Prepared by: Ismael K. Saeed Chapter One : Introduction, Single Line Diagram, Per Unit System and Network Matrices. Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed INTRODUCTION Modern Power Systems Power Producer generation station prime mover and generator step-up transformer Transmission System HV transmission lines switching stations circuit breakers transformers Distribution Utility distribution substations step-down transformers MV distribution feeders distribution transformers Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed System Control Network Protection Energy Management Systems Energy Control Centre computer control SCADA - Supervisory Control And Data Acquisition Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Practical power systems must be safe reliable economical System Analysis for system planning for system operations requires component modelling types of analysis transmission line performance power flow analysis economic generation scheduling fault and stability studies Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed System Modelling Systems are represented on a per-phase basis A single-phase representation is used for a balanced system The system is modelled as one phase of a Y-connected network Symmetrical components are used for unbalanced systems Unbalance systems may be caused by: generation, network components, loads, or unusual operating conditions such as faults The per-unit system of measurements is used Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Single Line Diagram A single line diagram or one-line diagram is a representation of the essentials of a system in a most simplified form. The purpose of one-line diagram is to supply in concise form the significant information about the system. Thus, short line feature can be applied to a large part of the grid network. Power Component Symbol = Generator Circuit breaker = Transformer Transmission line = Motor Feeder + load M = Busbar (substation) Power components and symbols Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Per-Unit Quantities Prepared by: Ismael K. Saeed

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Prepared by: Ismael K. Saeed Example The one-line diagram of a three-phase power system is shown Use a common base of 100 MVA and 22 kV at the generator draw an impedance diagram with all impedances marked in per-unit the manufacturer’s data for each apparatus is given as follows G: 90 MVA 22 kV 18% T1: 50 MVA 22/220 kV 10% L1: 48.4 ohms T2: 40 MVA 220/11 kV 6% T3: 40 MVA 22/110 kV 6.4% L2: 65.43 ohms T4: 40 MVA 110/11 kV 8% M: 65.5 MVA 10.45 kV 18.4% Ld: 57 MVA 10.45 kV 0.6 pf lag Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Since generator and Transformer voltage bases are the same as their rated values, their per-unit reactances on a 100MVA base are: Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Motor reactance on a 100 MVA, 11 kV base is Impedance bases for lines 1 and 2 are: Lines 1 and 2 per-unit reactances are: The load apparent power at 0.6 power factor lagging is given by Hence, the load impedance in ohms is Prepared by: Ismael K. Saeed

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Prepared by: Ismael K. Saeed Network Matrices Formation of Bus Admittance Matrix Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Consider node (bus) 1 that is connected to the nodes 2 and 3. Then applying KCL at this node we get In a similar way application of KCL at nodes 2, 3 and 4 results in the following equations Prepared by: Ismael K. Saeed

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Prepared by: Ismael K. Saeed Node Elimination by Kron Reduction Consider an equation of the form Consider an equation of the form where A is an ( n X n ) real or complex valued matrix, x and b are vectors in either Rn or Cn . assume that the b vector has a zero element in the n th row is given as We can then eliminate the kth row and kth column to obtain a reduced ( n - 1) number of equations of the form Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed   The elimination is performed using the following elementary operations 𝑎 𝑘𝑗 𝑛𝑒𝑤 = 𝑎 𝑘𝑗 𝑜𝑙𝑑 − 𝑎 𝑘𝑛 ∗ 𝑎 𝑛𝑗 𝑎 𝑛𝑛 In electrical power systems in which those nodes with zero current injections are eliminated are said to be KRON reduced. If Ip = 0 in the nodal equations of the N-bus system, we may directly calculate the elements of the new reduced bus admittance matrix by choosing Ypp as the pivot and by eliminating bus p using the formula where j and k take on all the integer values from 1 to N except p since row p and column p are to be eliminated. Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Example: The nodal admittance equations of the system in p.u values is using Y22 as the initial pivot, eliminate node 2 and the corresponding voltage v2 from 4 * 4 system. Solution: Prepared by: Ismael K. Saeed

Prepared by: Ismael K. Saeed Similar calculations yields the other elements of the KRON-reduced matrix (1) (3) (4) (1) (3) (4) Prepared by: Ismael K. Saeed