# 1. instability are: 1. Loss of load in some

1.
Introduction

Power System

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Power systems are large and complex
electrical networks designed to generate, transmit and distribute electrical
energy to different types of constantly varying loads. The transmission and
distribution of power is done through the alternating current (ac) system.
Since power loss is dictated by the formula I2R, transmission is
done at high voltage (HV) to increase transmission efficiency.

A well-designed power system has the
following characteristics:

1.
It can supply power practically
everywhere the customer demands.

2.
It can supply power to the
customer all the time.

3.
It can always supply the

4.
The power supplied is of good
quality.

5.
The power supplied is
economical.

6.
It satisfies the necessary
safety requirements.

The structure of the Mauritian power system
is shown in Figure 1.

Figure 1.1: Power System Structure in Mauritius (Source: CEB)

Power System Stability

Power system stability can be defined as
the ability of an electric power system to regain equilibrium in its state of
operation such that practically the entire system remains intact and all its system
variables stay bounded after being subjected to a physical disturbance.

Figure
1.2: Classification of Power System Stability

Voltage Stability

Voltage stability is the ability of a power
system to maintain steady voltages at all its buses after it is subjected to a
disturbance from an initial point of operation. Voltage stability is dependent
on the system’s ability to restore equilibrium between load demand and load
supply.

Some possible consequences of voltage
instability are:

1.
Loss of load in some parts of
the system.

2.
Tripping of transmission lines

3.
Loss of synchronism of
generators which may arise from the outages.

4.
Electric motors tend to run on
over speed when they are fed with higher voltages resulting in vibration and
mechanical damage. Over voltage may also cause insulation failure.

5.
Heating issues as a result of
drop in voltage and subsequent rise in current.

To avoid these nefarious effects, it is
important to keep the system voltage fluctuations to a minimum.

Voltage stability can be broken down into
two subcategories:

1.
Large disturbance voltage
stability

It refers to the
system’s ability to maintain steady voltages following large disturbances such
as system faults, loss of generation, or circuit contingencies.

2.
Small disturbance voltage
stability

It refers to the
system’s ability to maintain steady voltages when subjected to small agitations
such as incremental changes in system load.

Voltage Collapse

Voltage collapse is the process by which
the series of events associated with voltage instability leads to a blackout or
abnormally low voltages in a significant part of the power system.

Voltage Stability Timeframe

The timeframe of interest for voltage
stability can be divided into two parts, namely:

1.
Short-term voltage stability

2.
Long-term voltage stability

Short-term voltage stability

This involves the dynamics of fast acting
High Voltage Direct Current (HVDC) converters. The study period of interest is
in the order of several seconds, and analysis requires solution of appropriate
system differential equations.

Long-term voltage stability

This involves slower acting equipment such
as tap-changing transformers, thermostatically controlled loads, and generator
current limiters. The study period of interest is in the order of several
minutes. Long term simulations are required for analysis of system dynamic
performance.

Frequency Stability

Frequency stability refers to the ability
of a power system to maintain a steady frequency after a severe system upset
resulting in a significant imbalance between generation and load.

The resulting instability occurs in the
form of sustained frequency swings leading to tripping of generating units
and/or loads. In large interconnected power networks, the systems are split
into islands. Stability in this case depends on whether each island will reach
a state on operating equilibrium with minimal unintentional loss of load.

Generally, frequency stability problems are
associated with inadequacies in equipment responses, poor coordination of
control and protection equipment, or inadequate generation reserve. Frequency
stability may be a short-term phenomenon or a long-term phenomenon.

Frequency fluctuations can have the
following effects:

1.
Three phase ac motors run at
speeds that are directly proportional to the frequency. The variation of system
frequency affects the motor performance.

2.
The blades of stem and water
turbines are designed to turn at a pre-determined speed. Frequency variations
causes changes in that speed which results in excessive vibration and hence can
cause damage to the turbine blades.

3.
Frequency error may result in a
disaster in digital storage and retrieval process.

Automatic Voltage Regulators

Figure 1.3: Simple AVR System

The Automatic Voltage Regulator (AVR) is
used to control the reactive power on the generation side and hence the
terminal voltage on the load side. The main duty of the AVR is to maintain the
terminal voltage of the synchronous generator at a predetermined level. Figure 1.3 shows
a simplified representation of an AVR system.

When the reactive power load of the
generator increases, the terminal voltage level suffers a drop in magnitude.
This variation in voltage level is sensed through the potential transformer on
one phase, rectified and fed to a comparator. The comparator compares this
feedback signal to a DC setpoint Vref and outputs an error signal to
the amplifier. The amplifier acts based on the error signal and increases the
terminal voltage across the exciter. This results in an increase in the
generator field current and hence an increase in the emf generated. The outcome
is an increase in the reactive power which in turn increases back the terminal
voltage to the desired level.

2004)

Proportional Integral Derivative (PID)
Controllers

A PID (Proportional Integral Derivative)
controller is a popular, simple-to-use control algorithm used in the industry
to overcome various kinds of problems. Figure 1.4 shows
the PID controller in block form.

Figure 1.4: PID Controller
Representation

The PID controller can be represented
mathematically by the following equation in the s-domain (Laplace operator):

Or

Where

Kp is the proportional
constant term

Ki is the integral constant
term

Kd is the derivative constant
term

Ti is the integral action time
(also known as reset time)

Td is the derivative action
time (otherwise known as rate time)

A more general equation for the output of
the PID controller in time domain is:

Where u(t) is the controller output signal
and e(t) is the error signal.

Ant Colony Optimisation Algorithm

The Ant Colony Optimisation (ACO) algorithm
is a swarm intelligence based metaheuristic technique which mimics the
behaviour of ant species looking for food. A trail of pheromone built up by
previous ants on their way back from the food source to the colony helps the
ensuing ants to locate the best route to their food destination. Every ant
contributes to find a solution to the given problem; whereas the whole colony
work together to find the optimal solution. ACO has been used in many difficult
combinatorial optimisation problems such as the travelling salesman problem
(TSP), quadratic assignment problem, hydroelectric generation scheduling
problems, inter-alia. (Dorigo, et al.,
2006)

Whale Optimisation Algorithm

The Whale Optimisation Algorithm (WOA) is a
metaheuristic optimization algorithm that draws inspiration from the
predatorial movements of humpback whales in locating and hunting down their
preys. This behaviour, demonstrated in Figure 1.5, is
specifically observed in humpback whales and is termed as the bubble-net
feeding method in which the whales create bubbles along a spiral-shaped path
surrounding their prey. A mathematical model following the bubble-net feeding
method is derived in WOA to perform optimisation. (Mirjalili, et al., 2016)

Figure 1.5: Bubble-net feeding
behaviour of humpback whales (Mirjalili, 2016)