Planeamento
Aulas Teóricas
Introduction
Welcome notes
Online page
Course contents:
Power flow
Short circuit analysis
State estimation
Course structure
Lab tests (two)
Models for generator, line, load
Line as a two-port network
Linear model: Y, Z, H,
Symmetry and reciprocity
Two bus system
Writing the PF equation
Linearity and analyticity
How to solve it: Jacobi, Gauss
Fixed-point theorem
Iterative procedure: does it work?Power flow or load flow
Review of a power flow equation in complex notation
Power circles in a (p,q) plane
The importance of reactive support for power transfer
Generalization of the power flow equation for a system of any number of buses
Jacobi-Gauss procedure (how, why, when it works)
Classification of some bus types
A closed-form solution for a two bus system connected by a longitudinal impedance; feasible and infeasible solutions; the role of reactive power
Jacobi-Gauss and PV buses
Review of
PQ bus
PV bus
Reference bus
Slack bus
The two-bus system with a PV bus
Phase angles as “state” variables
Adapt Jacobi-Gauss method to handle PV buses
Review of the pu system and percentage
How to choose values for basis
Typical quantities
The pi model for linespower flow modelling
PF equation -- revisited
Current injections vs power injections
Models for generators as S injectors or S sources
Generators as P sources and V sources
Load Demand as negative power injections (strictly active loads)
Load demand as admittances (strictly passive loads)
Formulas
Demand as a function of voltage D=f(V)
The concept of elasticity epsilon
Epsilon for P and for Q
Range for elasticity
How to handle elasticity in a power flow program
Example for Jacobi-Gauss formulas
Basic introduction to transformer (xf)
Model for xf -- simple models
Steinmetz model
xfs and Newton
review of last class
Xfs and per unit system
Vsc and Zsc as a model for xfs
TCUL xfs and offline tap xfs
model for transformer with complex m
role of phase shifters
why model transformer with an admittance model
when a pi model is possible
symmetry and reciprocity
Special case of zero impedance transformer
Newton’s method for solving pf problems
principles of Newton method
problems encountered when applied to ps
Newton’s method applied to power flow eq
Newton’s method applied to power flow eq
Newton and Taylor
Formal iterative Newton’s algorithm
Geometric interpretation (could one do it better? success vs shape of f(x))
Why is it necessary to work with non-complex quantities for Newton’s method applied to pf equation
Separating complex equation into real part and imaginary part
Options: consider voltage as V. cis(teta); other options
Jacobian: ordering, properties. symmetry, real-valuedness, sparsity, positive-definiteness?
Importance of Jacobian properties on the success of Newton’s methods
Newton's method (continued)
Newton's method for solving pf eqs for voltage magnitude and phase angle
pf adjustments and fdlf
pf adjustments
Stott’s FDLP (continued)
Review of Stott’s FDLP
Advantage of physical properties
Loose physical interaction between P and Q
Convergence characteristics
Iteration and half iteration
Testing a load flow program
Special challenges posed by special network types
Tap changing -- how to handle that
PV buses converted to PQ and PQ reconverted to PV
Approach by retriangulating B”
MIL matrix inversion lemma
Approach by correction of voltages by a linear coefficient of sensitivity
Outages, security and contingencies
Generation outages
Introduction to branch outages
DCPF; contingency analysis
What is DCPF?
Derivation from FDLF method
What makes DCPF so important?
Analogy with DC circuits
How to compute losses with DCPF
Compensation by MIL
Analysis of MIL
Application to single line contingency
MIL and contingency analysis in the FDLF context
An introduction to circuit compensation methods
Loss sensitivities, radial networks
The role of losses in engineering planning
How to assess losses
How to compute losses traditionally (B method)
How to compute losses today
How to compute derivatives for losses (or sensitivities): traditionally and by using power flow results and power flow solution, including pf Jacobian
How to compute those derivatives using a power flow program and the definition of derivative
Example of a situation in which one could compute sensitivity of losses wrt power injections
Radial networks and distribution system
Why a radial structure for networks
Trees and the relationship parent-child
Power flow for distribution networks
Upward sweeping for currents
Downward sweeping for voltages
Convergence criterion
Review of power flow fundamentals; introduction to short circuit analysis
Review of power flow fundamentals
Symmetric system
Balanced loads
Line transposition
Banks of transformers
Concepts for fault analysis
Current, power, impedance, admittance
Meanings and importance
Thevenin and Norton
Example
DY-YD conversions
Fictitious nodes
Three phase fault analysis
Review of basics of short circuit
Short circuit at generators
Models of X”, X’, and X behind an emf
The importance of Z matrix
Computation of Isc based on Zmm
Computation of voltages at any node and currents in any branch of the network under short-circuit condition
How to obtain Z matrix
Y extended: why and how
Examples
Symmetrical components
Review of short-circuit time evolution for generators
Models with X, X’, and X”
The problem of unbalanced systems
The idea behind symmetrical components
Voltage and currents in dih
F (Fortescue) transformation, inverse, hermitian and orthogonality
Similarity transformation for diagonalizing
The demand for cyclical symmetry
General procedure to deal with unbalance: go to dih, work there, come back in the end
Examples of what the network and network element models look like in dih
Symmetrical components and LG short circuit
Symmetrical components and LG short circuit
Review of Symmetrical components (Fortescue transformation)
Basis (d,i,q),
Diagonlization of Zabc
Modelling of a generator in dih
LG short-cricuit Ia=Ea/Za
How to compute Za in dih
Traditional approach to get the series circuit arrangement of dih circuits
The role of Zn, derivation of formulas
Three Thevenin equivalents
Special case of h circuits
Example for a YY transformer
LL and LLG faults
Review of LG fault from abc to dih
Circuit arrangement for LL fault (also from abc to dih)
Circuit arrangement for LLG fault (also from abc to dih)
Role of neutral impedances and fault impedances
Choice of which phases to fault
Duality circuitry: phase a to ground and bc clear and bc to ground and a clear
Voltages and currents
Procedures for computation
Summary of the ideas behind symmetrical component analysis and use of three-phase symmetric analysis
Voltages and currents in a system under fault condition
Review of symmetric three-phase fault analysis
Impedance matrix
Driving point impedance and short-circuit
Voltages at bus m for a short-circuit at bus j
Currents thru branch mn for a shot circuit at bus j
Review of nonsymmetric faults
Review of dih circuit arrangement for LG fault
Impedance matrices for dih
Finding dih currents at the short-circuit
Finding dih voltages at bus m for dih injections at bus j
Finding dih currents thru branch mn for dih injections at bus j
Finding abc voltages and currents everyhere
What is still missing: the construction of Yih for Zih
Generators: Zi and Zh; and grounding Zn
Lines: Zh and Carson’s formulas
Transformers: YY0 example
Importance of Zn1 and Zn2
Case of Zn1 = infinity
Practical consequences for isolated neutralTransformer connections
The most common connections: YY, DY, YD, DD
Models
Phase shift due to D
0, 2, 4, ... for YY
1, 3, 5, ... for DY, YD
Roles: changes decrease in magnitude and balance in phase
When each is mostly used
Neutral isolated -- fault currents and fault voltages
Neutral solidly grounded -- fault currents and fault voltages
Neutral grounded thru a resistance or reactance
Compensated neutral (Petersen coil)
Transformers and neutrals
Revision: given V1abc and V2abc, compute I12a
Do it in two ways: (1) using Idih and Vdih (and Zdih) formulas
(2) using Ibac and Vabc (and Zabc)
Transformers and neutrals
Neutral resistances and neutral reactances
Importance of neutral -- examples
Connections in zig-zag for making neutral reactances
Performance and importance of zig-zag reactances
State Estimation
The problem of State Estimation
General formulation
Determined, undetermined and overdetermined problems
What’s the time if one has two watches
The role of sigma
Best estimates (mean square error)
Measurements and linear model constraints
z=Hx+e
Characteristics of e
What if e is not normal?
Compact formulation and W >0
Best linear estimate for x, x^
Distribution of e^2
Chi square distribution: characteristics
State estimation (continued)
State estimation revisited
Linear case
z is a linear combination of x
Model coefficients
x^ is a linear combination of z
Estimate coefficients
Errors and residues
Covariance for errors and covariance for residues
Errors normalized and squared are Chi square
Residues normalized by their standard deviation (obtained from the diagonal of computed covariance)
Sum of squared residues as Chi square
Test of hypothesis and significance level alpha
Difference between alpha=1% and alpha=5% and their impact on rejection (or not) of measurements
Rejection helps? How much can one reject?
Nonlinear state estimation
Nonlinear least-squares and lsqnonlin
What is x for power system state estimation?
Example of f1, f2, ...
How to solve it, gradient, and how to deliver the problem to a mathematician and get the solution back
Three winding transformers; Probabilistic fault analysis
Nonlinear state estimation (review)
Three winding transformer
Short circuit voltages
Star equivalent system
Fictitious node and possibly negative reactance branch
Probabilistic fault analysis
Causes of uncertainty
PDF for currents and voltage sags
The regulatory environment
An introduction to probabilistic power flow
Probabilistic power flow
Power flow for planning
Multiple scenario analysis
Peak load scenario
Does one need an off peak load scenario as well?
Other drivers for investment planning
Quality of service (duration and frequency of interruptions, voltage sags, )
System security and robustness
Capture of renewables
Scenario generation
Output analysis
Power flow as the working horse of system analysis
Power flow for reliability assessment
Load, equipment, resource availability as random quantities
Sampling techniques and Monte Carlo
Importance of autocorrelation and cross-correlation for sampling
Probabilistic power flow and radial networks
Simultaneity factor
Kirchhoff for random currents and random powers
Currents based on mu+2*sigma (95% certainty)
The apparent non-compliance with KCL
Example for a case of independent loadsUnbalanced power flow; DC transmission link
Unbalanced power flow
Two options: the choice of using the already-familiar abc-dih-abc approach
Example for a two-bus system
DC transmission link
Why a link and not a network
Advantages and disadvantages of DC links
Links for long distances and for offshore wind power plants
Problems and solutions
Review
General review