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In: Proceedings of the COMADEM '99; UK: Chipping Norton; 1999. pp. 77–85

Wiley & Sons; 2005

USA: ASA-SIAM; 2002

48 Recent Improvements of Power Plants Management and Technology

Bernard Tonderayi Mangara

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68285

#### Abstract

The graph theoretical analysis and a graph's characteristic polynomial are deployed as the basis for a system approach employed to develop a model for estimating the reliability index and evaluating the availability index for a coal-fired generating power station. In this research, the coal-fired generating station system is divided into six subsystems. Elementary to evaluating the reliability (estimate) of the said system is the consideration of all the sub-systems and their interrelations. Approximate reliability attributes of the graph are used to model the approximate reliability of the coal-fired generating station. Sub-system reliability is represented by the nodes in the graph, and the links represent the reliability of interrelations of these sub-systems. Computing a graph's characteristic polynomial using three different methods, namely, the linearly independent cycles, the figure equation and the adjacent matrix, the approximate reliability of the system is determined. Three methods are used, for comparison purposes, as estimating reliability is always an imperfect endeavour. The methodology proposed in this study is illustrated step-by-step with the help of examples.

Keywords: Reliability, coal-fired, steam, method, components

## 1. Introduction

Figure 1 shows the schematic diagram of a coal-fired generating station [1–3]. The energy conversion in a coal-fired generating station is as follows [1–3]. Coal is conveyed to a mill and the mill crushes the coal into fine powder, which is pulverized. Thereafter, the pulverized fuel is blown into the boiler where it mixes with a supply of pre-heated air for combustion. The said combustion of a mixture of pulverized fuel and pre-heated air in the boiler produces steam, at high temperatures and pressures, which is passed through the steam turbine. The boiler drives the steam turbine, which is coupled to the electricity generator. The generator then supplies the national electrical load. In spite of the advances in the design and materials in the last few

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Figure 1. Schematic view of a coal-fired generating station.

years, the efficiency of the most modern coal stations is in the region of 40%. Therefore, the 60% of energy rejected as heat forms the exhaust steam from the low pressure turbine, which is cooled to form condensate by the passage through the condenser of large quantities of sea- or river-water. If the station is located inland or if there is concern over the environmental effects of raising the temperature of the sea- or river-water, cooling towers are then used.

Steam power-stations operate on the Rankine cycle [1]. The Rankine cycle, in a steam powerstation, is modified to include super-heating, feed-water heating, and steam re-heating. To achieve high efficiency, the steam has to be used at maximum possible pressures and temperatures. Furthermore, for economic construction of turbines, the larger the size, the less the capital cost per unit of power output. Consequently, turbo-generator sets of 500MW and more have been used. For steam turbines 100MW and above, efficiency is increased by using an external heater to re-heat the steam, after it has been partially expanded. In addition, the re-heated steam is then returned to the turbine where it is expanded through the final stages of blading.

The study of reliability of complex systems such as steam power plants is of interest for power utility companies. This is so because the power utilities have to minimize operation and maintenance expenses while ensuring the reliability, safety, and security of supply to their national electrical load in order to remain competitive in the global market. In this study, complex systems are defined as large collections of interconnected components whose interactions lead to macroscopic behaviours [2, 3]. For complex systems, it is required to translate system reliability requirements into detailed specifications for all components that constitute the system. This process is often referred to as the reliability apportionment. During reliability apportionment, the reliability analyst has to perceive and develop the relationships between component, sub-system, and system reliabilities. The decisive role in this process is in understanding and quantifying the reliability importance of different parts of the equipment.

Steam power plant reliability encompasses a range of issues related to the design and analysis of these power plant generating networks. Furthermore, the said coal-fired thermal power plant components are prone to random failures. Generally, network models are comparatively simple, and yet quite generic. Varied applied problems of steam power plant environments can be modelled with networks [2, 3]. In the field of steam power plant reliability, the final goal of research is to provide design engineers with procedures to further improve the quality of designs whereupon reliability is a significant factor to take into account.

In this study, a system is defined as a bounded physical entity that achieves in its domain a defined objective through interaction of its components ([4], pp. 604). It follows from this definition of system that the following notation is going to be used for system reliability ([4], pp. 148):

Since the state variables XiðtÞ for i ¼ 1, 2, …:, n are binary, then

$$\Pr[X\_i(t)] = 0 \cdot \Pr(X\_i(t) = 0) + 1 \cdot \Pr(X\_i(t) = 1) = p\_i(t), \quad \text{for } i = 1, 2, \dots, n \tag{1}$$

Similarly the system reliability (at time t) is

years, the efficiency of the most modern coal stations is in the region of 40%. Therefore, the 60% of energy rejected as heat forms the exhaust steam from the low pressure turbine, which is cooled to form condensate by the passage through the condenser of large quantities of sea- or river-water. If the station is located inland or if there is concern over the environmental effects

Stack

Steam Turbine Energy

Boiler feed pump

Water & Steam Condensate

Condenser

Cooling Tower

Exhaust steam

Generator

Steam power-stations operate on the Rankine cycle [1]. The Rankine cycle, in a steam powerstation, is modified to include super-heating, feed-water heating, and steam re-heating. To achieve high efficiency, the steam has to be used at maximum possible pressures and temperatures. Furthermore, for economic construction of turbines, the larger the size, the less the capital cost per unit of power output. Consequently, turbo-generator sets of 500MW and more have been used. For steam turbines 100MW and above, efficiency is increased by using an external heater to re-heat the steam, after it has been partially expanded. In addition, the re-heated steam

The study of reliability of complex systems such as steam power plants is of interest for power utility companies. This is so because the power utilities have to minimize operation and maintenance expenses while ensuring the reliability, safety, and security of supply to their national electrical load in order to remain competitive in the global market. In this study, complex systems are defined as large collections of interconnected components whose interactions lead to macroscopic behaviours [2, 3]. For complex systems, it is required to translate system reliability requirements into detailed specifications for all components that constitute the system. This process is often referred to as the reliability apportionment. During reliability apportionment, the reliability analyst has to perceive and develop the relationships between component, sub-system, and system reliabilities. The decisive role in this process is in understanding and quantifying the reliability importance of different parts of the equipment.

Steam power plant reliability encompasses a range of issues related to the design and analysis of these power plant generating networks. Furthermore, the said coal-fired thermal power

of raising the temperature of the sea- or river-water, cooling towers are then used.

Forced draft fan

Steam

Precipitator (dust collector)

Burner

Pre-heated air

Boiler

Coal

Pulverising mill

Figure 1. Schematic view of a coal-fired generating station.

Pulverised fuel plus air

50 Recent Improvements of Power Plants Management and Technology

is then returned to the turbine where it is expanded through the final stages of blading.

$$p\_s(t) = E\left(\mathcal{Q}\left(X(t)\right)\right) \tag{2}$$

It can be shown that when the components are independent, the system reliability, psðtÞ, will be a function of the pi ðtÞ 0 s only. Hence, psðtÞ may be written as follows

$$p\_s(t) = h\left(p\_1(t), p\_2(t), \dots, p\_n(t)\right) = h\left(p(t)\right) \tag{3}$$

Unless stated otherwise, the letter h will be used to express system reliability in situations where the components are independent.

The size of the state space is one of the major impediments in steam power plant system reliability analysis. For a complex and large-scale power plant system, the number of system states is enormous. It can be noted that a system consisting of n components and each component with binary states (working or failed) has a total of 2<sup>n</sup> states. For example, if one considers a case when <sup>n</sup> is 300, the number of states is 2:<sup>04</sup> � 1090. For the preceding example, if one would analyse all the possible states individually in order to identify the contingencies that help to bring about the system unreliability, this would require much computational effort. Furthermore, it would be impractical for typical steam power plants. Therefore, there is need to choose a methodology which reduces the state space, and a subsequent selection and evaluation of contingencies. The graph theoretical analysis (GTA) method [5, 6] is chosen for this study.

Graph theory is a branch of mathematics which has existed for many years not only as an area of mathematical study but also as a tool for intuition and illustration [7]. Graphs can be used in wiring diagrams to represent the physical elements of an electrical circuit; a street map is also a graph with the streets as links (edges), intersections of streets as nodes (vertices), and street names as labels of the links (edges). In the above-mentioned cases, the graphs resemble the physical objects that they represent. Thus, the application (and sometimes the genesis) of the graph-theoretic ideas is immediate. Computer program flow diagrams and road maps with one way streets are examples of graphs that contain the concept of direction or flow to the links (edges); and these are called directed graphs. The applications of graphs and directed graphs in almost all areas of the physical sciences and mathematics have been known for half a century 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 availability index for a coal-fired generating power station.

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 between the present reliability and the design value.

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 given below [2, 3]:


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.

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