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Smartworld.asia Specworld.in Computer Methods in Power Systems 10EE71 CONTENTS Sl. No TOPICS PAGE NO. 1. UNIT - 1 NETWORK TOPOLOGY 04-18 2. UNIT - 2 NETWORK MATRICES 19-50 UNIT - 3 & 4 LOAD FLOW STUDIES 51-84 UNIT - 5 & 6 ECONOMIC OPERATION OF POWER SYSTEM 85-109 UNIT - 7 & 8 TRANSIENT STABILITY STUDIES 110-144 3. 4. 5. JNTU World Smartzworld.com Page 3 3 jntuworldupdates.org

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Smartworld.asia Specworld.in Computer Methods in Power Systems 10EE71 UNIT-1 NETWORK TOPOLOGY 1. INTRODUCTION The solution of a given linear network problem requires the formation of a set of equations describing the response of the network. The mathematical model so derived, must describe the characteristics of the individual network components, as well as the relationship which governs the interconnection of the individual components. In the bus frame of reference the variables are the node voltages and node currents. The independent variables in any reference frame can be either currents or voltages. Correspondingly, the coefficient matrix relating the dependent variables and the independent variables will be either an impedance or admittance matrix. The formulation of the appropriate relationships between the independent and dependent variables is an integral part of a digital computer program for the solution of power system problems. The formulation of the network equations in different frames of reference requires the knowledge of graph theory. Elementary graph theory concepts are presented here, followed by development of network equations in the bus frame of reference. 1.1 ELEMENTARY LINEAR GRAPH THEORY: IMPORTANT TERMS The geometrical interconnection of the various branches of a network is called the topology of the network. The connection of the network topology, shown by replacing all its elements by lines is called a graph. A linear graph consists of a set of objects called nodes and another set called elements such that each element is identified with an ordered pair of nodes. An element is defined as any line segment of the graph irrespective of the characteristics of the components involved. A graph in which a direction is assigned to each element is called an oriented graph or a directed graph. It is to be noted that the directions of currents in various elements are arbitrarily assigned and the network equations are derived, consistent with the assigned directions. Elements are indicated by numbers and the nodes by encircled numbers. The ground node is taken as the reference node. In electric networks the convention is to use associated directions for the voltage drops. This means the voltage drop in a branch is taken to be in the direction of the current through the branch. Hence, we need not mark the voltage polarities in the oriented graph. Connected Graph : This is a graph where at least one path (disregarding orientation) exists between any two nodes of the graph. A representative power system and its oriented graph are as shown in Fig 1, with: e = number of elements = 6 n = number of nodes = 4 b = number of branches = n-1 = 3 l = number of links = e-b = 3 Tree = T(1,2,3) and Co-tree = T(4,5,6) Sub-graph : sG is a sub-graph of G if the following conditions are satisfied: · sG is itself a graph · Every node of sG is also a node of G · Every branch of sG is a branch of G JNTU World Smartzworld.com Page 4 4 jntuworldupdates.org

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Smartworld.asia Specworld.in Computer Methods in Power Systems 10EE71 For eg., sG(1,2,3), sG(1,4,6), sG(2), sG(4,5,6), sG(3,4),.. are all valid sub-graphs of the oriented graph of Fig.1c. Loop : A sub-graph L of a graph G is a loop if · L is a connected sub-graph of G · Precisely two and not more/less than two branches are incident on each node in L In Fig 1c, the set{1,2,4} forms a loop, while the set{1,2,3,4,5} is not a valid, although the set(1,3,4,5) is a valid loop. The KVL (Kirchhoff‟s Voltage Law) for the loop is stated as follows: In any lumped network, the algebraic sum of the branch voltages around any of the loops is zero. Fig 1a. Single line diagram of a power system Fig 1b. Reactance diagram JNTU World Smartzworld.com Page 5 5 jntuworldupdates.org

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Smartworld.asia Specworld.in Computer Methods in Power Systems 10EE71 Fig 1c. Oriented Graph Cutset : It is a set of branches of a connected graph G which satisfies the following conditions : · The removal of all branches of the cutset causes the remaining graph to have two separate unconnected sub-graphs. · The removal of all but one of the branches of the set, leaves the remaining graph connected. Referring to Fig 1c, the set {3,5,6} constitutes a cutset since removal of them isolates node 3 from rest of the network, thus dividing the graph into two unconnected subgraphs. However, the set(2,4,6) is not a valid cutset! The KCL (Kirchhoff‟s Current Law) for the cutset is stated as follows: In any lumped network, the algebraic sum of all the branch currents traversing through the given cutset branches is zero. Tree: It is a connected sub-graph containing all the nodes of the graph G, but without any closed paths (loops). There is one and only one path between every pair of nodes in a tree. The elements of the tree are called twigs or branches. In a graph with n nodes, The number of branches: b = n-1 (1) For the graph of Fig 1c, some of the possible trees could be T(1,2,3), T(1,4,6), T(2,4,5), T(2,5,6), etc. Co-Tree : The set of branches of the original graph G, not included in the tree is called the co-tree. The co-tree could be connected or non-connected, closed or open. The branches of the co-tree are called links. By convention, the tree elements are shown as solid lines while the co-tree elements are shown by dotted lines as shown in Fig.1c for tree T(1,2,3). With e as the total number of elements, The number of links: l = e – b = e – n + 1 (2) For the graph of Fig 1c, the co-tree graphs corresponding to the various tree graphs are as shown in the table below: JNTU World Smartzworld.com Page 6 6 jntuworldupdates.org

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