外文原文及中文翻译
Modelling and Analysis of Electric Power Systems
Power Flow Analysis Fault Analysis Power Systems Dynamics and Stability
Preface
In the lectures three main topics are covered,i.e. ? Power flow an 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. The notes start with a derivation and discussion of the models of the most common power system components to be used in the power flow analysis.A derivation of the power ?ow equations based on physical considerations is then
given.The resulting non-linear equations are for realistic power systems of very large dimension and they have to be solved numerically.The most commonly used
techniques for solving these equations are reviewed.The role of power flow analysis in power system planning,operation,and analysis is discussed.
The next topic covered in these lecture notes is fault current calculations in power systems.A systematic approach to calculate fault currents in meshed,large power systems will be derived.The needed models will be given and the assumptions made when formulating these models discussed.It will be demonstrated that algebraic models can be used to calculate the dimensioning fault currents in a power system,and the mathematical analysis has similarities with the power ?ow analysis,soitis natural to put these two items in Part I of the notes.
In Part II the dynamic behaviour of the power system during and after disturbances(faults) will be studied.The concept of power system stability is de?ned,and different types of power system in stabilities are discussed.While the phenomena in Part I could be studied by algebraic equations,the description of the power system dynamics requires models based on differential 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 insight into these problems,bibliographies are given in the text.
Part I
Static Analysis
1 Introduction
This chapter gives a motivation why an algebraic model can be used to de scribe the power system in steady state.It is also motivated why an algebraic approach can be used to calculate fault currents in a power system.
A power system is predominantly in steady state operation or in a state 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 most of the variables are varying with the
time.However,these variations are most of the time so small that an algebraic,i.e.not time varying model of the 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 the system,and to study these dynamic models are needed.However,when it comes to calculate the fault current sin the system,steady state(static) model swith appropriate parameter values can be used.A fault current consists of two components,a transient part,and a steady state part,but since the transient part can be estimated from the steady state one,fault current analysis is commonly restricted to the calculation of the steady state fault currents.
1.1 Power Flow Analysis
It is of utmost importance to be able to calculate the voltages and currents that different parts of the power system are exposed to.This is essential not only in order to design the different power system components such as
generators,lines,transformers,shunt elements,etc.so that these can withstand the stresses they are exposed to during steady state operation without any risk of
damages.Furthermore,for an economical operation of the system the losses should be kept at a low value taking various constraint into account,and the risk that the system enters into unstable modes of operation must be supervised.In order to do this in a satisfactory way the state 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 and
reactive power flows can be calculated,and other relevant quantities can be calculated in the system.
Generally the power ?ow,or load ?ow,problem is formulated as a nonlinear set of equations
f (x, u, p)=0 (1.1)
where
f is an n-dimensional(non-linear)function
x is an n-dimensional vector containing the state variables,or states,as
components.These are the unknown voltage magnitudes and voltage angles of nodes in the system
u is a vector with(known) control outputs,e.g.voltages at generators with voltage control
p is a vector with the parameters of the network components,e.g.line reactances and resistances
The 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 with in the first part of these lectures.A necessary condition for eq.(1.1) to have a 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 the general case there is no unique solution,and there are also cases when no solution exists.
If the states x are known,all other system quantities of interest can be calculated from these and the known quantities,i.e. u and p.System quantities of interest are active and reactive power flows through lines and transformers,reactive power generation from synchronous machines,active and reactive power consumption by voltage dependent loads, etc.
As mentioned above,the functions f are non-linear,which makes the equations 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 information
?fabout the system.The Jacobian matrix whose elements are given by
?X (?f)ij??X?fi?Xj
(1.3)
can be used form any useful computations,and it is an important indicator of the system conditions.This will also be elaborate on.
1.2 Fault Current Analysis
In the lectures Elektrische Energiesysteme it was studied how to calculate fault currents,e.g.short circuit currents,for simple systems.This analysis will now be extended to deal with realistic systems including several generators,lines,loads,and other system components.Generators(synchronous machines) are important system components when calculating fault currents and their model will be elaborated on and discussed.
1.3 Literature
The material presented in these lectures constitutes only an introduction to the