ࡱ> 5x z [ sPWVEyH=q`  bjbjqPqP ֣::D 00004LLLh`<-6#vvvR\R\R\"""""""!1h3V"q `4TN\``"00veQ-R`00bv(D "`"6 Lh  jL,t8d -H-3dy 38  3  R\V]^`_R\R\R\""pR\R\R\-````dLL000000  Part II: Analysis Methods of Electrical Power Systems CONTENTS Chapter 1: Functions of Electrical Energy Systems 1.1. Introduction 1.2. Hierarchy and representation of electrical power systems 1.2.1. Transmission lines and apparatus 1.2.2. Transformers 1.2.3. Electric loads 1.2.4. Generators Chapter 2: Network Representation 2.1. Graphical and topological description of a network 2.1.1. Review of graph theory 2.2. Network global modeling: CIM model 2.3. Matrix representation of networks 2.3.1. Network matrices 2.3.1.1. Incidence matrix 2.3.1.2. Matrix of elementary network 2.3.1.3. Transfer matrix Chapter 3: Formation of Network Matrices 3.1. Formation of the Ybus matrix 3.2. Formation of the Zbus matrix 3.2.1. Adding branches 3.2.1.1. Calculation of Zqi terms 3.2.1.2. Calculation of Zqq terms 3.2.2. Adding cords 3.2.2.1. Calculation of augmented matrix elements 3.2.2.2. Elimination of fictitious node 3.3 Exercises 3.3.1 Exercise No. 12: Construction of Zbus matrix 3.3.2 Exercise No. 13: Construction of network matrices Chapter 4: Load Flow Calculations 4.1. Objectives 4.1.1. Definition of network state 4.1.2. Device current rating 4.1.3. Line losses 4.1.4. Strategy for adjustment and control 4.1.5. Optimizing power transfer capacity 4.2. Model of network elements 4.2.1. Lines and transformers 4.2.2. Generators and loads 4.2.3. Representations of voltage 4.3. Problem formulation 4.3.1. General equations 4.3.2. Simplified models 4.4. Solution methods 4.4.1. Gauss-Seidel method 4.4.2. Newton-Raphson method 4.4.3. Calculation of power flows 4.5. Software tools for network analysis 4.6. Appendix: principle of numerical iterative methods 4.6.1. Gauss-Seidel method 4.6.2. Newton-Raphson method 4.7 Exercises 4.7.1 Exercise No. 14: Load flow calculation 4.7.2 Exercise No. 15: Power flow 4.7.3 Exercise No. 16: Matrices and load flow Chapter 5: Transient Analysis Methods 5.1. Interest in transient analysis 5.2. Transient network analyzer 5.2.1. Principle of operation 5.2.2. Advantages and disadvantages 5.3. Method of traveling waves 5.3.1. Principle 5.3.2. Representation of a line (or cable) 5.3.3. Representation of a resistor 5.3.4. Representation of an inductor 5.3.5. Representation of a capacitor 5.3.6. Representation of a voltage source 5.3.7. Operating principle 5.3.8. Illustration example 5.4. Conclusions 5.5. Exercises 5.5.1. Exercise No. 17: Transient analysis on a line 5.5.2 Exercise No. 18. Matrices and transient analysis 5.5.3 Exercise No. 19. Transient analysis under lightning strike Chapter 6: Fault Current Calculations 6.1. Definition 6.2. Effects of short-circuit conditions 6.3. Common causes of faults 6.4. Importance of short-circuit current calculations 6.5. Types of short-circuits 6.6. Notion of short-circuit power 6.7. Polyphase balanced and unbalanced systems 6.7.1. Balanced three-phase systems 6.7.2. Complex representation 6.7.3. Symmetrical components 6.7.4. Powers in terms of symmetrical components 6.7.5. Symmetrical components and impedance/admittance matrices 6.7.6. Concept of circulating matrices 6.7.7. Case of the synchronous machines 6.7.8. Short-circuit current calculations 6.7.8.1. Single-phase-to-ground fault 6.7.8.2. Two-phase-to-ground fault 6.7.9. Other types of faults 6.8. Generalization of fault calculation in complex networks 6.9. Symmetrical (three-phase) faults 6.10. Symmetrical fault currents: systematic approach 6.11. Short-circuit power 6.12. Unsymmetrical fault current calculations 6.12.1. Generalization of symmetrical components 6.12.1.1. Positive sequence (direct) network 6.12.1.2. Negative sequence (inverse) network 6.12.1.3. Zero sequence (homopolar) network 6.12.2. Neutral and homopolar currents 6.12.3. Impedances of network components 6.12.3.1. Impedance of rotating machines 6.12.3.2. Impedance of lines and transformers 6.12.3.3. Homopolar impedance of lines 6.12.3.4. Homopolar impedance of transformers 6.12.4. Illustration example 6.12.5. Systematic calculation of unsymmetrical fault currents 6.13 Exercises 6.13.1 Exercise No. 20: Fault current in a simple network 6.13.2 Exercise No. 21: Symmetrical faults in a network Chapter 7: Stability Analysis of Power Systems 7.1. Objective 7.2. Introduction 7.3. Categories and classes of stability problems 7.4. The equation of motion 7.5. Simplified model of synchronous machine 7.6. Power-angle considerations at steady-state 7.7. Case of small perturbations 7.8. Transient stability 7.9. Application of equal-area criteria 7.9.1. Case of a short-circuit at generator terminals 7.9.2. Critical fault clearing time 7.9.3. Case of a short-circuit on a line 7.10. Case of a multi-machine system 7.11 Exercises 7.11.1 Exercise No. 22: Stability and critical fault clearing time Bibliography CHAPTER 1 Functions of Electrical Energy Systems 1.1. Introduction Electrical energy is produced in particular sites related to the nature of the primary energy source: mountain for hydroelectric plants; rivers for hydroelectric or nuclear installations; seaside for nuclear installations and the tidal power plants; Countryside and coal mines for the thermal plants. This energy is used in centers of consumption which are often located in places away from the generating plants. These include urban centers; industrial centers; steel and metallurgical processing plants; electrical railway systems; etc Since electric energy cannot be stored in large quantities, it is necessary to produce it, transmit it, and distribute it in real time to various customers for consumption. The role of the transmission network is to essentially carry the energy produced from various power plants to the load centers where it is consumed. From the operational point of view, we recall that the crucial role of the network is to allow the supply of power at every moment power required by the consumer under guaranteed frequency and voltage magnitudes. However, this constraint requires an adjustment of the generating machines and equipment so that: all apparatus operate in good conditions; the energy losses are minimized; the use of the spinning reserves is optimized; The limits of the network variables are respected under normal circumstances. While the network is operated such that the above constrained are met under normal circumstances through monitoring and adjustments, there exist however unexpected incidents such as, short-circuits; bad weather (e.g., lightning strikes); Unintentional tripping. The role of preventive maintenance and the security of the network are to assure that the above incidents should not lead to widespread power outage. The old electrical networks were oversized and thus redundant by their design, which took into account the requirements of security. Todays networks, however, are very often exploited under conditions close to their limits of operation because of high capital costs, stricter environmental and societal constraints (i.e., the acceptability building new transmission lines becoming increasingly problematic). The liberalization of the energy markets facilitated power transactions between many players, energy producers as well as consumers, who can be located in different territories. This led to an increase in the number and volume of energy transfers on the network that was originally designed to operate in a monopolistic mode. These power exchanges, which significantly increased after the introduction of market competition, are straining many parts of the transmission network. This situation of fragility, with respect to incidents being able to occur in the course of exploitation, has led the network operators to set up means of reacting in an adequate way at the time of critical situations for several decades (well before the advent of competition). The diagram of Figure 1.1 below illustrates the installation of these measures. These issues which are matters of analysis concern all the elements of the life of the network, from its long-term planning to the study of fast transient phenomena.  Translation: Necessity de decisions rapides: Necessity of rapid decisions Automatization des actions: Automated actions Control en temps reel: Real time control Resolution prealable de nombreux problems: Priority resolution of the numerous problems Figure 1.1. Strategic elements of network control. The list below shows the majority of the above subjects. The analytical methods developed in the chapters that follow will allow a precise and thorough study: network planning; reliability studies; simulation of operation; load forecasting and distribution; short-circuit analysis; high voltage transients; insulation coordination; protection and adjustment of relays; analysis of static and dynamic security; optimal reserve management; congestion management; etc Each one of the above subjects has its own time-constant and requires a resolution adapted to its time scale. Thus, the reinforcement of a network must be envisaged years in advance, while the elimination of a short-circuit must be carried out in a few milliseconds. The diagram of Figure 1.2 shows the various time scales that one meets in the life of an electric grid.  Time scale 109 = 10 years Long-term planning (strategies, scenarios), network reinforcement, maintenance. 106 = 10 days Load forecast, load distribution. 103 = 15 min Security study, controls, turbine monitoring and regulation. 100 = 1sec. Data transmission, state estimation, voltage and speed regulation. 10-3 = 1ms Protection: overvoltages (lightning, switching operation), short-circuits. 0 Figure 1.2 Time scale. 1.2. Hierarchy and representation of electrical power systems Each of the studies listed above requires a good knowledge of the topology of the network and characteristics of its elementary components. Topology can be described or represented by a diagram of the network which is generally a three-phase network. Its operation will in most cases be reduced to the study of the behavior of one of its phases, which allows it representation by a one-line diagram. Figures 1 .3a gives an illustration of such diagrams. Although electrical power networks are in general three-phase, the representation by one phase is the first information source used (by considering a balanced network). Schematically, one resorts to a representation known as one line-diagram (see Figure 1.3b). This representation is more compact but comprises a loss of information compared to the preceding representation, especially when the system under study not completely balanced. In this representation, one represents only the general structure of the network.  Figure 1.3a. Three-phase representation of a power network.  Figure 1.3b Equivalent single-phase network. However, it should be noted that a complete representation of the network would require for example a detailed description of the three-phase transformers, circuit breakers (compressed air, oil, etc...), line details (size, length, etc), insulator locations, the geometry of busbars, etc. This representation is of course not necessary for the majority of the studies quoted above. The unifilar representation will thus give us the essence of the "simplified" information including the various voltage levels in the network. The unifilar network includes, in addition to the connections between the various nodes of the network, information like line impedances, the power and the electromotive force (emf) of the generators, and the electric representation of the loads. When the study requires only the information of connection between the various nodes and the lines which compose the network, this latter can be represented by a graph as indicated in Figure 1.3c which schematizes the network shown in figures 1.3a and 1.3b. The parameters which make it possible to characterize the operation of the network are defined by their per-unit (p.u.) values which make it possible to fix the nominal values at a value equal to 1.  Figure 1.3c Graph corresponding to unifilar network of Fig. 1.3b. 1.2.1. Transmission Lines and Apparatus The transmission lines are defined by -model which characterizes lines of medium length whose parameters are resistance R, reactance X = L and susceptance B = C. In case of the long lines, one can always use their equivalent -model. The static compensators for voltage support, the shunt reactors, and the series capacitors for the long lines are defined by their admittance Yc or specific parameters. 1.2.2. Transformers Transformers which operate at their nominal turn ratio do not appear in a diagram where all the electric quantities are represented in per-unit values. On the other hand, transformers equipped with tap changers under load (or in vacuum) are represented in a specific way. In the unifilar diagram, they are denoted by the symbol shown in Figure 1.4 below.  Figure 1.4 Representation of transformer with tap changer. 1.2.3. Electric Loads The loads can be represented in several ways: Active power (P) and reactive power (Q): P = Re (VI*) Q = Im (VI*) where V and I are respectively the voltage and current phasors, and the symbol (*) indicates complex conjugate. This modeling, which is in polar form, is used in load flow calculations as one will see it in Chapter 4. Impedance: EMBED Equation.3 where EMBED Equation.3 This representation is often used in stability studies, and makes it possible to reduce the equivalent unifilar network to one that contains only generation nodes. Current sink: I = V/Z This representation is often used in modeling distribution networks that characterized by radial topological structures. It is worth noting that the above load representations are valid for steady-state analysis only ("static" mode). Thus, they do not take account of the dynamic characteristics of these loads. 1.2.4. Generators The generators, the majority of which consist of synchronous machines, are represented either by their equivalent circuit with active power production and internal voltage (P, V) in studies involving steady-state. Note that generators are generally equipped with voltage regulators which make it possible to maintain the voltage magnitude at their terminals. For the static studies, the power produced by electric generators is also considered constant. CHAPTER 2 Network Representation 2.1. Graphical and topological description of a network The studies quoted in Chapter 1 require the modeling of the networks. We begin this modeling with a description of the topology of these networks. The graph theory provides us useful elements to carry out this modeling. A graph is a concise manner of description of the bonds between topological entities which are representative points of a geometrical structure, called nodes, and of the connecting elements which connect these points. A graph can be a geometrical drawing which illustrates these connections graphically. However, its informational representation is characterized by a table which has the properties of binary matrices. The elements of these are 0 or 1 with possibly a sign when the graph known to have a direction, as it is in the general case of electrical networks. We will successively study the properties of graphs and those of their associated matrices. 2.1.1. Review of graph theory Graph: a drawing with defined points called nodes connected by elements called branches. Incidence: a node and a branch are known as incidents if the node is one of the terminals of the branch. Path: formed by connected edges in such a way that one has at most two incident branches at each node. Oriented graph: graph in which one assigns a direction in each branch. Connected graph: graph in which there exists a path between each pair of nodes. Circuit: closed path. Tree: graph containing all the nodes but no circuit. Branch: link of a tree. Cord: branch belonging to the graph but not to the tree. Co-tree: a group of links of a graph which do not belong to a tree. Cut: a group of links whose extraction causes the separation of the graph in several disjoined graphs. Fundamental circuits: group of independent circuits each containing only one cord. Fundamental cuts: group of cuts each containing only one branch. Proposition: Consider a directed graph containing n nodes and e links. When any tree is chosen, the number of branches is: b = n-1; the number of cords is: l = e-n+1; the number of fundamental circuits is: m = e-n+1; the number of fundamental cuts is: c = n-1; the chosen orientation of a circuit: that of the associated cord; Of a cut: that of the associated branch. Figures 2.1 illustrate these concepts on the graph defined in the example of network representation earlier in Chapter 1. Figures 2.1a, b, c, and d respectively show the network representation by a directed graph, a tree with cords and branches, fundamental circuits, and fundamental cuts.  Figure 2.1a Representation of a network of Fig 1.3 by a directed graph.  Figure 2.1b Tree with branches (1-4), and cords (5-7).  Figure 2.1c Fundamental circuits (E, F, G).  Figure 2.1d Fundamental cuts (A, B, C, D). Starting from a description of the network by a unifilar diagram and extraction of the graph which is the topological representation, it is possible to seek by specialized algorithms possible trees and associated cords, branches and circuits. As will be seen in the sections that follow, this description will allow the derivation of the network equations. 2.2. Network global modeling: CIM model Electrical power networks are inter-connected and one cannot study part of a system without having a sufficient knowledge of the neighboring systems. It is therefore essential to establish information exchange between the network operators both within the same electric utility company and between different companies. Any exchange of data is consistent with the interchange formats based on the models most commonly used in electric systems. Thus a model which is used by all companies internationally called "Common Information Model" (CIM) was created. CIM is a conceptual model which is developed under the aegis of the International Electrotechnical Commission (IEC) in a language of the type UML (Unified Modeling Language) [UML 03]. This model covers the whole data necessary to the study and exploitation of electric systems, including the operations of market between companies or producers and consumers. The complete model is of a great complexity and contains several sections. One particular section, which makes it possible to represent the data specific to the network elements and the types of calculation, will be described in the following chapters. The section under consideration contains the topology of the system, the electrical data of all the elements of the system (lines, transformers, circuit breakers, electric power generators) and load modeling. A detailed description of CIM model is beyond the framework of this work and interested readers may consult [AVA 06]. 2.3. Matrix representation of networks The formulation of the equations of network is based on the definition of a coherent and exact mathematical model which describes the characteristics of the individual components (machines, lines, transformers, loads) and the interconnection between these components. The matrix equation is a suitable model adapted to the mathematical treatment and processing under a systemic aspect. The matrix elements can be either impedances (when node voltages are written in terms of injected currents), or admittances (when injected currents are written in terms of node voltages). 2.3.1. Network Matrices The network can be described by three types of matrices: Elementary matrices (or primitive): these matrices describe the individual components by taking into account, if necessary, their electromagnetic (capacitive and inductive) couplings for lines having common or partial right-of-ways. They are of diagonal structure except for the components whose coupling is represented by non-diagonal elements; Incidence matrices: these matrices describe the interconnections between the various components of the network. The terms of these matrices are binary digits 1, 0, - 1, which represent the bond between branches and nodes of the network with their orientation; Transfer matrices: these matrices describe in a mathematical way the electric behavior of the mesh network. They are essentially impedance or admittance matrices which correspond to the nodes of the network (nodal matrices). The relation between the above three matrices can be described by the operational equation of Figure 2.2. The figure shows that the transfer matrix is obtained from a complex operation using the elementary matrix and the incidence matrix. This operation will studied in the following sections.  Translation: Matrice primitive: Elementary Matrix Matrice dincidence: Incidence Matrix Matrice de Transfer: Transfer Matrix Figure 2.2 Network matrices. 2.3.1.1. Incidence Matrix As indicated above, the incidence matrices characterize the relation between the network elements (generally called branches) and the nodes connecting these elements. 2.3.1.1.1. Incidence Matrix branches-nodes: A Definition: It is a matrix A with general term {aij} and dimension (e x n) such as: aij = 1 if branch i is incident with node j and is directed towards this node; aij = -1 if branch i is incident with node j and is directed away from this node; aij = 0 if branch i is non-incident with node j. Properties For every line i: EMBED Equation.3 Indeed on the same line corresponding to the branch referred by i, there are only two nonzero elements: The first corresponds to the starting node with value 1, and the second corresponds to the arrival node with the value - 1. The above property indicates that the number of rows of the matrix is lower than n. 2.3.1.1.2. Incidence matrix branches-access: A This corresponds to the incidence matrix branch-node in which the choice of a node of reference (for voltage) led to the removal of a column of the matrix A (in general the first). This matrix is of row n - 1. 2.3.1.1.3. Incidence matrix branches-fundamental cuts: B Definition: It is a matrix B of general term {bij} and dimension (e x b) such as: bij = + 1 if the ith branch belongs to the jth fundamental cut with same orientation; bij = - 1 if the ith branch belongs to the jth fundamental cut with opposite orientation; bij = 0 if the ith branch does not belong to the jth fundamental cut. Properties: Let the following sub-matrices of A and B be denoted by: Ab: branches/access, Ac: cords/access. Bb: fundamental branches/cuts, Bc: cords/fundamental cuts. Since there is an identity between the branches and the fundamental cuts, then the sub-matrix Bb is equal to the unity matrix I. Moreover one can notice that the product: Bc*Ab = incidence matrix cords/access Which is precisely the sub-matrix Ac, i.e., Bc*Ab = Ac The above yields Bc=Ac* Ab-1 Thus, one can build the matrix B from sub- matrices Ab and Ac of matrix A by the formula: EMBED Equation.3 2.3.1.1.4. Incidence matrix links-fundamental circuits: C Definition: It is a matrix C of general term {cij} and of dimension (e x m) such as: cij = + 1 if the ith link belongs to the jth fundamental circuit with same orientation; cij = - 1 if the ith link belongs to the jth fundamental circuit with opposite orientation; cij = 0 if the ith does not belong to the jth fundamental circuit. Properties: Let the following sub-matrices of C be denoted as follows: Cb: branches/fundamental circuits; Cc: cords/fundamental circuits. Since there is identity between the cords and fundamental circuit, the sub-matrix Cc is equal to the unity matrix I. Example of incidence matrices: If the graphs of Figures 2.1a - 2.1c are condensed into one graph as displayed in Figure 2.3 which shows the branches, cords, fundamental circuits and fundamental , one can easily build matrices A, B, and C corresponding to this graph:  Figure 2.3 Graph for the matrices A, B, C, of network.    2.3.1.2. Matrices of elementary network [ST 68] Definition: One calls "elementary network" the set of all components of the network including their electric and magnetic couplings. Each component is defined by its impedance zpq or admittance ypq = 1/zpq where subscripts p and q represent the starting and arrival nodes, respectively. Moreover, the generators are modeled by an electromotive force (emf) epq in series with internal impedance (Thevenin equivalent), or a current source Jpq in parallel with internal admittance (Norton equivalent). 2.3.1.2.1. Equation in terms of impedance Figure 2.4 below shows the Thevenin circuit (i.e., electromotive force in series with internal impedance) of a generator. The terminal voltage vpq is related to the current ipq, emf epq and impedance zpq as follows: vpq + epq = zpq. ipq (2.1)  Figure 2.4 Generator represented in impedance form. 2.3.1.2.2. Equation in terms of admittance Figure 2.5 shows the Norton equivalent circuit of a generator (i.e., current source in parallel with generator admittance). In here, the currents ipq and Jpq and the generator terminal voltage vpq are related by equation (2.2).  Figure 2.5 Generator represented in admittance form. ipq + Jpq = ypq.vpq (2.2) Jpq = - ypq.epq The matrix of the elementary network is a matrix whose diagonal elements correspond to the impedances of each link of the network. These impedances are referred to as self-impedances, and are denoted by four subscripts zpq,pq to indicate that it is the self-impedance of the link pq. On the other hand, the coupling impedances of between the links pq and rs, which represent the off-diagonal elements of the matrix (as illustrated in Figure 2.6), are denoted by zpq, rs.  Figure 2.6 Coupled elements. Since a transmission line is generally coupled with not more than two lines, there will be only few non-diagonal elements in the matrix of the elementary network. Figure 2.7 shows the current vector i, the voltage vector v, and the impedance z matrix of the elementary network, with diagonal and off-diagonal elements illustrated for column rs and row pq.  Figure 2.7 Voltage-current relations in an elementary network. Similarly, if one represents the current sources and electromotive forces by vectors e and j, one can obtain the equations of the elementary network below.   v+e = zi i + j = yv (2.3) y= z-1 Example of Elementary Network: Consider the network below in Figure 2.8 with 4 nodes and 5 links. It is assumed that there is coupling between line pairs 1-2 and 1-4, as indicated by the yellow arrows. For identification purposes, links 1 and 4 which are both connected in parallel between nodes 1 and 2 are denoted by indices (1) and (2), respectively.  Figure 2.8 Example of elementary network. The presentation of this network is shown below in the form of a connection table listing the values of the self-impedances of the links and the coupling impedances (if applicable). The impedances are given in per-unit values. linkAccessSelf- ImpedanceAccessMutual- Impedance11-2 (1)0.621-30.51-2 (1)0.133-40.541-2 (2)0.41-2 (1)0.252-40.2 Table 2.1 Network self- and mutual-impedances. If one adopts the classification of the links defined in table 2.1, we can then build matrix Z of impedance of the elementary network.  Figure 2.9 Elementary impedance matrix. When one modifies the numbering of the links (which obviously does not alter the operation model of the network), the matrix Z can be converted to diagonal submatrices. Such an alteration allows easier matrix operation, especially when determining its inverse. As in illustration, if the links 3 and 4 are exchanged, the resulting matrix is shown in Figure 2.10.  Figure 2.10 Elementary impedance matrix with exchange of links 3 and 4. The inverse of the matrix in Figure 2.10 is obtained by separately inverting a 3 x 3 matrix (first three rows and columns) and a 2 x 2 diagonal matrix (last two rows and columns). The result is shown in Figure 2.11.  Figure 2.11 Inverse of matrix Z shown in Figure 2.10. 2.3.1.3. Transfer Matrices 2.3.1.3.1. Nodal transfer matrices Consider a network containing N nodes, numbered 0, 1, 2, ..., N-1. Let node 0 be the reference node to which all the node voltages are referred to. Furthermore, let (E1, E2, EN-1) and (I1, I2, IN-1) respectively denote the node voltages and injected currents at nodes 1, 2, N-1, as illustrated in Figure 2.12. This notation allows one to bring back the analysis of the network to its individual components (such as they are seen outside) without taking account of the internal structure which will be represented by the transfer matrices.  Figure 2.12 Network with node voltages and injected currents. Thus the network of Figure 2.13 below can be modified be represented by a schematic as shown in Figure 2.14.  Figure 2.13 Example of 4-node network.  Figure 2.14 Modified schematic of Figure 2.13. We then define the vectors Ebus ={ Ep} and Ibus ={ Ip}whose elements contain the node voltage and injected currents at that node, respectively: EMBED Equation.3 and EMBED Equation.3 The operation of the network is then modeled by the relationship between these quantities through the nodal impedance matrix Zbus, or the nodal admittance matrix Ybus. These relationships are expressed by equations 2-4 and 2-5 below:   Of course there is a strong link between the transfer matrices, nodal impedance matrices or nodal admittance matrices, and incidence matrices. Consider the matrix equations of the elementary network: i + j = y v (2-6) Multiplying both sides of the above equation by the transpose At of matrix A, we obtain: At(i+j)=At.y.v ! At i + At j = At.y.v The first term Ati is the sum of the currents arriving at each node of the network is, according to Kirschoffs current law, equal to zero. The term Atj is the sum of the currents injected into each node. This latter is by definition equal to Ibus , the current injected into each node. Then the above equation reduces to : Ibus= At.y.v (2-7) Now let us calculate the total complex power injected into the network. This quantity is the same as that when the network is represented in the form of nodal voltages and currents or in its basic form. It follows that: P = ( Ibus )*t Ebus = j*t v ( 2-8) But since Ibus = At j, Then, ( Ibus )*t = ( At j )* t Since matrix A is composed of real numbers, it is equal to its conjugate (A = A*) and therefore: ( Ibus )*t = j * t A and thus Equation (2-8) becomes: j* t A Ebus = j*t v , (2-9) This is true regardless of the vector j and therefore implies that: v = A Ebus Since Ibus = At y then , At y A Ebus = Ybus . Ebus So in conclusion: Ybus=At y A (2-10) and Zbus= (Ybus)-1 (2-11) 2.3.1.3.1. Transfer matrices of meshes Consider a network that contains m fundamental circuits or meshes which carry currents i1, i 2, im , and e1 , e2,em represent the voltage sources inserted within each mesh. We define Ebus and Ibus and voltage and current vectors with general term Em ={ek} and Im ={ ik}, respectively.  EMBED Equation.3  and  EMBED Equation.3  The operation of the network in this reference frame of current and voltage is expressed using the equations that relate the mesh currents to the voltage sources inserted in each mesh. These relationships are expressed in matrix from by the mesh transfer impedances (or mesh impedance matrix) denoted by Zm, or by the mesh transfer admittances (or mesh admittance matrix) denoted by Ym. The relationship between Em and Im are expressed through these transfer matrices by Equations (2-11) and (2-12) below: As stated earlier, there is a strong relationship between these transfer matrices, matrices of the elementary network, and incidence matrices. To express these relations, we will consider the complex power injected into the network to be represented by transfer matrices or matrices of the elementary network. In the elementary network, the circuit equation in matrix form is: v + e = z .i (2-14) Taking into account the interconnections by multiplying both sides by Ct: Ct. v + Ct. e = Ct . z . i In the above equation, the first term is nil: Ct.v = 0 This is due to the fact that the sum of the voltages around a loop is equal to zero (i.e., Khitchoffs voltage law). Furthermore, the second term is equivalent to Em defined above: Ct.e =Em Hence, we conclude that: Em = Ct . e = Ct . z. i Expressing the conservation of power between the two frames of reference: [Imt]* Em = (i*)t. e and taking into account the previous relations between e and Em: [Imt]* . Ct . e= (i*)t. e This relationship is true regardless of the vector e. We deduce that: (i*)t = [Imt]* . Ct and therefore, i = {([Imt]* . Ct)t}* = (C . Imt*t)* = C* . Im = C. Im Note that the matrix C is a matrix that based on real numbers, thus equal to its complex conjugate. In conclusion: Em = (Ct. z .C). Im (2.16) Therefore Zm=Ct.z.C (2.17) And Ym = (Ct.z.C)-1 (2.18) In conclusion to this chapter, we can summarize in Table 2.2 below the relationship between the transfer matrices and matrices of the primitive (or elementary) network through the incidence matrix that basically represents the interconnections between the elements in the primitive network.  Table 2-2 Relationship among different matrices of a network CHAPTER 3 Formation of Network Matrices The previous chapter identified the network transfer matrix from concrete data of the network elements (i.e., line impedances, shunt connected devices, generator electromotive force, etc....). These transfer matrices are the basis for all models of the network, as will be seen in the following sections. However, the theoretical formulas for building the transfer matrices from the elementary network are rarely used in practice. Indeed, the network that we study at a given moment is result from an earlier network configuration by adding or removing certain elements (lines, cables, switchgear, generators ...). Hence, it would be very cumbersome to reconstruct the full transfer matrices without reflecting these changing situations. In this chapter, we present a systematic and progressive algorithm to build transfer matrices, especially when constructing the Zbus matrix. We will restrict our presentation to the most important and commonly used matrices; namely, the bus impedance matrix Zbus and bus admittance matrix Ybus. 3.1. Formation of the Ybus matrix In the previous chapter, we have seen that each network component can be represented by a circuit diagram that consists of a series elements and/or shunt elements. For example, a line or cable segment that is connected between nodes i and j can be represented by a quadripole as shown Figure 3.1 below. In here, the series admittance yij connects node i to node j, while the shunt admittances yiij and yjji respectively connect nodes i and j to the reference node. The diagram in Figure 3.2 shows a representation of three network components connecting nodes i and j, i and k, k and 1, respectively. This sub network sets the stage for the derivation of network and equations and related matrices.  Figure 3.1 Representation of line or cable segment.  Figure 3.. Representation of a sub-network with three series components Let E1, E2, ..., Ei, Ej be the phasor voltages at nodes 1,2, ... , i,... j, when a phasor current Ii is injected at node i. These voltages and current Ii are related by the Equation (3.1) below:  EMBED Equation.3  (3-1)  EMBED Equation.3  From the above relation, we can deduce the expressions of diagonal terms and off-diagonal terms of the bus admittance matrix Ybus as follows:  EMBED Equation.3  (3-2) Notes: If we call yii the some of all the admittances connecting node i to the reference node, yii =  ( yiij ), the the diagonal term Yii can be rewritten as:  EMBED Equation.3  (3-3) When the branch ij is electromagnetically coupled with several branches with indices rs, then:  EMBED Equation.3  (3-4) When multiple branches with indices rs are coupled with a branch ik, then:  EMBED Equation.3  (3-5) 3.2. Formation of the Zbus matrix The formation of the Ybus matrix is simple because it carries elements of direct admittances. Furthermore, since there are a limited number of connections between different network elements, the structure of this matrix is usually very sparse, i.e., there are a small percentage of matrix elements that are non-zero, which further simplifies its construction. However such a structural feature of the network does not simplify the construction of the Zbus matrix. In fact, the is often easier to derive Ybus, then used it to directly construct Zbus because matrix inversion is not more than solving a system of linear equations whose order is equal to that of the matrix. Solving a linear system by triangular factorization is simplified by the fact that the matrix has only a small number of non-zero terms. However, a direct matrix inversion of Ybus to obtain Zbus can be time consuming as these complex matrices of real networks tend to be very large. In addition, the structure of a network at a given moment is a result of a small number of changes to the previous state of the network by addition or deletion of some well identified components. It is therefore important to have a systematic and easy way to deduce the matrix Zbus of the modified network structure from that of the previous network structure without going through another matrix inversion process. To do so, consider an initial network with m nodes numbered from 1 to m as shown in Figure 3.3 below. Let Zbus (with dimension m x m) be the impedance matrix of this network. We will study the changes induced in this matrix by the two different additions illustrated in Figure 3.3 in red. Figure 3.3 Adding branch and/or cord to a network. 3.2.1. Adding branches In practice, the addition of a new branch corresponds to the extension of a line for new service in a distribution system, or a new transmission line to serve a new area. In general, this line is placed in a new right-of-way; hence no electromagnetic coupling with existing lines is expected. It may however share a certain part of its path with other lines which will lead us to consider the general case, although the latter is relatively rare. This branch is supposed to have a self-admittance ypq,pq or self-impedance zpq,pq in the elementary network. It may be coupled with a branch rs through a coupling admittance ypq, rs. This will transform a network with m nodes to a network with m+1 nodes. Similarly, it will transform the Zbus matrix with dimension (m x m) matrix into a new dimension (m +1) x (m +1). This new matrix is:  EMBED Equation.3  (3-6) The new node q is connected to node p of the previous network trough impedance Zqp. Next, we will discuss the calculation of all the newly added impedance elements in Equation (3.6) above. 3.2.1.1. Calculation of Zqi terms The impedance Zqi (i=1, 2, m with i `" q) is equal to the voltage Eq that appears at node q when we inject a current Ii = 1 ampere at node i: Eq = Zqi Ii = Zqi (3-7) Hence, the new off-diagonal elements of the new impedance matrix are equal to the corresponding node voltages as a result of the injected current at node i: E1= Z1i Ii = Z1i E2 = Z2i Ii = Z2i . . (3-8) . Ep = Zpi Ii = Zpi . . . Em = Zmi Ii = Zmi From the elementary network, we have:  EMBED Equation.3  and  EMBED Equation.3  But since Ip = Iq= 0, and ipq= Ip- Iq=0, we can conclude that:  EMBED Equation.3   EMBED Equation.3  Hence:  EMBED Equation.3  (3-9) and taking into account Equations (3.8):  EMBED Equation.3  (3-10) If the branch pq is coupled with any other branch, then: Zqi = Zpi for all i `" q. 3.2.1.2. Calculation of Zqq term Just as we defined the Zqi terms above, we define Zqq as the impedance that caraterizes the voltage Eqh that appears at node q when we inject a current Iq = 1 Ampere at the same node. The currents injected at all other nodes are set equal to Ii = 0. Hence: Eq = Zqq Iq = Zqq (3-11) while the voltages appearing at the other nodes are: E1 = Z1q Iq = Z1q E2 = Z2q Iq = Z2q . . . Ep = Zpq Iq = Zpq (3.12) . . . Em = Zmq Iq = Zmq However, according to the equations of elementary network:  EMBED Equation.3  and  EMBED Equation.3  Bust since Ip= 0, Iq=1, ipq= Ip- Iq= -1, from which we deduce that:  EMBED Equation.3  and:  EMBED Equation.3  Substituting vrs:  EMBED Equation.3  Taking into account Equations (3.12):  EMBED Equation.3  Finally:  EMBED Equation.3  (3-13) If the branch pq is coupled with any other then all terms ypq, rs are zero and: Zqq = Zpq + 1/ypq, pq = Zpq + zpq, pq (3-14) 3.2.2. Adding cords This operation corresponds to the strengthening of the network by the addition of a new line between two existing nodes. This addition does not affect the number of existing nodes and therefore the size and order of matrix, but the inclusion of the impedance of this new line changes all terms of Zbus. The calculation of the change will be made in two separate steps by taking advantage of the procedure outlined in the previous subsection: In the first step, insert between nodes p and q a node l and a fictitious voltage source el defined so that the current ipq is nil (ipq = 0). During this step, the size of the augmented Zbus matrix is increased by one unit due to the addition of node l; In the second step, the fictitious node l is eliminated and the added voltage source is short-circuited. This step reduces the size of the bus impedance matrix to its original value and changes the entire set of matrix elements. 3.2.2.1. Calculation of augmented matrix elements Consider Figure 3.4 where a node l is inserted between nodes p and q, and a voltage source e1 between nodes l and q. When branch pl is inserted in series with the voltage source e1, the Zbus matrix is modified as follows:  Figure 3.4 Addition of a line between two existing nodes.  EMBED Equation.3  (3-15) With regards to the calculation of terms Zlj (for j = 1m), we proceed just like the case of adding a branch by injecting a current Ii = 1 Amp at node i and setting all the injected currents at other node to 0: Ek = Zki Ii = Zki ( for k= 1.m) el = Zlj Il= Zli with: el = Ep- Eq - vpl As indicted above, the voltage source e1 must be chosen so that ipq= ipl = 0, which leads to:  EMBED Equation.3  from where w derives:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Therefore,  EMBED Equation.3  Finally, taking into account that the node l is fictitious: ypl, pl = ypq, pq and ypl, rs = ypq, rs we get:  EMBED Equation.3 , for i = 1 ... m. (3-16) Similarly we can calculate Zll:  EMBED Equation.3  (3-17) Having established the method for calculating all the terms of the augmented matrix, the next step is to reduce the matrix back to its original size. 3.2.2.2. Elimination of fictitious node Before elimination of the fictitious node, the line p is represented as:  EMBED Equation.3  (3-18) The current value is obtained from the line (m + 1):  EMBED Equation.3  Since el is a fictitious voltage source whose only usefulness was to allow the calculation of the elements of the augmented matrix, its value is zero. This allows us to calculate Il as a function of the terms of the augmented matrix:  EMBED Equation.3  Therefore:  EMBED Equation.3  Substituting this value in equation (3.18) results in:  EMBED Equation.3  This allows us to express the general term of the new Zbus matrix:  EMBED Equation.3  (3-19) Notes: In case where there is no electromagnetic coupling between branches pq and rs, i.e., ypq, rs = 0. Then, Zli = Zpi - Zqi, and Zll = Zpl - Zql The Zbus matrix is particularly important for short-circuit current calculations, as will be seen in the upcoming chapters. Table 3.1 below summarizes all formulas used for the construction of this matrix.  Table 3.1 Summary of construction of Zbus matrix. 3.3 Exercises 3.3.1 Exercise No. 12: Construction of Zbus matrix Consider the primitive network shown in Figure E.12. The two coupled branches 1 and 4 connecting nodes 1 and 2 are referred to as 1-2 (1) and 1-2 (2), respectively. The per-unit values of the self and mutual impedances of each branch of the network are given in Table E.12.  Figure E.12 Network under study. BranchSelf (pu)Mutuel (pu)AccessImpedanceAccessImpedance11-2(1)0,641-2(2)0,41-2(1)0,221-30,51-2(1)0,133-40,552-40,2 Table E.12 Self- and mutual- impedances (pu). Construct the Zbus matrix of the network. Determine the modified Zbus matrix after adding a branch between nodes p = 2 and q = 4. Assume this new branch has a self-impedance of 0.3 pu, and is coupled with branch 5 by a mutual impedance of 0.1 pu. 3.3.2 Exercise No 13: Construction of network matrices A network is defined by the graph in Figure E.13 below. The per-unit values of the impedances of the 5 branches are listed in Table E.13.  SHAPE \* MERGEFORMAT  Figure E.13 Graph of network under study. BranchNodesImpedanceCoupling1A-B0,05none2B-C0,1none3C-D1,1none4D-A0,04none5A-C0,1none Table E.12 Network impedances (pu). Let node A be the reference node, then derive the branches-access incidence matrix of the network. Determine the primitive matrix of the network. Assume the impedances listed in Table E.13 are pure reactors. To simplify calculations, ignore the imaginary operator "j". Use the "step by step" procedure to construct the transfer impedance matrix Zbus of the network. Calculate Zbus using following expression in terms of the incidence matrix A and primitive admittance matrix Y: Zbus = (A-1.Y.A)-1. Compare the results with those of question C) above. CHAPTER 4 Load flow calculations 4.1. Objectives 4.1.1. Definition of network state The objective of load flow calculation in a network is to determine the network status according to the connected loads and the distribution of consumption across all nodes in the network. This calculation is based on the assumption that the network is operating at steady-state and that the generators provide power in the form of AC sinusoidal, balanced three-phase voltages. The purpose of the calculation is to provide an accurate picture of active and reactive power flow in every element of the transmission network, as well as voltage levels at every node. 4.1.2. Calculation of current flow Knowing the value of the current flowing through each component of the network (line, cable or transformer) ensures that this does not exceed the current ratings of these components. Without this knowledge, excessive current flow may occur and this can lead to component overheating and even failure. 4.1.3. Line losses Similarly, knowledge of current flow provides an evaluation of power losses in lines and transformers. Excessive power loss may call for network reconfiguration in order to minimize losses on the entire network. 4.1.4. Strategy for adjustment and control The ability to determine node voltages and the active and reactive power that each generator should furnish in order to ensure power delivery to each load center, allows one to define guidelines for the regulation of every machine connected to the network. Such a calculation is also used in stability analysis. 4.1.5. Optimizing power transfer capacity Finally, it is possible from the algorithms we will cover in this chapter, to introduce the concept of constraint and objective function to optimize the power flow in order to increase the transfer capacity of available power in the network. 4.2. Model of network elements 4.2.1. Lines and transformers The model commonly used to represent lines and insulated cables is the  model of medium lines. Such a model allows us to adequately represent the capacitance effect on the transmission lines without much complication, especially when modeling large power networks (often with hundreds of nodes and thousands of lines and cables). This model illustrated the Figure 4.1 where the line conductance is neglected and the capacitance is represented by two shunt admittances. Similarly, transformers are modeled by their equivalent  model which allows the inclusion of iron losses as well as the magnetizing reactance. In this type of transformer model, the shunt admittances are different from each other for those transformers equipped with tap changes. Figure 4.1 Line, cable and transformer model (yji `" yji for transformers with tap changes). Starting from the impedances of these elements, we can construct the Ybus matrix of the network using the simple steps described in Chapter 3. This admittance matrix is composed of two types of elements: - The diagonal elements,  EMBED Equation.3  - The off-diagonal elements, Yij = - yij with yij being the admittance of the network element connected between nodes i and j. 4.2.2. Generators and loads The generators and loads are defined using a pair of fundamental quantities which include active power P, reactive power Q, voltage magnitude V, and voltage phase angle . When we describe a load at the so-called load bus, we represent it by the complex power it consumes, i.e., it active power and reactive power. When we describe a generator at a generator bus, we represent it by the active power it is scheduled to deliver and the magnitude of the voltage at that node. We also define the range of reactive power [Qmin, Qmax] that the generator is able to supply/absorb reactive power. An exception is made, however, for the most powerful machine connected to the network. The node at which this machine is connected to is called the swing bus (or reference bus) and serves as a reference for the phase angles of the voltages at all other buses in the network. We describe the swing bus by the voltage magnitude and phase angle  = 0o. Note that the generator connected at the swing bus must supply the active power needed to balance the load demand and system losses which are not know ahead of time. 4.2.3. Representation of voltage The voltage in any node i is represented by a complex quantity Ei that can be defined in polar coordinates  EMBED Equation.3 or in rectangular coordinates Ei = ei + j fi, where ei = Vi cos(i) and fi =Vi sin(i). The notation in polar coordinates (Vi and i) is the most used because these components are measurable quantities; hence, the latter notation will be exclusively used in this text, although other authors [STA 68], recommended to use either one or the other. 4.3. Problem formulation 4.3.1. General equations The power complex Si = Pi +j Qi injected to node i is given by the following equation by taking into account the relationship between the injected current Ii at node i in terms of the node voltages in the network and the elements of the bus admittance matrix Ybus:  EMBED Equation.3  (4-1) Equation (4.1) can be expanded:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Let us define the real and imaginary parts of each element of the bus admittance matrix as follows: Yii = Gii + j Hii and Yij = Gij + j Hij Now we can separate the real and imaginary parts of the complex power equation, above, and obtain the two basic expressions of active and reactive powers at node i:  EMBED Equation.3  (4-2) From these equations, we note that each node i is characterized by 4 electrical quantities: The active power Pi injected into the network at this node (this quantity will be negative for a load since it draws power from the network); The reactive power Qi injected into the network at this node (like Pi above, this quantity will be negative if drawn from the network); The magnitude of the voltage Vi at node i; The phase angle i of the voltage at node i. The pair of electrical quantities assigned to a bus depends on whether this node is a swing bus, load bus, or generator bus. As stated earlier, there are three types of bus: Generator bus where real power is injected into the network. In such a node, the quantities Pi and Vi are known while Qi and i are unknown. This node type is often referred to as  PV bus ; Load bus where consumer power is drawn from the network. In here, the quantities Pi and Qi are known while the unknowns are Vi and i. This node type is often referred to as  PQ bus ; Swing bus corresponds to the reference bus where an unknown quality of real power is injected into the network. In here, Vi and i ( = 0o) are given while Pi and Qi are unknown. This node type is often referred to as PV node although the real power is unknown. 4.3.2. Simplified models The mathematical problem defined by equations (4.2) is a very complex as it involves solving a large system of non-linear transcendental algebraic equations. However, electric utility companies were required to solve this problem in connection with the operation their network with rudimentary means rather than actual calculations. They therefore had to rely on simplified models involving the simplifying assumptions of electrical or mathematical nature. These assumptions are based on the difference in magnitude between the parameters of electrical lines. Indeed the frequency of 50Hz can be seen that the resistance r and capacitance c of a typical line are such that (1/c) << r << l, where l the inductance of the line. This fact led to neglecting the capacitance and resistance in both line and transformer models. Under these approximations, the network is composed of inductive reactances only; hence, the admittance between nodes i and j is simplified to yij = (1/j l(), and the conductance terms Gij in equations (4.2) can be ignored. 4.4. Solution methods Equations (4.2) apply to each node in the network. This means that for a network of n nodes, we have a system of 2n equations with 2n unknowns to solve. The unknowns differ from one node to another depending on the type of node: At a generator bus, quantities (Pi ,Vi) are known and (Qi , i) are unknown. At a load bus, quantities (Pi , Qi) are known and (Vi , i) are unknown. Finally, at the swing bus, quantities (Vi , i) are known and (Pi , Qi) are unknown. Solution methods that can be applied for solving these equations are iterative in nature because of the non-linearities involved. Two types of methods are commonly used: the Gauss-Seidel method and the Newton-Raphson method. The principle of each of these methods is described in the Appendix. 4.4.1. Gauss-Seidel method The idea of this method is to describe each node p with:  EMBED Equation.3  (4-3) We will describe this method assuming, for the clarity of the statement, that all nodes are considered load buses where Pp and Qp are known while Ep and p are unknown. The method is divided into 4 stages: Initialization: At the initial stage, the magnitude and phase angle of the voltage at each node is assumed to be known:  EMBED Equation.3 (or EMBED Equation.3 ) and EMBED Equation.3  for all nodes p = 1, 2, ... n; Relaxation: From the above initial values, we calculate the current flow into each node at iteration number 1 from Equations (4.1) and (4.3):  EMBED Equation.3  or  EMBED Equation.3  Hence:  EMBED Equation.3  Similarly,  EMBED Equation.3  . . .  EMBED Equation.3  From these values we can calculate the new voltages for the currents at the second iteration: EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , which in turn will enable us to calculate the new values of voltages, knowing:  EMBED Equation.3 ,  EMBED Equation.3 ,.  EMBED Equation.3 . The relaxation consists of running this process until the voltages converge to the solution set. Convergence test: This test consists of calculating the difference between all voltages calculated with iteration number (k) and those calculated at the next iteration (k+1). The maximum gap between two successive iterations of all node voltages is then compared to a predetermined tolerance value . If,  EMBED Equation.3  Then the solution is reached and the iteration process is halted. Acceleration of convergence: To speed up the convergence of this process, it is possible to establish a modality which we call the acceleration of convergence. The process of acceleration is as follows: at each step for node p with a newly calculated voltage EMBED Equation.3 , the value of EMBED Equation.3  is given by:  EMBED Equation.3  Where the coefficient  is called the relaxation coefficient, and is usually chosen between the values of 1 and 2. This coefficient can then amplify the reduction of differences between two iterations. However, it is imperative to carefully choose this coefficient as it heavily depends on the network under study. Indeed, an inappropriate value of  can cause divergence of the iterative process. Treatment of generator buses: At a generator bus, a generator is connected to that bus and the magnitude of the voltage Vg is set along with the injected real power Pg. The unknown quantities are the phase angle g of the voltage and the reactive power Qg that must be provided or absorbed by the generator in question, knowing that this reactive power will, according to the characteristics of the machine be between two extreme values: Qmin Qma , then set Qi = Qmax, and transform bus i into a load bus, If Qi < Qmax, then set Qi = Qmin and transform this bus into a load bus in which P and Q are fixed and the iterations continue. Simplified models: As indicated earlier, although the Newton Raphson method convergence very fast, the large number of operations performed at each iteration made the application of this method relatively expensive in terms of computing time in the past. While constraint has become less binding due to the increased computing power of modern computers, it is still important when the size of networks to be analyzed is very large. This is especially true when load flow calculations are a repeated step in large scale studies such as network stability optimal network operations in economic terms. It is then necessary to perform a many load flow calculations on the same network but under different operating conditions. This has led to the finding ways to reduce the computation time of each iteration, even if the number of iterations required for the convergence of the algorithm is higher. One of the first simplifications proposed by Ward and Hale was to ignore the off-diagonal submatrices M and N of the Jacobian matrix and retain only the diagonal submatrices H and L, i.e., update only the terms "Pi/"i and "Qi/"Vi . The advantage of such a simplification is that the resolution of linear system (4.5) is immediate because the matrix is diagonal. The disadvantage is that the number of iterations rises very quickly because the modified algorithm is far from a quadratic convergence. This simplified model is useful when one does not have sufficient computing power (which is unlikely the case these days). Fast decoupled load flow: This model is based on approximations induced by electrical engineering consideration on the terms of the equations (4.3). It is well know that the magnitude of the voltage affects mainly the reactive power, while the phase angle affects mainly the active power. Consequently: "Qi /" j << "Qi /"Vj "Pi/" Vj << "Pi/"j This notice leads us to consider that off-diagonal matrices of the Jacobian matrix N = 0 and M = 0. We then deduce that system of equations (4.5) is decoupled: "V = [ "Q/"V]-1 "Q and "  = ["P /" ]-1 "P This leads us to solving two systems of equations of order n instead a system of order 2n, thus to divide the computing time by a factor of 4. In addition, the following assumptions are justified when considering the order of the electrical quantities involved: cos(i  j) H"1, Gij sin((i  j) << Hij, Qi << Hii Vi These considerations indicate that the Jacobian elements are nearly constant relative to the magnitude and phase angle of the voltage, which implies that Jacobian matrix does not have to be inverted at each iteration. Thus, we can solve the system by matrix triangularization which should be performed only once at the beginning of the calculation, and only the resolution by backward substitution is performed at each iteration. The advantage of the decoupled load flow is that the need for memory space and computation time is considerably reduced. However, one should take precautions when considering networks in which there are lines with a high degree of compensation, i.e., where the ratio R/X is very high. In these networks, as elsewhere in heavily loaded networks, the convergence of iterative methods is very slow and sometimes they may diverge and not be able to reach a solution. 4.4.3. Calculation of power flows One of the important results of the calculation of load flow is evaluation of the flow of active and reactive power at both terminals of each element of the network, as shown in Figure 4.2 below. If we consider an element that connects nodes i and j, we can write the complex power that passes through this element as a function of voltages Ei and Ej at the terminals of this element.  Figure 4.2 Power flow at terminals of an element. The complex power flowing from nodes i to j is: Sij = yij (Ei-Ej)* Ei + (yijEi)* Ei (4-7a) The complex power that travels from nodes j to i is equal to: Sij = yij (Ej-Ei)* Ej + (yjiEj)* Ej Ei (4-7b) where Yij*= - yij* = Gij - jHij, and yij*= yji* = gij j hij = - jC(/2 (since the conductance gij is often negligible). Decomposing equations (4.7) into real and imaginary parts allows us to obtain the active and reactive powers: Sij = Pij +j Qij and Sji = Pji +j Qji From (4.7) Sij can be rewritten as: Sij = ( -Gij+ j Hij ) ( Ei*-Ej*)Ei +(-j hij)EiEi* with: Ei =Vi eji and Ej = Vj ejj) Sij = ( -Gij+ j (Hij-hij))Vi - (-Gij+j Hij) ViVj ej(i- j) ) The same applies for Sji: Sji = ( -Gij+ j (Hij-hij))Vj - (-Gij+j Hij) ViVj ej(j- i) ) Now the active and reactive powers expressions can be deduced: Pij= -Gij Vi +Gij ViVj cos (i- j) + HijViVj sin (i- j) Qij =(Hij-hij)Vi - (Hij ViVj cos (i- j)-Gij ViVj sin (i- j)) Likewise: Pji= -Gij Vj +Gij ViVj cos (j- i) + HijViVj sin (j- i) Qij =(Hij-hij)Vj - (Hij ViVj cos (j- i)-Gij ViVj sin (j- i)) But the power loss of line or transformer is given by the sum of the powers injected into nodes i and j: "Pij =Pij +Pji = -Gij[ (Vi+Vj) - 2 ViVj cos (i- j) ] "Qij =Qij +Qji = (Hij-hij) (Vi+Vj) - 2 Hij cos (i- j) After some mathematical manipulations, we obtain: "Pij = -Gij[ (Vi-Vj) +4ViVj sin( (i- j)/2) ] "Qij= Hij[ (Vi-Vj) +4ViVj sin( (i- j)/2)]  hij (Vi+Vj) The analysis of these equations shows that the losses are proportional to the square of the voltage drop; hence, adequate voltage regulation to minimize the loss in the line is warranted. After calculating the loss in each element of the network, the total losses are added to the swing bus generator:  EMBED Equation.3 , and  EMBED Equation.3  We must ensure that the machine connected to the swing bus is capable of delivering these additional powers in addition to the scheduled values. 4.5. Software tools for load flow analysis The calculation of load flow is often is essential and is often a pre-requisite for other power system studies. For example, software associated with stability analysis begins with a calculation of load flow to determine the initial state of equilibrium. Thus, a software tool that is dedicated specifically for stability studies, such as Eurostag [EUR 04], has such a function to initialize a stability study. As a consequence, this particular software package can be used for load flow analysis. Also tools for system security studies, planning, short circuit analysis, and system optimization utilize (or are based on) load flow analysis. 4.6. Appendix: principle of numerical iterative methods Several numerical methods are available of solving non-linear algebraic equations. In the context of this work, we restrict our presentation to the methods that can be applied to load flow calculations; namely, the Gauss-Seidel and the Newton-Raphson methods. We apply these methods for solving the equation F (x) = 0, where F(x) is a non-linear function of one variable. For convenience, the equation to be solved can be rewritten as: x= f(x) (4.7) 4.6.1. Gauss-Seidel method The principle of this method is rather simple. From an approximate initial value x(0), we construct a sequence of iterated values x(k) by applying equation (4.7): x(k+1) = f(x(k) ) This iteration, if it converges, will lead to a value x(") such that: x(") = f(x(")) This will be the solution of equation (4.7). It is possible to represent graphically the principle of this method by drawing on the same graph the function y = f (x) and y = x as shown in Figure 4.3. It is possible to accelerate the convergence by introducing the concept of relaxation we discussed earlier This principle introduces a relaxation factor  which combines the values of x(k) and that of xcalc obtained from the previous calculation, i.e., xcalc = f (x(k)) to calculate the value of x(k +1) using the formula: x(k+1) = ( xcalc + (1-( ) x(k) The coefficient  is usually such that 1 < < 2.  Figure 4.3 Direct iterative method of solving the equation x = f (x). 4.6.2. Newton-Raphson method This method takes into account of changes in the function F (x) by using its derivative. From an initial value we calculate an increase or decrease in x by x such as: F(x(0)+ "x) =0 To begin with, we expand F (x(0)+x) using Taylor series expansion around x(0) and keep only the first term: F(x(0)+ "x) = F(x(0)) +"x F (x(0)) = 0 From which: "x = - F(x(0))/ F (x(0)) Then we update x(0) by x(1) as follows: x(1) = x(0) - F(x(0))/ F (x(0)) The iteration continues as follows for the kth sequence: x(k+1) = x(k) - F(x(k))/ F (x(k)) until it converges to a value of x such that  EMBED Equation.3 , where  is a predetermined tolerance value. A graphical illustration of this iterative process is shown in Figure 4.4 below.  Figure 4.4. Newton-Raphson iterative method. The advantage of the Newton-Raphson relative to conventional relaxation methods is the speed of convergence due to the use of information provided by the derivative of F (x). It is possible to demonstrate that the convergence of this method is quadratic in nature while the relaxation methods converge in a linear fashion. This rapid convergence led to wide-spread applications of the Newton-Raphson method to solving non-linear algebraic equations. 4.7 Exercises 4.7.1 Exercise No. 14: Load flow calculations A power network consists of three buses and three lines. The per-unit values of the series impedance and shunt admittance of each line are given in Table E.14.1 below. LineNodesSeries impedanceShunt admittance11-20.+j0.10.+j0.22-30.+j0.20.+j0.31-30.+j0.20.+j0. Table E.14.1 Lin series impedance and shunt admittance. Nodes 1 and 2 are considered as P-V nodes with fixed voltages. Each generator at these buses can produce an active power 0 d" P d" 6 pu, and reactive power -1.5 d" Q d" 2.1 pu. Bus 3 is a PQ node at which the connected load consumes P = 3.6392 pu and Q = 0.5339 pu. The network operators have decided to establish the following: Fix the voltages at the generation nodes to V = 1.0 pu at node 1, and V = 1.1 pu at node 2; Active power generated at node 2: P = 5.3217 pu. Note that this is situation in not practical since generation at node 2 is greater that the entire load, hence the generator at node 1 has to driven in motor mode to balance the power! For the calculation, we consider the data presented in Table E.14.2 below. Perform an iteration of load flow using the Newton-Raphson method. BusTypeGenerationConsumptionVoltageLimits QPQPQVMin.Max.1Swing---1.0--2PV5.3217--1.1-1.52.13PQ--3.63920.5339-- 4.7.2 Exercise No. 15: Power flow A 132 kV network consists of 4 elements as shown in Figure E.15 below. Elements AB, AD, DC are overhead lines, while element BC is an underground cable. The electrical parameters of these elements are listed in Table E.15. We want to study this network under various operating conditions.  Figure E.15 Network under study. ParameterOverhead LineUnderground CableInductancel=2.1 mH/kml=0.6 mH/kmCapacitancec=15 nF/kmc=300 nF/kmResistancer=0,25 W/kmr=0.104 W/kmConductanceg=0g=0 Table E.15 Overhead and underground cable circuit parameters. Assume that all the 4 elements as dipoles (short lines) without loss. Build the admittance matrix Ybus when taking node A as the reference node. Calculate the matrix Zbus associated with the method step by step. Build the new Ybus matrix when taking all the line parameters into account, and taking and taking the earth (neutral) as reference. Now node A is maintained at the rated voltage of 132 kV (i.e., VA = 1 pu), while node B, C, D are load centers with the following active and reactive power consumption: NodeP (MW)Q (MVAR)B600C10020D5010 Calculate the voltages at the above load centers after 2 or 3 iterations of the Newton Raphson method (use Ybus calculated in B) above). Calculate the resulting power flow in each element. Calculate the total network losses in the power generated at node A. 4.7.3 Exercise No. 16: Matrices and load flow Figure E.16 shows a one-line diagram of an electric power transmission network that consists of 3 nodes and 3 identical overhead lines (each is 100 km long). The total series impedance and shunt admittance of each line are as follows: impedance Z = 0.02 + j 0.08(, admittance Y = j0.02 S.  SHAPE \* MERGEFORMAT  Figure E.16 Three-bus network under study. Provide an equivalent circuit of the network. Taking the earth as reference node, determine the bus admittance matrix Ybus of this network. We are now interested in the power flow under the operating conditions listed in the Table E.16 below, where P and Q represent the injected real and reactive powers into the network, while V and  correspond to the magnitude and phase angle of the voltage at each node. BusBus TypePQV1Swing1.040.02Load (PQ)0.50.13Generator (PV)-1.51.04 Table E.16 Input data for power flow. Express the active powers P2, P3 and reactive power Q2 as functions of V2, q2, q3. Using V2(0)=1, 2(0)= 3(0)=0 as initial values, calculate the values of V2, 2, 3 after the first iteration of the Newton-Raphson iterative methods. Then calculate the resulting value of the relative power Q3(1) injected at node 3 after this first iteration. CHAPTER 5 Transient Analysis Methods 5.1. Interest in transient analysis The ever increasing electric energy consumption over the past few decades has resulted in network operation close to the capacity limits of its components. Therefore, any overvoltage or overcurrent caused by an unintended topology or operating condition is likely to cause a major outage. Thus, protection devices must be designed with higher accuracy to overcome these potential problems. Different types of transients must be analyzed in terms of their magnitude and duration and their impact on different materials. Voltage transients can result from: - Lightning strikes; - Switching surges; - Ferro-resonance; - Momentary faults. A careful analysis must be undertaken on the severity of each of these types of transients in order to take adequate measures to avoid cascaded tripping of protection devices or equipment failure (e.g., surges arresters, circuit breakers, transformers). Whatever the situation, the calculation or simulation of transients is extensive because they involve both the inclusion of propagation phenomena on the lines as well as non-linearities of various network elements such as lightning arresters and transformers. The difficulty is compounded by the nature of the transient phenomenon which is the propagation of electromagnetic waves with very steep fronts of the order of microseconds. The primary analysis tool has been the Transient Network Analyzer (or TNA) which was widely used in the last three decades of the twentieth century. The TNA was quickly replaced by numerical methods based on Laplace and Fourier transformations, and by the method of finite differences. Methods based on Laplace or Fourier transforms soon proved to be not sufficiently accurate when representing the steep fronts of the waves encountered in electrical networks. The finite difference method is to decompose the time t into very small intervals t, and the length of each transmission line into very small segments of length x, to realize the first order approximation of the derivatives of voltage and current in time and in space. The result is a system of algebraic equations whose unknowns are the values of voltage and current at every time instant, and at each point along the line. This method is very accurate if the discretization is fine enough, but the need to represent changes in voltage or current with very steep fronts requires us to choose a very short time step (in the nanosecond range). As a consequence, the application of finite difference method tends to be very cumbersome and very demanding in terms of computation time. If the study is to be performed on a network with several lines, transformers and apparatus, this method becomes no applicable because of its complexity. Thereafter, an original method based on the analytical equations of wave propagation on the lines appeared. By first neglecting the resistance and conductance of the line, this method consists of splitiing the line in several sections, then reintroducing the resistance between the sections of the line. This method is simply called the method of traveling waves, and the software associated with it was published under the name Electro-Magnetic Transient Program (EMTP) by H. Dommel. This method and the EMTP are universally used nowadays. In the next section, we will describe the operation and use of the transient network analyzer since it is still used in some cases. This will be followed by a description of the method of traveling waves. 5.2. Transient network analyzer 5.2.1. Principle of operation The Transient Network Analyzer is an analogue representation of the network components. The representation of transformers, alternators, circuit breakers and inductors poses no particular because of their small scale representation. It is however important to respect the characteristics of electrical transients such as time constants, damping coefficients and frequencies of oscillation during the construction of these machines. It is through careful design that takes into account these phenomena at the level of inductors, resistors and capacitors that we have a good quality results during testing. The most delicate point in the construction of the analyzer is the representation of lines and cables. The line and cable have distributed elements, hence cannot be represented as such because this would require a lengthy cable. In fact, these elements are subdivided into more or less discrete cells, each of which represents several kilometers of line or cable, as shown in the diagram Figure 5.1 below.  Figure 5.1 Schematic diagram of a line (or cable) segment. 5.2.2. Advantages and disadvantages Operating a TNA is based on the principle of discretization of the line. This implies an increase in the number of cells thereby improving the accuracy of results, i.e., the representation of transient phenomena that are we are seeking to reproduce. However, the increase in the number of cells leads to the generation of parasitic modes; namely, oscillation phenomena due to the natural frequencies of the device not related to operation of the line. The superposition of these parasitic modes with the actual wave propagation on the line distorts the results by either capping or distorting the surge. It is then necessary to insert damping resistors in these cells to reduce the effect of these natural frequencies that may be within the bandwidth of the filter that is made up of the series connected cells. After succeeding in eliminating the effect of these parasitic oscillations, the operation of the analyzer can be automated and provides satisfactory results. It must be said that this type of device is expensive to build and maintain because the representation a small network requires the construction of over 600 three-phase cells. However, this type of tool is useful when one wants to study a typical network from a topological point of view because once the network model is built, is easy to conduct testing on this network. The TNA is a great tool for statistical studies on switching surges because on can easily plan several thousands of opening and closing sequences (in the order of 20 000 trials) and analyze the recorded data statistically. Finally, many engineers like this type of device for the analysis of nonlinear phenomena which are difficult to represent numerically. An example of this would be the effect of overvoltage on the saturation of magnetic circuits of the machines installed on the network. Indeed, by its analog nature, the TNA allows one to understand the physical meaning of transient phenomena more easily than the numerical methods that we study. However, a typical TNA is rather bulky and this is a serious drawback when one must consider several networks especially if different from each other. It is this problem that led electrical power companies, which did not build a TNA, to turn to numerical methods. 5.3. The method of traveling waves [SAB 73] 5.3.1. Principle Consider a lossless line of length , inductance and capacitance per unit length of L and C, respectively. The operating equations of this line can be written as: "V/ "x = L " I/" t ; "I/ "x = C "V/" t (5-1) where V and I respectively represent the voltage and current at any point of the line. Differentiating equations (5.1) with respect to distance x results in: "V/"x = LC "V/" t (5-2) "I/"x = LC "I/" t The solution of the above equations is of the form: V = F1 (x- V0t) + F2 (x + V0 t) (5-3) I = (-F1 (x- t) + F2 (x + V0 t))/Zc where V0=1/"(LC) is the speed of propagation of electromagnetic waves in air (or in the dielectric material for underground cable), and Zc ="(L/C) is the characteristic impedance of the line (or cable); Functions F1 and F2 depend on the boundary conditions at both ends of the line (or cable). From Equations (5.3), we can derive the relations between V and I as follows: V(x,t) Zc I (x,t) = 2 F1 (x - V0t) (5-4) V(x,t) + Zc I (x,t) = 2 F2 (x + V0t) The above relations describe the representation of a line (or cable) between two points. 5.3.2. Representation of a line (or cable) Let two nodes k and m of the network connected by a line as shown in Figure 5.2 below, where Vk and ikm respectively represent the voltage at node k and current flow at node k towards node m. Similarly, Vm and imk respectively represent the voltage at node m and current flow at node m towards node k. Let us define  =/V0 as the travel time of an electromagnetic wave over the length of this line.  Figure 5.2 Line (or cable) representation between points k and m. The current and voltage magnitudes at time t and distance x (measured from node m) are expressed by: V(x,t)  Zc I (x,t) = V(x- , t- )  Zc I (x-  , t- ) (5.5) V(x,t) + Zc I (x,t) = V(x+ , t- ) + Zc I (x+ , t -) Having chosen m as the reference of distance x along the line, then the voltage and current values at m (i.e., x = 0) are: V (0,t)= Vm (t), and I(0,t) = - imk (t), Substituting these in Equations (5.5.): Vm (t)  Zc imk (t) = V(, t- ) + Zc I ( , t -) (5-6) Furthermore, x =  corresponds to node k. Hence: V(, t) = Vk(t), and I( ,t) = ikm (t), This leads us to derive the relations of terminal voltages and currents using the above boundary conditions: Vm (t)  Zc imk (t) = Vk(t- ) + Zc ikm (t -) (5-7) Vk (t)  Zc ikm (t) = Vm(t- ) + Zc imk (t -) Equations (5.7) are fundamental to the representation of a line or cable when conducting transient analysis. These equations can be rewritten as shown below for convenience: ikm (t) = Vk (t)/Zc  (1/Zc){ Vm (t- ) + Zc imk (t -) } imk (t) = Vm (t)/Zc  (1/Zc){ Vk (t- ) + Zc ikm (t -) } By defining: Ik(t) =  (1/Zc){ Vm (t- ) + Zc imk (t -) } Im(t) =  (1/Zc){ Vk (t- ) + Zc ikm (t -) } We obtain Equations (5.9) below: ikm (t) = Vk (t)/Zc+ Ik(t- ) (5-9) imk (t) = Vm (t)/Zc+ Im(t- ) Equations (5.9) are the basis of the representation of the line when taking into account the propagation of traveling waves. Now let us divide time t into discrete steps t which are sub-multiples of  such that  = pt where p is an integer, as illustrated in Figure 5.3 below. Then at time ti = it, we have: ti   = ti - pt= ti-p  Figure 5.3 Time discretization. However at time ti, Ik(ti-p) and Im(ti-p) are known and may in fact be considered as current sources with known values. Thus, the line can be represented by the following equivalent circuit diagram in Figure 5.4. This diagram is that of a conventional electrical circuit powered by current sources.  Figure 5.4 Representation of a line between nodes k and m. At any time ti, it is possible to calculate quantities Vk , Vm, ikm, imk as a function of other quantities since the values of the current sources Ik(ti-p) and Im(ti-p) are known at that instant of time. This representation of transmission lines and cables in the form of an electric circuit is the basis of the method of traveling waves. To enable the modeling of a network, it is necessary to propose models for each element of the network. In particular, the resistance, inductance and capacitance will be addressed in the following paragraphs. 5.3.3. Representation of a resistor Figure 5.5 shows a resistor that is connected across node k and m, with Vk and Vm representing the terminal voltages, and ikm the current flow from node k to node m.  Figure 5.5 Representation of a resistance. The voltage and current defined above are related by Ohms law: Vk Vm= R ikm If the resistor R is represented by its conductance GR = 1/R, then: ikm= GR (Vk Vm) where GR=1/R 5.3.4. Representation of an inductor The voltage across an inductor L that is connected between nodes k and m is related to the current flowing through it by: Vk  Vm = L dikm/dt This equation can be integrated between times t and (t  t):  EMBED Equation.3  (5-10) In equation (5.10), the integral to the right can be approximated by the trapezoidal method of integration since the time interval t is sufficiently small: ikm (t) - ikm(t- t)= (t/2L)[Vk(t) - Vm(t)+ Vk(t-t) - Vm(t-t)] From which the current ikm(t) is deduced : ikm (t)= (t /2L)(Vk(t) - Vm(t)) + ikm(t- t)+(t /2L) (Vk(t- t) - Vm(t-t)) (5-11 ) If we let: Ikm (t)= ikm (t)+ (t /2L)(Vk(t) - Vm(t)) and YL = (t/2L) Equation (5.11) simples to: ikm (t)= YL (Vk(t) - Vm(t)) +Ikm (t-t) (5-12) where at time t, Ikm(t - t) is known and can therefore be considered as a current source. Therefore, and inductor can be modeled by an equivalent electrical circuit as shown in Figure 5.6 below.  Figure 5.6. Representation of an inductor 5.3.5. Representation of a capacitor The voltage-current equation of a capacitor C connected between nodes k and m is as follows:  EMBED Equation.3  (5-14) After integration between the time intervals t and (t  t), the above equation becomes:  EMBED Equation.3  (5-15) Similarly, the integral to the right of (5.14) can be approximated using the trapezoidal formula:  EMBED Equation.3  (5-16) Solving for ikm (t) yields:  EMBED Equation.3  (5-17) By defining:  EMBED Equation.3  and  EMBED Equation.3  Equation (5.16) simplifies to: ikm (t)= Yc ( Vk(t)-Vm(t)) + Ikm (t-t) (5-18) where ikm(t - t) is a known quantity at time t and can therefore be regarded as a current source. This allows us to represent a capacitor by the equivalent circuit shown in Figure 5.7 below.  Figure 5.7 Representation of a capacitor. 5.3.6. Representation of a voltage source The electrical network often contains a number of voltage sources that are inserted between a pair of nodes, say k and m. Any voltage source ekm with internal impedance Z can be converted to an equivalent current source by applying Norton's theorem as shown in Figure 5.8 below. In the latter diagram, Y = 1/Z and Jkm= -Y ekm.  Figure 5.8 Representation of a source of tension. 5.3.7. Operating principle The total time t during which one wishes to carry out a transient analysis on a network is discretized in time intervals t that are sufficiently small to represent the phenomena sought with suitable accuracy (e.g., when sizing protection devices). The duration of the time interval t is chosen such that it is less than the largest sub-multiple of the propagation time k of electromagnetic waves over all the lines of the network under study. For each discrete time ti, the network is fully represented by a set of dipoles, each of which is defined by its internal admittance and possibly a current source according to the circuit representations of the various elements we coved in the previous section. Hence, at time ti, the network is defined by admittances and current sources which are known at this time because it is possible to calculate these currents from the earlier calculations at time ti-1. We can therefore consider that the operation of the network is defined at time ti by the moment matrix of admittances Ybus and the vector current sources Ibus. The network topology is known at each moment by the state ("open" or "closed") of circuit breakers located between the lines, cables and other devices. Hence, it is possible to construct the matrix Ybus and vector Ibus for each network topology. We recall that the terms of the matrix Ybus are defined by:  EMBED Equation.3 , Yij = - yij Similarly, the general term Ibus is defined by:  EMBED Equation.3  where the terms Iik represent the values of current sources attached to lines and inductive and capacitive elements, while Jik values are Norton current sources derived from the voltage sources. The global matrix equation of the network is: Ybus V = Ibus where V is the vector of network node voltages. The voltage Vi in each node in the network may be obtained by solving the above linear equation (e.g., by Gauss method). Knowledge of these voltages allows one to deduce the current sources in the various network elements (according to equations [5.7] to [5.16]). This will in turn allow us to calculate the voltage at the instant ti +1 = ti + t, and the iteration continues until we reach the time tmax where the calculation ends. Notes: 1) Introducing resistive losses: The basic assumption of this method is to neglect the resistance of the line to able to solve analytically the equations of propagation. This assumption is incompatible with the study of the transitional arrangements as it leads neglecting the damping of the waves during the propagation. One practice to overcome this major drawback is to divide the line into several sections and insert a resistance between adjacent sections. The sum of these insertions should add up to the total line resistance. Figure 5.9 below illustrates this concept where the line of length  is divided into 3 sections, and half of the total line resistance is inserted between two adjacent sections.  Figure 5.9. Resistance insertion between line sections. One expects that dividing the line into a higher number of sections and inserting smaller fractions of the total line resistance leads to more accurate results. However, a numerical analysis showed that the results do not change significantly after splitting the line into 3 sections as illustrated above. 2) Introduction of elements with non-linear features: Network studies under transients often take into account nonlinear elements such as transformers. However, it is still possible to use this method by segmenting the non-linear curve of such elements and approximating each segment by straight lines. 3) Processing the matrix Ybus: It was indicated in the chapter devoted to the construction of the admittance matrix Ybus that it is sparse, i.e., the majority of its elements are zero. It is recommended to use a special method to store only the non-zero elements of the matrix rather than the entire matrix. Similarly, when calculating the node voltages, we solve the linear system of equations often by triangularisation of the matrix. Such an algorithm transforms the linear system into an equivalent system where the elements of the matrix below the main diagonal terms are all zero. The resolution of such a system of equations is then fast and straight forward. Triangularisation of a matrix is by far the longest a costliest in terms of computing time. It is therefore advisable to organize the sequence of calculations in such a way to avoid frequent changes in network topology, since this leads to new matrices each of which requires triangularisation. 5.3.8. Illustration example In this subsection, we will consider an example that illustrates the details of the method including the different stages involved in such a study. This study shows a way to reduce overvoltages along line by inserting a resistor RF to dampen the amplitude of the voltage wave before propagating on the line. In this case, the maneuver is to gradually close secondary switches by varying the closing resistance, and to close when the main switch when the wave is sufficiently dampened. The network under study consists of a line connected through the closing resistor RF to a generator with electromotive force E0, resistance R0, subtransient inductance L0 and capacitance to ground C0. This line is equipped at both ends with compensating inductors L1 and L2 as indicated in Figure 5.10a.   Figure 5.10a. Network under study. Figure 5.10b. Equivalent circuit of network in Figure 5.10a. The application of the methods described above leads to the equivalent circuit shown in Figure 5.10b at any instant ti within the interval time of study. The transformation of the voltage source E0 to a current source results leads to the circuit in Figure 5.10c.  Figure 5.10c. Transformation of voltage souce to current source. Finally, the above circuit is simplified to that shown in Figure 5.10d after combining admittances and current sources. This final circuit has 5 nodes (with node 0 being a reference for voltage) and contains only current sources and admittances. Figure 5.10d. Final equivalent circuit after combining admittances and current sources. Setting the node equations of this network results in the admittance matrix Ybus as described in Section 5.3.5:  EMBED Equation.3  Similarly, we can calculate the vectors I and J of the second part of the matrix equation at each instant ti of the interval time to study:  EMBED Equation.3  the value of each element of the above vectors will be explained below. At time instant ti, the solution of the matrix equation YbusV = I+J gives the value of the voltage Vk(ti) in each node (V1, V2, V3, V4). This will enable us to calculate the instantaneous currents in all branches of the circuit: i01 (ti)= -YL0V1 + I01 (ti-1) i20 (ti)= YC0V2 + I20 (ti-1) i12 (ti) = YR0 (V1 - V2) i30 (ti)= YL1V3 + I30 (ti-1 ) i34 (ti) = YZcV3 + I3 (ti-1-) i43 (ti)= YZcV4 + I4 (ti-1-) i04 (ti) = YL2V4 + I40 (ti-1) These values in turn allow us to calculate the values of the components of the vector I: I01(ti) = i10 (ti) -YL0V1 I02(ti) = - i20(ti)  YC0V2 I30(ti) = i30 (ti) +YL1V3 I30(ti) = i40(ti) +YL2V4 I3(ti) = -(YZcV4 (ti-) + i34 (ti-)) I4(ti) = (YZcV3(ti-) + i43 (ti-)) So for a given time step, we calculate the node voltages, then the branch currents, then identify current sources which are needed to calculate the voltages the next time around. 5.4. Conclusions This method is widely used today because it is very powerful in terms of computation time required which, despite the high speed of current microprocessors, is still appreciable when large scale studies are to be carried out in the framework of a statistical evaluations. In addition, it is simple to implement, and recent advances in modeling and simulations are providing means of taking into account the non-linear nature of power electronic switching devices. Finally, this is the base of the well-known EMTP software that is well publicized and maintained [05 DEG, 05 AH]. However, we stress a disadvantage that must be remembered, namely the difficulty to control the accuracy of the results due to the nature of the assumptions made in the development of the method. One should therefore be careful when analyzing the results and to ensure the validity the method before launching a full scale simulation study. 5.5. Exercises 5.5.1. Exercise No. 17: Transient analysis on a line Figure E.17 shows a circuit diagram of an induction furnace that is powered from a thermal power plant through a transformer and a transmission line. The generator is represented by a constant e.m.f. E0, resistance R0, inductance L0, and capacitance C0. The station transformer is represented by a series inductor L1 and shunt capacitance C1. The transmission line with parameters Zc and , is compensated by a shunt inductor Lc. Finally, the furnace is represented by inductor Lf and is fed through an underground cable with parameters Lc2 and 2.  Figure E.17 Equivalent circuit of network under study. Represent the above network by an electrical circuit diagram for transient analysis using the method of traveling waves. Propose a way to take into account the resistance of the line and cable. 5.5.2 Exercise No. 18: Matrices and transient analysis The network in Figure E.18.1 below is part of the French transmission system operating at 380 kV. The total inductance, resistance, and capacitance of each of the 11 lines are described in Table E.18.1. Ignore the conductance of each line. The generators can be modeled as shown in Figure E.18.2 with parameter values as shown in Table E.18.2.  Figure E.18.1 Network schematic. Line1234567891011Length (km)237228145276120809012015015580 EMBED Equation.DSMT4 0,260,250,160,30,130,090,10,0130,170,170.09Rdt()7168438336242736454624 EMBED Equation.DSMT4 2,52,41,52,91,20,80,91,20,160,160.08 Table E.18.1 Line parameter values.  Figure E.18.2 Generator circuit model. GeneratorG1G2G3R ()1,01,00,5L (H)0,140,30,06C (F)0,0250,010,03 Table E.18.2 Generator parameter values. Select a tree containing nodes 1 - 6, then give for the graph corresponding to this network the following: the network incidence matrix C, the fundamental circuits and two examples of cuts. Determine the nodal admittance matrix of the network when operating at 50 Hz (2f H" 300 rad/s). Use ground as reference. Assume a reduced network contains nodes 1, 2, 3, 4. Derive an equivalent circuit of this sub-network that is suitable for transient analysis. 5.5.3 Exercise No. 19: Transient analysis under lightning strike We wish to conduct a transient analysis on a line under a lightning strike. To do so, consider the circuit in Figure E.19 that represents a generator, a transmission line, a surge arrester Rp, and a capacitor C at the line terminal.  Figure E.19 Circuit diagram under study. The generator is represented by an emf E0, a series inductance L0 and shunt capacitance C0. The arrester is represented by a variable resistor and is placed at 1/3 of the line length from the generator end. Finally, the line parameters are as follows: length:  EMBED Equation.DSMT4  travel time  EMBED Equation.DSMT4  resistance  EMBED Equation.DSMT4  Represent the above network with a schematic that is suitable for transient analysis using the theory of traveling waves. CHAPTER 6 Fault Current Calculations 6.1. Definition A short-circuit current generally occurs at the closure of any electrical circuit on a zero or low value impedance. The most common case is where a short occurs between one or more phases of a network and ground. Another case is the accidental connection between two phases with negligible impedance. The resulting short-circuit current is often referred to as fault current since this represents an abnormal operation of the system and associated electrical equipment. Fault currents are characterized by their very high values. These currents are several times the nominal current (over one order of magnitude), depending on the overall driving point impedance of the system at the fault location. 6.2. Effects of short-circuit conditions The effects and consequences of short-circuits are extremely damaging to the material and equipment of the electrical network. In Indeed, the electrical equipment installed on power systems is not designed to withstand such very high current values. These currents will therefore cause excessive heating which can cause equipment destruction and fire hazards. Various malfunctions can be observed as a result of a fault, especially in measuring apparatus (e.g., current transformers) and telecommunication equipment. It is therefore essential to protect the equipment from the effects of short-circuits. This function is generally provided by a device that can sense and interrupt very high currents, namely, the circuit breaker. However, these breakers are associated with protection devices which are equipped with algorithms for the detection of faults and coordination with other protection devices (selectivity) within the network. 6.3. Common causes of faults Causes of a short-circuit may be diverse: Weather conditions (lightning strike, wind, storm, etc.) in these severe weather conditions, dielectric breakdown (following temporary overvoltages caused by lightning for example) or the partial material destruction of equipment can cause short-circuits; Aging of the material: this phenomenon leads to rupture or local failure which could result in short-circuits; Lack of electrical insulation: this defect is often at the root of some short-circuits where insulation is no longer able to maintain its function, thus leading to a contact between phases or between turns of a transformer for example. The accidental contact between a line air and vegetation (tree) is considered as a defect in insulation since the insulation distance between the line and the vegetation (earth) is no longer respected for whatever the reason (lack of maintenance or tree trimming); Lack of flexibility and human error: This type of maneuver or error often causes severe short-circuits. These can vary from the oblivion of a grounding during maintenance, to an operational error that causes a temporary overvoltage which triggers a dielectric breakdown; Vandalism: throwing an iron bar on an overhead line inevitably causes a short circuit; Animals: birds often cause short-circuits. The stork case of Portugal is famous. Indeed, in May 2000, the wings of a stork got stuck between two phases of an overhead line of Portuguese grid. An inadequate response of the protection system has led to a power failure encompassing approximately half of the Portuguese network. 6.4. Importance of short-circuit current calculations Knowledge of the current values that may be caused by short-circuits is essential for the design of protective equipment (breaker and protection). For circuit breakers, these current magnitudes are required for the dimensioning terms of the breaking capacity. The interrupting capacity of a breaker sufficient to interrupt the highest possible short-circuit current that may occur in the protected area. The calculation of short-circuit currents does not only imply knowledge of the current at the fault location, but also the distribution of these current flows on other lines and phases of the network and their impact on the voltages. These values allow us to choose the most suitable protection for each type of network. The short-circuit currents, although they give rise to when additional transitional schemes are often calculated using steady-state analysis, although they occur during a transient state. Hence, the remainder of this chapter is devoted to fault calculations using such calculations. On the other hand, with the exception of a balanced fault (short-circuit in all three phases), an unsymmetrical fault causes imbalance in the currents and voltages. Therefore, the calculation methods based on a perfectly balanced network which allows per-phase analysis is longer valid in this case. We therefore call for specific methods for addressing unsymmetrical or unbalanced circuits. The most popular of such methods is based on Fortescue transformation and is referred to as "symmetrical components". We therefore introduce and apply this method for calculating short-circuit currents. But before doing so, we will briefly review the treatment of multiphase unbalanced situations. 6.5. Types of short-circuits There are mainly two types of short-circuits: Symmetrical short-circuit: this consists of a balanced three-phase fault (either phase-to-phase-to-phase or phase-to-phase-to-phase-to-ground). These are illustrated in Figure 6.1. Asymmetrical short-circuit: this consists of the following unbalanced faults which are illustrated in Figure 6.2: Single-phase-to-ground; phase-to-phase (isolated) phase-to-phase-to-ground (grounded) For both types of short-circuits above, the faults may be bolted (i.e., with zero fault resistance or impedance) or with a fault impedance. The three-phase short-circuits generally generate higher fault currents. Therefore, the interrupting capacity of circuit breakers is often defined by the values of these currents.  Figure 6.1. Balanced short-circuit through fault impedance.   (a) (b) (c) Figure 6.2. Diagram of (a) phase-to-phase fault, b) phase-to-phase-to-ground fault, c) single-phase-to-ground fault. The impedance Zf shown in Figures 6.1 and 6.2 represents the fault impedance. If Zf = 0, then the short-circuit is said to be "bolted". It should be noted that the single-phase-to-ground fault is the most common type of fault in electrical power networks (about 60 to 70% of all types of faults). 6.6. Notion of short-circuit power The concept of short-circuit power is often used to illustrate the robustness of the network at a particular point. The higher this power, the more robust is the network seen from the point in question. This is illustrated in particular by low voltage variations when a fault occurs in this network. This concept is fictional since it combines the short-circuit current with that of the nominal voltage. The general expression of the short-circuit power is:  EMBED Equation.3  (6-1) Where U0 and Un correspond to the line-to-line voltage before the fault and the nominal value, respectively, and Icc is the short-circuit current. The short-circuit power can also provide an idea of the severity of a short-circuit current in a given network. As will be seen later, once the current terms court-circuit currents are known, it is very easy to determine the value of the short-circuit power at a given point on the network when using per-unit (pu) values. 6.7. Polyphase balanced and unbalanced systems As stated previously, polyphase networks are assumed to be balanced and symmetrical under normal steady-state operation where the generated power matches system load demand (including losses). Indeed, the networks are statistically considered balanced in practice, except for some abnormal operating conditions (such as unbalanced fault conditions). Hence, their study can therefore be reduced to a per-phase analysis which greatly simplifies the calculations involved in various studies, e.g., load flow, voltage drop, or system losses. When an asymmetrical fault or a pronounced imbalance occurs in the network, the per-phase is no longer applicable. In this case, we must represent and model the phenomena that occur in all three phases. These are usually lengthy and complicated calculations that power engineers wish to avoid. Thanks to the method of "Symmetrical components" or Fortescue method that allows considerable simplifications in the analysis of unbalanced circuits. These methods actually work on balanced networks via symmetrical transformations. The principle of these transformations is based on the fact that an unbalanced 3-phase network can be decomposed into 3 balanced systems each containing 3-phases that are referred to as the symmetrical components" of the unbalanced system. The method is general and applies to any polyphase system containing n phases. 6.7.1. Balanced three-phase systems  There are 3 separate balanced three-phase systems (for a given angular frequency , magnitude "G" and phase ). These systems called( "direct", "inverse" and "homopolar". By adopting the Fresnel representation (vector), these systems are represented by the following formulas: direct inverse homopolar (clockwise) (counter-clockwise) Figure 6.3. Three-phase systems (direct, inverse and homopolar) In the equations of these systems above, the variable "vector" g represents the electrical (voltage, current or emf), G amplitude and phase shift  relative to a given reference. In practice, the electromotive forces induced at the terminals of generators are balanced and represent a direct system. Hence, the normal steady-state system operates in a direct mode. 6.7.2. Complex representation The quantities representing the balanced system of Figure 6.3 can also be represented by complex numbers as: g=G ej or in the Anglo-Saxon literature as: g=G ( This notation defines an operator representing the complex phase shift of the three-phase quantities in relation to another, i.e., 2/3: a = ej2/3 = cos(2/3) + j sin (2/3) This operator is referred to as the "operator of 3-phase rotation . Therefore, we can conclude that: a2 = ej4/3 and: a3 =1 Hence: a + a2 + 1 = 0 representing a balanced system where the sum of the 3 quantities is equal to zero. We can then represent the balanced system using one variable, such g1: Balanced system direct:  EMBED Equation.3 ; a-1  EMBED Equation.3 ; a-2  EMBED Equation.3  ( EMBED Equation.3 ; a2  EMBED Equation.3 ; a  EMBED Equation.3 ) Balanced system inverse:  EMBED Equation.3 ; a  EMBED Equation.3 ; a2  EMBED Equation.3  6.7.3. Symmetrical components C.L. Fortescue has shown that any 3-phase asymmetrical system can be decomposed into: - A balanced three-phase direct: direct component; - A balanced three-phase inverse: inverse component; - A balanced three-phase homopolar: homopolar component. A three-phase sinusoidal system can be expressed as a linear combination of these three components. Let the vector of phase currents Ip and phase voltages Vp be written as:  EMBED Equation.3  (6-2) where currents (ia, ib, ic) are the unbalanced currents, (va, vb, vc) represent unbalanced voltages. These phase currents and voltages can be written as a linear combination of other symmetrical currents and voltages as follows:  EMBED Equation.3  (6-3) where the index "s" is associated with quantities representing the symmetrical components, and the index "p" representing the phase quantities. In the case of symmetrical components, Is and Vs are vectors containing components representative of direct inverse and homopolar (see Figure 6.3). The elements of the transformation matrix T in Equation (6-3) are constant (complex or real) and independent of time. This transformation allows us to express the unbalanced phase quantities in terms of other quantities that are symmetrical and balanced. Thus, each phase current or voltage can be expressed as a linear combination of direct, inverse and homopolar components. In the case of voltages:  EMBED Equation.3  (6-4a) where: Va,b,c are the voltages of phases a, b, and c, respectively; and Vd, i, h are the direct, inverse and homopolar symmetrical components, respectively. The same applies to the currents:  EMBED Equation.3  (6-4b) where: Ia,b,c are the currents of phases a, b, and c, respectively; and Id, i, h are the direct, inverse and homopolar symmetrical components, respectively. We therefore express a system of 3 unbalanced variables as a system of 9 variables belonging to balanced 3 balanced systems. Let us first take a look at the characteristics of a balanced system:  EMBED Equation.3 (6-5) Thus, the system of equations (6-4) becomes:  EMBED Equation.3  (6-6) We note that all quantities are expressed in terms of the symmetrical components of phase "a" (subscript). To simplify the notation, we can then remove this subscript for convenience, and Equation (6-6) becomes:  EMBED Equation.3  (6-7) The above system of equations can be conveniently written in matrix form:  EMBED Equation.3  and  EMBED Equation.3  (6-8a) Or in abbreviated form as:  EMBED Equation.3  et  EMBED Equation.3  (6-8b) where:  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3  and  EMBED Equation.3  The above relation expresses the phase components in terms of the symmetrical components. On the other hand, we can express the symmetrical components in terms of the phase components by inverting the transformation matrix T:  EMBED Equation.3  and:  EMBED Equation.3  where:  EMBED Equation.3  Or in the expanded form of Equation (6-7):  EMBED Equation.3  (6-9) 6.7.4. Powers in terms of symmetrical components The expression of the complex three-phase power is given by the following formula:  EMBED Equation.3  =  EMBED Equation.3  (6-10) where:  EMBED Equation.3  represents the complex conjugate vector of phase current; and  EMBED Equation.3  represents the transposed vector of phase voltages. Using the expressions given by equation (6-3), we deduce:  EMBED Equation.3  But the product:  EMBED Equation.3  = 3 [I] where [I] is the identity matrix. Therefore:  EMBED Equation.3  (6-11) The expression (6-11) shows that the total power of a 3-phase system is equal to 3 times the sum of the powers of symmetrical components. This also shows pseudo power invariance. 6.7.5. Symmetrical components and impedance/admittance matrices The vectors of phase voltages and the injected phase currents at different nodes are related by the impedance/admittance matrix as follows:  EMBED Equation.3  (6-12) Expressing the phase quantities in terms of symmetrical components according to equation (6-3) yields:  EMBED Equation.3  Therefore:  EMBED Equation.3 (6-13a) Hence:  EMBED Equation.3  and  EMBED Equation.3  (6-13b) The above impedance matrix EMBED Equation.3 and admittance  EMBED Equation.3 correspond to the system matrices transformed in symmetrical components. These relate the vectors of the symmetrical components of voltages and currents. This transformation allows the diagonalization of matrices, hence the decoupling of systems decoupled. This property shows the importance of symmetrical components as it simplifies the calculations associated with unbalanced circuits such as unsymmetrical faults. 6.7.6. Notion of circulating matrices Consider a system of 3 impedances (or 3 admittances), coupled or not, with a symmetry of order 3, i.e., insensitive to a circular permutation at its terminals. This system is characterized by a cyclic impedance matrix (or admittance). From the relation (6-12), we have:  EMBED Equation.3  with:  EMBED Equation.3  and  EMBED Equation.3  (6-14) When these matrices are transformed to symmetrical components via the relations (6-13), they become diagonal. In the case of impedance we have:  EMBED Equation.3  (6-15) Obviously the transformed admittance is also diagonal. The expression of the above impedance may be written as:  EMBED Equation.3  Where  EMBED Equation.3  It can be seen that the transformation of Fortescue of any cyclic matrix results in a diagonal matrix. This transformation decomposes a three-phase system into a linear combination of three independent and balanced systems which are much easier to analyze. This method applies not only to 3-phase systems, but also to systems with a higher number of phases. Thus, the relationship Vs = [Zs] Is results in a decoupled system when transformed into symmetrical components:  EMBED Equation.3  Thus, if the cyclic impedances are fed by an unbalanced set of currents, we obtain 3 single-phase systems where each is characterized by a single cyclic impedance: direct Zd, inverse Zi, and homopolar Zh. This clearly shows the decoupling resulting form the Fortescue transformation. 6.7.7. Case of synchronous machines A synchronous machine under balanced load condition is characterized by complete decoupling between the phases. Under these conditions, the terminal voltages of a synchronous generator are expressed by:  EMBED Equation.3  (6-16) where:  EMBED Equation.3 : vector of the e.m.f of the synchronous machine;  EMBED Equation.3 : vector of terminal voltage of the generator;  EMBED Equation.3 : cyclic characteristic impedance matrix (not diagonal) of the synchronous machine;  EMBED Equation.3 : vector of phase currents of the synchronous machine. As stated earlier, in the case of three-phase synchronous machines, the induced electromotive forces  EMBED Equation.3 are balanced and considered direct in terms of symmetrical components. The inverse and homopolar symmetrical components are therefore void. On the other hand, the currents  EMBED Equation.3 and voltages  EMBED Equation.3 can be unbalanced if the machine feeds an unbalanced load or network. By applying the transformations given by the relationship (6-8b) on phase voltages and currents, we get:  EMBED Equation.3  (6-17) Multiplying the equation (6-17) on both sides by [T]-1, we obtain:  EMBED Equation.3  (6-18a) which results in:  EMBED Equation.3  (6-18b) The product term of the above equation EMBED Equation.3  can be written as :  EMBED Equation.3  =  EMBED Equation.3  (6-18c) Hence:  EMBED Equation.3 =  EMBED Equation.3  (6-18d) The relationship [6.18d] shows once again that the electromotive force of a synchronous machine contains only the direct component, i.e., it has no inverse component nor homopolar component. Knowing that the matrix EMBED Equation.3  is diagonal, equation [6.16] finally simplifies to:  EMBED Equation.3  (6-19a) and therefore:  EMBED Equation.3  (6-19b) As expected, the decoupling between the 3 systems is clearly noted in the above equation. 6.7.8. Short-circuit current calculations To illustrate the application of the method of symmetrical components for the calculation of short-circuit currents, particularly in asymmetrical cases, we will first illustrate this method on a simple case of a synchronous machine under no load. We then generalized the method for complex networks. 6.7.8.1. Single-phase-to-ground fault 6.7.8.1.1. Case of a short-circuit through an impedance Let a short-circuit occur between phase a and ground as illustrated in Figure 6.4 below. The current circulating in phase Ia is denoted by Isc during the fault. In the other phases, the current Ib and Ic are obviously zero. In this case, we assume that the following quantities are known: E: e.m.f. vecotor of synchronous machine;  EMBED Equation.3 : diagonal impedance matrix consisting of direct, inverse, and homopolar impedances of synchronous machine; Zf: fault impedance.  Figure 6.4. Phase-to-ground fault at the terminals of an alternator. The vector of the phase currents for this particular fault is equal to:  EMBED Equation.3  The symmetrical components of the current are computed by:  EMBED Equation.3  Substituting the elements on the right yields:  EMBED Equation.3  (6-20a) Therefore:  EMBED Equation.3  Since:  EMBED Equation.3  The transformed voltages are calculated as follows:  EMBED Equation.3  (6-20b) On the other hand:  EMBED Equation.3  Then,  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  Finally we have:  EMBED Equation.3  (6-20c) where Icc is he short circuit current. The resulting voltages at the generator terminals are calculated next. The relation between phase b voltage and the 3 sequence voltages is:  EMBED Equation.3  by replacing the values of  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  of expression [6.20b], we find:  EMBED Equation.3  We further replace the value of  EMBED Equation.3  has the expression given by [6.20c] and obtain:  EMBED Equation.3  The voltage  EMBED Equation.3  is obtained in a similar way. The voltage  EMBED Equation.3  can be obtained using the relation:  EMBED Equation.3  6.7.8.1.2. Case of a bolted phase-to-ground fault A bolted fault is characterized by zero fault impedance. Setting Zf = 0 in the expressions of short-circuit current and voltages derived in the previous case yields:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  NOTE : If we calculate the sum  EMBED Equation.3 + EMBED Equation.3 , we obtain:  EMBED Equation.3 + EMBED Equation.3 =3 EMBED Equation.3  knowing that:  EMBED Equation.3 = EMBED Equation.3  we deduce that:  EMBED Equation.3 + EMBED Equation.3 =3( EMBED Equation.3 ) = EMBED Equation.3  Therefore:  EMBED Equation.3  The latter expression shows that it is possible to determine the magnitude of the homopolar impedance of a circuit (alternator and other equipment) in question. This is achieved by measuring the vector sum of the two unfaulted phase voltages and dividing the result by the short-circuit current. This method to determine the homopolar impedance is valid only for single-line-to-ground bolted faults. 6.7.8.2. Phase-to-phase-to-ground fault  Let a short-circuit occur between phases b, c and ground as illustrated in Figure 6.5 below. Note that the fault being considered in bolted (i.e., Zf = 0), but the procedure can easily be expanded to include a fault impedance. Figure 6.5. Bolted two-phase-to-ground fault at the terminals of an alternator. This short-circuit is therefore characterized by the following phase quantities:  EMBED Equation.3  and:  EMBED Equation.3   EMBED Equation.3  The objective is to calculate the short-circuit current Icc and that the voltage of phase a. We begin with expression [6.19b] and calculate the homopolar current component:  EMBED Equation.3  Then we calculate the direct, inverse and homopolar voltage components:  EMBED Equation.3  Knowing that  EMBED Equation.3  and  EMBED Equation.3  are zero, we deduce:  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  The fact that:  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  results in:  EMBED Equation.3  Similarly,  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  Leads to:  EMBED Equation.3  The above expression gives us the direct current component as a function of short-circuit current:  EMBED Equation.3  This expression can also be re-written as follows:  EMBED Equation.3  Since,  EMBED Equation.3 =0, we obtain:  EMBED Equation.3  Finally we determine the short-circuit current in terms of known quantities:  EMBED Equation.3  (6-21) For the phase a voltage, we have:  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 =3 EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 . Substituting the short-circuit current in the above expression yields:  EMBED Equation.3  (6-22) NOTE: Let us calculate the current of phase b:  EMBED Equation.3  After replacing the direct, inverse and homopolar current values, and diving both sides of the equation by the short-circuit current, we obtain:  EMBED Equation.3  (6-23) The above expression indicates that the current in the faulted phases (b or c) can be higher than the short-circuit current. It is therefore essential to take precautions in this regard to avoid any overheating, especially by measuring the current in one the two phases in question. NOTE Consider the ratio of the unfaulted phase voltage over the short-circuit current:  EMBED Equation.3  Therefore, we can obtain the magnitude of the homopolar impedance Zh simply by measuring the voltage Va and short-circuit current Icc (being careful of course with the faulted phase currents). This is one of the methods that may be used to determine experimentally the homopolar impedance of a circuit. Once again, this is valid only when this type of fault is bolted. 6.7.9. Other types of faults To avoid burdening this chapter with detailed calculations for other types of faults, we limit the presentation to the two previous cases. However, the method described previously also applies to the isolated phase-to-phase fault and to a symmetrical fault. Of course, calculating the short-circuit current in the latter type of fault can be achieved without using the method of symmetrical components. But it is worth noting that symmetrical components are general, and therefore apply also for the symmetrical case (where the inverse and homopolar components are no-existent). 6.8. Generalization of fault calculation in complex networks The examples presented in the previous section illustrate the application of the method of symmetrical components for a simple circuit that consists of one synchronous machine operating under no load condition. However, electricity networks are much more complex and consist of hundreds or thousands of components including synchronous machines, overhead lines, underground cables, transformers, and loads. These systems are therefore of great dimension, especially when considering the fact that the loads are continuously changing. In the sections that follow, we generalize the concepts presented above for the case of complex networks. We will deal with the short-circuit calculations for these networks in the case of symmetrical and asymmetrical faults. We first begin with a symmetrical fault, and then introduce a systematic calculation method for asymmetrical faults. We will see that this generalization always describes the network in question by its Theremin equivalent looking from the faulted point in the system. Of course, in the case of asymmetrical faults, the will be crucial to have the 3 direct, inverse and homopolar equivalent networks. 6.9. Three-phase symmetrical fault current calculations We will first introduce the concept, then, generalize to large networks using network matrices, including transfer matrices. The method of calculation in this case is based on the application of the Thevenin method. Indeed, the Thevenin theorem, along with the superposition principle in linear systems, is very useful when attempting to determine changes in the currents and voltages as a result of adding and impedance between two points in the electrical circuit. This method allows to the short-circuit as a change that the network configuration. We can then easily apply the method of superposition of two network states: the state before the fault and the state after the fault. Thevenin's theorem states that "changes affecting voltages and currents in a linear network as a result of adding an impedance between two nodes in a network are identical to the voltages and currents caused by an e.m.f. (place behind series impedance) of the same amplitude and phase angle of the voltage before the change in the network with all other voltage sources short-circuited. "  Figure 6.6. Fault impedance and Thevenin equivalent circuit. In the circuit of Figure 6.6 above, the fault can be represented by the addition of the branch impedance Zf (fault impedance) to the Thevenin equivalent circuit (ETH in series with ZTH). The current resulting from this change is the short-circuit or fault current. This current is given by:  EMBED Equation.3  (6-23) with:  EMBED Equation.3 : Thevenin voltage identical to the pre-fault voltage  EMBED Equation.3 at the fault location;  EMBED Equation.3 : Thevenin impedance of the network. This corresponds to the impedance measured for the fault point with all voltage sources short-circuited;  EMBED Equation.3 : fault impedance. In the case of a bolted fault (Zf = 0), the above expression simplifies to:  EMBED Equation.3  The problem is then to determine the Thevenin equivalent circuit of the network when looking from the fault location. On this basis, we will address the general problem of calculating systematical short-circuit currents in power systems. The resulting fault currents in the branches calculated by this method are variations caused by the short-circuit (i.e., after adding the branch Zf). The resulting current  EMBED Equation.3  in each branch of the network is thus the sum of current values prior to the fault  EMBED Equation.3  and the variation  EMBED Equation.3  caused by the fault, i.e.,:  EMBED Equation.3  (6-24) Likewise, the voltage  EMBED Equation.3 at node i is equal to the sum of the voltage before the fault  EMBED Equation.3  and the variation caused by the fault:  EMBED Equation.3  (6-25) 6.10. Symmetrical fault current: systematic approach When it comes to fault calculations on large networks, the Thevenin equivalent become difficult to obtain, especially for the impedance  EMBED Equation.3 . It is therefore essential to have a systematic calculation approach that can be incorporated into any model. This approach is based on the use of the transfer matrices of the network, including the  EMBED Equation.3  matrix presented in the previous chapters. In a given network, the injected current represented by the vector  EMBED Equation.3  and node voltages represented by the vector  EMBED Equation.3  are bound by the following relationship:  EMBED Equation.3  (6-26a) where  EMBED Equation.3  represents the transfer impedance matrix of the network;  EMBED Equation.3  being the vector of node voltages, and  EMBED Equation.3  being the vector of injected currents. These are illustrated in Figure 6.7 below.  Figure 6.7. Network transfer impedance matrix. We can now generalize the above method based on Thevenin theorem: Consider a network that is represented by its transfer impedance matrix EMBED Equation.3 . It should be noted that the matrix  EMBED Equation.3 must also integrate reactances of the synchronous machines in the network. Indeed, these reactances have an impact on the magnitude of the fault current as well as resulting current flow through various paths. The changes in currents and voltages following a short-circuit at node r of the network can be expressed by:  EMBED Equation.3  (6-26b) In the above expression, the vector of injected change in currents  EMBED Equation.3 contains only one value which corresponds to fault current at node r, i.e., the vector  EMBED Equation.3 is written as:  EMBED Equation.3  (6-27) By convention, the injected current  EMBED Equation.3 is in the opposite direction of short-circuit current EMBED Equation.3 :  EMBED Equation.3 = EMBED Equation.3  The transfer impedance matrix is expressed by  EMBED Equation.3  where  EMBED Equation.3 is the transfer admittance admittances of the network. By substituting equations [6.27] and [6.28] in [6.26b], we obtain:  EMBED Equation.3  6-(29) And according to the relation [6.25], we have:  EMBED Equation.3  (6-30) Finally, the post-fault node voltages are computed by:  EMBED Equation.3  for  EMBED Equation.3  (6-31) Knowing that  EMBED Equation.3 , we therefore have  EMBED Equation.3 , which gives us:  EMBED Equation.3  (6-32) The relation [6.32] is similar to the relation [6.23] after substituting:  EMBED Equation.3  and  EMBED Equation.3 . Thus, the Thvenin equivalent impedance  EMBED Equation.3  is obtained directly from the transfer impedance matrix EMBED Equation.3 . Therefore, one can substitute  EMBED Equation.3  in place of the Thevenin equivalent impedance at node r, as illustrated in Figure 6.8 below.  Figure 6.8. Thvenin equivalent circuit at node r. Finally the post-fault node voltages are computed by:  EMBED Equation.3  for  EMBED Equation.3  (6-33) Relations [6.33] are general and apply to any balanced three-phase fault occurring in the network. Relation [6.32] is used to calculate the fault current as a function of the Thevenin equivalent impedance at the fault point, the voltage before the fault at that point, and the fault impedance at that point. 6.11. Expression of short-circuit current and short-circuit power We saw in section 6.6 that the short-circuit power is calculated using the relation:  EMBED Equation.3  Knowing that the short-circuit current is given by the relation [6.32] and assuming a bolted fault (Zf = 0):  EMBED Equation.3  We then have:  EMBED Equation.3  (6-34) Expressing the relation [6.34] in per-unit values:  EMBED Equation.3  Knowing that the voltage is at or near it nominal value (i.e., base or reference voltage), we can approximate the above relation as follows:  EMBED Equation.3  (6-35) Where  EMBED Equation.3  is the per-unit (pu) value of the Thvenin impedance. We therefore write:  EMBED Equation.3 , The above relationship indicates that the per-unit value of the short-circuit power at a node is equal to the inverse of the per-unit value of the Thevenin impedance at that node. 6.12. Unsymmetrical fault current calculations 6.12.1. Generalization of symmetrical components As mentioned in section 6.8, electricity networks are often large. Hence the calculation of short-circuit current in the case of asymmetrical faults requires a generalization of the method of symmetrical components. This ensures to always transform the network into 3 balanced equivalent networks corresponding to direct, inverse and homopolar components. When an asymmetrical fault occurs within a network, we always begin by determining the 3 equivalent symmetrical networks (direct, inverse and homopolar) viewed from the fault, and each network is composed of only its corresponding sequence. 6.12.1.1. Equivalent network of direct system This contains all the components of the network of origin: Electromotive forces of the synchronous machines, Impedances of different components (cables, transformers, etc.) This network characterizes the direct current  EMBED Equation.3  and direct voltage  EMBED Equation.3 components. 6.12.1.2. Equivalent network of inverse system Typically, this network is similar to the network of the direct sequence above except: It does not contain electromotive forces; One needs to replace the impedances of different components by their inverse values. So this is a purely passive network which characterizes the inverse current  EMBED Equation.3  and inverse voltage  EMBED Equation.3  components. 6.12.1.3. Equivalent network of homopolar system The determination of this equivalent network particularly depends on the existence of path for zero-sequence (or homopolar) currents in the network under study. Hence this network is often different from the two earlier networks, and like the inverse network, it is passive with no voltage sources. Indeed, this network often contains fewer components or parts and the value of the homopolar impedances are quite different from the direct or inverse impedances. Most importantly, the network connection depends on how the transformers are connected (Delta, grounded-Wye, or ungrounded-Wye) and whether other components are connected to ground. In short, the homoploar network is very dependent on the paths for the flow of homopolar currents. The following subsections provide some illustrations on this subject. 6.12.2. The neutral and homopolar currents To illustrate the conditions of existence of homopolar currents, we simply consider a system of three-phase impedances connected in Wye as shown below.  Figure 6.9. Neutral current and homopolar current In Figure 6.9, we have:  EMBED Equation.3  Note that if the neutral is disconnected, then the sum of phase currents is zero. The consequence is that the homopolar current is forced to be zero, even in case of imbalance. In this case, only the direct and inverse current components ( EMBED Equation.3 and  EMBED Equation.3 ) will be present. We can therefore conclude the following: A homopolar system can exist only in the presence of a connected neutral. In case where the loads are connected in a Delta, we have of course the sum of current zero. Therefore, the homopolar current cannot exist. However, in this case, homopolar currents can circulate within the branches of the Delta connection. This situation occurs particularly in the case of transformers where the primary side is connected in grounded-Wye (thus allowing hompolar current flow) and the secondary side is connected in Delta/ 6.12.3. Impedances of network components To determine the equivalent networks representing the direct, inverse and hompolar systems, we need to know the network elements and their individual sequence impedances. The components that come into consideration are often rotating machines, transformers and power lines. Among these elements, lines and transformers are static while the machines are active devices. The static aspect is manifested by direct and inverse components. On the other hand, homopolar components are dependent on couplings (for transformers) and ways of dealing with the neutrals (when they exist). In what follows, we examine schematic equivalents of the symmetrical components of these various network elements. 6.12.3.1. Impedance of rotating machines Rotating machines can have 3 different impedances ( EMBED Equation.3 , EMBED Equation.3  and  EMBED Equation.3 ). These impedances are essentially inductive reactances, although we refer to these as impedances. For the direct and inverse impedances, they depend on the reactance corresponding to each system mode (direct or inverse). In cases of synchronous machines, for a system of direct current, the field caused by armature reaction turns synchronously with the rotor field; hence we have the effective reactance that corresponds to the synchronous reactance. The impedance of the direct system is simply the synchronous reactance. For a system of inverse currents, the field of armature reaction runs in reverse relative to the rotor field, hence the effective reactance that corresponds to the reactance of the inverse system. The latter can have a lower than the reactance of the direct system. In case of a short-circuit, the first moments of fault current involve the subtransient reactance of the machine. In practice, we often ignore the subtransient current because it is dampened in just few cycles. Hence, we usually consider either the stead-state or transient state. The latter (involving the transient reactance) is used to calculate the short-circuit current that is consistent with the cumulative operating time of protective relays and circuit breakers. These transient reactances depend on the type machine (round or salient-pole rotor). The homopolar impedance of a synchronous generator depends on how whether the neutral of the machine is either directly connoted to ground or through a grounding impedance (which can be reactance or resistive). The value of this impedance is generally lower than that of the inverse or direct impedance. These different impedances can be obtained either analytically or experimentally. 6.12.3.2. Impedances of lines and transformers Lines and transformers are static elements. As a result, the direct and inverse impedances are identical since they are insensitive to phase permutations. In the case of transformers, these correspond to the leakage impedances that can be obtained from short-circuit tests (Kapp diagram). Therefore:  EMBED Equation.3 = EMBED Equation.3  For the homopolar impedance, the situation is different for lines and transformers. 6.12.3.3. Homopolar impedance of lines For lines, the homopolar impedance depends on the return paths of homopolar currents. This path is a combination of neutral wires and the earth. Hence it depends on soil type, moisture and other empirical factors. For these reasons, the exact value of this impedance is not easy to obtain; hence, we often resort to approximate or typical values. 6.12.3.4. Homopolar impedance of transformers Three-phase transformers play a fundamental role in transmission (or blockage) of homopolar currents. This is made possible by two features: the type of connection (Wye / Delata) and the method of grounding the neutral (in case of Wye connection). Indeed, the mode of grounding dictates the existence of homopolar currents, and the connection type determines whether there is circulation of these currents (from one side of the transformer to another). We will therefore note, in what follows, that size of the network equivalent corresponding to the homopolar system is often determined by the type of transformer connection and their grounding method (if any). The homopolar impedance of a transformer varies widely as it can from one order of magnitude larger than the direct or inverse impedance (in case there is a free flow of homopolar current), to infinity (in case there is no path for homopolar current). We usually determine these impedances experimentally. The methodology is similar to that of determining the series impedance (through a short-circuit test). The difference here is to feed the transformer with a homopolar voltage source and measure the resulting current. Figure 6.10 below illustrates the concept for measuring the homopolar impedance of a transformer. Thus, the homoplar impedance can be directly obtained by:  EMBED Equation.3   Figure 6.10. Schematic for measuring the homopolar impedance of transformers. We now consider some common transformer connections to illustrate the possible values of this impedance. 6.12.3.4.1. Wye-Wye connection with both neutrals grounded(Yn/Yn) Grounding the two neutrals guarantees the free movement of homopolar current from one side of the transformer to the other. These currents are limited only by the leakage impedance of the transformer. Therefore, this impedance is equal to the homopolar impedance, as well as the direct and inverse impedances:  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  The equivalent homopolar circuit of this particular connection is shown in Figure 6.11 below. PrimarySecondaryequivalentEMBED Word.Picture.8 EMBED Word.Picture.8 Figure 6.11. Equivalent homopolar circuit for Yn/Yn transformer connection. 6.12.3.4.2. Wye-Wye connection with one grounded neutral (Yn/Y) This connection, where the primary neutral is grounded and the secondary neutral is floating (ungrounded), offers no path to the flow of homopolar currents. From the secondary side, the homopolar current sees an open circuit. Therefore, the impedance looking from the secondary side is infinite. From the primary side, one sees the magnetizing impedance of the transformer. This latter has a very high value when compared to the short-circuit impedance. The equivalent homopolar diagram for this connection is shown in Figure 6.12 below. PrimarySecondaryequivalent EMBED Word.Picture.8  EMBED Word.Picture.8   EMBED Word.Picture.8  Figure 6.12. Equivalent homopolar circuit for Yn/Y transformer connection. As the impedance of the transformer under no load is very high, we often neglect the excitation current flowing through this impedance. We can therefore consider it as having a nearly infinite value. The equivalent circuit for the homopolar currents would then be a open circuit on both sides of the transformer. 6.12.3.4.3. Wye-Delta connection Two cases may arise depending on whether the neutral of the Wye connection is grounded or not (i.e., either Yn/D or Y/D). Knowing that the homopolar current can circulate in the windings only if the neutral of the Wye is grounded, we deduce that in case of (Y/D) connection, the homopolar currents have no path to flow. This means an open circuit, or the equivalent impedance on both sides is infinite. With the neutral grounded (Yn/D), the homopolar currents can flow on the primary side. The induced homopolar currents on the secondary side are able to flow in the mesh formed by the Delta connection, but cannot exit to the secondary lines. These homopolar currents will therefore be trapped within the secondary windings. On the other hand, the homopolar currents in the primary side (Yn) are free to flow in the lines and will be limited only by the short-circuit impedance of the transformer. The equivalent circuit seen from the secondary side by the homopolar currents is an open circuit as these currents do not leave the Delta. The equivalent homopolar diagram for the Yn/D transformer connection is shown in Figure 6.13 below: PrimarySecondaryequivalent EMBED Word.Picture.8   SHAPE \* MERGEFORMAT  Figure 6.13. Equivalent homopolar circuit for Yn/D transformer connection. 6.12.4. Example of generalization on a complex network Considering the network shown in Figure 6.14 below.  Figure 6.14. Example network with a fault at node C The network contains two generators, two transformers and one power transmission line. The fault occurs at node C. We will therefore determine the equivalent diagrams for the direct, inverse and homopolar system seen from the fault location C. For simplicity, we assume that all impedances consist of pure reactances. The equivalent circuit of the direct system for a fault at node C is shown in Figure 6.15a below:  (b) Figure 6.15. a) Schematic of direct system; b) Thevenin equivalent seen from node C. Figure 6.15b shows the Thevenin equivalent circuit looking from node C. Thus, the Thvenin impedance  EMBED Equation.3 in this case is equal to the equivalent impedance ( EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 ) in parallel with ( EMBED Equation.3 + EMBED Equation.3 ). The Thvenin voltage is equal to the voltage at node C before the fault occurs. Similarly, the equivalent circuit for the inverse system is shown below, along with the Thevenin equivalent looking from node C.  Figure 6.16. Equivalent circuit of the inverse system. The equivalent diagram of the inverse system contains the same number of impedances. Only the values are different, since this corresponds to the inverse values. The main difference lies in the absence of e.m.f. for this system. Finally, the equivalent diagram in the homopolar system and its Thevenin equivalent are shown below:  Figure 6.17. Equivalent circuit for homopolar system. As shown in Figure 6.17, the circuit opens on the Delta side of transformer 1. The general equations given by [6.19b] can now be implemented on these equivalent circuits. For other transformer connections, one can always apply the rules set out for the flow of homopolar currents. The reader may also refer to literature that provides the equivalent diagrams for all types of possible connections. 6.12.5. Systematic calculation of asymmetrical fault currents We will now generalize the systematic calculations presented in section 6.10 that were associated with asymmetrical faults. The concepts presented above will be used to determine the network equivalent circuits. These equivalent networks will be modeled by the transfer matrices, including ZBus. We then have to deal with three ZBus matrices: - The network equivalent of direct system:  EMBED Equation.3 ; - The network equivalent of inverse system:  EMBED Equation.3 ; - The network equivalent homopolar system:  EMBED Equation.3 . Once the equivalent networks of these three systems are determined, we can construct these transfer matrices (YBus or ZBus) for each equivalent network. Since they are used most often in short-circuit calculations, the ZBus matrices can be built either step-by-step, or by inversion of the admittance matrix YBus. The admittance matrices YBus can again be constructed easily and directly by the method described in the previous chapters. With the admittance matrices YBus, we will have EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  representing the direct, inverse and homopolar systems, respectively. For each system, and therefore each network, there will be a voltage-current relationship:  EMBED Equation.3  Therefore, for the direct system:  EMBED Equation.3  (6-36a) for the inverse system:  EMBED Equation.3  (6-36b) and for the homopolar system:  EMBED Equation.3  (6-36c) where,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 : the vector of node voltages (direct, inverse and homopolar);  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 : the vector of injected current, (direct, inverse and homopolar). We can also group the 3 equations [6.36a] [6.36b] and [6.36c] into a single system:  EMBED Equation.3  (6-37) where:  EMBED Equation.3 : vector of node voltages in symmetrical components;  EMBED Equation.3 : vector currents injected into symmetrical components;  EMBED Equation.3 : impedances transfer matrix in symmetrical components. The system [6.37] is therefore of dimension 3n x 3n where:  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  Each element of  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  is:  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  To determine the short-circuit currents and resulting voltages, we apply the relations [6.32] and [6.33] for each sequence. Once the symmetrical component values of current and voltage are calculated, we can then return to the phase values by applying the Fortescue transformation. Consider an asymmetrical fault at node j as illustrated in the Figure below. We can express the fault currents and voltages in node j on all 3 phases (a, b and c) by:  EMBED Equation.3 , and  EMBED Equation.3   Figure 6.18. Fault at node j through fault impedance. In the above notation, the subscript p indicates phase quantities. The relationship between these two vectors at the faulted node j is:  EMBED Equation.3  (6-38) Where the matrix [Zf] represents the fault impedance. We can represent all types of short-circuits of Figure 3 by using this matrix, and its elements will then depend on the type of fault. The transformation in symmetrical components gives:  EMBED Equation.3  (6-39a) And  EMBED Equation.3  (6-39b) With [Yf] representing the fault admittance matrix (inverse of [Zf]). Applying the principle of equation [6.33] (i.e., superposition principle and Thevenin theorem), we obtain:  EMBED Equation.3  (6-40) The vector  EMBED Equation.3 does not contain the fault current (at node j). It is of the form:  EMBED Equation.3  We know  EMBED Equation.3  corresponding to the voltage before the fault and, EMBED Equation.3  corresponding to the transfer impedance matrices of the network, but do not know  EMBED Equation.3  because it is the fault current. We must therefore proceed in the same way as the calculation of symmetrical faults (relation [6.33]) but in symmetrical components:  EMBED Equation.3  (6-41) Eliminating,  EMBED Equation.3  from [6.41] above by replacing it with the expression given by [6.39a]:  EMBED Equation.3  (6-42) The relation [6.42] allows us to calculate the fault current:  EMBED Equation.3  (6-43) By replacing the current value,  EMBED Equation.3 in [6.41], we obtain the voltage at any node i of the network for a short-circuit at node j:  EMBED Equation.3  (6-44a)  EMBED Equation.3  (6-44b) Expressions [6.43] and [6.44] are general and can be applied to a short-circuit at any node in the network. These formulas allow calculation of symmetrical components of current and voltage. The phase values are then obtained by Fortescue transformation. Voltages before the fault are assumed to be known as they can be derived from load flow calculation. More generally, we can consider that these voltages are balanced and close to their assigned values. It may be noted that these formulas are based on the existence of the fault matrix. This may not be defined in some cases, but the fault admittance matrix  EMBED Equation.3 is. In such a case, we have to express the relations [6.43] and [6.44] according to the admittance matrix instead of  EMBED Equation.3 : - Current at fault location:  EMBED Equation.3  - Voltage at fault location:  EMBED Equation.3  - Voltages at other nodes in the network:  EMBED Equation.3  where I is the identity matrix. 6.13 Exercises 6.13.1 Exercise No. 20: Fault current in a simple network Two identical generators A1 and A2 are interconnected by a 225 kV high-voltage line through two identical transformers T1 and T2 as shown in Figure E.20 below. The line can be represented by its series inductance (ignore the resistance, conductance and capacitance).  Figure E.20 One-line diagram of network under study. The characteristics and parameters of these components follow: Generators: apparent power rating Sa = 100 MVA, nominal line voltage Un = 20 kV, subtransient reactance xd = 12%. Transformers (Delta on low voltage -Wye grounded on high voltage): apparent power rating Sa = 100 MVA, nominal voltages 20 kV/225 kV, reactance EMBED Equation.DSMT4 . The ground impedance is negligible. Transmission line: direct, inverse, and homopolar reactances  EMBED Equation.DSMT4 . The objective of this exercise is to study the fault current when a short circuit occurs at the left end of the line. Assume that there is no load on the network and the voltages are equal to their nominal values at the initial moment of the fault. A symmetrical bolted fault (Z = 0) occurs at bus A. Draw an equivalent circuit that represents this condition immediately after the faults occurs. Calculate the short circuit current and the current that flows through the line and transformer T1. Now a bolted fault (Z = 0) occurs at bus B between a phase and ground. Draw an equivalent circuit that represents this condition immediately after the faults occurs. Calculate the fault current. Calculate the current flow in each phase of the transmission line. To limit the fault current calculated in the second case above, the two transformer neutrals are connected to ground by an inductor whose reactance is x = 5% (based on 100 MVA) Repeat question B) after taking the above inductor into account. Calculate the voltage across these inductors (i.e., the rise in the potential of the transformer neutral). 6.13.2 Exercise No. 21: Symmetrical fault on a network (source: ELGER) Consider the three-phase balanced network of Figure E.21.1. The network consists of two generators G1 and G2, two transformers T1 and T2 (connecting the generators to the high voltage lines), three lines L1, L2 and L3 and two loads at nodes 1 and 3. A symmetrical bolted fault (Zf = 0) occurs at node 3.  Figure E.21.1 Network under study. Use Thevenin theorem to calculate: The fault current Icc 3 to 4 cycles after the default; The short-circuit power; The circulating currents in the network lines during the fault; The voltage at nodes 1 and 2 during the fault. Repeat questions a-d above using the Zbus matrix. System Data: Lines: The 3 lines are identical and can be represented by the  model in Figure E.21.2 where the impedance and admittance values are given in per-unit (base apparent power 50 MVA, and base voltage 120 kV).  Figure E.21.2 Line model Transformers and generators apparent powers: - Alternators: SG1 = 100 MVA, SG2 = 200 MVA; - Transformers: ST1 = 100 MVA, ST2 = 200 MVA. Generator reactances - Direct axis transient reactance G1: xd = 0.2 pu - Direct axis transient reactance G2: xd = 0.2 pu Transformer reactances: - Leakage reactance T1: xT1 = 0.1 p.u; - Leakage reactance T2: xT2 = 0.1 p.u. Load complex power (based on 50 MVA, 120 kV) SD1 = p1 + jq1 = (1 + j 0.5) p.u SD3 = p3 + jq3 = (0.5 + j 0.0) p.u CHAPTER 7 Stability Analysis of Power Systems 7.1. Objective The purpose of this chapter is to introduce and explain the phenomenon of stability in power networks. We discuss the equations of motion of the rotating masses and the criterion of equal areas, with the notion of critical time to isolate the fault in case of a machine connected to a network of infinite power. Then we will highlight directions for studies of the so-called classical stability in the case a multi-machine network. 7.2. Introduction One important aspect of the design and operation of electrical networks is to maintain the security of the system at any moment. The operator who operates the network ensures that the system can withstand the failure of one or more components of the network without exceeding the safety limits. He or she must also integrate preventive function in order to have enough time to optimize the operation of the network, and ensure a swift and certain recovery to a normal operating state. This analysis can be viewed from two aspects: static and dynamic. In static security, we consider the electrical system at steady-state and ignore the transient state. This case it is based on the fact that in order to ensure a good operation of a power grid, it is essential that the voltage in different parts of the network and the power (or current) flow is maintained within the acceptable limits. In dynamic security, we consider the dynamic evolution of the electrical system in response to an any disturbance (such as short-circuit, circuit tripping, loss of load, etc.). Dynamic security is thus defined as the system's ability to "survive" the transient (not electromagnetic) and the dynamic that can occur after disturbance. Dynamic security is mainly based on assessing the stability of the system. Thus, decisions and actions are taken based on the results of the evaluation of the stability and dynamic behavior of the system. This chapter is devoted to stability and particularly to angular stability. We will review the basic definitions and the models for assessment of stability. 7.3. Categories and classes of stability problems There are several forms of stability. While historically angular stability (with reference to the timing generators) has been the dominant problem of stability of electricity grids. With the evolution of electrical networks and the tendency to operate them near their physical limits, other forms of stability have emerged. These include voltage stability and frequency stability, or oscillatory stability. Figure 7.1 summarizes these different forms of stability [KUN 04]. These categories are analyzed for a better understanding of the phenomena involved and means to tackle them. In fact, stability of the electrical system as a whole remains essentially the main issue.  Translation: Stabilite des reseaux electrique: Stability of electrical networks Stabilte angulair: Angular stability Stabilite de tension: Voltage stability Stabilite de frequence: Frequency stability Stabilite transitoir: Transient stability Court terme: Short term Long terme: Long term Figure 7.1. Different forms of network stability In this chapter, we are interested that the angular stability. The interested reader may refer to the literature to become familiar with other concepts and categories of stability and their definitions. Thus, for the remainder of the chapter, by stability we mean "angular stability. However, as mentioned earlier, angular stability has transient and dynamic aspects. The historical aspect static stability is often incorporated either in dynamic and small disturbances, or simply treated as a matter of balancing generation with the load. Transient stability is defined by large and sudden disturbances, such as short-circuits, while dynamic stability is defined in relation to small perturbations and slow electromechanical oscillations. As angular stability is related to the synchronous machine which plays a fundamental role in understanding this type of phenomenon. Indeed, most of the electric power generated is provided by synchronous type machines at the synchronous frequency of 50 Hz or 60 Hz (see Figure 7.2). Therefore, the synchronous machine representation and modeling plays a vital role in the analysis of stability. We assume that the reader is familiar with electrical machines in general, including the theory of synchronous machines. Hence, we will not address this theory except on the simplified model used to analyze the transient stability.  Translation: Reseau electrique: Electric grid Vitesse: Speed Bouche de regulation de tension: Voltage regulation Bouche de regulation vitess/puissance: Speed/power regulation Bouche damortissement: Damping regulation Turbine: Turbine Figure 7.2. General schematic of a synchronous generator connected to the grid . 7.4. The equation of motion The basic equations describing the reaction of rotating masses of synchronous machines to various disturbances are related to the inertia of the synchronous machine, and describe the resulting imbalance between the electromagnetic torque and mechanical torque of these machines. This imbalance may be expressed by the following relationship:  EMBED Equation.3  (7.1) - a: acceleration torque N.m; - m: mechanical torque N.m; - e: electromagnetic torque N.m. This equation applies for both generators and motors. However, for generators, m and e are positive while in the case of motors they are negative. In the case of generators, m is the torque that mechanically produced by the turbine in the direction of rotation on the shaft of the machine. This allows the rotor to accelerate in the positive direction of rotation. The electromagnetic torque e created by the interaction of the magnetic flux of the rotor and stator, opposes the mechanical torque and corresponds to the electric power across the air gap of the machine. In the case of motors, e corresponds to the air gap power provided by the network and m is the opposite torque provided by the mechanical load and mechanical losses due to friction. In what follows, we focus exclusively on the synchronous generator. Under steady-state conditions, the accelerating torque is zero because m = e. In this case, there is no acceleration or deceleration of moving masses. Thus, the speed is constant and corresponds to synchronous speed. The equation of motion of masses in the case of synchronous machines is based on equation [7.1]. When an imbalance is product between e and m , there is an acceleration (or deceleration) of the rotating masses. The latter is expressed as the product of the moment of inertia of the masses by its angular acceleration, i.e.,  EMBED Equation.3  (7-2) where: - J: moment of inertia of all rotating masses including generator and turbine shafts (kg/m2); - m: angular velocity of rotor (mech rad/ s). In fact, equation [7.2] also contains an additional term opposing the mechanical torque. This term encompasses all torques corresponding to friction associated with rotation masses (mechanical) and the various windings and magnetic losses (electromagnetic), such as the damper windings of synchronous machine. This torque also opposes the rotation, so the above equation is modified as follows:  EMBED Equation.3  (7-3) where Drm is the damping torque coefficient. This additional torque is relatively small and is often neglected when compared to mechanical torque m and electric torque e. However, for some studies, this damping torque is taken into account, particularly in the study of mechanical oscillations. In what follows, and in the context of transient stability, we do will not consider this additional torque. The relationship between the torques and powers is given by:  EMBED Equation.3  (W), mechanical power supplied by the turbine;  EMBED Equation.3  (W), electromagnetic power. Then:  EMBED Equation.3  (7-4) where Jm represents the angular momentum of the rotor. One can then introduce the concept of relative inertia constant H (in per-unit value) that frequently used in stability studies. This is defined as the kinetic energy stored at the reference speed (i.e., synchronous speed) normalized to the reference power or apparent power rating (Sref) of the machine; - H = kinetic energy of rotating masses at reference speed / Sref. Therefore:  EMBED Equation.3  (7-5) The moment of inertia J can then be expressed in terms of H:  EMBED Equation.3  Substituting for J in equation [7.2], we get:  EMBED Equation.3  (7-7) Considering the fact that ref = Sref/m , we can write the expression [7.7] in actual values as well as per-unit values. In per-unit (pu) values, the equation becomes:  EMBED Equation.3  (pu) (7-7) gm and ge are mechanical and electrical torques in per-unit values. Similarly, the equation can be rewritten as follows:  EMBED Equation.3  where pe and pm are the electric and mechanical powers in per-unit values. The angular velocities m and ms are in mechanical radian per second. But since the equation contains the ratio of these velocities, they can be indifferently expressed in mechanical or electrical radians (or degrees) per second. Indeed, the magnetic field in the rotor (produced by excitation current in the rotor winding rotor) rotates at the speed of the rotor, and therefore at re angular velocity. Similarly, the stator magnetic field rotates at an electrical angular velocity s. Therefore:  EMBED Equation.3  (pu) (7-8) The expression [7.8] using power is more preferred than expression [7.7] using torque as it is more convenient to work with powers than torques. This expression can also be derived is a different way. Starting with the following relationships among angular velocities:  EMBED Equation.3  where Np is the number of magnetic poles of the machine. This expression allows us to make the transition between the angular velocity expressed in mechanical radians per second, and angular velocity expressed in electrical radians per second. In considering  the angular position of rotor in electrical radians with respect to the synchronously rotating frame of reference, we have:  EMBED Equation.3  and therefore:  EMBED Equation.3  Equation [7.8] becomes:  EMBED Equation.3  (pu) (7-9) where  is in radians. This equation is known as the equation of motion and is the basis for both angular and transient stability analysis. The solution of this equation gives us the temporal evolution of the rotor angle , often called the internal angle of the machine, and can track the behavior of the machine against synchronism when a fault occurs in the network. One further consideration to substitute s = 2f where f is the frequency of network. The equation [7.9a] becomes:  EMBED Equation.3  (pu) (7-9b) NOTE: The quantity H is in per-unit value with the rated power of the machine as a base. H can be expressed as (MJ / MVA) or (W.sec / VA) or any other coherent expression using SI units. However, when multiple machines are connected to the network electric, we consider only on single MVA base. Then the H of each machine (if it is indicated on the basis of the power of the individual machine) has to be converted to the new power base of the system. Since equation [7.9] is expressed in per-unit values, H can also be expressed in seconds, while the powers are in (pu) and the angle  in radians. The conversion of Hg, i based on the power base Sbase, i of machine i to a the common power base of the network Sbase , is achieved by a simple change of base:  EMBED Equation.3  (7-9c) where Hi is now based on the network power base Sbase. 7.5. Simplified model of synchronous machine At steady-state there is generally a balance between the mechanical power and electrical power from the alternator. There is no acceleration and the machine runs at synchronous speed. This corresponds to some equilibrium value of the internal angle . When a fault occurs in a network or when there is a sudden load change, the electric power suddenly changes. Of course, the mechanical power is expected to offset this balance by acting on the turbine. This is the role of speed and power regulator (i.e., governor control). This action is mechanical and has basically its own time constants that are naturally slower than electrical time constants. For this reason, it is often assumed that immediately after the fault, mechanical power Pm remains constant (in the equation of motion). Indeed, the turbine has not yet had time to respond to this imbalance in the earliest moments of imbalance. This momentary imbalance will be reflected in the rotor speed which will accelerate or decelerate depending on the sign of this imbalance. In fact, if mechanical power is greater than the electric power, the excess energy will be transformed into kinetic energy in the rotor. This causes the rotor to accelerate and therefore deviate from its synchronous speed. This type of phenomenon occurs mainly during a sudden change in the network (e.g., short-circuit, rapid increase or decrease of the load, etc.). These phenomena are accompanied by electromagnetic transients, which will impact the models used in synchronous machine. Indeed, there are various models of synchronous machines, from the simplest to the most complicated, depending on the phenomenon to be studied and the desired degree of accuracy. As we have stated above, we consider that the reader is familiar with the theory of synchronous machine. However, to simply explain some phenomena related to transient stability (but also to static stability), we will used a simplified model (the most simple) consisting of a transient reactance Xd' behind a fixed transient e.m.f. E'. Remember that at steady-state, the reactance in question is the synchronous reactance Xs and the e.m.f. is the steady-state E. In case where a simple model can be used in transient stability analysis, let us further assume that the machine rotor is round, and the rotor flux does not change (i.e., the impact of the change of rotor speed on induced voltages is negligible). Such a simple machine model is shown in Figure 7.3 below. The relation between the transient e.m.f., transient reactance, generator current and terminal voltage is described by equation (7-10).  EMBED Equation.3  (7-10)  Figure 7.3. Circuit and phasor diagram of simplified machine model. For a more comprehensive study, and with the aid of various tools and software, one can use a more detailed model of the machine, including salient-pole machines. 7.6. Power-angle considerations at steady-state The generator voltage is adjusted through the excitation system. We recall that the excitation winding is fed by a constant current. This current flow generates a magnetic field as it is rotated to the turbine; it induces an e.m.f. in the stator windings. Considering the equation [7.10] in case of steady-state (with stator winding resistance neglected), we have:  EMBED Equation.3  (7-11) where Xd is the direct axis reactance at steady-state, which of course corresponds to the synchronous reactance. The terminal voltage Vs is taken as reference and the internal angle  is defined as the angle of E with respect to Vs. The complex power per phase is defined as:  EMBED Equation.3  (7-12) Equation [7.12] can be re-written as follows:  EMBED Equation.3  Separating the real and imaginary parts, we obtain:  EMBED Equation.3  (7-13) We can therefore deduce the real and reactive powers per phase: - The active power (real part of S):  EMBED Equation.3  (7-14) - Reactive power (imaginary part of S):  EMBED Equation.3  (7-15) These expressions are similar to those we could have obtained for a transfer of energy between 2 nodes of a network linked by a line modeled by its reactance Xl (when the effects and capacitive and resistance of the line are ignored). 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Indeed, the active power given by expression [7.14] is the power produced by the machine, hence corresponds to Pe in the equation of motion [7.9a]. Similarly, the reactive power given by expression [7.15] highlights the close relationship between the reactive power Q and the e.m.f. E of the machine and therefore the excitation system. Consider expression [7.14] more closely. If we increase Pe by acting on the turbine while Vs and E are kept constant (Xd remains constant), we note that the angle  increases with Pe. However, this relationship is not linear since it is a sinusoidal function. There is also a point that corresponds to a maximum limit to the increase in power of the machine. The maximum power is obtained when the angle  = 90. Hence:  EMBED Equation.3  (7-17) This limit is known as the static stability limit of the machine. Figure 7.4 below illustrates this characteristic P = f (). If the machine is requested to an increase its output power beyond Pmax, it cannot remain synchronized with the rest of the network.  Figure 7.4. P () characteristic of a synchronous machine. In Figure 7.4, we can observe that by increasing the power from P0 to P1 there will a corresponding increase of the angle from 0 to 1. Beyond the static stability limit Pmax (limit) corresponding to  > 90, the machine is considered unstable (out of synchronism). For a given power in Equation (7.14), there are two angles that satisfy this expression, an angle of less than limit and an angle greater than limit. Only the angles that are less than limit result in a stable operation. Note that equation [7.14] shows that the power transfer also depends on the reactance of the machine. If the machine is connected to a network of infinite power (i.e., infinite bus) through a power line, the power transfer between the machine and the infinite bus will also depend on the total reactance between the two points (with resistive and capacitive effects neglected). To illustrate this phenomenon, consider the simplified representation of the diagram in Figure 7.5 below:  Figure 7.5. Generator connected to infinite bus through a transformer and power line. In this simple network, the generator is connected to the infinite network through a transformer and a transmission line. The transformer is represented by its leakage reactance, the line by its series reactance, and the network by equivalent reactance. The node R represents the infinite bus where the voltage is constant at all time. For simplification purposes, the resistive and capacitive components are neglected. The voltages Vs an node S and Vr at node R will be out of phase by an angle that is induced by the current I and the total reactance between these two nodes. This angle is often referred to as the transmission angle. If the transmission angle is ' and the internal angle , the angle between the e.m.f. E and Vr is ' =  +  . The expression of the power transfer becomes:  EMBED Equation.3  (7-17) The corresponding phasor diagram of this example is shown in Figure 7.6. We note that the expression [7.17] has the same structure as expression [7.14] and therefore reflects the transfer of power between two networks, i.e., between the machine e.m.f. and the infinite bus. These relations [7.14] and [7.17] also show that under the simplified conditions outlined above, the transfer of power between two nodes of network with fixed voltages depends on the angular difference between these voltages and the transfer reactance between the two nodes. These relations P () are known as  power-angle relations, and are fundamental in the analysis and understanding of stability phenomena. This simple network will also be used to further illustrate the phenomena of transient stability in the sections that follow.  Figure 7.6. Phasor diagram corresponding to Figure 7.5. 7.7. Case of small perturbations Consider a synchronous machine that connected to a network of infinite power. We will examine the case where the machine is subject to small variations due for example to slow variations in load. In this illustration of static stability, the actions of the voltage and speed regulators will not be considered. Equation [7.9a] is written as follows in the initial state:  EMBED Equation.3  (7-19) This equation shows a nonlinear relationship between the equation of motion and the internal angle . However, because small perturbations result in small changes, we can examine the behavior of the machine by linearizing the equation of motion around the equilibrium position of the rotor. This point is represented by the initial position of the internal angle . Let this initial value be denoted by 0. A linear analysis of a small deviation from this position due to a small disturbance is then translated into: 0 =  + . At steady-state,  = 0 and Pm = Pe. It follows that:  EMBED Equation.3  (7-20) This equation expresses the effect of the deviation from the equilibrium position. Since the mechanical power is assumed constant, pm = 0. For low values of  the difference in electric power becomes:  EMBED Equation.3  (7-21) Substituting these power deviations in (7-20) yields:  EMBED Equation.3  (7-22) The equation to solve is then:  EMBED Equation.3  (7-23) This is an equation of 2nd order in  which has solutions of the form ert. Considering that the losses are neglected, so for the machine movement to be stable, it is necessary that the response is decreasing, i.e., r2 d" 0. The above solution will have two roots in the complex plane, so the movement is oscillatory but it is dampened. Otherwise, if r2 > 0, the answer is a growing exponential that leads instability. In reality, the roots of equation [7.22] depend on the term pmaxcosd0. This term refers to the derivative of the electric power at the initial operating 0. This term is referred to as the "synchronizing power" in relation to the synchronizing torque. The conditions that allow a stable movement (r2 EMBED Equation.3 0) correspond to a positive synchronizing power and therefore an angle  between 0 and 90 (left side of the curve of the Figure 7.4). Static stability studies are performed as illustrated above. Indeed, the static stability can be defined as the network capacity, and therefore synchronous generators which are connected to it, to maintain synchronism during disturbances of low amplitudes. 7.8. Transient stability As indicated earlier, the network suffers disturbances which are more or less severe on a regular basis. From the point of view of stability, low amplitude perturbations can be treated in the studies of static stability. However, severe disturbances (such as a fault, a sudden loss of a significant load or generation) are treated in the context of transient stability studies. Indeed, this type of disturbance often results in strong reactions in the network particularly in alternators. This leads to large variations of different variables (voltage, power, internal angle, etc.). In extreme cases, part of the network, if not all, becomes unstable. Transient stability is defined in relation to the network capacity to keep the machines synchronized as a result of severe disturbance. The system then settles to a new point of stable operation once the disturbance has disappeared. The linearization in this case is not appropriate. Stability is therefore analyzed based on the solution of the equation of motion [7.9a]. The study of transient stability is relatively complex and requires various levels of more or less complex modeling. This complexity increases with the accuracy of the models and size of the network. One will have to use a software tool that is dedicated to perform stability studies. However, there is a simple method that allows a rapid prediction of transient stability with simplifying assumptions. This method is known as the method of equal areas. The equal area method is used for the case of a machine connected to an infinite bus, or a network with two machines. The method is not applicable for complex multi-machine systems that require high levels of modeling and high precision. This method is based on the exploitation of Figure 7.4 to predict the stability of the machine following a perturbation. Indeed, the energy stored in the masses of rotating machines can be interpreted graphically in Figure 7.4 before and after the disturbance. This method is therefore important to understand the phenomena related to transient stability. To explain the method of stability analysis, consider again a synchronous machine connected to an infinite bus through a line. Phenomena occurring in the synchronous machine before, during and after the disturbance are in the heart of this analysis. The transitional issues are considered very fast and the various regulators of the machine have no time to act. From the equation of motion [7.9a], we can deduce:  EMBED Equation.3  (7-24) Multiplying both sides of this equation by the term (2 d/dt) results in:  EMBED Equation.3  This equation can be written as:  EMBED Equation.3  Integration of this equation gives:  EMBED Equation.3  (7-25) Therefore:  EMBED Equation.3  (7-25b) The change in speed (compared to the synchronous speed) expressed by the term d/dt is initially zero (equilibrium position). The position of the internal angle  corresponding to the initial position is designated by 0. When the system undergoes a disturbance, the imbalance between mechanical power and electric power expressed by (pm - pe) will induce a change in the term d/dt and therefore a deviation in the internal angle 0. However, to ensure stability, we need the term d/dt to become zero once the disturbance has disappeared (after some time). We deduce therefore a criterion for stability as follows:  EMBED Equation.3  To better understand this equation, we will illustrate a graph of the curve P = f (). To do this, consider again Figure 7.4 but taking into account some form of disturbance such as a step in mechanical power. In this example, the initial operating point (before the disturbance) corresponds to an internal angle d=d0 and Pm0=Pe0. This starting point is shown in Figure 7.7 with 'a'. So when the mechanical power undergoes a sudden change, it moves from an initial state Pm0 to a new state Pm1 (see Figure 7.7). At this point, an imbalance occurs between the mechanical power and electric power. As the mechanical power of the new Pm1 state is greater than the electric power, the power Pa becomes positive and the rotor undergoes acceleration. However, because of mechanical inertia, the angle  can not adjust instantaneously. The angle  remains at its 0 initial position during the first moments of the disturbance, which is reflected in Figure 7.7 by an instantaneous transition point 'a' to point 'a'. The rotor acceleration creates an increase of the angle . This increase will continue until the value  = d1 corresponds to a point of equality between the mechanical power and electric power (point "b"). The excess energy stored during the acceleration phase can be expressed by:  Figure 7.7. Graphic representation of the method of equal areas.  EMBED Equation.3  (7-27a) However, stabilization has not yet been reached because although acceleration is theoretically zero, the rotor speed is exceeding the synchronous speed. The rotor cannot stop at point "b" because of mechanical inertias. The angle  then continues to increase. But for values of  > 1, the mechanical power becomes lower than the electric power, hence the accelerating power becomes negative. The rotor will then slow to return to synchronous speed until stabilization at point "b". During this phase, the excursion of the angle  can be up to a maximum value of  = 2. The rotor loses the energy it has accumulated during acceleration. Going back to point "b" is accompanied by oscillations around this point (between 0 and 2). The various damping components present in the machines will dampen these oscillations and eventually stabilize at  = 1 which corresponds to the new state of stable equilibrium. During the deceleration phase of the rotor, the energy lost (or restored) is:  EMBED Equation.3  (7-27b) For a system in which we have neglected the losses, the stored energy during the acceleration phase is equal to that lost during deceleration. The corresponding areas should be equal (A1=A2) according to the criterion of equal areas. This method determines the maximum excursion of the angle  and thus the stability without resorting to the numerical solution of differential equations related to the equation of motion. The criterion of equal areas is therefore interesting as it determines quickly the maximum amount of mechanical power which can be added to the system described above. For such a sudden change in power, stability is maintained only if the area A2 is at least equal to the area A1 situated above the line Pm1. Indeed, if A2<A1, stability will be lost because the angle  becomes above d3, pm1>pe and the rotor will continue to accelerate rather than decelerate. The angle 3 can be seen in this case, as the maximum angle and it may be beyond 90. The excursion of the internal angle  can therefore go beyond 90 but only during the transient period. The above reasoning assumes a sudden change of mechanical power and analyzes its affect on transient stability. In practice, the mechanical power changes rather slowly. However, the same reasoning can be applied to an abrupt variation of electric power. Such variation may be caused by a short circuit, loss of a generator, or simply loss of a line. So it is this type of problem that we are particularly interested in. 7.9. Application of equal-area criteria During the process of analysis of transient stability, it is often assumed that the disturbance disappears. Indeed, protective devices detect the failure and circuit breakers open and isolate the failure of the system. We distinguished 3 states that fall in the process of stability analysis: - The state before the occurrence of the disturbance; - The state during the disturbance; - The state after the disturbance is cleared. To illustrate this process, we will consider the network below (Figure 7.8) where a synchronous machine is connected to a network of infinite power through a transformer and two parallel lines. The simplifying assumptions described above on machine and other components are kept the same. 7.9.1. Case of a short circuit at generator terminals Consider a three-phase bolted fault at the alternator terminals (node S in Figure 7.8). Before the fault, the operating conditions is steady and the equilibrium position of the internal angle is  = 0 and mechanical power of the machine is constant and equal to Pm.  Figure 7.8. Machine connected to a network with a fault at node S. When this type of fault occurs at the terminals of the machine, the voltage at this point is zero. The electrical power will then also be zero and of no power will be transmitted to the network during the fault. Assume that the fault clears quickly and the system is back to its initial configuration. Let us look at the machine stability before, during and after the fault using the method of equal areas. Before the failure, the machine is at the equilibrium point "a" corresponding to Pe = Pm and  = 0 (see Figure 7.9). During the failure, due to the cancellation of the electric power produced by the machine, the operating point on the curve Pe( ) moves from a to a . Due to the inertia of rotating masse, the angle  can not change instantaneously. The mechanical power becomes greater than the electric power, thus causing acceleration of the rotor (positive acceleration power). The rotor gains kinetic energy and the internal angle  increases. Suppose the fault is cleared when  = 1, (at point 'b' in Figure 7.9), the machine suddenly returns to the initial setup and the point of operation on the power-angle curve are found at "c". At this point, the power becomes greater than the mechanical power. The rotor decelerates, returning the energy stored in the acceleration phase. During this phase, the rotor speed is still higher than the synchronous speed; the angle  continues to increase until the addition of energy (represented by the area A1) is fully restored. The operating point in the power-angle curve is then "d" which is characterized by the equality of areas A1=A2. In this respect, the rotor speed is equal to the synchronous speed and  = 2, the maximum value of the internal angle. But the power is still higher than the mechanical power. The rotor will continue to deceleration and speed drops below the synchronous speed. The internal angle  will then decrease and the point of operation on the curve will follow the "descending" path from point "d" to a minimum angle, through "c" and "a". This minimum angle is governed by the criterion of equal areas after the fault. The rotor will then oscillate around its initial value 0 so indefinitely in the absence of damping with constant amplitude and at the natural frequency of the rotor. However, different damping factors exist in reality and these oscillations will decay and disappear after a certain time, and the rotor return to its original position 0.  Figure 7.9. Iluustration of method of equal areas. 7.9.2. Critical fault clearing time The value of the corresponding internal angle at the time the fault clears plays an important role in the outcome of stability analysis. Indeed, the area A1 depends on the time between the occurrence of the fault and its clearance (or its removal by protective devices). If this time exceeds a certain limit beyond which the area A1 becomes greater than the area A2, stability will be lost. This time limit is critical clearing time CCT. This time also corresponds to a "critical angle cr" for which A1 = A2 as illustrated in Figure 10 below.  Figure 7.10. Critical clearing angle. The angle and time of clearing of critical faults can be determined from the curve in Figure 7.10. The area A1 is determined by:  EMBED Equation.3  (7-27a) The area A2 is determined by:  EMBED Equation.3  (7-27b) The criterion of equal areas gives us:  EMBED Equation.3  By separating the terms of cr , we get:  EMBED Equation.3  (7-28) On the other hand, from the power-angle curve, we have: dmax = p - d0 (7-29a) and: pm = pmaxsin d0 (7-29b) By replacing max and Pmax and by their expressions [7.29a] and [7.29b] in equation [7.28], we obtain the expression of cr as:  EMBED Equation.3 tttu!u%uDuLuoupuuuuuuuuuuuuuuvvv*v>v@vvvvvvvZwjwtwwwww6xHxjxxxxxxxxxxx#jGI hARhDCJUVaJhARhDCJaJjhARhDCJUaJhDCJaJhDh0kCJH*aJhDh?CJH*aJh h CJH*aJh?CJaJh CJaJhr'h0kCJaJ4xxxxxyyyzz2z4zdzjzzzzzzzz6{^{`{d{f{{{{{{{{{|P|b|ɮ~ririɮɮ]]hDh0kCJH*aJh6CJaJh?h0k6CJaJh?h?6CJaJjhARhDCJUaJh?6CJaJhhCJH*aJh?CJaJhr'hCJaJhCJaJhr'h0kCJaJhARhDCJaJjhARhDCJUaJ!jhARhDCJEHUaJ$b|f|h|r|t|||||||}}$}T}x}}}6~`~d~j~~~~~~~"$&(BDZ|4VX€dfrٽͱűٽٽٽٽٽٽhz9CJaJh h?CJH*aJhDhDCJH*aJh CJaJhDCJaJhDh0kCJH*aJhr'h0kCJaJhCJaJhDh?CJH*aJh?CJaJ< 08ׂ͂45W`opz΃؃2PƄʄ  휄h3"h3"CJH*aJhkhkCJH*aJh3"CJaJhz9CJaJhz96CJaJh?h0k6CJaJhk6CJaJh?h?6CJaJjhARhDCJUaJh?CJaJhr'h0kCJaJhkCJaJ4oȊHČ.|,n$a$gd9{$a$gdj$a$gd9{$a$gd3"$a$gdS}$a$gd?ABHISY† ,3<AknoLr(BñØÐÐ݅݅݅݅݅݅݅݅݅݅݅}rhr'hjCJaJhjCJaJhr'h3"CJaJh3"CJaJh9{CJaJ!j3h3"h3"CJEHUaJ#jLM hARh3"CJUVaJhARh9{CJaJjhARh9{CJUaJh3"CJaJhr'h0kCJaJh9{CJaJh?CJaJ,B!?nƊNJȊɊՊ '*Crs؋ڋ܋FHŒČyhz9CJaJh?CJaJh?h0k5CJaJhqg5CJaJhr'h0kCJaJh3"h?6CJaJh?h?6CJaJh?h0k6CJaJhj6CJaJh9{CJaJjDhARh9{CJUaJhr'hjCJaJhjCJaJ,Čƌ2p(J\Ž HЏ$.0Fb ,.NZ\^fhjprtđ巯h9{CJaJh9{h0kCJH*aJh9{h?CJH*aJh)CJaJhjCJaJh?CJaJhr'h0kCJaJ!jhARh9{CJEHUaJ#jGI hARh9{CJUVaJhARh9{CJaJjhARh9{CJUaJ0đƑȑʑDHZfz|~$*,.RTºp§^#jGI hARh9{CJUVaJ!j\hARh9{CJEHUaJ#jGI hARh9{CJUVaJh9{h?CJH*aJh?CJaJh)CJaJhr'h0kCJaJh9{CJaJhARh9{CJaJjhARh9{CJUaJ!jhARh9{CJEHUaJ#jGI hARh9{CJUVaJ!TVXnؔڔܔޔ&*F&46lɹԧԎɂɹvh````T`h)h)CJH*aJh)CJaJh)h0k6CJH*aJh)h0k6CJaJh)h0kCJH*aJh9{CJaJ!jhARh9{CJEHUaJ#jGI hARh9{CJUVaJh?CJaJh)CJaJhr'h0kCJaJhARh9{CJaJjhARh9{CJUaJ!jhARh9{CJEHUaJ nB؛ ֢ע+˦*XZΨ$a$gd9{gd9{ $1$7$8$a$gd? 1$7$8$gdr'$a$gd9{$a$gdS}.08T(@Bt~˜Șʘ̘ΘИvxbnrںګړhKCJaJh9{h)CJH*aJh3 nCJaJhARh9{CJH*OJQJaJhARh9{CJOJQJaJhARh9{CJH*aJh9{CJaJhARh9{CJH*aJhARh9{CJaJh?CJaJh)CJaJhr'h0kCJaJ-šĚƚʚBDF|֛؛ڛ'2LUX\zڜݜ  庯znhKB*CJaJph h?h?5B*CJaJph h?h0k5B*CJaJphhr'hKCJaJhKCJaJhr'h0kCJaJh?CJaJ!jhARh9{CJEHUaJ#jGI hARh9{CJUVaJhARh9{CJaJjhARh9{CJUaJ%4AK^qÝٝ)GI\xΞ֞8>NPTmʟޟ ;EJfˠ BRdxy~𱽱h0khc6B*CJaJphhH"B*CJaJphhc6B*CJaJphh0khKB*CJaJphh?B*CJaJphhKB*CJaJphh0kh0kB*CJaJph@~̡͡١ߡ ?MRSZ^Ӣ 02MWXeУڣP[_fhi{Ĥ*+Cau-Ljk̽hz9B*CJaJphh0khz9B*CJaJphh8NB*CJaJphhc6B*CJaJphh0kh0kB*CJaJphhH"B*CJaJphFʦ˦̦ͦߦ"#$  "(*,PRؽ؃teS#jGI hARhPCJUVaJh0khPB*CJaJphhPhPB*CJaJphhPB*CJaJph!j]hARhPCJEHUaJ#jGI hARhPCJUVaJhARhPCJaJjhARhPCJUaJh9{B*CJaJphh0kh0kB*CJaJphh8NB*CJaJphhH"B*CJaJphRTVX\ƨȨʨ̨ΨШҨڨBDFHP~Ǹߡ~vǸǸߡdSߡKǸh8NCJaJ!jhARhPCJEHUaJ#jGI hARhPCJUVaJhPCJaJ!jV hARhPCJEHUaJ#jGI hARhPCJUVaJhARhPCJaJhH"B*CJaJphh0kh0kB*CJaJphh8NB*CJaJphhPB*CJaJphjhARhPCJUaJ!j hARhPCJEHUaJΨШ®FH8YZn() $`a$gdL8F $`a$gdP $1$7$8$a$gdH"$a$gdPgdP $1$7$8$a$gd?ΩЩҩԩ&8DZBƫ2p&Nn] hPhH"B*CJH*aJph hPh0kB*CJH*aJph h8Nh0kB*CJH*aJphhPB*CJaJphhH"B*CJaJphh8NB*CJaJphh0kh0kB*CJaJphjhARhPCJUaJ!jhARhPCJEHUaJ#jGI hARhPCJUVaJhARhPCJaJ!NPftxܭ"`®Į&dʯ6^"8djln˿׿氥sc[h8NCJaJhARhPCJH*OJQJaJhARhPCJOJQJaJ!jhARhPCJEHUaJ#jGI hARhPCJUVaJhARhPCJaJjhARhPCJUaJh8NB*CJaJphhPB*CJaJphh0kh0kB*CJaJphhH"B*CJaJphhPB*CJH*aJph#nvx|~3<QRSU[efhiȲײ*+ʾʲʾʲʾwʲwʲwfʲw h[hH"B*CJH*aJphh[B*CJaJphh0khPB*CJaJph hPhPB*CJH*aJphhPhPB*CJaJphhH"B*CJaJphh8NB*CJaJphh0kh0kB*CJaJphhARhPCJaJhARhPCJH*aJh8NCJaJhPCJaJ'+,7Vw<~δ Bܵ4r *Vp÷Էԫ}l hH"h0k6B*CJaJphjhARhPCJUaJhARhPCJH*OJQJaJhARhPCJOJQJaJhz9B*CJaJphh[B*CJaJph hPh0kB*CJH*aJphhH"B*CJaJphh0kh0kB*CJaJph h[h0kB*CJH*aJph$6789KLMNYZ|ҸԸ&dz|~xlxlx`xlxTxEhL8FhL8FB*CJaJphhL8FB*CJaJphh[B*CJaJphhH"B*CJaJphh0kh0kB*CJaJphhPB*CJaJph!j2hARhPCJEHUaJ#jHI hARhPCJUVaJhARhPCJaJjhARhPCJUaJ hH"hH"6B*CJaJph hH"h0k6B*CJaJphh8N6B*CJaJph̺κ ,8^x̻8t,x@Bz<BlnӸĬĬĸĸĀĸĸĀĀĸĬӬĸĸ hL8Fh0kB*CJH*aJphh0khz9B*CJaJphhz9B*CJaJphh[B*CJaJphhH"B*CJaJphh0kh0kB*CJaJphhL8FB*CJaJph hL8FhL8FB*CJH*aJphh0khL8FB*CJaJph2()A[p  !@2pd"ʸʛwwhARhL8FCJH*aJha-B*CJaJphh[B*CJaJphhL8FB*CJaJph!jv5hARhL8FCJEHUaJ#jHI hARhL8FCJUVaJhARhL8FCJaJjhARhL8FCJUaJhH"B*CJaJphh0kh0kB*CJaJph." 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"? r? ? p@ $H$a$gd& $H$a$gd*Y $ 41$7$8$a$gd) $H$a$gd*Y :<>@x|~(2JLNTVʹʓpʟdʓUI>h*Y h&CJaJh*Y h&CJH*aJh*Y h&CJOJQJaJhz9B*CJaJph!j{hARh*Y CJEHUaJ#jHI hARh*Y CJUVaJh)B*CJaJphh1B*CJaJphh1B*CJH*aJph h*Y h0kB*CJH*aJphh0kh0kB*CJaJphh*Y B*CJaJphhARh*Y CJaJjhARh*Y CJUaJVdf$bǸ֝֝Ǹo`jh*Y h&CJUaJ h*Y h0kB*CJH*aJphh1B*CJaJph h1h0kB*CJH*aJphh*Y h&CJH*aJh*Y h)B*CJaJphh*Y h0kB*CJaJphh*Y h&B*CJaJphh*Y h&CJaJh*Y h&CJH*OJQJaJh*Y h&CJOJQJaJ< < < < 4< S< T< l< < < < < < < < < < < < < < = = = = > > > > sjsY h&h0kB*CJH*aJphh&CJH*aJhARh&CJH*aJhARh&CJaJh)B*CJaJphh1B*CJaJphh0kh0kB*CJaJphh*Y h0kB*CJaJphh*Y h&CJaJjh*Y h&CJUaJU!jl~h*Y h&CJEHUaJ#jHI h*Y h&CJUVaJ (7-30) To find the critical time corresponding to this critical angle, we will need to solve the equation of motion. This time can be calculated by considering the phase angle during the fault, i.e., between the time of occurrence of the fault t0 and time of its clearance tcr (phase 0 - cr). During this phase, the power is zero (Pe = 0), and the equation of motion simplifies to:  EMBED Equation.3  (7-31) By integrating this equation we have:  EMBED Equation.3  (7-32) The integration of the equation [7.32a] gives us the internal angle as a function of time:  EMBED Equation.3  (7-32) If  = cr, this corresponds to t = CCT, we have:  EMBED Equation.3  (7-33) hence the critical time to eliminate the fault is:  EMBED Equation.3  (7-34) Note that if the fault is removed beyond the critical time (A1> A2), when the operating point is at "d", the kinetic energy stored during the acceleration phase would not be totally returned to the system (the extra energy is not zero), the rotor speed is still higher than the synchronous speed and angle  will continue to increase (instead of decrease). Beyond point "d", the electrical power is less than the mechanical power. This will accelerate the rotor resulting in the loss of synchronism. 7.9.3. Case of a short circuit on a line Consider again the network of Figure 7.8 but in the case a three-phase short circuit occurring in the middle of the line 2 as illustrated in Figure 7.11 below.  Figure 7.11. Machine connected to a network of infinite power with a short circuit at node F We have the 3 following states: State before the fault: d=d0 and Pe= Pm , curve C0 of Figure 7.12; State during the fault: curve C2 of Figure 7.12. The amplitude of this curve is smaller than the initial situation as the equivalent reactance becomes larger. The maximum transmitted power is therefore lower; State after removal of the fault: curve C1 of Figure 7.12, the machine is connected to node R through the transformer and line 1 only (line 2 is open). For all the 3 states, the mechanical power is considered constant. The curves of the 3 states are shown in Figure 7.12.  Figure 7.12. Pe-d curves and criterion of equal areas. Before the fault occurs, the operating point is a on C0 curve (Figure 7.12). Upon the occurrence of the fault, the point of operation moves to a on C2 curve. Due to the inertia of rotating masses, the angle  can not change instantaneously. Mechanical power becomes greater than the electric power, hence causing an acceleration of the rotor (i.e., positive acceleration power). As the rotor accelerates, it gains kinetic energy, and the internal angle increases until the fault is removed at  = 1 corresponding to operating point b of C2 curve. When the fault is cleared by removing line 2, we switch to C1 curve, thus the operating point will jump from b' to c. At this point, the electrical power becomes greater than the mechanical power. The rotor decelerates thus releasing the stored kinetic energy gained during the acceleration phase. During this phase, the rotor speed is still higher than the synchronous speed; hence the angle  continues to increase until that the additional energy (represented by the area A1) is fully returned. The operating point in the power-angle curve is then d characterized by equality of areas A1 = A2. At point d, the rotor speed is equal to the synchronous speed and  = 2, the maximum value of the internal angle. But electric power is always greater than the mechanical power. Hence, the rotor will continue to slow and its speed falls below the synchronous speed. The internal angle  will then decrease to an operating point on the curve C1 and will follow the "descending path from d to a minimum angle. The minimum angle is governed by the criterion of equal areas corresponding to the situation after the fault. The rotor will then oscillate around the operating point c. Different damping will stabilize around the machine a new equilibrium point corresponding to the intersection of Pm and C1. As in the previous case, if the fault is eliminated beyond the critical time, the speed and the angle will continuously grow, leading to loss of synchronism. 7.10. Case of a multi-machine system The stability analysis described above is based on the case of a machine connected to a network of infinite power. The criterion of equal areas applies only to this case and to the case of two machines. In addition, we have neglected the effects regulators which have a significant impact in the process of stability especially in the dynamic post-fault phase and oscillations that follow. However, in larger networks electrical, several hundreds of machines may be connected to the system and these machines are of various types and technologies. A stability study for a multi-machine system requires taking into account all the components of the system. For a better accuracy, the components and the generators in particular must be modeled in detail including regulations. The study of stability becomes complex in such real cases. Several computer software tools are available for these studies, and allow various levels of details in modeling. Among them, one can cite EUROSTAG software developed by Electricit de France and Tractebel. However, if we want to focus only on the very first moments after the occurrence of fault, it is always possible to simplify the modeling and thus the complexity of calculation by taking the same simplifying assumptions as those used on generators during the presentation of transient stability above: Each machine can be modeled by a transient reactance behind a constant emf ( EMBED Equation.3 ). Voltage excitation is considered constant during and after the fault. The rotors are also considered made of smooth poles; Mechanical power supplied by the turbine remains constant during the study period; Damping torques are neglected; The mechanical rotor angle of each machine coincides with the internal electric angle  of the machine. In addition to these assumptions on the machines, we can add another on load buses. We can model the loads by equivalent admittances between the nodes in question and ground. This modeling makes it possible to reduce nodes by not keeping the internal nodes of the generators in the study of the stability during the fault and after the fault is cleared. This reduction is achieved by eliminating the load buses by Kron reduction. Thus, the equation of motion for each machine i can be written as:  EMBED Equation.3  (7.35) where Hi is given in per-unit based on the network apparent power base (see equation [7.9c]). In this case, the electrical power is by the power flow equations:  EMBED Equation.3  (7.37) with:  EMBED Equation.3 : power of machine i;  EMBED Equation.3 : magnitude of transient e.m.f of machine i;  EMBED Equation.3 : magnitude of transient e.m.f. of machine j;  EMBED Equation.3 : magnitude and angle of element ij of the matrix Ybus ,  EMBED Equation.3  - EMBED Equation.3 : angle of e.m.f.  EMBED Equation.3  ( EMBED Equation.3 ); > > > > > F> H> L> p> r> t> > > > > > ? ? ? ? ? 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The lines and transformers can be modeled by pure reactors. The synchronous machine has a round rotor, and assume that its emf E' is maintained at a value equal to 1 pu.  Figure E.22 Network with fault at node L. The data of the network are given below in per unit on a common base. Line reactances: Xl1 = Xl2 = 0.3 p.u Transformer reactance: Xt = 0.1 p.u Voltage at infinit bus: Vr = 1 p.u Synchronous machine transient reactance: X'd = 0.2 p.u Machine inertia constant: H = 5MJ/MVA. Mechanical power of machine: Pm = 0.8 pu (assume constant) Starting at equilibrium (Pm = Pe), a three-phase short circuit occurs at node L. This short-circuit is immediately removed by protective devices and the system is restored to its original state. Determine the equation of motion for this network before the fault. Calculate the internal angle of the machine before the fault. Calculate the critical time to remove the fault. Now a three-phase short-circuit occurs at point E in the middle of line L2 (see Figure E.22.2). The short-circuit is removed by protective devices without re-engagement, thus disconnecting the faulted line from the system.  Figure E.22.2 Network with fault at node E. Calculate the electric power during the fault and after the fault is cleared. Determine the equation of motion corresponding to situations during and after the fault. BIBLIOGRAPHY [AVA 06] Avan J., K. HUNTER, CSPM minimum data requirements in terms of IEC CIM, in March 2006. [DEG 05] DEGNI T., A. ENGLER, OSIK O., "Modeling of Distributed Energy Resources with ATP-EMTP ", European EMTP-ATP Users Group Meeting and Conference, Warsaw, Poland, 12-14 September 2005. [71 ELG] ELGERD OI, Electric Energy system Theory: An Introduction, Mc Graw Hill, Library of Congress Catalog Card Number 77-114445, 1971. [EUR 04] EUROSTAG, User Guide, Electricit de France - Tractebel October 2004. [GRA 94] JJ Grainger, WD Stevenson Jr., Power System Analysis, McGraw-Hill, New York, 1994. [KUN 04] P. Kundur, J. PASERBA, Vitet S., "Definition and Overview on Classification of Power System Stability ", IEEE On Beharfof / Joint ClGRE Task Force on Stability Terms and Definitions, IEEE Transactions on Power Systems, vol. 19, No. 2, May 2004. [KUN 94] P. Kundur, Power System Stability and Control, The EPRI Series Power Engineering, McGraw-Hill, New York, 1993. [AH 05] Mahseredjian J., DENNETIRE S., Dub L., KHODABAKHIAN GRIN LAJOIE-BL, "On a new approach for the simulation of transients in power systems IPST Conference, 2005. [PAC 99] H. SAADAT Power System Analysis, McGraw-Hill, New York, 1999. 196 lines and electricity grids 2 [SAB 73] SABONNADIERE JC, P. AURIOL, "Over-voltages in maneuvering the HV and EHV networks, "Revue gnrale de l'lectricit, t.2, No. 11, p. 715 -- 738, 1973. [STA 68] GW STAGG, EL ABIAD AM, Computer methods in power systems analysis, Mc Graw Hill Book, New York, 1968. [TIX 99] P. TIXADOR, components of electrical networks, the TS course, ENSIEG, 1999. [UML 03] Available at: fr.wikipedia.org / wiki / Unified_Modeling_ Language, 2003.     2 . . 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V s =V   d V   i V   h () F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qOle CompObjqObjInfoEquation Native a< I s =I   d I   i I   h () F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q_1240227287F-v-vOle CompObjqObjInfoEquation Native _1240227288F-v-vOle CompObjq T[]=111a 2 a1aa 2 1[] F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qgЗ V s =ObjInfoEquation Native _1240227289F-v-vOle T[] -1 .V p F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qgV I s =T[] -1 .I pCompObjqObjInfoEquation Native _1240227290F-v-v F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qY- T[] "1 =131aa 2 1a 2 a111Ole CompObjqObjInfoEquation Native [] FMicrosoft Equation 3.0 DS Equation Equation.39q pu"  V d   =13V a +aV b +a 2_1303194212F-v-vOle CompObjfObjInfoEquation Native _1240227292F-v-vOle CompObjq V c  ()      V i   =13V a +a 2 V b +a  V c ()    V h   =13V a +V b +V c ()            {             and       I d   =13I a +aI b +a 2 I c   ()     I i   =13I a +a 2 I b +aI c ()       I h   =13I a +I b +I c ()            { F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qObjInfoEquation Native _1240227293F-v-vOle  S=P+jQ=V a I a* +V b I b* +V c I c* F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qCompObjqObjInfoEquation Native `_1240227294FpwxpwxD, V pT I p* F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q&`I\h I p*Ole CompObjqObjInfoEquation Native B_1240227295FpwxpwxOle CompObjqObjInfo F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q&01 V pT F"Microsoft diteur d'quations 3.Equation Native B_1240227296FpwxpwxOle CompObjq   !"#&)*+,-./036789:;<=>ADEHKLORUX[^_`abcdgjklmnqtuvwx{~0 DS Equation Equation.39qWx S=P+jQ=T[].V s () T .T[].I s () * =V sT .T[] T .T[] * .I s*ObjInfoEquation Native s_1240227297FpwxpwxOle  F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qN(|X T[] T .T[] * FMicrosoft Equation 3.0 DS EqCompObj qObjInfo Equation Native  j_1303194128FpwxpwxOle CompObjfObjInfoEquation Native Fuation Equation.39q *o"}  S=P+jQ=3V sT .I s* =3V d I d* +3V i I i* +3V h I h*_1296185286nFpwxpwxOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q­H.d* V p =Z[] . I p                and              Equation Native _1296185368FpwxpwxOle $CompObj%f        I p =Y[] . V p                     FMicrosoft Equation 3.0 DS Equation Equation.39qW#ObjInfo'Equation Native (_1296185351FpwxpwxOle 1,* T[] . V s =Z[] . T[] . I s          and                T[] . I s =Y[] . T[] . V s                      FMicrosoft Equation 3.0 DS Equation Equation.39qIh,}* . V s =T[] -1  .Z[] . T[] . CompObj2fObjInfo4Equation Native 5e_1240227302FpwxpzI s                and                     I s =T[]  -1 .Y[] . T[] . V s                    Ole ?CompObj@qObjInfoBEquation Native C F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q`6d Z[] s =T[] -1  .Z[] . T[] F"Microsoft diteur d'quations 3._1240227303FpzpzOle FCompObjGqObjInfoI0 DS Equation Equation.39q47 Y[] s =T[] -1  .Y[] . T[] F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qEquation Native J_1240227304FpzpzOle MCompObjNqObjInfoPEquation Native QE_1240227305 FpzpzOle S)(_, Z[] s F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q) " Y[] sCompObjTqObjInfoVEquation Native WE_1296187123Fpzpz FMicrosoft Equation 3.0 DS Equation Equation.39q­N2}0 V p =Z[] . I p                aOle YCompObjZfObjInfo\Equation Native ]nd                     I p =Y[] . V p                      F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q_1240227308 FpzpzOle eCompObjfqObjInfoh6I  Z[]=Z A Z B Z C Z C Z A Z B Z B Z C Z A ()Equation Native iR_1240227309 FpzpzOle oCompObj  pq F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q6  Y[]=Y A Y B Y C Y C Y A Y B Y B Y C Y AObjInfo rEquation Native sR_1240227310Fpz`2}Ole y () F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qZ Z s []=T[] "1 Z[] T[] = CompObjzqObjInfo|Equation Native }_1240227311F`2}`2}Z A +Z B +Z c 000Z A +a 2 Z B +aZ c 000Z A +aZ B +a 2 Z C () F"Microsoft diteur d'quations 3.Ole CompObjqObjInfoEquation Native 0 DS Equation Equation.39qL Z s []=Z d 000Z i 000Z h ()    FMicrosoft Equation 3.0 DS Eq_1303198417F`2}`2}Ole CompObjfObjInfouation Equation.39q _])}' Z d : direct cyclic impedance                   Z i : inverse cyclic impedance  Equation Native {_1240227314'F`2}`2}Ole CompObj q               Z h : homopolar cyclic impedance            F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qObjInfo!Equation Native _1240227315$F`2}`2}Ole p  V d V i V h ()=Z d 000Z i 000Z h () I d I i I h ()CompObj#%qObjInfo&Equation Native _1240227316",)F`2}` F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qy0! V p =E p "Z[].I p F"Microsoft diteur d'quations 3.Ole CompObj(*qObjInfo+Equation Native =0 DS Equation Equation.39q!P E p F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q!< V p_1240227317.F``Ole CompObj-/qObjInfo0Equation Native =_12402273183F`2}`2}Ole CompObj24q F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q#; Z[] F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qObjInfo5Equation Native ?_12402273198F``Ole CompObj79qObjInfo:Equation Native =_12402273206@=F``!T\ I p F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qH T[].V s =E p "Ole CompObj<>qObjInfo?Equation Native Z[].T[].I s F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q0 V s =T[] "1 . E _1240227321BF``Ole CompObjACqObjInfoDEquation Native _1240227322;OGF``Ole CompObjFHqp "T[] "1 . Z[] . T[] . 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.lS V d F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q!p V h F"Microsoft diteur d'quations 3.CompObjqObjInfoEquation Native =_1240227389F0E0EOle CompObjqObjInfoEquation Native b0 DS Equation Equation.39qFGR " Z h .I h F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q_1240227390F0E0EOle CompObjqObjInfoEquation Native _1240227391F0E0EOle CompObjq,$- E a "Z d .I d ="13Z h I cc F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qObjInfoEquation Native _1240227392F0E0Ole @" I d =E a Z d +13Z h Z d I cc F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qCompObjqObjInfoEquation Native _1240227393F00 I d ="131+Z h Z i ()I cc F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qOle CompObjqObjInfoEquation Native =!PO I a F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qd I d +I i +I h =0_1240227394F00Ole CompObjqObjInfoEquation Native _1240227395F00Ole CompObjq F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qL I cc =3E a Z d +Z h +Z d . Z h Z iObjInfoEquation Native _1240227396F00Ole  F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q! V a F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qCompObjqObjInfoEquation Native =_1240227397F00Ole CompObjqObjInfoEquation Native =!z,3 V d F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q!l V i_1240227398F00Ole CompObjqObjInfoEquation Native =_1240227399F00Ole CompObjq F"Microsoft diteur d'quations 3.0 DS Equation Equation.39q!p V h F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qObjInfoEquation Native =_1240227400F00Ole CompObjqObjInfoEquation Native =_1240227401F00!p V h F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qF/$ "3.Z h I hOle CompObjqObjInfoEquation Native b_1240227402F00Ole CompObjqObjInfo F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qB(7 "Z h I cc F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qEquation Native ^_1240227403F00Ole CompObjqObjInfoEquation Native _1240227404F0'0'Ole   V a =3E a 1+Z d Z h +Z d Z i F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qCompObjqObjInfoEquation Native _1240227405F0'0' I b =a 2 I d +aI i +I h F"Microsoft diteur d'quations 3.0 DS Equation Equation.39qOle CompObjqObjInfoEquation Native            " % * - . / 2 7 < ? 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