Power System Analysis
Power System AnalysisPower Flow AnalysisFault AnalysisPower System Dynamics and StabilityLecture 227-0526-00, ITET ETH ZürichGöran AnderssonEEH - Power Systems LaboratoryETH ZürichSeptember 2012
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ContentsPrefaceIviiStatic Analysis11 Introduction1.1 Power Flow Analysis . . . . . . . . . . . . . . . . . . . . . . .1.2 Fault Current Analysis . . . . . . . . . . . . . . . . . . . . . .1.3 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Network Models2.1 Lines and Cables . . . . . . . . . . . . .2.2 Transformers . . . . . . . . . . . . . . .2.2.1 In-Phase Transformers . . . . . .2.2.2 Phase-Shifting Transformers . . .2.2.3 Unified Branch Model . . . . . .2.3 Shunt Elements . . . . . . . . . . . . . .2.4 Loads . . . . . . . . . . . . . . . . . . .2.5 Generators . . . . . . . . . . . . . . . .2.5.1 Stator Current Heating Limit . .2.5.2 Field Current Heating Limit . .2.5.3 Stator End Region Heating Limit3 Active and Reactive Power Flows3.1 Transmission Lines . . . . . . . . . . . .3.2 In-phase Transformers . . . . . . . . . .3.3 Phase-Shifting Transformer with akm 3.4 Unified Power Flow Equations . . . . . .1.4 Nodal Formulation of the Network Equations.1133.569101214161718182020.2121232425275 Basic Power Flow Problem315.1 Basic Bus Types . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Equality and Inequality Constraints . . . . . . . . . . . . . . 32iii
ivContents5.3Problem Solvability . . . . . . . . . . . . . . . . . . . . . . . .346 Solution of the Power Flow Problem6.1 Solution by Gauss-Seidel Iteration . . . . . . . . . . . . .6.2 Newton-Raphson Method . . . . . . . . . . . . . . . . . .6.2.1 One-dimensional case . . . . . . . . . . . . . . . .6.2.2 Quadratic Convergence . . . . . . . . . . . . . . .6.2.3 Multidimensional Case . . . . . . . . . . . . . . . .6.3 Newton-Raphson applied to the Power Flow Equations . .6.4 P θ QU Decoupling . . . . . . . . . . . . . . . . . . . . .6.5 Approximative Solutions of the Power Flow Problem . . .6.5.1 Linearization . . . . . . . . . . . . . . . . . . . . .6.5.2 Matrix Formulation of DC Power Flow Equations.37373940414244454949527 Fault Analysis7.1 Transients on a transmission line . . . . .7.2 Short circuit of a synchronous machine . .7.3 Algorithms for short circuit studies . . . .7.3.1 Generator model . . . . . . . . . .7.3.2 Simplifications . . . . . . . . . . .7.3.3 Solving the linear system equations7.3.4 The superposition technique . . . .7.3.5 The Takahashi method . . . . . . .576163666666676971II.Power System Dynamics and Stability778 Classification and Definitions of Power System Stability8.1 Dynamics in Power Systems . . . . . . . . . . . . . . . . . .8.1.1 Classification of Dynamics . . . . . . . . . . . . . . .8.1.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . .8.2 Power System Stability . . . . . . . . . . . . . . . . . . . . .8.2.1 Definition of Stability . . . . . . . . . . . . . . . . .8.2.2 Classification of Power System Stability . . . . . . .8.3 Literature on Power System Dynamics and Stability . . . .79808081828284879 Synchronous Machine Models9.1 Design and Operating Principle . . . . . .9.1.1 Rotor Types . . . . . . . . . . . .9.1.2 Stator Field . . . . . . . . . . . . .9.1.3 Magnetic Torque . . . . . . . . . .9.2 Stationary Operation . . . . . . . . . . . .9.2.1 Stationary Single Phase Equivalent9.2.2 Phasor diagram . . . . . . . . . . .8989909194959597. . . . . . . . . . . . . . . . . . . . .Circuit. . . . .
vContents9.310 The10.110.210.39.2.3 Operational Limits . . . . . . . . . . . . .Dynamic Operation . . . . . . . . . . . . . . . .9.3.1 Transient Single Phase Equivalent Circuit9.3.2 Simplified Mechanical Model . . . . . . . 98. 100. 100. 100Swing Equation103Derivation of the Swing Equation . . . . . . . . . . . . . . . . 103Analysis of the Swing Equation . . . . . . . . . . . . . . . . . 105Swing Equation as System of First Order Differential Equations10611 Power Swings in a Simple System11.1 The Swing Equation and its Solutions11.1.1 Qualitative Analysis . . . . . .11.1.2 Stable and Unstable Solutions .11.2 Equal Area Criterion . . . . . . . . . .11.3 Lyapunov Stability Criterion . . . . .11.4 Small Signal Analysis . . . . . . . . .11.5 Methods to Improve System Stability.10910911111312112312412712 Power Oscillations in Multi-Machine Systems13112.1 Classical Model for Systems with Several Machines . . . . . . 13112.2 General Model for Electro–Mechanical Oscillations . . . . . . 13413 Voltage Stability13.1 Mechanisms of Voltage Instability . . .13.1.1 Long Term Voltage Instability13.1.2 Short Term Voltage Instability13.2 Simple Systems for Analysis of Voltage13.3 Analysis of Voltage Stability . . . . . .13.3.1 Stability Indicators . . . . . . .13.3.2 Analysis of Simple System . . . . . . . . . . . . . . . . . .Stability. . . . . . . . . . . . . . . .14 Control of Electric Power Systems14.1 Control of Active Power and Frequency . . . .14.1.1 Spinning reserve . . . . . . . . . . . . .14.1.2 Supplementary Reserves . . . . . . . . .14.1.3 Back-Up Reserves . . . . . . . . . . . .14.2 Control of Reactive Power and Voltage . . . . .14.2.1 Reactive Power Control . . . . . . . . .14.2.2 Voltage Control . . . . . . . . . . . . . .14.3 Supervisory Control of Electric Power SystemsA Phase-Shifting Transformers.137137138138139142143144.153. 155. 156. 158. 159. 159. 159. 160. 162165
viB Protections in Electric Power SystemsB.1 Design of Protections . . . . . . . . . .B.2 Distance Protections . . . . . . . . . .B.2.1 General Principles . . . . . . .B.2.2 Automatic Re-Closure . . . . .B.3 Out of Step Protections . . . . . . . .B.4 System Protections . . . . . . . . . . .Contents.169169171171173174174
PrefaceThese notes are intended to be used in the lecture Power System Analysis (Lecture number ETH Zürich 227-0526-00) (Modellierung und Analyseelektrischer Netze) given at ETH Zürich in Information Technology andElectrical Engineering. In these lectures three main topics are covered, i.e. 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 systems. In appendicesbrief overviews of phase-shifting transformers and power system protectionsare given.The notes start with a derivation and discussion of the models of the mostcommon power system components to be used in the power flow analysis.A derivation of the power flow equations based on physical considerations isthen given. 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 isdiscussed.The next topic covered in these lecture notes is fault current calculations 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 discussed.It will be demonstrated that algebraic models can be used to calculate thedimensioning fault currents in a power system, and the mathematical analysis has similarities with the power flow analysis, so it is natural to put thesetwo items in Part I of the notes.In Part II the dynamic behaviour of the power system during and afterdisturbances (faults) will be studied. The concept of power system stabilityis defined, and different types of power system instabilities are discussed.While the phenomena in Part I could be studied by algebraic equations,the description of the power system dynamics requires models based ondifferential equations.These lecture notes provide only a basic introduction to the topics above.To facilitate for readers who want to get a deeper knowledge of and insightinto these problems, bibliographies are given in the text.vii
viiiPrefaceI want to thank numerous assistants, PhD students, and collaborators ofPower Systems Laboratory at ETH Zürich, who have contributed in variousways to these lecture notes.Zürich, September 2012Göran Andersson
Part IStatic Analysis1
1IntroductionThis chapter gives a motivation why an algebraic model can be used to describe the power system in steady state. It is also motivated why an algebraicapproach can be used to calculate fault currents in a power system.APOWER SYSTEM is predominantly in steady state operation or in astate that could with sufficient accuracy be regarded as steady state.In 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, i.e. not time varying model ofthe power system is justified.A short circuit in a power system is clearly not a steady state condition.Such 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 calculation of the steady statefault currents.1.1Power Flow AnalysisIt is of utmost importance to be able to calculate the voltages and currentsthat different parts of the power system are exposed to. This is essentialnot 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 economical operation 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 thestate of the system, i.e. all (complex) voltages of all nodes in the system,must be known. With these known, all currents, and hence all active and1
21. Introductionreactive power flows can be calculated, and other relevant quantities can becalculated in the system.Generally the power flow, or load flow, problem is formulated as a nonlinear set of equationsf (x, u, p) 0(1.1)wheref is an n-dimensional (non-linear) functionx is an n-dimensional vector containing the state variables, or states, ascomponents. These are the unknown voltage magnitudes and voltageangles of nodes in the systemu is a vector with (known) control outputs, e.g. voltages at generators withvoltage controlp is a vector with the parameters of the network components, e.g. linereactances and resistancesThe power flow problem consists in formulating the equations f in eq. (1.1)and then solving these with respect to x. This will be the subject dealt within the first part of these lectures. A necessary condition for eq. (1.1) to havea physically meaningful solution is that f and x have the same dimension,i.e. 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 exists.If the states x are known, all other system quantities of interest canbe calculated from these and the known quantities, i.e. u and p. Systemquantities of interest are active and reactive power flows through lines andtransformers, reactive power generation from synchronous machines, activeand reactive power consumption by voltage dependent loads, etc.As mentioned above, the functions f are non-linear, which makes theequations harder to solve. For the solution of the equations, the linearization f x y x(1.2)is quite often used and solved. These equations give also very useful infor fmation about the system. The Jacobian matrix, whose elements are xgiven by f fi (1.3) x ij xjcan be used for many useful computations, and it is an important indicatorof the system conditions. This will also be elaborated on.
1.2. Fault Current Analysis1.23Fault Current AnalysisIn the lectures Elektrische Energiesysteme it was studied how to calculatefault currents, e.g. short circuit currents, for simple systems. This analysiswill now be extended to deal with realistic systems including several generators, lines, loads, and other system components. Generators (synchronousmachines) are important system components when calculating fault currentsand their modelling will be elaborated on and discussed.1.3LiteratureThe material presented in these lectures constitutes only an introductionto the subject. Further studies can be recommended in the following textbooks:1. Power Systems Analysis, second edition, by Artur R. Bergen and VijayVittal. (Prentice Hall Inc., 2000, ISBN 0-13-691990-1, 619 pages)2. Computational Methods for Large Sparse Power Systems, An objectoriented approach, by S.A. Soma, S.A. Khaparde, S. Pandit (KluwerAcademic Publishers, 2002, ISBN 0-7923-7591-2, 333 pages)3. Electric Energy Systems - Analysis and Operation. A. Gómez-Expósito,A.J. Conejo, C. Cañizares (Editors), (CRC Press, Boca Raton, Florida,2009, ISBN 978-0-8493-7365-7)4. Power System Stability and Control, P. Kundur, (McGraw-Hill, NewYork, 1994. ISBN 0-07-035958-X)5. Power System State Estimation: Theory and Implementation, A. Abur,A. Gómez-Expósito (Marcel Dekker, 2004, ISBN 0-8247-5570-7)
41. Introduction
2Network 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 problem.ALL ANALYSIS in the engineering sciences starts with the formulationof appropriate models. A model, and in power system analysis we almost 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 physical or engineered component or system. Hence, depending on the purposeof the analysis different models of the same physical system or componentsmight be valid. It is recalled that the general model of a transmission linewas given by the telegraph equation, which is a partial differential equation,and by assuming stationary sinusoidal conditions the long line equations,ordinary differential equations, were obtained. By solving these equationsand restricting the interest to the conditions at the ends of the lines, thelumped-circuit line models (π-models) were obtained, which is an algebraicmodel. This gives us three different models each valid for different purposes.In principle, the complete telegraph equations could be used when studying the steady state conditions at the network nodes. The solution wouldthen include the initial switching transients along the lines, and the steadystate solution would then be the solution after the transients have decayed.However, such a solution would contain a lot more information than wantedand, furthermore, it would require a lot of computational effort. An algebraic formulation with the lumped-circuit line model would give the sameresult with a much simpler model at a lower computational cost.In the above example it is quite obvious which model is the appropriateone, but in many engineering studies the selection of the “correct” modelis often the most difficult part of the study. It is good engineering practiceto use as simple models as possible, but of course not too simple. If toocomplicated models are used, the analysis and computations would be unnecessarily cumbersome. Furthermore, generally more complicated modelsneed more parameters for their definition, and to get reliable values of theserequires often extensive work.5
62. Network ModelsiR dxuL dxG dxi diC dx u dudxFigure 2.1. Equivalent circuit of a line element of length dxFigure 2.1. Equivalent circuit of a line element of length dxIn the subsequent sections algebraic models of the most common powersystem components suitable for power flow calculations will be derived. Ifnot explicitly stated, symmetrical three-phase conditions are assumed in thefollowing.2.1Lines and CablesThe equivalent π-model of a transmission line section was derived in the lectures Electric Power Systems (Elektrische Energiesysteme), 227-0122-00L.The general distributed model is characterized by the series parametersR′ series resistance/km per phase (Ω/km)X ′ series reactance/km per phase (Ω/km)and the shunt parametersB ′ shunt susceptance/km per phase (siemens/km)G′ shunt conductance/km per phase (siemens/km)as depicted in Figure 2.1. The parameters above are specific for the lineor cable configuration and are dependent on conductors and geometricalarrangements.From the circuit in Figure 2.1 the telegraph equation is derived, and fromthis the lumped-circuit line model for symmetrical steady state conditions,Figure 2.2. This model is frequently referred to as the π-model, and it ischaracterized by the parametersZkm Rkm jXkm series impedance (Ω)sh Gsh jB sh shunt admittance (siemens)Ykmkmkm11In Figure 2.2 the two shunt elements are assumed to be equal, which is true for
72.1. Lines and CableszkmI kmI mkkmshykmshykmFigure 2.2. Lumped-circuit model (π-model) of a transmission linebetween nodes k and m.Note. In the following most analysis will be made in the p.u. system. Forimpedances and admittances, capital letters indicate that the quantity is expressed in ohms or siemens, and lower case letters that they are expressedin p.u.Note. In these lecture notes complex quantities are not explicitly markedas underlined. This means that instead of writing z km we will write zkmwhen this quantity is complex. However, it should be clear from the contextif a quantity is real or complex. Furthermore, we will not always use specifictype settings for vectors. Quite often vectors will be denoted by bold face typesetting, but not always. It should also be clear from the context if a quantityis a vector or a scalar.When formulating the network equations the node admittance matrixwill be used and the series admittance of the line model is needed 1ykm zkm gkm jbkmwithgkm rkm x2km2rkmandbkm xkm2 x2rkmkm(2.1)(2.2)(2.3)homogenous lines, i.e. a line with equal values of the line parameters along its length, butthis might not be true in the general case. In such a case the shunt elements are replacedshshshshby Ykmand Ymkwith Ykm6 Ymkwith obvious notation. A general model is presented insect. 2.2.3, which takes asymmetric conditions into account.
82. Network ModelsFor actual transmission lines the series reactance xkm and the series resistance rkm are both positive, and consequently gkm is positive and bkm issh and the shunt conductance g sh arenegative. The shunt susceptance ykmkmsh is soboth positive for real line sections. In many cases the value of gkmsmall that it could be neglected.The complex currents Ikm and Imk in Figure 2.2 can be expressed asfunctions of the complex voltages
Power flow analysis Fault current calculations Power systems dynamics and stability In Part I of these notes the two first items are covered, while Part II gives an introduction to dynamics and stability in power systems. In appendices brief overviews of phase-shifting transformers and power system protections are given.
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