Power System Analysis

1 Power System AnalysisPower Flow AnalysisFault AnalysisPower System Dynamics and StabilityLecture 227-0526-00, ITET ETH Z urichG oran AnderssonEEH - Power Systems LaboratoryETH Z urichSeptember 2012iiContentsPrefaceviiI Static Analysis11 Power Flow Analysis .. Fault Current Analysis .. Literature .. 32 Network Lines and Cables .. Transformers .. In-Phase Transformers .. Phase-Shifting Transformers .. Unified Branch Model .. Shunt Elements .. Loads .. Generators .. Stator Current Heating Limit .. Field Current Heating Limit .. Stator End Region Heating Limit .. 203 Active and Reactive Power Transmission Lines .. In-phase Transformers .. Phase-Shifting Transformer withakm= 1.. Unified Power Flow equations .. 254 Nodal Formulation of the Network Equations275 Basic Power Flow Basic Bus Types .. Equality and Inequality Constraints .. Problem Solvability .. 346 Solution of the Power Flow Solution by Gauss-Seidel Iteration.

2 Newton-Raphson Method .. One-dimensional case .. Quadratic Convergence .. Multidimensional Case .. Newton-Raphson applied to the Power Flow equations .. QUDecoupling .. Approximative Solutions of the Power Flow Problem .. Linearization .. Matrix Formulation of DC Power Flow equations .. 527 Fault Transients on a transmission line .. Short circuit of a synchronous machine .. Algorithms for short circuit studies .. Generator model .. Simplifications .. Solving the linear System equations .. The superposition technique .. The Takahashi method .. 71II Power System Dynamics and Stability778 Classification and Definitions of Power System Stability Dynamics in Power Systems .. Classification of Dynamics .. Modelling .. Power System Stability .. Definition of Stability .. Classification of Power System Stability .. Literature on Power System Dynamics and Stability ..879 Synchronous Machine Design and Operating Principle.

3 Rotor Types .. Stator Field .. Magnetic Torque .. Stationary Operation .. Stationary Single Phase Equivalent Circuit .. Phasor diagram .. Operational Limits .. Dynamic Operation .. Transient Single Phase Equivalent Circuit .. Simplified Mechanical Model .. 10010 The Swing Derivation of the Swing Equation .. Analysis of the Swing Equation .. Swing Equation as System of First Order Differential Equations10611 Power Swings in a Simple The Swing Equation and its Solutions .. Qualitative Analysis .. Stable and Unstable Solutions .. Equal Area Criterion .. Lyapunov Stability Criterion .. Small Signal Analysis .. Methods to Improve System Stability .. 12712 Power Oscillations in Multi-Machine Classical Model for Systems with Several Machines .. General Model for Electro Mechanical Oscillations .. 13413 Voltage Mechanisms of Voltage Instability .. Long Term Voltage Instability .. Short Term Voltage Instability.

4 Simple Systems for Analysis of Voltage Stability .. Analysis of Voltage Stability .. Stability Indicators .. Analysis of Simple System .. 14414 Control of Electric Power Control of Active Power and Frequency .. Spinning reserve .. Supplementary Reserves .. Back-Up Reserves .. Control of Reactive Power and Voltage .. Reactive Power Control .. Voltage Control .. Supervisory Control of Electric Power Systems .. 162A Phase-Shifting Transformers165viContentsB Protections in Electric Power Design of Protections .. Distance Protections .. General Principles .. Automatic Re-Closure .. Out of Step Protections .. System Protections .. 174 PrefaceThese notes are intended to be used in the lecturePower System Analy-sis(Lecture number ETH Z urich 227-0526-00) (Modellierung und Analyseelektrischer Netze) given at ETH Z urich in Information Technology andElectrical Engineering. In these lectures three main topics are covered, Power flow Analysis Fault current calculations Power systems dynamics and stabilityIn Part I of these notes the two first items are covered, while Part II givesan introduction to dynamics and stability in Power appendicesbrief overviews of phase-shifting transformers and Power System protectionsare notes start with a derivation and discussion of the models of the mostcommon Power System components to be used in the Power flow derivation of the Power flow equations based on physical considerations isthen given.

5 The resulting non-linear equations are for realistic Power systemsof very large dimension and they have to be solved numerically. The mostcommonly used techniques for solving these equations are reviewed. The roleof Power flow Analysis in Power System planning, operation, and Analysis next topic covered in these lecture notes is fault current calcula-tions in Power systems. A systematic approach to calculate fault currentsin meshed, large Power systems will be derived. The needed models will begiven and the assumptions made when formulating these models will be demonstrated that algebraic models can be used to calculate thedimensioning fault currents in a Power System , and the mathematical analy-sis has similarities with the Power flow Analysis , so it is natural to put thesetwo items in Part I of the Part II the dynamic behaviour of the Power System during and afterdisturbances (faults) will be studied. The concept of powersystem stabilityis defined, and different types of Power System instabilities are the phenomena in Part I could be studied by algebraic equations ,the description of the Power System dynamics requires models based ondifferential lecture notes provide only a basic introduction to thetopics facilitate for readers who want to get a deeper knowledge of and insightinto these problems, bibliographies are given in the want to thank numerous assistants, PhD students, and collaborators ofPower Systems Laboratory at ETH Z urich, who have contributed in variousways to these lecture urich, September 2012G oran AnderssonPart IStatic Analysis11 IntroductionThis chapter gives a motivation why an algebraic model can be used to de-scribe the Power System in steady state.

6 It is also motivated why an algebraicapproach can be used to calculate fault currents in a Power System is predominantly in steady state operation or in astate that could with sufficient accuracy be regarded as steady a Power System there are always small load changes, switching actions,and other transients occurring so that in a strict mathematical sense mostof the variables are varying with the time. However, these variations aremost of the time so small that an algebraic, not time varying model ofthe Power System is short circuit in a Power System is clearly not a steady an event can start a variety of different dynamic phenomena in thesystem, and to study these dynamic models are needed. However, whenit comes to calculate the fault currents in the System , steady state (static)models with appropriate parameter values can be used. A fault currentconsists of two components, a transient part, and a steady state part, butsince the transient part can be estimated from the steady state one, faultcurrent Analysis is commonly restricted to the calculationof the steady statefault Power Flow AnalysisIt is of utmost importance to be able to calculate the voltages and currentsthat different parts of the Power System are exposed to.

7 This isessentialnot only in order to design the different Power System components suchas generators, lines, transformers, shunt elements, etc. so that these canwithstand the stresses they are exposed to during steady state operationwithout any risk of damages. Furthermore, for an economicaloperation ofthe System the losses should be kept at a low value taking various constraintsinto account, and the risk that the System enters into unstable modes ofoperation must be supervised. In order to do this in a satisfactory way thestateof the System , all (complex) voltages of all nodes in thesystem,must be known. With these known, all currents, and hence all active and121. Introductionreactive Power flows can be calculated, and other relevant quantities can becalculated in the the Power flow, or load flow, problem is formulated as a non-linear set of equationsf(x,u,p) = 0( )wherefis ann-dimensional (non-linear) functionxis ann-dimensional vector containing the state variables, or states, ascomponents.

8 These are the unknown voltage magnitudes and voltageangles of nodes in the systemuis a vector with (known) control outputs, voltages at generators withvoltage controlpis a vector with the parameters of the network components, linereactances and resistancesThe Power flow problem consists in formulating the equationsfin eq. ( )and then solving these with respect tox. This will be the subject dealt within the first part of these lectures. A necessary condition foreq. ( ) to havea physically meaningful solution is thatfandxhave the same dimension, that we have the same number of unknowns as equations . But in thegeneral case there is no unique solution, and there are also cases when nosolution the statesxare known, all other System quantities of interest canbe calculated from these and the known quantities, Systemquantities of interest are active and reactive Power flows through lines andtransformers, reactive Power generation from synchronousmachines, activeand reactive Power consumption by voltage dependent loads, mentioned above, the functionsfare non-linear, which makes theequations harder to solve.

9 For the solution of the equations , the linearization f x x= y( )is quite often used and solved. These equations give also very useful infor-mation about the System . The Jacobian matrix f x, whose elements aregiven by f x ij= fi xj( )can be used for many useful computations, and it is an important indicatorof the System conditions. This will also be elaborated Fault Current Fault Current AnalysisIn the lecturesElektrische Energiesystemeit was studied how to calculatefault currents, short circuit currents, for simple systems. This analysiswill now be extended to deal with realistic systems including several gener-ators, lines, loads, and other System components. Generators (synchronousmachines) are important System components when calculating fault currentsand their modelling will be elaborated on and LiteratureThe material presented in these lectures constitutes only an introductionto the subject. Further studies can be recommended in the following Systems Analysis , second edition, by Artur R.

10 Bergen and VijayVittal. (Prentice Hall Inc., 2000, ISBN 0-13-691990-1, 619pages) Methods for Large Sparse Power Systems, An objectoriented approach, by Soma, Khaparde, S. Pandit (KluwerAcademic Publishers, 2002, ISBN 0-7923-7591-2, 333 pages) Energy Systems - Analysis and Operation. A. G omez-Exp osito, Conejo, C. Ca nizares (Editors), (CRC Press, Boca Raton, Florida,2009, ISBN 978-0-8493-7365-7) System Stability and Control, P. Kundur, (McGraw-Hill, NewYork, 1994. ISBN 0-07-035958-X) System State Estimation: Theory and Implementation, A. Abur,A. G omez-Exp osito (Marcel Dekker, 2004, ISBN 0-8247-5570-7)41. Introduction2 Network ModelsIn this chapter models of the most common network elements suitable forpower flow Analysis are derived. These models will be used in the subsequentchapters when formulating the Power flow Analysis in the engineering sciences starts with the formulationof appropriate models. A model, and in Power System analysiswe al-most invariably then mean a mathematical model, is a set of equations orrelations, which appropriately describes the interactions between differentquantities in the time frame studied and with the desired accuracy of a phys-ical or engineered component or System .