# Projects:2015s1-10 Lagrangian Modelling of Synchronous Machines

## Project Member

Student:

Christopher Bowley

Supervisors:

Andrew Allison

David Vowles

## Introduction

When dealing with a large electrical system, its always been a difficult tasks to determine the stability. The electrical and mechanical system can be expressed in term of energy domain. Modelling a system in energy domain can reduces the complexity of the solutions, as solving it does not required detailed solutions compared to the conventional model. This project serve as a proof of the concept of energy domain modelling.

## Aim

The aim of this project is to try to create a Lagrangian model of a synchronous machine and determine whether such a model is viable by using it to solve a simple stability problem. If the model is viable then it could be applied to solve more complicated stability problems and analyse real world power systems.

## Technical Background

There two aspects of technical background covered in this project, the behaviour of the synchronous machine and the energy domain modelling using the Lagrangian method.

Synchronous machine

Synchronous machines are type of machine that can be use either as a motor, converting electrical power to mechanical power or a generator, converting mechanical power to electrical power. This machine consists of stator and rotor. The rotor will produces magnetic field either by using permanent magnet or by passing constant DC current through a set of coils of wire. When acting as a motor, the three phase AC power is supplied to the stator causing the rotor to rotate at synchronous speed. When the machine acting as generator the rotor will be driven by a mechanical force that spun at near synchronous machine, and the rotating magnetic field will cause a three phase voltage generated in the stator windings. In this project the synchronous machine will be using as a generator and it behaviour will be examined.

Lagrangian Methods

Within a closed system energy can neither be destroyed nor created, it can only change from one form to another which this is the basis of the Lagrangian methods. The basic of the Lagrangian function without the consideration of losses had come out with the following equation.

ℒ = T - V

Where, ℒ is the lagrangian, T is the kinetic energy and V is the potential energy.

## Method

Short-circuit and open-circuit test

The stator resistance will be measured by having the average value of all the phase stator resistance. A 5% value is then added considering the skin effect, this value is then halved to obtain the stator resistance.

In the open circuit test the output of the generator is left open, this resulting to no current flow and the value of the output V and E to be equal. This allow to the value of E can be measured directly. This will give a linear relationship at low levels of If but at high levels magnetic saturation results in lowered values of E.

Inductances measurement

The procedures of this experiment is as follow . At a given notch of the rotor (notch 1), a reduced voltage is applied to phase A of the machine stator. The idea of applying a reduced voltage is to avoid saturation on the magnetic circuit. The current, voltage and the voltage phase angle fed in phase A are measured and recorded. As all the other phases are magnetically coupled to each other, the voltage and the phase angle induced in phase B and C by phase A are measured and recorded. The position of the rotor is then changed according to the notches labelled. The values of current and the voltages are measured and recorded for every notches of the machine’s rotor. The complete procedures above are repeated for phases B and C of the machine stator. After acquiring all these measurement, the values of all the inductances are calculated using the basic circuit theory analysis.

Lagrangian Analysis

The Lagrangian can be thought as the sum of overall energies contained within all the components in the electrical system. The Lagrangian equation for an electrical system can be obtained by summing all the energies of the component in an electrical circuit.

One of the key component in Lagrangian analysis is the Euler Lagrange equation, which is an ODE that describes the behaviour of the system that is defined by the Lagrangian equation.

The final key component in Lagrangian analysis is the constraints. If there exists some external forces acting on the system such as a voltage or a current source then the Euler Lagrange equation is not equal to zero. The Euler Lagrange will be equal to some unknown multiplier times the outcome of applier the Euler Lagrange to that constraint.

There will be multiple variables within a Lagrangian analysis, in order to deal with this problem and make the calculation appears more systematic and neat, the Del operator is used. Del operator is a vector that performs the Euler Lagrange operation for every defined variables. Del operator also will be applied to the constraints by differentiating it with respect to the defined variables

## Project outcome

The Lagrangian analysis had be done to solve some simple cases that can give us good understanding before proceed to analyse the Synchronous Machine. The simple cases had been analyse are single constant inductor, mutually coupled circuit and time varying inductor. All of these simple cases can be analysed successfully except the time varying inductor. In time varying inductor, there is a hurdle in solving the equation, where there is a derivative inside the equation which make it difficult to solve.

We also had compared the result obtained from the measurement and the result obtained from analysing the single constant inductor. Both the calculation and the measurement give exactly the same answer.

We also had discover that, choosing a suitable constraint is the most important thing in analysing the electrical system using the Lagrangian method. For the single constant inductor, we had try to applying both type of constraint, current source and voltage source. The result obtained by having the current source as the constraint give us a right answer while using the voltage source is the other way around.

## Conclusion

1. For a simple electrical systems Lagrangian method provide a viable alternative aside from the conventional circuit analysis techniques.

2. Constraints must be chosen correctly in order to obtain sensible equations to analyse the system.

3. Time varying circuit elements give a significant hurdle for Lagrangian analysis.