 \ 0 \ 0 \\ \mathbf{r}\_{21} \ 0 \ \mathbf{r}\_{23} \ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 0 \ \mathbf{r}\_{35} \ \mathbf{r}\_{36} \\ 0 \ 0 \ 0 \ 0 \ 0 \ 1 \ 0 \\ 0 \ 0 \ 0 \ 0 \ 0 \ 1 \end{bmatrix} \right) \right) \tag{15}$$

For identical units (i.e., for illustration purposes only) (i.e., R<sup>1</sup> ¼ R<sup>2</sup> ¼ R<sup>3</sup> ¼ R<sup>4</sup> ¼ R<sup>5</sup> ¼ R<sup>6</sup> ¼ R) with the link reliability assumed to be unity (i.e., rij ¼ 1), Eq. (15) simplifies to:

$$R\_{system} = |\left(R \times (-R^5 + R^3 + R)\right)|\tag{16}$$

where j•j denotes the absolute value and det is the matrix determinant.

For constant unit failure rate, substituting <sup>R</sup>ðtÞ ¼ <sup>e</sup>�λ<sup>t</sup> into Eq. (16) yields:

$$R\_{system}(t) = |e^{-\lambda t}(e^{-\lambda t} + e^{-3\lambda t} - e^{-5\lambda t})|\tag{17}$$

where RsystemðtÞ is the coal-fired generating station reliability at time t and λ is the unit constant failure rate.

## 6. Results

<sup>R</sup>ðGÞ ¼ <sup>x</sup><sup>6</sup> <sup>þ</sup> <sup>x</sup><sup>5</sup> <sup>þ</sup> <sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>x</sup><sup>3</sup> <sup>ð</sup>13<sup>Þ</sup>

�3λ<sup>t</sup> <sup>ð</sup>14<sup>Þ</sup>

For constant unit failure rate, substituting <sup>R</sup>ðtÞ ¼ <sup>e</sup>�λ<sup>t</sup> into Eq. (13) yields:

�6λ<sup>t</sup> <sup>þ</sup> <sup>e</sup>

Cycles in the sub-graph Coefficients

<sup>r</sup><sup>12</sup> ! <sup>r</sup><sup>21</sup> <sup>2</sup> pieces, c<sup>1</sup> ¼ ð�1<sup>Þ</sup>

<sup>r</sup><sup>12</sup> ! <sup>r</sup><sup>23</sup> ! <sup>r</sup><sup>35</sup> ! <sup>r</sup><sup>54</sup> ! <sup>r</sup><sup>42</sup> ! <sup>r</sup><sup>21</sup> <sup>6</sup> pieces, c<sup>2</sup> ¼ ð�1<sup>Þ</sup>

<sup>r</sup><sup>23</sup> ! <sup>r</sup><sup>35</sup> ! <sup>r</sup><sup>54</sup> ! <sup>r</sup><sup>42</sup> <sup>4</sup> pieces, c<sup>3</sup> ¼ ð�1<sup>Þ</sup>

where RsystemðtÞ is the coal-fired generating station reliability at time t and λ is the unit constant

The digraph for the coal-fired generating station of Figure 3 characterizes the visual representation of the system and its interdependence [2, 3]. The adjacent matrix procedure converts the digraph into a mathematical form, and the structure function is a mathematical model that helps to determine the reliability index [2, 3]. It may be noted here that the development of a structure function is not merely the determinant of the matrix. The structure function is developed in such a manner that no information regarding the system reliability is lost [2, 3].

Using the adjacent matrix method, the reliability of the structure of Figure 3 is determined

¼ R6 ðR3 R4 R5 r12 r21 þ R1 r23 r35 r42 r54�R<sup>1</sup> R2 R3 R4 R5Þ

For identical units (i.e., for illustration purposes only) (i.e., R<sup>1</sup> ¼ R<sup>2</sup> ¼ R<sup>3</sup> ¼ R<sup>4</sup> ¼ R<sup>5</sup> ¼ R<sup>6</sup> ¼ R)

<sup>R</sup> � ð�R<sup>5</sup> <sup>þ</sup> <sup>R</sup><sup>3</sup> <sup>þ</sup> <sup>R</sup><sup>Þ</sup>

1

0

BBBBBBBBB@

�

0 r12 0000 r21 0 r23 000 0000r35 r36 0 r42 0000 000r54 0 0 000000

1

9

>>>>>>>>>=

>>>>>>>>>;

ð15Þ

<sup>2</sup> <sup>¼</sup> <sup>1</sup>

<sup>6</sup> <sup>¼</sup> <sup>1</sup>

<sup>4</sup> <sup>¼</sup> <sup>1</sup>

CCCCCCCCCA

j ð16Þ

CCCCCCCCCA �

�5λ<sup>t</sup> <sup>þ</sup> <sup>e</sup>

�4λ<sup>t</sup> <sup>þ</sup> <sup>e</sup>

RsystemðtÞ ¼ e

For this purpose, a step-by-step procedure is proposed in Section 4.3.

R1 00000 0 R2 0000 00R3 000 000R4 0 0 0000R5 0 00000R6

with the link reliability assumed to be unity (i.e., rij ¼ 1), Eq. (15) simplifies to:

�

Rsystem ¼ j

where j•j denotes the absolute value and det is the matrix determinant.

�

0

8

>>>>>>>>><

>>>>>>>>>:

BBBBBBBBB@

5.3. Illustration of the adjacent matrix procedure

Table 1. Linearly directed cycles and their coefficients.

60 Recent Improvements of Power Plants Management and Technology

failure rate.

using Eq. (9) as [2–6]:

Rsystem ¼ det

#### 6.1. The approximate system reliability

The system reliability value of the coal-fired generating station of Figure 1 as illustrated by Eqs. (11), (14), and (17), respectively, is plotted as shown in Figure 5. It should be noted that the results in Figure 5 have been obtained assuming that the components are identical and assuming constant unit failure rate. This is only for the example to illustrate the methods. In practice, components are not identical, and different failure rates could be used for each component, which is the real-life scenario. The results of the approximation of the system reliability for the three methods which are shown in Figure 5 have three bounds: (1) the lower; (2) the in-between; (3) and the upper. The lower bound is given by the linearly independent cycles method; the in-between bound by the adjacent matrix method and then the upper bound by the figure equation method. Figure 5 reveals that the approximate system reliability value starts to decrease gradually with time as is expected. There are various reasons for this

Figure 5. Coal-fired generating station system reliability.

phenomenon. Some of the possible causes are: (1) the ageing effects of the system; and (2) the unavailability of some of the components that constitute the sub-systems of the thermal power plant. The thermal power plant actual or real-time performance system reliability, when available, is compared with the design system reliability (approximate/estimate) by way of the graph-theoretical analysis. The main objective of this comparison is to optimize the design system reliability (estimate).

#### 6.2. The Birnbaum's measure of structural importance

In 1969, Birnbaum advocated the measure for the structural importance of component i as follows [3, 4]:

$$B\_{\mathcal{Q}}(i) = \frac{\eta\_{\mathcal{Q}}(i)}{2^{n-1}} \tag{18}$$

The measure of structural importance, B∅ðiÞ, proposed by Birnbaum and illustrated in Eq. (18), shows the comparative portion of the 2<sup>n</sup>�<sup>1</sup> possible state vectors, that is, <sup>ð</sup>∙i, xÞ.These possible sate vectors form critical path vectors for component i. Birnbaum's measure of structural importance, as expressed in Eq. (18), can be used to partially rank components constituting a system in accordance with the size of B∅ðiÞ.

Figure 6 shows the results for Birnbaum's structural importance for the components in the coal-fired generating station of Figure 1. Figure 6 illustrates how much worse the coal-fired generating station system reliability would be if component i would fail. Figure 6 shows that eight components (N2, N6, L12, L21, L23, L35, L42, and L54) have the same Birnbaum structural importance (i.e., 3.12%) with four components (N1, N3, N4, and N5) having a negligibly small Birnbaum structural importance (i.e., 0%). Birnbaum's measure of structural importance only takes into account the system structure function and not the lifetime distributions of the components. Therefore, it is relatively easy to calculate and is in general used in the design

Figure 6. Birnbaum's measure of structural importance for components in the thermal power plant.

phase or when the lifetime distributions of components are not known. Birnbaum's measure of structural importance is also an alternative when the more advanced measures would be too time-consuming to compute or difficult to use.

#### 6.3. The Birnbaum's measure of reliability importance

phenomenon. Some of the possible causes are: (1) the ageing effects of the system; and (2) the unavailability of some of the components that constitute the sub-systems of the thermal power plant. The thermal power plant actual or real-time performance system reliability, when available, is compared with the design system reliability (approximate/estimate) by way of the graph-theoretical analysis. The main objective of this comparison is to optimize the design

In 1969, Birnbaum advocated the measure for the structural importance of component i as

<sup>B</sup>∅ðiÞ ¼ <sup>η</sup>∅ði<sup>Þ</sup>

The measure of structural importance, B∅ðiÞ, proposed by Birnbaum and illustrated in Eq. (18), shows the comparative portion of the 2<sup>n</sup>�<sup>1</sup> possible state vectors, that is, <sup>ð</sup>∙i, xÞ.These possible sate vectors form critical path vectors for component i. Birnbaum's measure of structural importance, as expressed in Eq. (18), can be used to partially rank components constituting a

Figure 6 shows the results for Birnbaum's structural importance for the components in the coal-fired generating station of Figure 1. Figure 6 illustrates how much worse the coal-fired generating station system reliability would be if component i would fail. Figure 6 shows that eight components (N2, N6, L12, L21, L23, L35, L42, and L54) have the same Birnbaum structural importance (i.e., 3.12%) with four components (N1, N3, N4, and N5) having a negligibly small Birnbaum structural importance (i.e., 0%). Birnbaum's measure of structural importance only takes into account the system structure function and not the lifetime distributions of the components. Therefore, it is relatively easy to calculate and is in general used in the design

Figure 6. Birnbaum's measure of structural importance for components in the thermal power plant.

<sup>2</sup><sup>n</sup>�<sup>1</sup> <sup>ð</sup>18<sup>Þ</sup>

system reliability (estimate).

follows [3, 4]:

6.2. The Birnbaum's measure of structural importance

62 Recent Improvements of Power Plants Management and Technology

system in accordance with the size of B∅ðiÞ.

In 1969, Birnbaum proposed the measure of the reliability importance of component i at time t as follows [3, 4]:

$$I^{B}(i|t) = \frac{\partial h\left(p(t)\right)}{\partial p\_i(t)}\text{ for }i = 1, \ 2, \ \dots, n\tag{19}$$

In order to obtain Birnbaum's measure of the reliability importance of component i at time t. A partial derivative of the system reliability with respect to pi ðtÞ is taken. Birnbaum's measure of the reliability importance is a specific case of sensitivity analysis that was used in various engineering applications for ages [4]. A large I <sup>B</sup>ðijt<sup>Þ</sup> results in a comparatively large change in the system reliability at time t, for a small change in the reliability of component i. Birnbaum's measure of the reliability importance depends on the component reliabilities at various points in time. Therefore, it gives perhaps a more global view of component importance.

Figures 7 and 8 show that all components except (N1, N3, N4, and N5) have the same Birnbaum's measure of the reliability importance. We deduce how the components behave with respect to time in Figures 7 and 8.

Figure 7. Birnbaum's measure of reliability importance for components: N2, N6, L12, and L21.

Figure 8. Birnbaum's measure of reliability importance for components: L23, L35, L42, and L54.

## 7. Conclusion

The graph-theoretical analysis procedure was used for the analysis of several reliability parameters of a steam power plant, in this chapter. Figure 5 illustrates the system reliability as a function of time with failure rates assumed to follow exponential time distributions. It is worth noting that the results of Figure 5 have been attained assuming identical components and constant unit failure rates. This is only for the example to illustrate the methods. In practice, components are not identical and different failure rates could be used for each component, which is the real-life scenario.

The structure function model for the steam power plant developed in this chapter represents its structural information, including its systems, their sub-systems, their components and their interconnections. The procedure transforms a real-life steam power plant into the following representations: its block (see Figure 1); its system structure digraph (see Figure 2); and then finally its system reliability digraph (see Figure 3). The structure function of the coal-fired generating station embodies all probable composites of its components and subsystems at a specific state of hierarchy. These composites and interactions create a method which can be used to analyse the structure and the function of the parameters that are dependent on the structure. The said methodology allows for either a top-down or bottomup analysis and design of various systems, sub-systems, and their interconnections. The model enables one to determine optimal maintenance strategies that will ascertain upper bound reliability of the thermal power station. Power plant managers can benefit from use of the model for the analysis of the reliability of the thermal power plants. The procedure permits changes in the model to make it plant-specific as well as design-specific.

In general, a collection of components performing a specific task or function is referred to as a system. It goes without saying that in a system, some components are more important for the system reliability than others. For example, in a system, if a component is in series with the rest of the components, it is a cut set of order one (1). A component which is a cut set of order one (1) is in general more important than a component that is a member of a cut set of higher order ([4]: pp. 149–150). In Section 6, Birnbaum's measure of structural importance and Birnbaum's measure of the reliability importance have been defined and discussed. Component importance measures may be used to rank the components, that is, to arrange the components in ascending or descending order of importance. Component importance measures may also be used for classification of importance, that is, to allocate the components into two or more groups, according to some pre-set criteria.

Systems usually consist of multiple components. The components constituting a system are not necessarily equally important for the performance (reliability, availability, risk, and throughput) of the constituent system. Ordinarily limited resources are available to design, enhance and/or maintain such a system efficiently. Nevertheless, for complex and large systems, it may be too tedious, or not even possible, to develop a formal optimal strategy. In analogous situations, it is advantageous for one to allocate resources in accordance of how important the components are to the system. Furthermore, it is desirable to concentrate the resources on the subset of components that are most important to the system ([11]: pp. 49–53). Therefore, the notion of component importance measures (also called sensitivity).

A basic problem that faces the reliability engineer in attempting to achieve maximum reliability for a large and complex system is that of evaluating the relative importance of the various components constituting the system. Thus, in reliability, a component importance measure evaluates the relative importance of individual components or group of components in that system. This relative importance can be determined based on the system structure, component reliability and/or component lifetime distributions. Measuring the relative importance of components may allow the engineer to: (1) determine which of these components deserve additional research and warrant development in order to improve the overall system reliability under cost (and/or effort) constraints; and (2) find the component that caused the failure of a system. By using importance measures, it is possible to draw conclusions about which components are the most important to improve in order to achieve better reliability of the whole system. In Sections 6.2 and 6.3, we have provided Birnbaum's measure of structural importance and Birnbaum's measure of the reliability importance for that specific purpose and given the results thereof.

## Author details

7. Conclusion

The graph-theoretical analysis procedure was used for the analysis of several reliability parameters of a steam power plant, in this chapter. Figure 5 illustrates the system reliability as a function of time with failure rates assumed to follow exponential time distributions. It is worth noting that the results of Figure 5 have been attained assuming identical components and constant unit failure rates. This is only for the example to illustrate the methods. In practice, components are not identical

The structure function model for the steam power plant developed in this chapter represents its structural information, including its systems, their sub-systems, their components and their interconnections. The procedure transforms a real-life steam power plant into the following representations: its block (see Figure 1); its system structure digraph (see Figure 2); and then finally its system reliability digraph (see Figure 3). The structure function of the coal-fired generating station embodies all probable composites of its components and subsystems at a specific state of hierarchy. These composites and interactions create a method which can be used to analyse the structure and the function of the parameters that are dependent on the structure. The said methodology allows for either a top-down or bottomup analysis and design of various systems, sub-systems, and their interconnections. The model enables one to determine optimal maintenance strategies that will ascertain upper bound reliability of the thermal power station. Power plant managers can benefit from use of the model for the analysis of the reliability of the thermal power plants. The procedure

and different failure rates could be used for each component, which is the real-life scenario.

Figure 8. Birnbaum's measure of reliability importance for components: L23, L35, L42, and L54.

64 Recent Improvements of Power Plants Management and Technology

permits changes in the model to make it plant-specific as well as design-specific.

Bernard Tonderayi Mangara

Address all correspondence to: aragnam@gmail.com

Central University of Technology, Free State (CUT), Bloemfontein, South Africa
