## Introduction

Welcome notes

Online page

Course contents:

Power flow

Short circuit analysis

State estimation

Course structure

Lab tests (two)

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 lines

## power 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

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

Newton's method for solving pf eqs for voltage real part and imaginary part
Presentation of Tinney's paper on pf and Newton's method
Gaussian elimination, triangularization, LU factors, bifactorization for Matlab's A\b
The question of no. of eqs and no. of variables
Adjustments necessary for pf solutions (eg voltage specifications, reactive injection, xf taps)

PV nodes that may convert to false PQ nodes
false PQ nodes that may convert to PV nodes
adjustments based on intermediate pf solutions such as tap changing xfs
Fast Decoupled Load Flow and B Stott
Going thru Sott's paper
Why Newton-like methods work
Nonlinearity and successive linearization
Linearity

## Stott’s FDLP (continued)

Review of Stott’s FDLP

Loose physical interaction between P and Q

Convergence characteristics

Iteration and half iteration

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

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

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

Line transposition

Banks of transformers

Concepts for fault analysis

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 neutral

## Transformer 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

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 loads

## Unbalanced 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

Why a link and not a network