2. System structure function for the coal-fired generating station

or more years [7]. In this study, graph-theoretic ideas are applied to some of the fundamental topics in power plant engineering. While there are many such applications, we shall focus on only using graph-theoretic ideas for estimating the reliability index and evaluating the avail-

Graph theory has been successfully used to model many different types of systems, inclusive of coal-based steam power plants [2, 3, 5, 6]. The GTA modelling requires the large and complex systems, such as the steam power plant network, to be reduced and divided into sub-systems for convenience of the analysis procedure. The GTA model simulates the inheritances and interdependencies of the sub-systems of the coal-fired generating station in addition to giving a quantitative measure of the system reliability. The GTA procedure is composed of three steps namely: (1) digraph representation; (2) matrix representation; and (3) development of a permanent structure function. The quantitative measure of the steam power plant system reliability enables the design engineer to determine the similarity or dissimilarity

The GTA procedure is used here to model the entire system of a coal-fired generating station, as shown in Figure 1. The system is divided into six sub-systems ðNi : i ¼ 1, 2, …:6Þ which are

The identified and above-mentioned sub-systems for the steam power plant of Figure 1 are displayed in Figure 2 [2, 3]. The discourse on the said six sub-systems follows in Section2.

L42

Water + Steam

N2

Steam

L23

L21

L12

Pre-heated air

Pulverised fuel + Air

N1

Figure 2. System structure digraph for a coal-fired generation station.

N4

N3

L36

Energy

Condensate

L54

L35

Exhaust steam

N6

N5

Output Power

ability index for a coal-fired generating power station.

52 Recent Improvements of Power Plants Management and Technology

between the present reliability and the design value.

given below [2, 3]: N1: The coal system; N2: The boiler system; N3: The steam turbine;

N6: The generator.

N4: The boiler feed pump; N5: The cooling system; and

> Input Coal

In this research, consideration is made of systems of components that satisfy the following hypothesis ([2, 3], [4], pp. 123):

Systems that are composed of n components are denoted as systems of order n. The constituent components are assumed to be numbered consecutively from 1 to n. The study is confined to situations where it suffices to distinguish between only two states, a functioning state and a failed state. The preceding study limitation applies to each component as well as to the system itself. The state of component i, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …: <sup>n</sup> can then be described by a binary<sup>1</sup> variable (function) xi, where:

$$\mathbf{x}\_{i} = \begin{cases} 1, \text{ if component i is functioning} \\ 0, \text{ if component i is in a failed state} \end{cases} \tag{4}$$

Then, vector X ¼ ðx1, x2, …:, xnÞ denotes the states of all components. Vector X ¼ ðx1, x2, …:, xnÞ is known as the component state vector. In addition, the assumption is that if one knows all of the n components' states, then it follows that they also know the state of the system, that is whether it is functioning or failed. The system state is determined completely by and is an inevitable consequence of the states of the components that constitute the system.

In a similar way, the system state can then be delineated using a binary function: ∅ðXÞ ¼ ∅ðx1, x2,…:, xnÞ, where:

$$\mathfrak{D}(X) = \begin{cases} 1, \text{ if the system is functions} \\ 0, \text{ if the system is in a failed state} \end{cases} \tag{5}$$

and where ∅ðXÞ is called the structure function of the system or just the structure (e.g., the system structure digraph for a coal-fired generation station as shown in Figure 2). Each unique system corresponds to a unique structure function ∅ðXÞ. Thus, one also talks about structures instead of systems.

The performance of a coal-fired generating power station is a function of its basic structure (i.e., the layout and design), availability, safety and security, dependability, and other regulatory aspects. Availability here is defined as the ability of an item (under combined aspects of its reliability, maintainability, and maintenance support) to perform its required function at a stated instant of time or over a stated period of time (BS 4778) ([4], pp. 599). Understanding of its structure function will help in the improvement in performance, design, and maintenance planning. A mathematical model using the graph theory and matrix method is developed to evaluate the performance of a coal-based steam power plant in the subsequent sections.

<sup>1</sup> In this context a binary variable (function) is a variable (function) that can take only the two values, 0 or 1.
