2 NETWORK FORMULATION

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Erika Wilkinson
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1 NTWRK FRMUATN NTRDUCTRY RMARKS For performing any power system studies on the digital computer, the first step is to construct a suitable mathematical model of the power system network The mathematical model of the power system network for the steady state analysis is the network equations which describes, both, the characteristics of the individual elements of the power system network as well as their interconnections Usually data available from any power company or electricity board are in the form of primitive network matrix The primitive network matrix, while giving the complete information regarding the characteristics of the individual elements of the power system network, does not give any information about their interconnections And hence, the primitive network (ie, corresponding primitive network matrix) must be transformed either by singular or by nonsingular transformations to the network (matrix) equations as the network equations which describes both the characteristics as well as the interconnections of the different elements, are the suitable mathematical model of the power system network The primitive network is a set of uncoupled elements of the network These network equations can be formed either in the bus frame of reference, or loop frame of reference (sometimes also in the branch frame of reference) using either impedance or admittance parameters We have discussed in this chapter, the different methods of formulating these network equations and the corresponding network matrices We have shown that in certain situations, some of these network equations and thus the corresponding network matrices can be developed directly by inspections The network matrices can also be developed by performing open circuit or short circuit tests or by graph theoretic approach Since graph theory plays an important role both in the development of network matrices as well as in the analysis of power systems, we have briefly described graph theory in this chapter NTWRK QUATNS The mathematical model of the power system network is the network equations which can be established in Bus (ie, model), loop (ie, mesh) or branch frame of reference n the bus frame of reference, the performance of the interconnected network is described by n linear independent equations where n is the number of nodes and n is the number of buses n counting the number of buses, the reference node (which is normally the ground) is always neglected And hence, for n-node systems, the number of buses are equal to n Therefore, the number of linear independent equations are equal to the number of 4

2 NTWRK FRMUATN 5 buses Thus the performance equation in the bus frame of reference in the admittance form is given by BUS = Y BUS BUS () where BUS is the vector of injected bus currents (ie, external current sources), it is positive when flowing towards the bus and it is negative if flowing away from the bus BUS is the vector of bus voltages (ie, nodal voltages) measured from the reference node (which is normally ground) and Y BUS is the bus admittance matrix of the power system network The equation () can be written as follows : È Î n = ÈY Y Y Y Y Y Î Y Y Y, n-, n- ( n-), ( n-), ( n-), ( n-) È Î n () Here the nth bus is the reference bus and hence there are n linear independent equations to be solved Now Y BUS is a non-singular square matrix of the order (n ) (n ) and hence it has a unique inverse ie, Y BUS = Z BUS Thus the performance equations in the bus frame of reference in the impedance form will be BUS = [Z ] () BUS BUS Z BUS is again of the order (n )(n ) for a system having n nodes (ie, n buses) Here, in the formulation of Y BUS or Z BUS matrices, ground is included and is taken as reference node (in this case we have taken nth node as the ground ie, reference node) And therefore, the elements of the Y BUS or Z BUS matrix will include the shunt connections such as transformer magnetising circuit, static loads, and line charging etc The bus voltages ie, BUS are also measured with respect to ground which is the reference node However, if ground is not included in the development of Y BUS or Z BUS matrices a bus known as slack or swing bus is taken as reference bus and all the variables are measured wrt this reference bus Since the shunt connections between buses and the ground are not included in the elements of Z BUS or Y BUS matrices, their effect is taken into account by treating them as external current sources which will be discussed later Assuming the slack bus voltage as S, the performance equations in this case will be BUS = [Y ]( _ ) (4) BUS BUS S or _ BUS S = [Z BUS] (5) BUS For n nodes systems (including ground as one of the node), there will be n linear independent equations to be solved The performance equations in the loop frame of reference, will be given by

3 6 ADVANCD PWR SYSTM ANAYSS AND DYNAMCS = È Î (6) Here we obtain l number of linear independent equations l is the number of loops ie, meshes and loop is the vector of the basic loop voltage ie, resultant voltage sources in the loop loop is the vector of unknown loop ie, mesh currents Z loop is the loop impedance matrix of the order l l However, Z loop is a non-singular square matrix which has an unique inverse, ie, - = and hence network equation in the loop frame of reference in the admittance form is = È Î (7) The performance equation in the branch frame of reference is expressed as shown below: BR = Z BR BR (8) or BR = Y BR BR (9) Here Z BR (where Z BR = Y BR ) is the branch impedance matrix of the branches of the tree of the connected power system network and is of the order b b where b is the number of branches BR is the vector of branch voltages and BR is the vector of current through the branches There will be n ( = b) linear independent equations to be solved Now we discuss the methods to develop the impedance and admittance matrices in the different reference frames Development of Bus Admittance and mpedance Matrices Bus admittance matrix is developed by applying Kirchhoff s current law at every bus n this way, systematic nodal equations are written for every node except for the reference node which is normally the ground node Assuming ground as the reference node, let us write the nodal equations using Kirchhoff s current law for the following power system network (Fig ) Here and are the external current sources at the bus and n the nodal formulation, all the voltage sources with the series impedances, which is usually the case in the power system network, are replaced by the equivalent current sources with shunt impedance by the following method : Fig (a) Power system network

4 NTWRK FRMUATN 7 Fig (b) Corresponding graph The two sources (refer to Fig ) are equivalent if (i) g = S Z S and (ii) Z g = Z S (0) (a) deal voltage source Fig (b) deal current source Now, going back to Fig () and applying Kirchhoff s current law at the buses, and, we will obtain the following nodal equations, = V Y + (V V )Y =(V V ) Y + V Y + (V V ) Y () = V Y + (V V ) Y Here V, V and V are the voltages of buses, and respectively wrt the reference bus (which is ground in this case) and these voltages are known as the bus voltages Moreover, for this network Y ij = Y ji ie, Y = Y and so on, as the network elements are linear and bilateral Arranging equations, and and combining the coefficients of bus voltages (ie, V, V and V ), we obtain the following equations: = (Y + Y ) V Y V = Y V + [Y + Y + Y ] V Y V () = Y V + [Y + Y ]V et Y + Y = Y Y + Y + Y = Y and Y + Y = Y () Substituting relation () in equation (), we obtain = Y V Y V

5 8 ADVANCD PWR SYSTM ANAYSS AND DYNAMCS = Y V + Y V Y V = Y V + Y V (4) These equations can be written in the following matrix form: P = P Q N M Y Y Y Y Y Y Y Q P V V V QP (5) From the above it is clear that any diagonal element (say Y = Y + Y ) is the sum of the admittances of the elements terminating at that node and the off diagonal element is always negative of the admittance of the elements between the adjacent nodes (say between node and, off diagonal element is Y ) The above equations can be written in the more general form for any power system network with n buses (ie, n + nodes) Thus, we have, é ù éy Y Y Y ù V = ë n û and in the symbolic form, we have n Y Y Y Y ëy Y Y Y n n n nn n û N M V V n Q P (6) BUS = Y V BUS BUS (7) Now for n + nodes system, we obtain n linear independent nodal equations ie, the number of linear independent equations is one less than the number of nodes Thus, the Y BUS matrix is : Y BUS = éy Y Y Y n ù Y Y Y Yn ëyn Yn Y n Ynn û (8) As we have seen, its diagonal element Y ij is the sum of admittances of the elements terminating at the node i These elements are known as short circuit driving point admittances and they correspond to self admittances Similarly off diagonal elements Y ij are the negative of the admittances of elements connected between nodes i and j These are known as the short circuit transfer admittances and they correspond to the mutual admittance Thus Y BUS matrix can be found by inspection provided mutual coupling between the elements of the given power system network (known as primitive network) is neglected The elements of Y BUS (ie, short circuit parameters) can also be obtained as follows: The diagonal elements Y ii (short circuit driving point admittances) are found by applying a unit voltage source between the i-th bus and the reference node which is ground normally and measuring or calculating its (ie, ith bus) current while keeping all other buses short circuited Similarly, the off diagonal elements Y ij (short circuit transfer admittance) are found

6 NTWRK FRMUATN 9 by applying a unit voltage source between the ith bus and the ground and measuring or calculating the current of jth bus while keeping all other buses short circuited (ie, grounded) We will show later using Graph theoretic formulation that the elements of Y BUS matrix can also be found by graph theoretic approach ie, Y BUS = A[Y] A T where A is the reduced (or bus) incidence matrix and Y is the primitive admittance matrix We shall discuss this later in detail Thus we have shown that the Y BUS matrix can be obtained directly by inspection from a given primitive network provided mutual coupling between the elements is neglected which is usually the case in most of the power system studies such as load flow studies etc n addition, the elements of Y BUS matrix can also be obtained by short circuit test and also through a graph theoretic approach Though the Y BUS matrix can be developed directly by inspection (ie, there is a direct correspondence between the element of Y BUS matrix and the corresponding power system network), this is not true in the case of Z BUS matrix There is no direct relation between the elements of Z BUS matrix and the corresponding power system network We can find the elements of Z BUS matrix by the following method: by finding inverse of Y BUS matrix and thus we obtain Z BUS = Y BUS by open circuit test The performance equation of the power system network in the bus frame of reference using Z BUS is: where BUS = Z BUS BUS (9) Z BUS = éz Zn ù Z Z Zn ëzn Z n Znnû The diagonal elements Z ii are known as the open circuit driving point impedance and they are determined by applying a unit current source between the ith bus and the ground (ie, reference node) and measuring the voltage of the same bus ie, ith bus while keeping all other buses open circuited The off-diagonal elements Z ij are known as open circuit transfer impedances and they are found by applying a unit current source between the ith bus and the ground and measuring or calculating the voltage of jth bus while keeping other buses open circuited Z BUS matrix can also be found by building algorithm Several algorithms are available in the literature but all of them have nearly the common approach n this case from the given primitive network (matrix), the elements of Z BUS matrix are developed starting from the reference node known as the slack bus, in steps, by adding one element (either a branch or a link) at a time and in this way Z BUS matrix is developed by building algorithm We shall discuss the building algorithm for Z BUS matrix later

7 0 ADVANCD PWR SYSTM ANAYSS AND DYNAMCS Development of oop mpedance and Admittance Matrices n this case, only the elements of loop impedance matrix ie, Z loop can be found by inspection and hence Z loop matrix has direct correspondence with the given primitive network just like the Y BUS matrix To illustrate the method to develop Z loop matrix, let us take the following example: The elements of Z loop matrix are developed by applying Kirchhoff s voltage laws and writing loop ie, mesh equations of the given power system network However, if there is any current source in the network, it must be transformed to the equivalent voltage source with series impedance et us take the example of Fig and write the loop ie, mesh equations using Kirchhoff s voltage laws: Z f Z d Z e Z a Z b Z c + a + b + c Ref Node Fig Corresponding group Tree branches ; links Fig 4 a b = Z + Z + + Z b b g b g b g + + b + g b + g + b + g a b d = Z Z Z c b c e = Zd Z e + Zf (0) By collecting the coefficients of current (ie, loop current, and which are also called mesh currents), we obtain a b = Z + Z + Z Z + Z b b a b dg b d Z b bzb Zc Zeg Z e d + e + d d + e + fi () = c =Z Z Z Z Z

8 NTWRK FRMUATN et, = is the loop voltage of loop ie, resultant voltage source in the loop a b = is the loop voltage of loop ie, resultant voltage source in the loop b and let, c Za + Zb + Zd = Z Zb + Zc + Ze = Z Zd + Ze + Zf = Z By this substitution, the loop equation () becomes: = Z Z + Z b d = _ Z + Z + Z b e = Z + Z + Z () d e et us also define Z ij for the impedance common to loop i and j With this definition we will get: Z b = Z = Z = Z Z d = Z = Z = Z Z e = Z = Z = Z () With the substitution of equation () in the equation (), we get = Z Z + Z = Z + Z + Z and in the matrix form, equation (4) becomes as shown, =Z + Z + Z (4) P Q P = Z Z Z _ Z Z Z Z Z Z QP QP (5) The above equations (ie, equation (5)) can be written in most general form as follows: n QP = Z Z Z Z Z Z n n Z Z Z n n nn and in the symbolic form the equation (6) becomes = È Î Z loop loop loop QP n QP (6)

9 ADVANCD PWR SYSTM ANAYSS AND DYNAMCS The diagonal elements of Z loop matrix ie, Z ii is the self loop impedance and is equal to the sum of impedance of the elements in the loop i Similarly the off diagonal elements Z ij which is known as the mutual impedance, is equal to the impedance of the elements common to loop i and j The elements Z ij is positive if loop i and j agrees and it is negative if loop i and j does not agree This result is clear from the example shown above where Z = Z a + Z b + Z d = sum of the impedances of elements in loop and such is the case with Z and Z And also Z = Z b impedance of the element common to loop and where the loops do no agree and Z = Z d = imp common to loop and where the loops agree Note, in the case of Y BUS matrix, all its off diagonal elements are negative From this it is clear, that the elements of Z loop matrix can also be found directly by inspection from a given power system network (ie, primitive network) and its order is l l where l is the number of links, ie, loops or meshes where and From graph theoretic point of view (which will be shown later in this chapter), Z loop = B[Z]B T The inverse of Z loop matrix is Y loop, ie, Z = primitive impedance matrix B = basic loop incidence matrix Y loop = Z - loop The Y loop can only be found by inverting Z loop matrix and thus there is no direct connection between Y loop matrix and the actual network We shall give later in this chapter, development of these network matrices from the graph theoretic approach Primitive Network The data obtained from electricity boards or the power companies is in the form of primitive network (or primitive network matrix) Primitive network is a set of uncoupled elements which gives information regarding the characteristics of individual elements only The primitive network can be represented in the impedance form as well as in the admittance form The performance equation of any element p q in the impedance form will be (see Fig 5) Here v iq i pq e pq Z pq Fig 5 = voltage across the element p q = current through the element p q = voltage source in series with the element p q = self impedance of the element p q

10 NTWRK FRMUATN and p and q are the bus (ie, nodal) voltages of the buses (ie, nodes) p and q respectively Assuming p at higher potential, we have + e Z i = q p pq pq pq ie, p q + epq = Z pq i pq ie, v + e = Z pq i pq (7) pq pq where v pq = p q voltages across the element p q Similarly in the admittance form, the performance equation will be as in Fig 6 Fig 6 Here j pq is the current source in parallel with the element p q and Y pq is the self admittance of the element p q Now (from Fig 6), i pq + j pq = Y pq v pq (8) where v pq is the voltage drop across the element p q From equation (7), we get, i pq = v pq + e pq Z Z pq = Y pq v pq + Y pq e pq where Y pq = /Z pq Hence i pq Y pq e pq = Y pq v pq (9) Comparing the similar equations (9) and (8), we obtain, pq j pq = Y pq e pq (0) The performance equations of the primitive network can be derived from equations (7) and (8) above and hence for the entire network, variables become column vectors and parameters become matrices Thus the performance equations in the impedance and the admittance form for the complete network will be as follows: v + e = [ Z] i and i + j = [ Y] v () Here the diagonal elements of [Z] and [Y] viz Z pq pq or Y pq pq are the self impedances/ admittances for the element p q and the off diagonal elements Z pq, rs or Y pq, rs are mutual impedances/admittances, between elements p q and r s The primitive admittance matrix [Y] is obtained by finding inverse of [Z] However, if there is no mutual coupling between the elements, the matrices Y or Z will be diagonal

11 4 ADVANCD PWR SYSTM ANAYSS AND DYNAMCS GRAPH THRY n order to describe the geometrical structure of the network, it is sufficient to replace the different power system components (of the corresponding power system network) such as generators, transformers and transmission lines etc by a single line element irrespective of the characteristics of the power system components The geometrical interconnection of these line elements (of the corresponding power system network) is known as a graph (rather linear graph as the graph means always a linear graph) Hence the graph G = (V, ), consists of a set of objects V = (v, v, ) called nodes or vertices and another set of objects = (e, e, ) called elements or edges A subgraph is a subset of the graph A path is a subgraph of the connected elements with not more than two elements incident at a node A node ie, vertex and an edge ie, element are incident if the node is the end point of the edge A loose-end is the vertex with not more than one element incident to it A graph is connected if and only if there exists a path between every pair of nodes A single edge or a single node is a connected graph f every edge of the graph is assigned a direction, the graph is termed as an oriented graph The number of edges incident to a vertex with self loop counted twice is called the degree of the vertex ie, node A tree is a subgraph containing all the nodes of the original graph but no closed path and hence the tree has the following properties: Tree is a subgraph containing all the vertex of the original graph Tree is connected Tree does not contain any closed path The total number of trees in a simply connected graph = det Y with every element assigned to an admittance of mhos Here Y is a bus admittance matrix To illustrate these points, let us take the following example: 5 4 Fig 7 Power system network Fig 8 (a) riented graph of the network of Fig 7

12 NTWRK FRMUATN Fig 8 (b) Tree and cotree of the corresponding graph of Fig 8(a) Notes : Thick lines indicate a tree and hence tree branches are shown as Dotted lines indicate cotree and hence the elements of cotree ie, links are shown as The elements of the tree are known as branches The elements of the original graph, not included in the tree, form a subgraph which may not necessarily be connected and is known as cotree The cotree is a complement of a tree The elements of cotree are called links For example (Fig 8), the total number of elements in the graph = e = 9, the total number of nodes = n = 6, the number of branches of the corresponding tree = b = 5 = n, and the total number of links = l = 4 = e n + = e b, where b is the number of branches = n Basic oops : Since the tree is a connected subgraph containing all the vertices but no closed path, a closed path ie, loop will be created when a link is added to the tree Thus the addition of each link will create one or more closed path ie, loop oop which contains only one link, is independent and is called a basic loop or fundamental circuit The number of basic loops is equal to the number of links ie, meshes l (where l = e n + ) and it has the same orientation as that of the corresponding link as shown in Fig 9 Fig 9 Basic loops of the network shown in Fig 7 Note : By adding the link 7 to the tree, a closed loop A is formed which has the same orientation as that of the corresponding link 7 Similarly by adding links 6, 8 and 9, the corresponding basic loops B, D and C having the same orientation as that of the links are formed As the number of links is four, there are only four basic loops in this case Cut set : Cut-set is the minimal set of the elements in the connected graph which when removed, divides is the graph in only two parts ie, two connected subgraphs Basic cutsets are those which contain only one branch and hence the number of basic cutsets is equal to the number of branches b (b = n ) and it has the same orientation as that of the corresponding branch as shown in Fig 0

Power System Analysis

Power System Analysis BY A. P U R N A C H A N D E R A S S I S T A N T P R O F E S S O R D E P A R T M E N T O F E E E A C E E N G I N E E R I N G C O L L E G E Course Objectives: 1. To understand and develop