Design of Reliable Network Topology for the North Refinery of Baiiji with Minimum Cost

Musaria K. Mahmood1, and Fawzi M. Al-Naima2

1Department of Electrical Engineering, College of Engineering, Tikrit University, Tikrit, Iraq 2College of Engineering, Al-Nahrain University, Baghdad, Iraq

E-mail: Musariaoja@yahoo.com, fawzi.alnaima@ieee.org

Abstract.

Oil refinery industry in Iraq is in evolution toward the modernization of many material and production procedures. This work is about the design of reliable communication network (Intranet) for oil refinery industry taking as case study the North Oil Refineries of Baiji (NORB) in Iraq. The design of the communication network has a major goal the increase of the network reliability using a link addition procedure to find the optimal solution with minimum cost. This paper deals with performance evaluation of industrial plant Intranet used to interconnect several Distributed Control Systems (DCS) and the evaluation of real-time parameters describing the behavior of the network. Intranet can also be used for other applications than the control systems such as security monitoring system, communication network between different locations and all additional future applications.

1. INTRODUCTION

Process control is a discipline that deals with controlling the output of a specific industrial process. Complex chemical processes are controlled by complete, complex control systems such as Supervisory Control And Data Acquisition system (SCADA), Distributed Control System (DCS), and Programmable Logic Controller (PLC).

The increased use of computer network in control systems and the use of Internet as communication backbone have brought benefits in communication and for the process control in general. The ability to share information, making decision concerning production and distribution has been greatly improved [1]. New information technologies have recently been introduced for control system based on open standards like Visual Basic, C#, Java, Open Database Connectivity (ODBC), Ethernet communication and finally the use of internet based control system. Internet or web based control system uses some internet protocols in the

application layer to monitor and control plants at a distance. Older control systems were based on closed protocols implemented by vendors of SCADA, DCS, and PLC systems. The Internet, however, has its limitations when compared to a traditional control system in terms of functionality, performance, security and reliability. Nevertheless, recent emerging web technologies are promising to overcome many of these limitations, and are helping the Internet to evolve into a highly graphical, interactive and collaborative environment [2,3].

The use of Internet web browser to monitor and control industrial plans began in electric power industry. Since the year 2000, many works in the field of web based supervisory and control for power systems were published especially for SCADA systems.

Qiu and Gooi in 2000 [4], and Qiu et al. in 2002 [5], proposed the implementation of a Web-Based SCADA display systems for access via Internet based on the client/server architecture.

Stojkovic and Vukasovic in 2006 [6] proposed to upgrade the old SCADA system of the power system in Montenegro. The new system is based on the use of web services to increase the possibilities of the old Energy Management System (EMS). Ole for Process Control (OPC) server is used with open source Hypertext Preprocessor (PHP), Visual Basic for the web interface server side programming, and MySQL as data management system.

By the nature of SCADA system which is a control system that connect sites located in large distance, it can be implicated in the field of oil pipelines as proposed by Trung in 1995 [7]. Siddiq in 2006 [8] studied a SCADA system for the Iraqi-Turkish pipeline based on the use of Internet as communication backbone.

Web based DCS system is very little investigated except in certain newly published works. This is due to the fact that with DCS we have to deal with big data which are very difficult to manage. Also the nature of DCS systems which are used to control plants in a located region with all data available in-site. Xiong et al. in 2003 [9] was examined the compatibility of high speed communication between DCS and existing RTU in SCADA system. This work can be considered as an attempt to integrate DCS systems to modern SCADA system which will give big possibility to integrate all control types into one innovative control system integrating SCADA, PLC, DCS systems together.

Craig and Befus in 2006 [10] investigated the integration of DCS into existing SCADA of the power corporation in Canada. The design and implementation of a SCADA system for a central heating and power plant using Siemens equipment was investigated by Iacob et al. in 2009 [11].

A distributed real-time production supervisory designing conception based on OPC and Web technique was presented by Jing and Meng in 2009 [12]. The OPC technique and its application in the DCS of cement enterprise are introduced, the seamless connection between the application and plant, and the Web distribution of production data are realized.

Endi et al. in 2010 [13] presented a new technique based on the use of three tier client server model for process automation and data monitoring. [13] Presents the State-of-the-Art and recent trends of SCADA system Architecture. The integration of PLC, DCS into networked SCADA system is presented.

The present work can be considered as first step toward the design of complete monitoring system in refinery industry based on the use of Internet protocols which takes the advantage of Internet/intranet technologies and the World Wide Web to monitor and control industrial processes.

2. CONTROL SYSTEMS IN REFINERY INDUSTRY

Industrial control system (ICS) is a general term that encompasses several types of control systems used in industrial plants, including SCADA, DCS, and PLC. The basis for different types of ICS is the same; data are collected from field, transferred via communication network to be projected in a control center. Control center can be located near the field as in DCS and PLC or at long distance as in the SCADA system. The ICS monitors data from field, tracks alarm conditions, and executes immediate control to keep in good functionality the industrial operations.

The evolution of ICS was begun with SCADA system. SCADA is a process control system that enables a system operator to monitor and control processes distributed among various separated remote sites [14]. SCADA systems have evolved in parallel with the growth and sophistication of modern computing technology. Three generations of SCADA systems appear [15]:

A- First generation – Monolithic or centralized architecture: It is the first generation of SCADA system presented in Fig. 1. The concept of computing in general centered on “mainframe” systems. Networks were generally non-existent, and each centralized system stood alone.

B- Second Generation – Distributed architecture: SCADA systems took

advantage of developments and improvement in system miniaturization and Local Area Networking (LAN) technology to distribute the processing across

Fig. 1 Centralized SCADA

multiple systems. Multiple stations, each with a specific function, were connected to a LAN and shared information with each other in real-time. Some served as operator interfaces, providing the human-machine interface (HMI) for system operators. Still others served as calculation processors or database servers. The distribution of individual SCADA system functions across multiple systems provided more processing power for the system as a whole than would have been available in a single processor. Fig. 2 depicts typical second generation SCADA architecture [15].

C- Third generation- Networked SCADA architecture: The current generation of SCADA master station architecture is closely related to that of the second generation, with the primary difference being that of open system architecture rather than a vendor controlled, proprietary environment. The major improvement in the third generation is that of opening the system architecture, utilizing open standards and protocols and making it possible to distribute SCADA functionality across a Wide Area Network (WAN) and not just a LAN. Open standards eliminate a number of the limitations of previous generations of SCADA systems. Another improvement of the third

generation SCADA systems comes from the use of WAN protocols such as the Internet Protocol (IP) for communication between the master station and communications equipment [16]. Fig. 3 represents a networked SCADA system. With this generation the use of Intranet/ Internet as communication backbone is made possible by the use of open protocols like TCP/IP.

Fig. 2 Distributed SCADA

Fig. 3 Networked SCADA

Distributed Control Systems (DCS) came into existence as presented in Fig. 4, where control functions are distributed into many different microprocessors located throughout the facility and linked together via a communication or data highway. The operator interface is located in a central location such as a control room. The

operator interface includes color graphics with dynamic process data, face plate displays, trend data, and an alarm summary. DCS systems are computer-based control systems in which the main components are located in different places. These components interact with each other via a LAN [17]. Today the dominating control system in refinery industry is the DCS system.

Fig. 4 Distributed Control System (DCS)

A programmable logic controller (PLC) is a digital computer used for automation of electromechanical processes, such as control of machinery on factory assembly lines [18]. Modern PLCs can be programmed in variety of ways, from ladder logic to more traditional programming languages such as BASIC and C. With the advance in technology, it is possible today to control very complicated operations with PLC.

3. GEOGRAPHICAL LOCATION OF COMMUNICATION CENTERS

NORB is a large industrial complex located on various sites scattered around/ and out of the city of Baiji, north of Baghdad in Salahuddin province. The normal capacity of the company is 300.000 Barrels/day. This complex comprises six industrial blocks in an area of about 10 × 6 km in Baiji region as presented in fig. 5: Salahuddin 1, Salahuddin 2, North, Lube Oil, Plastic making plant, and Carbon black plant.

Also, there are six other external refineries outside Baiji complex which are part of the same direction of principal refineries: Sinniyah, Haditha, Kirkuk, Quyarah, Kasik, and Aljazeera. All these refineries are functioning under the same direction and are technically sported by unified sections. Each refinery unit has many DCS systems controlling several chemical processes. The communication network will

be implemented by fiber optical communication. High-capacity fiber-optic networks make it possible to satisfy network throughput constraints with significantly fewer lines than required with older technology.

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6

7 5

4

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The direction Of refineries

Central server

Salah-Aldine1Refinery

NorthRefinery

Salah-Aldine 2Refinery

OilRefinery

Tanks group2

Tanks group 1

500 m

500 m

3 K

m30

0 m

300

m

2 Km

Fig 5. Geographical location of refineries blocks

4. NETWORK DESIGN APPROACHES

Consider each refinery as communication center. We have 6 industrial blocks, connected to the central control room (which will represent block 7). For each refinery there are many DCS systems, which will be networked via appropriate Ethernet to one local database server. This local server will be the front of communication node. The cost of each links is a direct function of link length because same technology will be used for all links. The cost matrix (CM) gives the cost of each links. The probability matrix (PM) represents the associated probability of "link in good operation" for all possible links. For CM lines and column represent node numbers. For diagonal is composed by "0" because it is the same node (ex link 1-1, 4-4,…). Each cost is in fact the distance between the two nodes connected by this link ( in Km).

𝐶𝐶𝐶𝐶 =

⎣⎢⎢⎢⎢⎢⎡

0.0 2.0 2.3 2.6 5.6 3.1 6.12.0 0.0 0.3 0.6 3.6 1.1 4.12.3 0.3 0.0 0.3 3.3 0.8 3.82.6 0.6 0.3 0.0 3.0 0.5 3.55.6 3.6 3.3 3.0 0.0 3.5 0.53.1 1.1 0.8 0.5 3.5 0.0 4.06.1 4.1 3.8 3.5 0.5 4.0 0.0 ⎦

⎥⎥⎥⎥⎥⎤

(1)

𝑃𝑃𝐶𝐶 = � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑓𝑓𝑝𝑝𝑝𝑝 𝑠𝑠𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑙𝑙𝑙𝑙𝑠𝑠 = 0.9� (2)

Our job is to develop design procedures that yield a good network- that is, at low cost with high reliability. The problem is actually called the topological design problem and is an abstraction of the complete design problem. We assume that network design is composed of four phases:

(a) Backbone design (BD),

(b) Local access design (LAD),

(c) Local area network within the building (LAN),

(d) Enhancement Phase (EP).

LAD and LAN using in petroleum industry are conform to critical control system standards to achieve the need of high redundant network for chemical process. Here we focus our attention on backbone design and the enhancement stages. Phase A1 is the BD stage, whereas phase A2 is augmentation with additional arcs to improve performance under constraint(s) representing EP stage. Each new arc in general increases the reliability, cost, and throughput and may decrease data delay.

4.1 Design of a backbone network using Prim’s algorithms

A connected graph with the smallest number of arcs is a spanning tree. In general, a complete network (all edges possible) with n nodes will have nn−2 spanning trees, each with 𝑠𝑠𝑠𝑠=(𝑙𝑙 − 1) edges (arcs). For example, for the seven-node network of Fig.5 there are 77−2 = 75 = 16807 possible spanning trees with (7-1)= 6 arcs for each tree. Prim's algorithm is used to find the best spanning tree (communication backbone). A simple statement of Prim’s algorithm is to select an initial edge of minimum weight, by first ordering all edges by weight. Subsequent edges are selected as minimum weight edges from the set of edges that is connected to a node of the tree but does not form a circuit (loop). The process is

continued until (n−1) edges have been selected. We wish to design a network that connects seven locations represented by the graph nodes of Fig. 5. We begin our design by finding the lowest-cost spanning tree using Prim’s algorithm. To start, we order the edge costs as shown in the first column of Table. 1, In increasing order (from minimum to maximum). The algorithm proceeds as follows until 6 edges forming a tree are selected from the 21 possible edges to form the lowest-cost spanning tree. Select 2-3, select 3-4, select 4-6, select 5-7, we cannot select 2-4, we cannot select 3-6, we cannot select 2-6, select 1-2, we cannot select 1-3, we cannot select 1-4, select 4-5. The minimum cost spanning tree is represented in Fig.6 composed by (n-1=6) edges and connect all node.

Table 1 Prim’s Algorithm for Minimum Cost

Link (edge) Cost Link reliability

Selected step nb.

Deleted Step nb.

1 2-3 0.3 0.9 1 - 2 3-4 0.3 = 2 - 3 4-6 0.5 = 3 - 4 5-7 0.5 = 4 - 5 2-4 0.6 = - 5 6 3-6 0.8 = - 6 7 2-6 1.1 = - 7 8 1-2 2 = 8 - 9 1-3 2.3 = - 9 10 1-4 2.6 = - 10 11 4-5 3 = 11 - 12 1-6 3.1 = 13 3-5 3.3 = 14 4-7 3.5 =

The backbone is found to be the same network given in fig. 5.

4.2 The Enhancement Phase

The all-terminal reliability of a spanning tree is easy to compute since removal of any one edge disconnects at least one terminal pair. Thus, each edge is one of the (n−1) cut sets of the spanning tree, and all the (n−1) edges are the single tie set. All edges of the spanning tree must work for all the terminals to be connected, and the all-terminal reliability of a spanning tree with n nodes and (n−1) independent branches with probabilities 𝑝𝑝𝑖𝑖 is the probability that all branches are good.

Rall = ∏ 𝑝𝑝𝑖𝑖n−1i=1 (3)

If all the branch reliabilities are independent and identical, 𝑝𝑝𝑖𝑖 = 𝑝𝑝 then

𝑅𝑅𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑝𝑝𝑛𝑛−1 (4)

The associated reliability of the spanning tree of fig. 5 of the case study is:

𝑅𝑅𝑠𝑠𝑠𝑠 = 𝑅𝑅𝐴𝐴1 = 𝑝𝑝6 = 0.96 = 0.531441 (5)

Thus, for identical branches, all the spanning trees have the same reliability (equal to 0.9 for example).

Now the first-stage design of phase A1 is finish (design of backbone using Prim's algorithm), and the improvement stage A2 will begin. We will analysis the addition of one edge to the spanning tree in various location and finding the best solution for the final design.

In general for network with (N) nodes and (e) arcs or edge the possible number of links for complete graph will be:

𝑠𝑠𝑐𝑐 = �𝑁𝑁2� = 𝑁𝑁(𝑁𝑁−1)2

(6)

The number of edge in spanning tree is

𝑠𝑠𝑠𝑠 = (𝑁𝑁 − 1) (7)

Then after the design of A1 stage there are number of remaining edge equal to:

𝑠𝑠𝑟𝑟 = 𝑠𝑠𝑐𝑐 − 𝑠𝑠𝑠𝑠 = 𝑁𝑁(𝑁𝑁−1)2

− (𝑁𝑁 − 1) = (𝑁𝑁−1)(𝑁𝑁−2)2

(8)

In our case study: 𝑠𝑠𝑐𝑐 = 21, 𝑠𝑠𝑠𝑠 = 6, 𝑠𝑠𝑟𝑟 = 15. we have 7 nodes, 21 possible edges in the complete graph, 6 edges in the spanning tree, and 15 possible additional arcs that can be assigned during the enhancement phase. Three attractive choices for enhancement are (T1, T2, T3) presented in fig. 6, gives three possible improved topologies. For all these topologies the reliability will be better than that found for spanning tree but one will be the best.

Using the decomposition (edge-factoring) theorem, we can write the reliability of the enhanced network 𝑅𝑅𝐴𝐴2 as:

𝑅𝑅𝐴𝐴2 = 𝑃𝑃(𝑋𝑋)𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋) + 𝑃𝑃(𝑋𝑋′)𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋 ′) (9)

The term 𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋 ′) is the reliability of the network with added edge X failed (open-circuited) = 𝑅𝑅𝐴𝐴1 , that is, what it was before we added the enhancement edge.

1

7

6

5

4

3

21

7

6

5

4

3

2

(a) Topology T1 (b) Topology T2

1

7

6

5

4

3

2

(c) Topology T3

Fig. 6 Three improved topologies.

If P(X) is denoted by 𝑝𝑝𝑥𝑥, then:

𝑃𝑃(𝑋𝑋 ′) = 1 − 𝑝𝑝𝑥𝑥. (10)

Lastly, 𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋) is the reliability of the network with X good, that is, with both nodes for edge X merged (one could say that edge X was “shorted,” which simplifies the computation). Eq. (9) becomes:

𝑅𝑅𝐴𝐴2 = 𝑝𝑝𝑥𝑥 𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋 𝑠𝑠ℎ𝑝𝑝𝑝𝑝𝑝𝑝𝑠𝑠𝑜𝑜) + (1− 𝑝𝑝𝑥𝑥)𝑅𝑅𝐴𝐴1 (11)

Evaluation of P (Network\X shorted) for T1, T2, T3 gives the Fig. 7for transformed graph.

2

6 5

6

1,43

2

7

1,5

6 4

3

(a) T1 topology with 1-5 edge shorted (b) T2 topology with 1-4 edge shorted

5

1

7

4

3

2,6

(c) T3 topology 2-6 edge shorted

Fig. 7 Evaluation of the X shorted term.

Calculation of T1 reliability

𝑃𝑃2−3−4 = 𝑝𝑝2 (𝑝𝑝𝑁𝑁𝑝𝑝 𝑠𝑠𝑜𝑜𝑒𝑒𝑠𝑠𝑠𝑠 𝑝𝑝𝑙𝑙 𝑠𝑠𝑠𝑠𝑝𝑝𝑝𝑝𝑠𝑠)

To find the associated reliability of triangle 2-4-1(5) we have to use graph reduction technique:

For simplification consider link 1 = link 2-4, 2 = link 4-1(5), 3 = link 2-1(5).

We begin by using event-space methods to calculate the all-terminal reliability. The events are E1 = 123, E2 = 1′23, E3 = 12′3, E4 = 123′, E5 = 1′2′3, E6 = 1′23′, E7 = 12′3′, and E8 = 1′2′3′. By inspection, we see that the good events (which connect node 2 to 4 and 2 to 1(5)) are, namely, E1, E2, E3, and E4. Note that any event with two or more edge failures isolates the vertex connected to these two edges and is a cut set.

𝑃𝑃2−4−1(5) = 𝑃𝑃(𝐸𝐸1 + 𝐸𝐸2 + 𝐸𝐸3 + 𝐸𝐸4) (12)

𝑃𝑃2−4−1(5) = 𝑃𝑃(123) + 𝑃𝑃(1′23) + 𝑃𝑃(12′3) + 𝑃𝑃(123′)

𝑃𝑃2−4−1(5) = 𝑝𝑝4 + 2(1− 𝑝𝑝)𝑝𝑝3 + (1 − 𝑝𝑝2)𝑝𝑝2

Combining (12) with other links in Fig.7-a gives the result:

𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋 𝑠𝑠ℎ𝑝𝑝𝑝𝑝𝑝𝑝𝑠𝑠𝑜𝑜) = [𝑝𝑝4 + 2(1 − 𝑝𝑝)𝑝𝑝3 + (1 − 𝑝𝑝2)𝑝𝑝2] × 𝑝𝑝 × 𝑝𝑝

If p = 0.9 then

P(Network\X shorted )= 0.774198. (13)

Equation (3), (7), (11) give the result:

𝑅𝑅𝑇𝑇1 = 0.7499223 (14)

𝑝𝑝𝑝𝑝𝑝𝑝𝑠𝑠𝑝𝑝 𝑐𝑐𝑝𝑝𝑠𝑠𝑝𝑝 𝑝𝑝𝑓𝑓 𝑇𝑇1 = 10.7 𝐾𝐾𝑠𝑠 (15)

Calculation of T2 reliability

To find P (Network\X shorted), of T2 see Fig. 4-b, the same principal is used to compute 𝑅𝑅2−3−1(4) and then to find the improved reliability and total cost:

𝑅𝑅𝑇𝑇2 = 0.6908733

𝑇𝑇𝑝𝑝𝑝𝑝𝑠𝑠𝑝𝑝 𝑐𝑐𝑝𝑝𝑠𝑠𝑝𝑝 𝑝𝑝𝑓𝑓 𝑇𝑇2 = 8.6 𝐾𝐾𝑠𝑠

Calculation of T3 reliability

As presented in Fig. 4-c the graph will be composed of series-parallel links.

Node 4 is connected to 3 and then to 2(6) in series, and exist an edge in parallel connected directly 4 to 2(6).

𝑃𝑃4−3−2(6) = 𝑝𝑝 × 𝑝𝑝 = 𝑝𝑝2 two edges in series

𝑃𝑃4−3−2(6) = 1 − [(1 − 𝑝𝑝)(1 − 𝑝𝑝2)] previous result in parallel with link 4-2 (6)

This will gives the following results:

𝑅𝑅𝑇𝑇3 = 0.715149

𝑇𝑇𝑝𝑝𝑝𝑝𝑠𝑠𝑝𝑝 𝑐𝑐𝑝𝑝𝑠𝑠𝑝𝑝 = 7.4 𝐾𝐾𝑠𝑠

If we summarize all result for the three improved topology in table 2 we have: T1 is better in reliability but with higher cost then the two other topologies. T2 gives the

minimum reliability. T3 seem to be good compromise between higher reliability and low cost. In the three cases we have an improvement in reliability.

Table 2 reliability – cost table

Net. Reliability 𝑅𝑅𝑎𝑎𝑎𝑎𝑎𝑎 Cost (Km) Spanning tree A1 0.531441 6.6 Improved T1 0.749922 10.7 Improved T2 0.690873 8.6 Improved T3 0.715149 7.4

Any of topologies given in Fig. 3 (a, b, c) still to be not reliable because violation of principal of network reliability exist that "each network node must have at least 2 links connected to it to have an alternate route in case of failure of link connected to it". This we must add another edge to T1 for example link between node 7 and node 6. Now we have a network T11 conform to requirement for good network topology Fig. 5 (a). To compute the reliability of this topology (T11), T1 is the A1 stage for this new topology "before improvement". Fig. 8 -b represent T11 with new edge shorted.

From Fig. 8 (c) we have another representation of the graph with node 6 shorted to node7. By the use of decomposition (edge factoring) method we will find the reliability of this graph expanded about edge 5 (link 4-5).

𝑅𝑅6(7),1 = 𝑅𝑅61 is the reliability for node 6 (7) to 1.

𝑅𝑅61 = [𝑃𝑃(5) × 𝑃𝑃(𝐺𝐺1) + 𝑃𝑃(5′) × 𝑃𝑃(𝐺𝐺2)] (16)

𝐺𝐺1 is the graph result when edge 5 is good (P(5)=1), then node 4 and 5 will be the same (shorted).

𝐺𝐺2 is the graph result when edge 5 is bad (P(5)=0), then no link between node 4 and node 5 (opened).

𝑃𝑃(5′) = 1 − 𝑃𝑃(5)

Also node 1, 2, 3, 4 are in series gives one edge (called 8) connecting node 4 to node 1 Fig. 5-(d and e).

𝑃𝑃(𝐺𝐺1) = 𝑃𝑃[(1 + 2)] × 𝑃𝑃[(8 + 3)] (17)

2

3

2

1

6

7 5

4

3

1

6,7

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(a) T11 topology (b) Nodes 6 and 7 shorted

4

1

2

6,7

5

3

Edge1

Edge6

Edge4

Edge3

Edge5

Edge2

Edge7

(c) Another representation of node 6 and 7 shorted

4

14,56,71

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6,7

(d) Edge 5 opened (G2) (e) Edge 5 shorted (G1)

Fig. 8 new improved topology

𝑃𝑃(𝐺𝐺2) = 𝑃𝑃[(1.8) + (2.3)] (18)

From the principal of probability theory we have:

𝑃𝑃(𝑠𝑠 + 𝑝𝑝) = 𝑃𝑃(𝑠𝑠) + 𝑃𝑃(𝑝𝑝) − 𝑃𝑃(𝑠𝑠) × 𝑃𝑃(𝑝𝑝) = 1 − [𝑃𝑃(𝑠𝑠′).𝑃𝑃(𝑝𝑝′) (19)

𝑃𝑃(𝑠𝑠. 𝑝𝑝) = 𝑃𝑃(𝑠𝑠) × 𝑃𝑃(𝑝𝑝\𝑠𝑠) (20)

In the case of independent event (18) will be:

𝑃𝑃(𝑠𝑠. 𝑝𝑝) = 𝑃𝑃(𝑠𝑠) × 𝑃𝑃(𝑝𝑝)

𝑃𝑃(𝐺𝐺1) = [𝑃𝑃(1) + 𝑃𝑃(2) − 𝑃𝑃(1) × 𝑃𝑃(2)] × [𝑃𝑃(8) + 𝑃𝑃(3)− 𝑃𝑃(8) × 𝑃𝑃(3) (21)

Two edges in series (for example 4, 6) give overall probability equal to:

𝑃𝑃4 𝑠𝑠𝑠𝑠𝑟𝑟𝑖𝑖𝑠𝑠𝑠𝑠 𝑤𝑤𝑖𝑖𝑤𝑤ℎ 6 = 𝑃𝑃(4)𝑃𝑃(6)1−𝑃𝑃(4′)𝑃𝑃(6′)

(22)

With P(4')=1-P(4).

To calculate 4 in series with 6 in series with 7 we proceed by step

𝑃𝑃46=0.9×0.9

1−0.01=0.81818

Now this resultant in series with edge 7

𝑃𝑃(8) = 𝑃𝑃4−1 =0.9 × 0.818181 − 0.018182 =

0.7363620.981818 = 0.749998

From (19), we have:

𝑃𝑃(𝐺𝐺1) = [0.9 + 0.9 − 0.9 × 0.9] × [0.749998 + 0.9 − 0.749998 × 0.9]

𝑃𝑃(𝐺𝐺1) = [0.99] × [0.975] = 0.96525

From equation (16) we have:

𝑃𝑃(𝐺𝐺2) = [𝑃𝑃(1.8)] + [𝑃𝑃(2.3)] − 𝑃𝑃(1.8)𝑃𝑃(2.3)

𝑃𝑃(𝐺𝐺2) = [𝑃𝑃(1)𝑃𝑃(8)] + [𝑃𝑃(2)𝑃𝑃(3)] − 𝑃𝑃(1)𝑃𝑃(8)𝑃𝑃(2)𝑃𝑃(3)

𝑃𝑃(𝐺𝐺2) = 0.674998 + 0.81 − 0.546749 = 0.938349

From the above and equation (4) we have

𝑅𝑅 = 𝑃𝑃(𝑙𝑙𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙 𝑇𝑇11\𝑠𝑠𝑜𝑜𝑒𝑒𝑠𝑠 6 − 7 𝑒𝑒𝑝𝑝𝑝𝑝𝑜𝑜 =

0.9 × 0.96525 + 0.1 × 0.938349 = 0.9625599 (23)

To find the total reliability of the improved network T11 we have to calculate equation (7):

𝑅𝑅𝐴𝐴2 = 𝑃𝑃(𝑋𝑋)𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋) + 𝑃𝑃(𝑋𝑋′)𝑃𝑃(𝑁𝑁𝑠𝑠𝑝𝑝𝑁𝑁𝑝𝑝𝑝𝑝𝑙𝑙\𝑋𝑋 ′)

𝑅𝑅𝐴𝐴2 𝑜𝑜𝑜𝑜 𝑇𝑇11 = 0.9 × 0.9625599 + 0.1 × 0.749922 = 0.94129611

Then we have the reliability of T11 of Fig. 5(a) = 0.94129611. With cost = 13.7 Km.

5. CONCLUSION:

The above calculation has as evident result the following points:

(1) The addition of any link (edge) will improve the reliability. (2) The choice must be made as trade off of reliability and cost. (3) The final network is conforming to reliability rule by having at least two

edge connected to each node. (4) The assumption of edge with same reliability was made. In case where each

edge has different reliability, the same calculation procedure must be followed.

(5) The choice of best topology will be subject to other consideration like the investment made by the industrial plant.

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