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Chapter 1 Fault Current Limiter

1.1 Introduction

A Fault Current Limiter (FCL) is a device which limits the prospective fault current when a

fault occurs in a power system. Fault current limiting devices have been gaining interest in recent

times to combat large fault currents. Also low rating circuit breakers can be used in this case as

they are very cost effective compared to breaker replacement. The term FCL is generally applied

to superconducting devices, whereas non-superconducting FCL devices are typically termed

Fault Current Controllers.

Depending on the function and the main characteristics of each FCL type, we compare to

what extent each type can fulfill the main requirements of an ideal limiter. The major

requirements of a fault current limiter device are:

• Fast and effective current limitation

• Quick and automatic recovery

• Capability to limit the fault current within its first peak

• Fail-safe and reliable operation

• Low AC loss and voltage drop and low impedance in the normal state

• No requirement for sensors or other control devices

• Capability to perform gradual change of impedance from normal state to fault state

and vice-versa.

• Small, compact and light weight.

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1.2 The different Current Limiter approaches

A few approaches of limiting fault current are discussed.

The approaches are

1. Superconductor Technologies.

2.Non-Superconducting Technologies.

3. Resonant circuit devices.

4. Switched devices.

1.2.1 Superconducting Technologies

The concept of using superconductors to carry electric power and to limit peak currents has

been around since the discovery of superconductors and the realization that they possess highly

non-linear properties. More specifically, the current limiting behavior depends on their nonlinear

response to temperature, current and magnetic field variations. Increasing any of these three

parameters can cause a transition between the superconducting and the normal conducting

regime. The current increase can cause a section of superconductor to become so resistive that

the heat generated cannot be removed locally. This excess heat is transferred along the

conductor, causing the temperature of adjacent sections to increase. The combined current and

temperature can cause these regions to become normal and also generate heat. The term

“quench” is commonly used to describe the propagation of the normal zone through a

superconductor. Once initiated, the quench process is often rapid and uncontrolled; however, the

extent of the normal region and the temperature rise in the materials can be predicted. Thus, the

quench process can be used in the design into a superconducting component.

Several hundred patents exist showing theoretical ways in which this phenomenon might be

used to control fault currents in the electric power grid. However, efforts to develop the concepts

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into commercially viable product have culminated in only a few practical designs and even fewer

working prototypes. Many of these designs have shortcomings (e.g. size, performance,

reliability, recovery under load, or cost) that hinder them from reaching full commercial

potential. Most SFCL designs use the aforementioned quench behavior to limit fault currents

within the first cycle. However, that is often where the similarities end as each SFCL design has

its own methods of sustaining the limiting action once the superconductor becomes resistive.

Described below are brief & simplified overviews of SFCLs, categorized into three types.

The first two types depend on the quenching action of superconductors and the third type uses

DC HTS magnet windings to saturate an iron core.

1.2.1.1 Resistive SFCL

Resistive SFCL utilize the superconducting material as the main current carrying conductor

under normal grid operation. The principle of their operation is shown in Fig1.1

Fig.1.1. Resistive SCFC with parallel impedance

When a fault occurs, the current increases and causes the superconductor to quench thereby

increasing its resistance exponentially. The current level at which the quench occurs is

determined by the operating temperature, and the amount and type of the superconductor. The

rapid increase in resistance produces a voltage across the superconductor and causes the current

to transfer to a shunt, which is a combined inductor and resistor. The shunt limits the voltage

increase across the superconductor during a quench. In essence, the superconductor acts like a

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switch with millisecond response that initiates the transition of the load current to the shunt

impedance.

Fig.1.2. Normalized plot of voltage and current in a superconductor at constant temperature and magnetic field.

The above Fig. 1.2 shows voltage across RSC as a function of ration of current through the

device, Iline , to the "critical current" , Ic which is define as the current at which a voltage drop

of 1V /cm is observed along the conductor. The grid characteristic of the resistive SFCL after a

quench is determined by the shunt element. Thus, because the shunt is typically quite reactive, a

resistive SFCL typically introduces significant inductance into the power system during a fault.

During the transition period when current is being transferred from the superconductor to the

shunt, the voltage across the combined element (shown in the normalized plot above) is typically

higher than it is after the current has transitioned into the shunt. The dynamics of this process

depend on the two elements and their mutual inductance. The quench process in resistive SFCLs

results in heat that must be carried away from the superconducting element by the cryogenic

cooling system. Typically, there is a momentary temperature rise in the superconducting element

that causes a loss of superconductivity until the cryogenic system can restore the operating

temperature. This period of time, known as the recovery time, is a critical parameter for utility

systems (which may see multiple fault events occurring close together in time) and is a key

distinguishing characteristic among various SFCL designs.

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1.2.1.2 Shielded-Core SFCL

One of the first SFCL designs developed for grid deployment was the shielded-core design,

a variation of the resistive type of limiter that allows the HTS cryogenic environment to remain

mechanically isolated from the rest of the circuit. An electrical connection is made between the

line and the HTS element through mutual coupling of AC coils via a magnetic field. Basically,

the device resembles a transformer with the secondary side shunted by an HTS element.

During a fault, increased current on the secondary causes the HTS element to quench,

resulting in a voltage increase across L1 that opposes the fault current.

Fig.1.3. Configuration of a Shielded-Core SFCL

The principle of the Shielded-Core SFCL and its function is similar to a transformer with a

short-circuited secondary winding. As long as the induced current in the superconducting

cylinder is lower than the critical current, the superconducting cylinder serves as a perfect

magnetic shielding for the primary winding. The flux in the iron core is negligible and the

residual impedance is low. With increasing short-circuit current the current in the

superconducting cylinder exceeds the critical current and the cylinder starts to develop a

resistance.

Although the superconductor in the shielded-core design has to re-cool after a limiting

action just like the resistive type, non-uniform heating of the superconductor (i.e. hot spots) is

easier to avoid through optimization of the turns ratio. A major drawback of the shielded-core

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technology is that it is approximately four times the size and weight of purely resistive SFCLs.

Although prototypes of shielded-core designs have worked well, their size and weight have

limited grid deployment.

1.2.1.3 Saturable -Core SFCL

Saturable core SFCLs utilizes the dynamic behavior of the magnetic properties of iron

(nonlinear permeability of the iron core, thus no problem of recovery time) to change the

inductive reactance of the ac line. The concept (shown in Fig. 1.4. and Fig. 1.5.) utilizes two iron

core and two AC windings for each phase. The AC windings are made of conventional

conductors, wrapped around the core to form an inductance in series with the AC line. A

constant current superconducting winding on the iron core provides a magnetic bias. . At the

same time, the high temperature superconducting coils are supplied by a DC source & do not

have AC power loss.

Fig.1.4. Circuit & Characteristics of a Saturable-Core SFCL

Fig.1.5. Electrical configuration of a Saturated core type SCFCL

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Under nominal grid conditions (when the AC current does not exceed the maximum rating

of the local system), the HTS(High Temperature Superconductor) coil fully saturates the iron so

that it has a relative permeability of one. To the AC coils, the iron acts like air, so the AC

impedance (inductive reactance) is similar to that of an air-core reactor. Under fault conditions,

the negative and positive current peaks force the core out of saturation, resulting in increased line

impedance during part of each half cycle. The result is a considerable reduction in peak fault

current.

Essentially, the saturable-core SFCL is a variable-inductance iron-core reactor that has the

impedance of an air-core reactor under normal grid conditions and very high impedance during

fault events. Unlike resistive SFCLs, which may require time between limiting actions to cool

the superconducting components, the saturable-core approach can manage several actions in

succession because the superconductor does not quench. However, a major drawback of

saturable-core SFCL technology is the volume and weight associated with the heavy iron core;

however, manufacturers hope to improve this issue in future prototypes. Zenergy has recently

tested a prototype saturable-core SFCL based on an entirely new design concept that is four

times smaller than its predecessor. GridON, an Israeli-based startup company, is in the process of

developing saturable-core concept intent on reducing size and weight to more accommodating

levels for commercial use.

1.2.2 Non-Superconducting Technologies

FCLs that do not rely on superconducting materials to perform the limiting action

include current-limiting fuses, solid-state devices, and a number of novel technologies that

are either available for commercial use or are in development.

1.2.2.1 Current Limiting Fuses

Fig. 1.6.shows the circuit of using a fuse as a limiter. The basic idea is that when the switch

opens it generates a voltage arc and the current in the arc is transferred to the silver sand fuse.

Once the fuse melts the current is transferred to the resistor across it.

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Fig.1.6. Fuse based FCL

The main thing to be considered with this approach is the mechanism associated with the

operation of the bypass switch. The other method used, based on this approach, is the current

limiting protector which is mainly used in the distribution systems. In this method the sensing

and the switch operation mechanism are separated from each other.

This circuit consists of silver sand fuse and a copper conductor which is parallel to the fuse

and has s series of notches. The fuse melts upon fault; this is accomplished by high arc voltage

which limits current. The melted fuse and the notches on the copper conductor limit the fault

current. The sensor detects the fault current.

1.2.2.2 Solid State FCLs

Solid-state FCLs utilize actively controlled power electronics to perform or initiate the

limiting action. Typically, the power electronics switch the fault current into a limiting branch or

open the circuit in such a way that the characteristics simulate a circuit breaker. One topology

being developed is the solid-state FCL circuit breaker (SSFCL-CB) shown in the Fig. 1.7. below,

a device that utilizes a diode rectifier bridge circuit with a thyristor-based commutating circuit

that can be switched off during a fault event. SSFCL that utilize thyristors in the bridge circuit

rather than diodes is currently in development which will provide more control of the fault

current through adjustment of the firing angles of the thyristors. EPRI is currently developing an

SSFCL that utilizes thyristors in the bridge circuit rather than diodes.Because the fault current is

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controlled dynamically through the use of power electronics, the SSFCL is referred to as a fault-

current controller (FCC).In addition to SSFCL-CB and FCC, designs that use solid-state

switching to direct fault current into inductive coils or detuned LC series circuits arealso being

pursued.

Fig.1.7. Principle of a Solid-State Fault-Current-Limiting Circuit Breaker Based on Diodes

There are advantages and disadvantages associated with each design. Solid-state devices

offer the advantage of additional control of fault currents not available with other technologies

and have the potential to be realized in small geometries. However, issues with solid-state

devices include the reliability of switching electronics to sense fault levels and respond

accordingly as well as possible distortion of the current waveform due to the nature of switching

electronics. To date, no full-scale prototype of a solid-state FCL has achieved grid deployment.

1.2.3 Resonant circuit devices

The circuit of the resonant limiter is shown in Fig. 1.8.During normal operation the switch S

is open and the effective impedance of the circuit is very low as the capacitor and inductor in

both sides are in series.

During fault the effective impedance increases, this becomes equal to equivalent reactance

in series with equivalent resistance. Due to high resistance the fault current is limited to a lower

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value. Due to switching action there might be a spark across the switch because of the voltage

appearing across one of the capacitors. If one of the components, the capacitor or reactor

becomes faulty, the current limiting action of the circuit will no longer be valid. The cost and

large size are its main drawbacks.

Fig.1.8. Resonant Circuit Limiter

1.2.4 Switched devices

In this method the devices used as FCL consist of commutation of the current from an

opening of the switch to very high impedance element, which is in parallel to the switch. There

are three types of switched devices; vacuum switches, fuses and forced chemical devices. In the

vacuum switched method, the device is set into operation by artificially making the current zero

by discharging a precharged capacitor through a second interrupting device.

In the second method, when there is fault the contacts of the protective switch open quickly.

The low voltage arc is formed during opening of the contacts because they open in the vacuum.

The interaction of the magnetic field and the arc causes the arc length to increase to a point

where the arc no longer exists. Due to low voltage across the shunt capacitor the arc rapidly goes

to zero. The fault current is transferred to the capacitor branch and the capacitor voltage rises;

this voltage across the capacitor is limited by the voltage drop across the bypass resistor. The

main advantage of this device is that the continuous current is carried by the device, the voltage

rating can be increased by placing the switches in series, the coils and the power supply can be

isolated from the line voltage and the current commutation is not polarity sensitive. The main

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drawback of this approach is that the current to be interrupted depends on the sensor response

and the operating time of the mechanism.

1.3 A Brief Comparison of FCL Technologies

A generalized comparison of the various SFCL technologies is given in Table 1.1 below. It

provides a general sense of operational capability, performance, and size. Much of the

information provided here is derived from an EPRI survey conducted in 2005.

Table 1.1. Comparison of the General Characteristics of various FCL Technologies

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Chapter 2 Structure and Performance Feature of Saturable Iron core SFCL

2.1 Introduction

Although has many advantages, the harmonic in the superconducting coil during a limiting

action, heavy weight, high costs, and huge volume are always the major obstacles for the

SICSFCL’s application. In recent years, a great breakthrough in design that not only can avoid

harmonics but also more compact and much cheaper rate has been achieved. More information

about the design is presented in the following paragraphs.

Fig.2.1. SICSFCL system structure

Fig. 2.1 shows the structure of the SICSFCL. It is composed of a reactor system, a

cryogenic system, a DC bias system and a monitoring system. The reactor system is the main

functional part. The cryogenic system provides low-temperature circumstance for the

superconducting coil. The DC bias system has three basic functions: fully magnetizing iron

cores during normal condition, cutting off magnetization circuit if a fault current occurs, and

fast recovering from high impedance to low after the fault is cleared. While the monitoring

system carries out the online monitoring of the status of whole system.

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2.2 Reactor System

The reactor system is consisted of iron core, superconducting coil, AC coil and some

subsidiaries. Usually the volume of reactor system is bulky. Innopower proposed their own

reactor structure by using three-phasesix iron cores to reduce its bulk. That is to say, three

single phase saturated reactors use one single superconducting coil as shown in Fig. 2.2.

Fig.2.2. A three phase six iron cores diagram a) AC coils,

b) cryostat containing superconducting coil, c)iron cores

2.3 Cryogenic System

The superconducting coil must operate under low temperature environment which

provided by the cryogenic system. There are two types of cryogenic system: closed-loop and

open-loop. To easy maintain and achieve high reliability, an open-loop cryogenic system is

adopted for the SICSFCL in Modern design. The cryogenic system is consisted of dewar,

liquid nitrogen (LN2) tank, nitrogen bleeding pump, control unit, and accessories such as

LN2pipes, valves, sensors for pressure and temperature, etc. Fig.2.3 shows the schematic

drawing of cryogenic system.

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Fig.2.3. Diagram of cryogenic system

2.4 DC Bias System

As previously described in fundamental principle, iron cores are driven into saturation

by DC current which could cause little the inductance under normal grid conditions.Under

fault conditions, the DC current is cut-offso that the two coils are out of saturation. This

results in a increased impedance that could simultaneously protect the DC bias system.In

order to protect superconducting coil from high induced voltage caused by cutting-off DC

current during fault condition, energy release circuit is used to suppress the inductive

voltage.

Fig.2.4. Diagram of DC bias systems

Fig. 2.4 is the schematic diagram of DC bias system, which includes DC source, switch,

current sensor and energy release circuit etc. DC source supplies exciting current for the

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superconducting coil. Current sensor is used to detect and determine the fault current.The DC

bias system has the following operating characteristics

• At the steady state, the DC source output constant exciting current.

• The switchers can cut-off DC current in 5 milliseconds once a fault current is

detected. The energy release circuit absorbs the energy of the superconducting

coil in dozens of milliseconds.

• After the clearance of fault current, the DC bias system recovers to normal DC

current within hundreds of milliseconds. The SICSFCL turns to low impedance

before the breaker close.

2.5 Monitoring System

Monitoring system’s major function is to control the DC bias system and cryogenic

system, to monitor the current of grid, and to communicate with the relay protection system.

A NI processor is used as the controller unit for DC bias system. A PLC is used to control the

cryogenic system. NI processor has shorter response time than PLC.

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Chapter 3 Design & Operation of Saturable Core SFCL

3.1 Introduction

There is renewed interest in superconducting FCLs due to the advantages offered by

high-temperature superconductors (HTS) using liquid nitrogen over their low-temperature

counterparts. This predominantly arises from an order of magnitude of savings in cryogenic

equipment costs, and a reduction in operational costs by a factor of up to 100. Liquid

nitrogen systems are also less complex compared to liquid helium systems.

The prototype saturable magnetic core-type FCL considered in this study is constructed

using HTS tapes. The application of HTS tapes in non-quench type SFCLs is most desirable

from commercial and operational viewpoints; they are the most robust, practicable, and

reliable forms available compared to other HTS forms. A description of the design principles

applied to an HTS saturable core-type SFCL is provided. The concepts behind development

of a time-domain model for the SFCL are also presented, followed by simulation results in

relation to the operation of the SFCL for typical medium-voltage (MV) distribution

substation applications.

3.2 Structure of a Saturated Core FCL

The structure of the saturated magnetic core HTS FCL, shown in Fig. 3.1 following, is

composed of two magnetic cores and a superconducting dc bias coil. Although the

permeability of the magnetic core is nonlinear, in order to analyze theoretically, the ψ – i

curve of both ac windings of FCL can be described approximately by five linear segments

with different slopes, as shown in Fig. 3.2. Curve 1 illustrates the magnetization curve of

single core without the dc bias, where Ic is a critical saturated current and ψc is the critical

saturated flux linkage. Curves 2 and 3 plot the magnetization curves of two cores,

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respectively, when a dc bias current Id is imposed. Curve 4 is the ψ – i curve of the saturated

core FCL, which is achieved by summation of curves 2 and 3.

Fig.3.1. Simulation circuit for a saturated core FCL; 1,2) AC windings; 3,4) Magnetic Cores; 5)

Superconducting DC coils

Fig.3.2. ψ – i curve of saturated core FCL

In the normal operation, both magnetic cores are driven into saturation by the dc current.

The saturated core FCL works in line segment I, so the impedance of the FCL is very low. When

a short-circuit current occurs, the rapidly increasing ac current drives both magnetic cores out of

saturation alternatively during one ac cycle, hence, the saturated core FCL works in line segment

II, the impedance of the FCL becomes so large that the fault current is limited. However, if the

ac current increases unceasingly, the magnetic cores will be saturated reversely, when the FCL

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works in line segment III and will lose the capability of limiting current, due to the low

impedance.

3.3 Design Principles of a Saturable Core SFCL

In discussing the principles of operation of any FCL device it is convenient to introduce the

concepts of insertion impedance and fault impedance. Insertion impedance is defined as the AC

terminal impedance (of any FCL device) during normal un-faulted conditions. The manner in

which the impedance increases during a fault is typically device dependent; however, for most

FCL technologies the increase is continuous and not as a step function. Hence, fault impedance

is best defined as the steady state equivalent impedance that would result in the same fault

current limiting effect. Saturated steel core FCL meets the fundamental FCL requirements of a

low insertion impedance and high fault impedance through the change in permeability between

saturated and unsaturated states of the steel core. This concept can be demonstrated through an

analysis of the magnetization properties of typical steel core material. As an example, we

consider the properties of M3 laminated electrical steel shown in the Fig. 3.3

Fig.3.3. Material Properties of M3 Laminated Steel

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In a saturated core FCL a DC current carrying winding is used to bias the core. The number

of ampere-turns supplied by the biasing winding needs to be high enough to sufficiently saturate

the core. A separate winding is then used to carry the normal AC load current. During steady

state un-faulted conditions the low current in the AC winding results in the flux density

oscillating through a minor loop and not the full B-H loop. The top right hand corner of the Fig.

2.1 above shows the region where M3 laminated steel saturates. During un-faulted conditions the

flux density is oscillating within this “Steady state” region. As can be seen in the figure, the

permeability of the material at this field intensity is approximately equal to the permeability of

air; hence, the impedance of the AC winding is equivalent to that of an air-cored inductor.

During a fault event the rising current causes the flux density to oscillate through a much larger

region of the B-H loop. As the core de-saturates and moves into the “Faulted state” region shown

in the figure, the permeability of the core material increases. Hence, the impedance of the AC

winding also increases, subsequently limiting the fault current.

A schematic diagram of superconducting fault current limiter is shown below in the Fig. 3.4

Fig.3.4. Schematic of a SISFCL

Here two magnetic cores are required to cater for each half cycle of the AC fault current. Each

core carries a coil in which the AC line current iac flows. An HTS coil carrying a DC bias current is

also wound on each core as illustrated, where the DC bias current Idc is common to both coils.

Magneto Motive Force (MMF) produced by the superconducting dc winding is given by NdcIdc=

2(2W+2h) Hdc where Ndc is the number of turns in the DC winding, Idc is the current in the DC 19

winding, W is the average core width, h is the average core height and Hdc is the magnetic field

intensity due to the DC current. The FCL inductance under normal power system conditions needs to

be as small as possible. Therefore, the normal operating point is selected such that Hdc>>Hknee ,

where the saturation knee point is indicated inFig. 2.5.

Fig.3.5. Generalized B-H curve

When a fault current flows through the ac winding, each core is alternatively taken out of

saturation in each half cycle of as a result of the relatively large MMF caused by the fault current. To

ensure that the FCL offers a large inductance during a fault, the MMF cancelation in each core

should be such that

(Hdc+Hknee) l >nacIF> (Hdc+Hknee) l

whereHdc>>Hknee, l is the length of the flux path around the core, nac is the number of turns in

the ac winding, and IFis the peak value of the fault current at which the FCL is designed to operate.

The conceptual design of the single-phase configuration has been extended to develop a three-phase

FCL, where a single DC winding is used to control all AC phases. The design is innovative in that

only one superconducting coil is used for all six of the magnetic cores.

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3.4 J-A Hysteresis Model

3.4.1 Non-linear Magnetic Theory

While the magnetic quantities analogous to voltage (V) and current (I) are magneto-motive

force (F) and magnetic flux (Φ), the behavior of a saturable magnetic core is typically described

in terms of its magnetic flux density (B) vs. magnetic field intensity (H). This relationship takes

the form of a bi-stable sigmoid, as shown in the following figure.

Fig.3.6. A model B-H Curve

Several figures of merit for a device can be taken from its B-H curve, the most important of

which are the saturation level, the retentivity (or remanence) and the coercivity (or coercive

force). The saturation level (Bsat) is the magnetization level at which all of the domains'

magnetic moments are pointing in the direction of the external field vector. This level, a constant

for the material, represents the maximum field density the material can sustain. The remanence is

defined as the amount of magnetism that remains in the material after the external forces have

been removed. On the B-H curve, it is the point at which the curve crosses the abscissa (H equal

to zero). The coercive force is defined as the external magnetic field required to return the

magnetism of the solid to zero. On the B-H curve, it is the point at which the curve crosses the

ordinate (B equal to zero). These parameters relate physical occurrences within a core to its

magnetic response and provide limits by which mathematical relationships can be derived to

model the magnetic behavior within the core. 21

The relationship between B and H is defined by the permeability (μ) of the material. For

magnetic materials, the permeability (μ) can be related to the permeability of free space (μ0) by

the relative permeability (μr). A particular class of magnetic materials (saturable) can exhibit

highly nonlinear relative permeabilities. The relationship between these quantities is shown in

the following equation

𝐵𝐵 = 𝜇𝜇𝜇𝜇 = 𝜇𝜇0𝜇𝜇𝑟𝑟𝜇𝜇 = 𝜇𝜇0(𝜇𝜇 + 𝑀𝑀) ( 3.1 )

The parameter M in the equation refers to the magnetism or field intensity within the

material that is contributed by the magnetic domains. The variation in permeability (μ) is a result

of the changing magnetism (M) of the solid as energy from the applied field (H) is absorbed into

the domains. Since the relative permeability (μr) is dependent on the ratio of applied and resident

fields (magnetic susceptibility), the field density equation may be expanded into the more

general relationship given by the final relationship, in which the relative permeability is

incorporated into the internal field intensity component (M).

If a sample of magnetic material is examined on a per domain basis, the differential field

strength around any given domain will be somewhat larger than expected due to its proximity to

the remainder of the domains in the material. In effect, a given domain experiences the magnetic

influence of the averaged total magnetism of the solid, since the orientation of any given domain

may be random. At this point, it is appropriate to define effective magnetic field intensity (Heff),

existing within the solid, that is the sum of the applied field (H) and some averaged contribution

from the magnetism (M) of the surrounding domains. The proposed equation adjusts the

percentage of bulk magnetism (M) added to the applied field intensity (H) through the scaling

coefficient alpha (α) which typically has a value around 10–3. The modified relationship for the

magnetic field intensity experienced by a single domain is given by the equation,

𝜇𝜇𝑒𝑒𝑒𝑒𝑒𝑒 = 𝜇𝜇 + 𝛼𝛼𝑀𝑀 ( 3.2 )

If a magnetic material was able to return all of the magnetic energy that was input, the

resulting magnetization curve would take the form of a single valued sigmoid (equivalent to the 22

center line of the hysteresis loop shown in the model B-H curve in Fig. 3.6. This curve, referred

to as the anhysteric magnetization curve, represents the ideal or lossless magnetization of a

material. The function that was chosen to model this semi-empirical representation was

developed by Langevin. The parameters used to calculate this quantity are the effective field

strength (Heff) given by the above equation, the saturation level (Ms), and the shaping coefficient

gamma (a), which adjusts the slope of the curve according to the magnetic hardness of the

material. The phenomenological representation of anhysteric magnetization (Man) proposed by

Langevin is defined by the equation,

𝑀𝑀𝑎𝑎𝑎𝑎 = 𝑀𝑀𝑠𝑠(coth�𝜇𝜇𝑒𝑒𝑎𝑎 � −

𝑎𝑎𝜇𝜇𝑒𝑒

) ( 3.3 )

3.4.2 J-A Hysteresis Model Analysis

The Jiles-Atherton model is a physically based model that includes the different mechanisms

that take place at magnetization of a ferromagnetic material. The theory is based on the existence

of magnetic domains, which are separated by domain walls. The motion of these walls,

especially translation, bending, and pinning, is shown to be closely related to hysteresis. The

magnetization M is represented as the sum of the irreversible magnetization Mirr due to domain

wall displacement and the reversible magnetization Mrev due to domain wall bending.

𝑀𝑀 = 𝑀𝑀𝑟𝑟𝑒𝑒𝑟𝑟 + 𝑀𝑀𝑖𝑖𝑟𝑟𝑟𝑟 ( 3.4 )

The rate of change of the irreversible part of the magnetization is given by,

𝑑𝑑𝑀𝑀𝑖𝑖𝑟𝑟𝑟𝑟

𝑑𝑑𝜇𝜇 =𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀𝑖𝑖𝑟𝑟𝑟𝑟

𝑘𝑘𝑘𝑘 − 𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀𝑖𝑖𝑟𝑟𝑟𝑟 ) ( 3.5 )

Where δ=sign(dH/dt) is a directional parameter, +1 for dH/dt>0 and -1 for dH/dt<0, k is

parameter defining the pinning site density of domain walls. It is assumed to be the major

contribution to hysteresis

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The component of reversible magnetization reduces the difference between the prevailing

irreversible magnetization Mirr and the anhysteretic magnetization Man at the given field

strength. This can be expressed as,

𝑀𝑀𝑟𝑟𝑒𝑒𝑟𝑟 = 𝑐𝑐(𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀𝑖𝑖𝑟𝑟𝑟𝑟 ) ( 3.6 )

Where c is a domain flexing parameter, defining the amount of reversible magnetization due

to wall bowing and reversal rotation, included in the magnetization process.

The hysteresis differential equation for reversible susceptibility can be expressed as

𝑑𝑑𝑀𝑀𝑟𝑟𝑒𝑒𝑟𝑟

𝑑𝑑𝜇𝜇 = 𝑐𝑐(𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎

𝑑𝑑𝜇𝜇 −𝑑𝑑𝑀𝑀𝑖𝑖𝑟𝑟𝑟𝑟

𝑑𝑑𝜇𝜇 ) ( 3.7 )

Combining the equations we obtain,

𝑑𝑑𝑀𝑀𝑑𝑑𝜇𝜇 = (1 − 𝑐𝑐)

𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀

𝑠𝑠𝑖𝑖𝑠𝑠𝑎𝑎 �𝑑𝑑𝜇𝜇𝑑𝑑𝑑𝑑� 𝑘𝑘(1− 𝑐𝑐)− 𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀)

+ 𝑐𝑐𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎

𝑑𝑑𝜇𝜇 ( 3.8 )

Or it can be derived as,

𝑑𝑑𝑀𝑀𝑑𝑑𝜇𝜇 = (1− 𝑐𝑐)𝑘𝑘

𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀

𝑠𝑠𝑖𝑖𝑠𝑠𝑎𝑎 �𝑑𝑑𝜇𝜇𝑑𝑑𝑑𝑑� 𝑘𝑘(1 − 𝑐𝑐) − 𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀)


𝑑𝑑𝜇𝜇 ( 3.9 )

Where,

𝑘𝑘 = �0 𝑖𝑖𝑒𝑒 𝑠𝑠𝑖𝑖𝑠𝑠𝑎𝑎 �𝑑𝑑𝜇𝜇𝑑𝑑𝑑𝑑 �

(𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀)

1,𝑜𝑜𝑑𝑑ℎ𝑒𝑒𝑟𝑟𝑒𝑒𝑖𝑖𝑠𝑠𝑒𝑒� ≤ 0

Now Man is function of He i.e.𝑀𝑀𝑎𝑎𝑎𝑎 = 𝑒𝑒(𝜇𝜇𝑒𝑒), and hence to obtain 𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎𝑑𝑑𝜇𝜇

we need to consider,

24

𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎

𝑑𝑑𝜇𝜇 =𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎

𝑑𝑑𝜇𝜇𝑒𝑒𝑑𝑑𝜇𝜇𝑒𝑒𝑑𝑑𝜇𝜇 ( 3.10 )

And,

𝑑𝑑𝜇𝜇𝑒𝑒𝑑𝑑𝜇𝜇 = (1 + 𝛼𝛼

𝑑𝑑𝑀𝑀𝑑𝑑𝜇𝜇) ( 3.11 )

Hence using the equation we can finally write the following relation,

𝑑𝑑𝑀𝑀𝑑𝑑𝜇𝜇 =

(1 − 𝑐𝑐) 𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀

𝑠𝑠𝑖𝑖𝑠𝑠𝑎𝑎 �𝑑𝑑𝜇𝜇𝑑𝑑𝑑𝑑 �𝑘𝑘(1−𝑐𝑐)−𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 −𝑀𝑀)+ 𝑐𝑐 𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎

𝑑𝑑𝜇𝜇𝑒𝑒

(1 − 𝛼𝛼𝑐𝑐) 𝑑𝑑𝑀𝑀𝑎𝑎𝑎𝑎𝑑𝑑𝜇𝜇𝑒𝑒

( 3.12 )

3.4.3 Simulation of Hysteresis Curves following the J-A Model

The above equations have been verified using MATLAB programming as well as

SIMULINK. To obtain the hysteresis curve the value of the parameters viz. Ms, c, a, k and α are

needed. These parameters vary from material to material as it is reflected by their hysteresis

curves. The values of the parameters are shown in table format before every hysteresis curve.

3.4.3.1 Example solution of the model equation

Some example solutions of the model equations of hysteresis are shown in the following

figures for various values of the hysteresis parameters. These results give some indication of the

range of hysteresis loops which can be obtained from solving the hysteresis equations. It can be

seen that the model is not just restricted to soft magnetic materials, but also can be used for hard

magnetic materials.

25

Sample Core1: Fe 1.0c

Ms 1.5e6

A 1800

K 1800

Α 1.4e-3

C 0.14

Sample Core 2: Fe 0.8c

Ms 1.6e6

A 1000

K 700

Α 1.4e-3

C 0.22

26


Ms 1.6e6

A 972

K 672

Α 1.4e-3

C 0.14


Ms 1.6e6

A 1010

K 455

Α 1.8e-3

C 0.21

27


Ms 1.6e6

A 1085

K 320

Α 2.0e-3

C 0.3

Sample Core 6: oldcore-thin

Ms 1.4e6

A 1085

K 50

Α 1.1e-5

C 0.2

28

Sample Core 7: oldcore-fat

Ms 1.5e6

A 2000

K 4000

Α 0.001

C 0.1

Sample Core 8: oldcore-medium

Ms 1.7e6

A 1000

K 500

Α 0.001

C 0.1

The hysteresis loops of a wide range of magnetic materials can be modeled using the

equations given above. Hysteresis parameters a, α, k and c can be determined from experimental

hysteresis measurements, and then used to model the hysteresis curves using the theory of

hysteresis. The B-H curves of different sample cores are shown in Fig. 3.7

29

Fig.3.7. B-H curves of different sample cores

3.5 Application of J-A Model in Saturable Core SFCL

Fig.3.8. Circuit diagram of SCFCL

The network circuit diagram is shown in the Fig. 3.8. It consists of a SFCL, a resistive load

R, a variable resistor Rv, an AC voltage source Vs(t) as power supply and a DC magnetizing

current source Id. If we label the cores by numbers 1 and 2, respectively, and suppose that the 30

turns number of AC winding and DC winding on each core are wc and wd, the mean magnetic

path length is l and cross-sectional area of iron core is A, then, according to Ampere’s circuit

law:

𝑒𝑒𝑑𝑑𝐼𝐼𝑑𝑑 − 𝑒𝑒𝑐𝑐𝑖𝑖 = 𝜇𝜇1𝑙𝑙 ( 3.13 )

𝑒𝑒𝑑𝑑𝐼𝐼𝑑𝑑 + 𝑒𝑒𝑐𝑐𝑖𝑖 = 𝜇𝜇2𝑙𝑙 ( 3.14 )

Hence,

𝑑𝑑𝜇𝜇1

𝑑𝑑𝑑𝑑 = −𝑒𝑒𝑐𝑐𝑙𝑙𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑

( 3.15 )

𝑑𝑑𝜇𝜇2

𝑑𝑑𝑑𝑑 =𝑒𝑒𝑐𝑐𝑙𝑙𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑

( 3.16 )

Also we know from magnetics, B = μ0 (H + M)

In the first core the flux B and the current i do not follow the Faraday’s Law of electro

magnetism as because the increase of current decreases the flux (since we are considering the net

flux to be in the opposite direction of the flux created by the AC current) and hence the emf

induced is positive. Hence it can be written,

𝑒𝑒1 = 𝑒𝑒𝑐𝑐𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑒𝑒𝑐𝑐𝐴𝐴

𝑑𝑑𝐵𝐵𝑑𝑑𝑑𝑑 ( 3.17 )

Therefore,

𝑒𝑒1 = 𝜇𝜇0𝑒𝑒𝑐𝑐𝐴𝐴𝑑𝑑(𝜇𝜇1 + 𝑀𝑀1)

𝑑𝑑𝑑𝑑 = 𝜇𝜇0𝑒𝑒𝑐𝑐𝐴𝐴𝑑𝑑𝜇𝜇1

𝑑𝑑𝑑𝑑 �1 +𝑑𝑑𝑀𝑀1

𝑑𝑑𝜇𝜇1� = −

𝜇𝜇0𝑒𝑒𝑐𝑐2𝐴𝐴𝑙𝑙

𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 (1 +

𝑑𝑑𝑀𝑀1

𝑑𝑑𝜇𝜇1) ( 3.18 )

Now if 31

𝐿𝐿 =𝜇𝜇0𝑒𝑒𝑐𝑐2𝐴𝐴

𝑙𝑙 ( 3.19 )

𝑢𝑢1 = −𝑒𝑒1 = 𝐿𝐿𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 (1 +

𝑑𝑑𝑀𝑀1

𝑑𝑑𝜇𝜇1) ( 3.20 )

But in the second core the flux B increases as the current i increase and it follows the

Faraday’s Law of electromagnetism and the emf induced in the second coil is negative to the

supply. Hence,

𝑒𝑒2 = −𝑒𝑒𝑐𝑐𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = −𝑒𝑒𝑐𝑐𝐴𝐴

𝑑𝑑𝐵𝐵𝑑𝑑𝑑𝑑 ( 3.21 )

Therefore,

𝑒𝑒2 = −𝜇𝜇0𝑒𝑒𝑐𝑐𝐴𝐴𝑑𝑑(𝜇𝜇2 + 𝑀𝑀2)

𝑑𝑑𝑑𝑑 = −𝜇𝜇0𝑒𝑒𝑐𝑐𝐴𝐴𝑑𝑑𝜇𝜇2

𝑑𝑑𝑑𝑑 �1 +𝑑𝑑𝑀𝑀2

𝑑𝑑𝜇𝜇2�

= −𝜇𝜇0𝑒𝑒𝑐𝑐2𝐴𝐴

𝑙𝑙𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 (1 +

𝑑𝑑𝑀𝑀2

𝑑𝑑𝜇𝜇2)

( 3.22 )

Hence,

𝑢𝑢2 = −𝑒𝑒2 = 𝐿𝐿𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 �1 +

𝑑𝑑𝑀𝑀2

𝑑𝑑𝜇𝜇2� ( 3.23 )

Now based on Kirchhoff’s voltage law,

𝑉𝑉𝑠𝑠(𝑑𝑑) = 𝑢𝑢1 + 𝑢𝑢2 + 𝑢𝑢𝑅𝑅 = 𝑢𝑢1 + 𝑢𝑢2 + 𝑖𝑖𝑅𝑅 ( 3.24 )

Therefore,

32

𝑉𝑉𝑠𝑠(𝑑𝑑) = 𝐿𝐿𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 �2 +

𝑑𝑑𝑀𝑀1

𝑑𝑑𝜇𝜇1+𝑑𝑑𝑀𝑀2

𝑑𝑑𝜇𝜇2�+ 𝑖𝑖𝑅𝑅 ( 3.25 )

Or,

𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑 =

𝑉𝑉𝑠𝑠(𝑑𝑑) − 𝑖𝑖𝑅𝑅

𝐿𝐿 �2 + 𝑑𝑑𝑀𝑀1𝑑𝑑𝜇𝜇1

+ 𝑑𝑑𝑀𝑀2𝑑𝑑𝜇𝜇2

� ( 3.26 )

From the above equation the value of the current i is obtained. Using this value we can

numerically work out the voltage u of each winding and obtain the voltage curve through the

simulation program MATLAB. Likewise, based on the rated current and the short resistance, we

can calculate the AC winding’s voltage under normal operation and the limited current under

fault condition respectively.

33

Chapter 4 Global Optimization Techniques: Genetic Algorithm

4.1 Introduction

The design of power systems often requires the incorporation of fault current limiter

elements such as SISFCL for successful fault limiting applications. The unique, often highly

restrictive constraints placed on the design of these electromagnetic elements by the system

performances and environmental requirements associated with fault limiting applications very

often lead to the need for optimization as part of the design process. While a variety of

optimization techniques are available to the electromagnetic design engineer, many of the

traditional techniques prove unsuitable when confronted with real world applications. This :is

because conventional functional optimization techniques are either based on greedy, local

optimization methods such as gradient methods or consist of random walk solution space

searches. In either case, these conventional techniques are often poorly suited to the task of

optimization when high dimensional, multi-modal functional domains are involved. In addition,

traditional optimization techniques usually require the object function to be, at the very least,

continuous and, in many cases differentiable, placing severe limitations on the form and content

of the object function. This paper discusses a radically different and relatively new functional

optimization methodology known as Genetic Algorithm (GA) optimization that overcomes the

above- mentioned problems of the conventional techniques.

4.2 Global vs. Local Optimisation

Genetic Algorithms are classified as global optimizers while more familiar techniques such

its conjugate gradient and the quasi- newton methods are generally classified as local

techniques.The distinction between local and global search techniques is that the local techniques

produce results that are highly dependent on the starting point or initial guess while global

methods are largely independent of the initial conditions. In addition, local techniques tend to be

highly coupled to the solution domain. This tight coupling enables the local methods to take

advantage of the solution space characteristics resulting in relatively fast convergence to a

34

local maxima. However, the tight solution space coupling also places constraints on the solution

domain such as differentiability and/or continuity, constraintsthat can be hard or even impossible

to realize in practice. The global techniques on the other hand, are largely independent of and

place few constraints on the solution domain. This absence of constraints means that the global

methods are much more robust when faced with ill-behaved solution spaces. .In particular, global

techniques are much better at dealing with solution spaces having discontinuities, constrained

parameters, and/or large numbers of dimensions with many potential local maxima. The

downside to the global method’s general independence from the solution space is that they

cannot take advantage of local solution space characteristics such as gradients, during the

search process resulting in generally slower convergence than the local techniques.

Often, however, in design problems convergence rate is not nearly as important as getting a

solution. Having found a solution the ultimate goal in design is to find the best solution or global

maxima. In these applications, global methods are favoured over local methods. Global

techniques either yield global or near global maxima instead of local maxima and often find

useful solutions where local techniques cannot. Global methods are particularly useful when

dealing with new problems in which the nature of the solution space is relatively unknown. Of

the global techniques GA optimizers can readily handle discontinuous and non- differentiable

functions. They are also well suited for constrained optimization problems. Genetic Algorithms

are considerably more efficient and provide much faster convergence than random searches. In

addition, they are easily programmed and readily implemented.

4.3 Genetic Algorithms Overview

A genetic algorithm is a search technique used in computing to find true or approximate

solutions to optimization & search problems. They are implemented as a computer simulation in

which a population of abstract representations (called chromosomes or the genotype or the

genome) of candidate solution (called individuals, captures, or phenotypes) to an optimization

problem evolves toward better solutions.

35

Genetic Algorithm (GA) optimizers are robust, stochastic search methods modelled on the

principles and concepts of natural selection and evolution. As an optimizer, the powerful

heuristic of GA is effective at solving complex, combinatorial and related problems. GA

optimizers are particularly effective when the goal is to find an approximate global maxima in

high dimension, multi-modal function domains in a near optimal manner. GA differ from more

conventional techniques in that

• They operate on a group (or population) of trial solutions in parallel

• They normally operate on a coding of the function parameters (chromosome) rather than

on the parameters themselves

• They use simple, stochastic operators (selection, crossover, and mutation) to explore the

solution domain in search of an optimal solution.

4.4 Key terms in Genetic algorithms

• Individual: Any possible solution that may be occurred is known as individual.

• Population: Group of all individuals is known as population.

• Search Space: The term Search Space means all possible solution to the problem.

• Gene: Encoded form of a parameter being optimized is known as Gene.

• Chromosome: The complete set of genes (parameters) which uniquely describe an

individual or Blueprint for an individual.

• Genome: Collection of all chromosomes for an individual is known as Genome.

• Locus: The position of a gene on the chromosome.

36

• Fitness: A value that we are trying to maximize.

Fig. 1.1 briefly presents a list of some of the commonly encountered terminology in the CA

field as it relates to the optimization problem.

Fig.4.1. Genetic Algorithm commonly used terminology

4.5 Nonlinear Constraint Solver Algorithm

4.5.1 Augmented Lagrangian Genetic Algorithm

By default, the genetic algorithm uses the Augmented Lagrangian Genetic Algorithm

(ALGA) to solve nonlinear constraint problems without integer constraints. The optimization

problem solved by the ALGA algorithm is

min𝑥𝑥𝑒𝑒(𝑥𝑥) ( 4.1 )

37

such that

𝐶𝐶𝑖𝑖(𝑥𝑥) ≤ 0, i = 1 … … m

𝐶𝐶𝑒𝑒𝐶𝐶𝑖𝑖(𝑥𝑥) = 0, 𝑖𝑖 = 𝑚𝑚 + 1 … .𝑚𝑚𝑑𝑑

𝐴𝐴.𝑥𝑥 ≤ 𝑏𝑏

𝐴𝐴𝑒𝑒𝐶𝐶. 𝑥𝑥 = 𝑏𝑏𝑒𝑒𝐶𝐶

𝑙𝑙𝑏𝑏 ≤ 𝑥𝑥 ≤ 𝑢𝑢𝑏𝑏

( 4.2 )

Where C(x) represents the nonlinear inequality constraints, Ceq(x) represents the equality

constraints, m is the number of nonlinear inequality constraints, and mt is the total number of

nonlinear constraints.

The Augmented Lagrangian Genetic Algorithm (ALGA) attempts to solve a nonlinear

optimization problem with nonlinear constraints, linear constraints, and bounds. In this approach,

bounds and linear constraints are handled separately from nonlinear constraints. A sub problem

is formulated by combining the fitness function and nonlinear constraint function using the

Lagrangian and the penalty parameters. A sequence of such optimization problems are

approximately minimized using the genetic algorithm such that the linear constraints and bounds

are satisfied. A sub problem formulation is defined as

Θ(𝑥𝑥,𝜆𝜆, 𝑠𝑠, 𝜌𝜌) = f(x) −�𝜆𝜆𝑖𝑖𝑠𝑠𝑖𝑖 log(𝑠𝑠𝑖𝑖 − 𝑐𝑐𝑖𝑖(𝑥𝑥)) + � 𝜆𝜆𝑖𝑖𝑐𝑐𝑒𝑒𝐶𝐶𝑖𝑖(𝑥𝑥) +𝜌𝜌2� 𝑐𝑐𝑒𝑒𝐶𝐶𝑖𝑖(𝑥𝑥)2𝑚𝑚𝑑𝑑

𝑖𝑖=𝑚𝑚+1

𝑚𝑚𝑑𝑑

𝑖𝑖=𝑚𝑚+1

𝑚𝑚

𝑖𝑖=1

( 4.3 )

Where

• The components λi of the vector λ are nonnegative and are known as Lagrange

multiplier estimates

• The elements si of the vector s are nonnegative shifts 38

• ρ is the positive penalty parameter.

The algorithm begins by using an initial value for the penalty parameter (Initial Penalty).

The genetic algorithm minimizes a sequence of sub problems, each of which is an approximation

of the original problem. Each sub problem has a fixed value of λ, s, and ρ. When the sub problem

is minimized to a required accuracy and satisfies feasibility conditions, the Lagrangian estimates

are updated. Otherwise, the penalty parameter is increased by a penalty factor (Penalty Factor).

This results in a new sub problem formulation and minimization problem. These steps are

repeated until the stopping criteria are met.

Each sub problem solution represents one generation. The number of function evaluations

per generation is therefore much higher when using nonlinear constraints than otherwise.

4.5.2 Penalty Algorithm

In its evaluation of the fitness of an individual, ga computes a penalty value as follows

• If the individual is feasible, the penalty function is the fitness function.

• If the individual is infeasible, the penalty function is the maximum fitness function

among feasible members of the population, plus a sum of the constraint violations of

the (infeasible) individual.

4.6 GA Process and Strategies

In general a GA optimizer must be able to perform five basic tasks

• Encode the solution parameters in the form of chromosome.

• Initialize a starting population.

• Evaluate and assign fitness values to individuals in the population.

39

• Perform reproduction through the fitness weighted selection of individuals from the

population

• Perform recombination and mutation to produce members of the next generation.

The base logic of the algorithm is that gens of powerful individuals are based to be carried over

next generation and others are based to be detached in next generation. In natural selection, a

human born, grows, and dies. These stages of human life correspond to the different operators in

the algorithm. GA operators related these stages can be explained as the following

4.6.1 Initial population

Initial population is determined in two ways. One of them is to form random individuals as

initial population size within their limits. The second way is to form initial population of the

certain individuals. The fitness values of individuals within population are obtained to be put in

objective function the formed individuals. The individual number within initial population is

randomly determined by Eq. 4.4

𝑒𝑒𝑖𝑖𝑖𝑖 = 𝑒𝑒𝑚𝑚𝑖𝑖𝑎𝑎 ,𝑖𝑖 + 𝑟𝑟𝑎𝑎𝑎𝑎𝑑𝑑(0,1) × (𝑒𝑒𝑚𝑚𝑎𝑎𝑥𝑥 ,𝑖𝑖 − 𝑒𝑒𝑚𝑚𝑖𝑖𝑎𝑎 ,𝑖𝑖 ) ( 4.4 )

where the parameters wmin,jand wmax,jshow the minimum and maximum of the variable wj.

4.6.2 Fitness scaling

The scaling stage is one of GA operators. The scaling prevents algorithm to get stuck on a

local point. There are different scaling methods as rank scales, top scales, and shift linear scales.

In this study, the better individuals than individual having average fitness value are selected and

can be formulated as follows:

40

𝐹𝐹𝑎𝑎𝑟𝑟𝑒𝑒 =∑ 𝐹𝐹𝑖𝑖𝑁𝑁𝑘𝑘𝑖𝑖=1𝑁𝑁𝑘𝑘

( 4.5 )

Where Fave, Nk, and Fi represent the average fitness value within population, the number of

individuals within population, and the fitness value of ith individual, respectively.

4.6.3 Selection

A number of selection strategies have been developed and utilized for Genetic Algorithm

optimization. These strategies are generally classified as either stochastic or deterministic.

4.6.3.1 Population Decimation

The simplest of the deterministic strategies is Population Decimation. In Population

Decimation, individuals are ranked from largest to smallest according to their fitness values. An

arbitrary minimum fitness is chosen as a cutoff pointand anyindividual witha lower fitness than

the minimum is removed from the population. The remaining individuals arethen used to

generate the new generation through random pairing. The pairing and application of GA

operators are repeated until the new generation is filled.

4.6.3.2 Proportionate Selection

. The most popular of the stochastic selection strategies is Proportionate Selection,

sometimes called roulette wheel selection. In Proportionate Selection, individuals are selected

based on a probability of selection given in equation 4.6 where f(parent)i is the fitness of the ith

parent

𝑃𝑃𝑠𝑠𝑒𝑒𝑙𝑙𝑒𝑒𝑐𝑐𝑑𝑑𝑖𝑖𝑜𝑜𝑎𝑎 =𝑒𝑒(𝑝𝑝𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎𝑑𝑑)𝑖𝑖∑ 𝑒𝑒(𝑝𝑝𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎𝑑𝑑)𝑖𝑖𝑖𝑖

( 4.6 )

41

Fig.4.2. Proportionate Selection represented as a roulette wheel with spaces on the wheel proportional to an

individual’s relative fitness.

Fig. 4.2 depicts the Proportionate Selection process as a roulette wheel where individuals are

assigned a space on the wheel that is proportional to their relative fitness. The wheel is spun and

the individual pointed to at the end of the spin is the individual selected.

4.6.3.3 Tournament Selection

A second popular strategy (and perhaps among the most effective for many applications)is

TournamentSelection. In Tournament Selection, a sub-population of N individualsischosenat

randomfromthe population.Theindividuals ofthissub- population competeonthebasisoftheir

fitness.The individual in thesub-population with thehighestfitness wins thetournament and

becomestheselected individual.All of thesub-population members are then placed back into the

general population and the process can be repeated.

4.6.4 Crossover

In this stage, a child is produced to be crossed the parents. New individuals same as the

determined number are produced to be used the crossing method with the scattered parameter

from parents selected via the tournament method.

42

The value of 1 and 0 as gen number of an individual is randomly produced. If the value is 1,

then gen is taken from mother, the value is 0, then gen is taken from father and thus the child is

produced.

• Cross: 1 0 1 1 0

• Mother: a b c d e

• Father: x y z u w

• Child: a y c d w

4.6.5 Mutation

The mutation operator provides a means of exploring portions of the solution surface that

are not represented in the genetic makeup of the current population. If p >pmutation , an element

in the string making up the chromosome is randomly selected and changed. In the case of binary

encoding this amounts to selecting a bit from the chromosome string and inverting; it. In other

words, a “1” becomes a “0” and a “0” becomes a “1”. If higher order alphabets are used, slightly

more complicated forms of mutation are required.Fig. represents mutation.

Fig.4.3. The mutation operator randomly modifies elements within the chromosome.

4.6.6 The stopping criterion of algorithm

There are many criterions for stopping algorithm. Some of these are the fitness value, time,

and iteration number. In this study, function and nonlinear constraint tolerance are preferred as

the stopping criterion. Fig. 4.4 represents a simple flow chart of genetic algorithm.

43

Fig.4.4. Flow chart of GA

Initial Population Constraint Check

Fitness Evaluation

Selection Select Parent#1 Select Parent#2

Perform Crossover

(with p = pcross)

End

Perform Mutation

(with p = pmutation)

Until population

Is full

Termination criteria is met

Fitness Evaluation

Constraint Check Replace Population

44

Fig. 4.5 gives a screenshot of GA solver App available in MATLAB Global Optimisation

Toolbox.

Fig.4.5. GA solver App

45

Chapter 5 Optimisation of SISFCL design on 50V level

5.1 Introduction

The model of Saturated Iron core Superconducting fault current limiter is optimised by using

Genetic Algorithm. Fault resistance, no. of ac turns (wc), no of dc turns (wd), dc biasing current

(Id) are taken as variable parameters.

5.2 Problem Formulation

The problem of SFCL designed is formulated using JA Hysteresis model with nonlinear

objective function that has equality and inequality constraints. The proposed objective function

includes minimisation of both ac fault current and voltage induced across dc coilimproving

system stability and voltage profile. JA Hysteresis model is used because systemdesign based on

this model can justify SFCL device behaviour on both small voltage and medium voltage level.

5.2.1 Objective Function

The objective of this optimisation is to minimise both ac fault current as well as the dc

voltage induced across dc coil. So, the objective function consists of two terms. The first term is

ac fault current and the second term is dc coil induced voltage. The objective function is square

root of summation of squares of these two terms.

𝑀𝑀𝑖𝑖𝑎𝑎(�𝑑𝑑𝑒𝑒𝑟𝑟𝑚𝑚12 + 𝑑𝑑𝑒𝑒𝑟𝑟𝑚𝑚22 ) ( 5.1 )

Including terms of the objective functions as explained as follows

46

5.2.1.1 Term1

It signifies the peak value ofac current that is obtained when there is a fault in system. It is

obtained by solving the differential equation

𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑

=𝑉𝑉𝑠𝑠(𝑑𝑑) − 𝑖𝑖𝑅𝑅

𝐿𝐿 �2 + 𝑑𝑑𝑀𝑀1𝑑𝑑𝜇𝜇1

+ 𝑑𝑑𝑀𝑀2𝑑𝑑𝜇𝜇2

� ( 5.2 )

Where L is the inductance in H and it is formulated as 𝐿𝐿 = 𝜇𝜇0𝑒𝑒𝑐𝑐2𝐴𝐴𝑙𝑙

where𝜇𝜇0 is absolute

permeability, wc is no. of ac turns in each core, A is net areaand l is the mean magnetic path

length. 𝑑𝑑𝑀𝑀1𝑑𝑑𝜇𝜇1

and𝑑𝑑𝑀𝑀2𝑑𝑑𝜇𝜇2

represent the susceptibilities of core 1 and core 2 which is given by the

following general expression

𝑑𝑑𝑀𝑀𝑑𝑑𝜇𝜇

= (1 − 𝑐𝑐)𝑘𝑘𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀

𝑠𝑠𝑖𝑖𝑠𝑠𝑎𝑎 �𝑑𝑑𝜇𝜇𝑑𝑑𝑑𝑑�𝑘𝑘(1 − 𝑐𝑐) − 𝛼𝛼(𝑀𝑀𝑎𝑎𝑎𝑎 − 𝑀𝑀)


𝑑𝑑𝜇𝜇 ( 5.3 )

Where 𝑘𝑘 = 𝑠𝑠𝑖𝑖𝑠𝑠𝑎𝑎 �𝑑𝑑𝜇𝜇𝑑𝑑𝑑𝑑� , +1 if 𝑑𝑑𝜇𝜇

𝑑𝑑𝑑𝑑≥ 0,−1 if 𝑑𝑑𝜇𝜇

𝑑𝑑𝑑𝑑≤ 0

k is parameter defining the pinning site density of domain walls, Man is anhysteretic

magnetization at a given field strength, Man=Ms (coth(He/a)-a/He ),Ms is the saturation level, c

is a domain flexing parameter, defining the amount of reversible magnetization due to wall

bowing and reversal rotation, included in the magnetization process. α is a scaling coefficient,.

He is the effective field strength, He=H+ αM.

47

5.2.1.2 Term 2

From JA hysteresis model it is cleared that a voltage is inducted across the biasing coil

during the fault limiting action of SFCL which should be minimum in order to operate the FCL

satisfactorily. Term 2 designates the peak voltage induced across dc coil. It can be formulated as

𝑉𝑉𝑑𝑑𝑐𝑐 = µ0𝐴𝐴𝑊𝑊𝑑𝑑𝑐𝑐 (𝑑𝑑𝜇𝜇1

𝑑𝑑𝑑𝑑�1 +

𝑑𝑑𝑀𝑀1

𝑑𝑑𝑑𝑑� +

𝑑𝑑𝜇𝜇2

𝑑𝑑𝑑𝑑�1 +

𝑑𝑑𝑀𝑀2

𝑑𝑑𝑑𝑑�) ( 5.4 )

Where wdcis the no of turnsin the dc coils, 𝑑𝑑𝜇𝜇1𝑑𝑑𝑑𝑑

and 𝑑𝑑𝜇𝜇2𝑑𝑑𝑑𝑑

can be derived as

𝑑𝑑𝜇𝜇1

𝑑𝑑𝑑𝑑 = −𝑊𝑊𝑐𝑐𝑙𝑙𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑

( 5.5 )

𝑑𝑑𝜇𝜇2

𝑑𝑑𝑑𝑑 = +𝑊𝑊𝑐𝑐𝑙𝑙𝑑𝑑𝑖𝑖𝑑𝑑𝑑𝑑

( 5.6 )

Where wcisthe no of terms in the ac coils in each core.

5.2.2 Constraints

5.2.2.1 Magnetic field density constraint

Normal operating constraint

For the successful operation of fault current limiter, it should be saturated during normal

operation condition so that it can offer low impedance and does not limit the normal operating

current flowing in the circuit. So, to ensure this the effective ampere turns in both cores should

be enough to place the operating point (Bopt) above the saturation limit (Bsat) . This saturation

level generally depends upon the core materials used.

48

𝐵𝐵𝑜𝑜𝑝𝑝𝑑𝑑 ≤ 𝐵𝐵𝑆𝑆𝑎𝑎𝑑𝑑 ( 5.7 )

Fault constraint

During the fault condition the ampere turns created by the large fault current oppose the dc

ampere turns created by the dc biasing current in such a manner that the operating point comes in

linear zone in BH curve. As a result of it the permeability of the cores increases resulting in

increasing flux linkages which in turns increases the impedance offered by the cores during fault

condition. But if the value of ac ampere turns created by the fault current is high enough, it will

make the net effective ampere turns of the cores to be operated on reverse saturation condition.

So, the fault constraints consist of

𝐵𝐵𝑜𝑜𝑝𝑝𝑑𝑑 ≤ 𝐵𝐵𝑠𝑠𝑎𝑎𝑑𝑑 ( 5.8 )

And

𝐵𝐵𝑜𝑜𝑝𝑝𝑑𝑑 ≥ −𝐵𝐵𝑠𝑠𝑎𝑎𝑑𝑑 ( 5.9 )

5.2.2.2 Voltage constraint

According to the law of KVL, the applied voltage is equal to the voltage across SFCL and

the losses on various parts of the circuit. So, the voltage across SFCL (𝑉𝑉𝑆𝑆𝐹𝐹𝐶𝐶𝐿𝐿) is always less than

the applied voltage (𝑉𝑉𝑎𝑎𝑝𝑝𝑝𝑝𝑙𝑙𝑖𝑖𝑒𝑒𝑑𝑑 ).

𝑉𝑉𝑆𝑆𝐹𝐹𝐶𝐶𝐿𝐿 ≤ 𝑉𝑉𝑎𝑎𝑝𝑝𝑝𝑝𝑙𝑙𝑖𝑖𝑒𝑒𝑑𝑑 ( 5.10 )

5.3 Optimisation using GA Solver

The objective function described earlier is formed in MATLAB. R-K fourth order method is

utilized. Both normal and fault condition are taken into account and constraint function is also

programmed in MATLAB. The objective function and constraint function are optimised using 49

MATLAB GA SOLVER. Though the problem is a multiobjective optimization problem, it is

converted into a single objective optimization problem in order to include the advantages of

applying constraints to its generation population. The programs used for the simulation are given

in the Appendix, on Pages 89 – 96.

In the models of saturated iron core superconducting fault current limiter variable

parameters of the optimization procedure are accepted to be: fault resistance (Rf), no of turns in

ac windings (wc), no of turns in dc windings (wd) and dc biasing current (Id).

• Knowing that the fault resistance is one of the most important quantities of the

saturated core SFCL design, the first idea is to use its value as variable parameter

during the optimization procedure—Model 1.

• Among the other interesting factors, the no of turns in ac windings is used as

variable parameter during the optimization procedure—Model 2.

• The no of turns in dc windings is effecting the net effective flux which decides the

point of operation in B-H curve, it is selected to use as a variable parameter during

the optimization procedure—Model 3.

• As dc biasing current provides the necessary bias for operation, the fourth idea is to

use it as a variable parameter during the optimization procedure—Model 4..

• Among the four variable parameters the next idea is to vary three of them, no of ac

turns, no of dc turns and dc biasing current while fault resistance is kept constant —

Model-5.

• The last optimization consists of varying all four parameters during the optimization

procedure— Model-6.

50

Table 5.1. core hysteresis parameters used

Ms c k Α a

1.7e6 0.1 500 0.001 1000

Table 5.2. Options used for GA

Population

Population type Double vector

Population size 20

Creation function Constraint dependent

Initial population Default

Initial score Default

Initial range [0;1]

Fitness scaling Scaling function Rank

Selection Selection function Stochastic uniform

Reproduction Elite count 2

Crossover function 0.8

Mutation Mutation function Constraint dependent

Crossover Crossover function Scattered

Migration

Direction Forward

Fraction 0.2

51

Interval 20

Constraint

parameters

Initial penalty 10

Penalty factor 100

Hybrid function Hybrid function None

Stopping criteria

Generations 100

Time limit Infinity

Stall generations 50

Stall time limit Infinity

Function tolerance 1e-4

Nonlinear constraint

tolerance

1e-4

5.4 Optimisation Results

5.4.1 Model 1

In this model fault resistance value is optimised while keeping other parameters fixed. The

circuit is simulated for circuit resistance R=5 ohm for normal condition. A good choice of initial

range of the varying parameter is very important for successful optimisation process. Here fault

resistance is varying between 0.1 and 0.99. Optimum value of parameter is 0.9890 and value of

the peak fault at this value is 47.178A and the value of the voltage induced across dc coil is

25.64 V. The values of constant parameters are shown in the table below

52

Table 5.3. Value of constant parameters for Model 1

Id wd wc V0 length freq A Rn

15A 100 20 50V 0.56m 50Hz 45.4e-4 5ohms

The following figures are obtained during optimization process.

It is observed from the above figures that increment of fault resistance limits the fault

current. Table 5.4 describes four cases including the optimised one.

Table 5.4. Different cases in Model 1 optimisation

Cases Rf(ohms)

Case 1 0.2

Case 2 0.4

Fig.5.2. IfvsRf Fig.5.1. Rf, If variations with iterations

53

Case 3 0.8

Rfopti 0.9890

Table 5.4 is represented by different response described below

Fig.5.6. B vs time for Core2 Fig.5.5. B vs time for Core1

Fig.5.4. Vdcvs time Fig.5.3. circuit current vs time

54

It is observed that among the cases considered Rfopti=0.9890 ohm gives the optimum value of

peak fault current as well as voltage induced across dc coils. If fault resistance increases further

fault current decreases but net effective flux produced in the cores increases and a condition is

reached when the cores don’t come out of saturation in fault condition.. Hence the constraint

function isn’t satisfied and optimisation gets stopped.

5.4.2 Model 2

In Model 2, no of turns in ac winding is optimised while keeping fault resistance, dc

winding turns and dc biasing current fixed. The ac winding turns no is optimised between 20 and

50. The values of constant parameters are shown in the table below

Table 5.5. Value of constant parameters for Model 2

Id wd Rf V0 length Freq A Rn

15A 100 0.1ohms 50V 0.56ms 50Hz 45.4e-4 5ohms

Theoretical optimum value is obtained at wc= 50, at this value the of peak fault current is

29.2A and peak value of voltage across dc coil is 77.72V.The variation of peak fault current with

wc is shownin Fig.5.7.Fig. 5.8describes the variations of both fault current and ac winding turns

with no. of iteration during optimization process.

Fig.5.8. If, wc variation with iteration Fig.5.7. Ifvswc

55

Following Table 5.6 describes four cases including the optimization condition

Table 5.6 Different cases in Model 2 optimisation

Cases Wc

Case 1 25

Case 2 30

Case 3 35

wcopti 50

Table 5.6 can be represented by different responses described below.From the above

responses it is observed that wcopti=50 give the best fault limiting condition satisfying all

constraints. Though further increase in no of ac winding turns can reduce the peak fault current

but it is uneconomical to keep high no. of ac winding turns beyond the range considered for

optimization because it can impose a limitation on system design. Also it can be noted that as no

of ac winding turns increases beyond 50, cores will not be in saturation region during normal

operating conditions which violets the normal magnetic flux density constraint described in

chapter. However if the no. of ac winding is reduced below 20, cores will get saturated in reverse

direction during fault which will violate fault magnetic flux density conditions.

56

Fig.5.9. vdcvs time

.

Fig.5.12. B vs time for core 2 Fig.5.11. B vs time for core1

Fig.5.10. circuit current vs time

57

5.4.3 Model 3

Model 3 is optimised where the dc biasing current Wd is varying parameter and all the other

parameters are kept constant for the sake of simplicity. The range of optimization is considered

between 80 to 120.The values of other constant parameters are shown in the Table 5.7 below

Table 5.7.Value of constant parameters for Model 3

Id Wc Rf V0 length freq A Rn

15A 20 0.1ohms 50V 0.56m 50Hz 45.4e-4 5ohms

It is evident from the above figures that peak value of fault current is less affected by the

variation of dc winding turns(as described by Fig. 5.13) but the magnetic characteristics of the

cores are highly affected. Theoretical optimum value is achieved at wd =82 and at this condition

the value of the peak fault current and peak value of voltage induced across dc coil is 99.87A and

187.8V. Fig. 5.14describes the variations of fault current and dc no of turns at every iteration

during optimization process.

Following Table 5.8describes four cases including the optimization condition.

Fig.5.14. If, wc variations with iterations Fig.5.13. Ifvswc

58

Table 5.8. Different cases in Model 3 optimisation

Cases wd

Case 1 100

Case 2 110

Case 3 120

wdopti 82

Table 5.8can be represented by different responses described below

Fig.5.16 If vs time Fig.5.15. vdcvs time

59

It is observed from Fig. 5.17that core1 is on the verge of reverse saturation condition during

fault operation. If no of dc turns decreases further peak fault current will decrease slightly but the

net effective flux in fault condition will cause reverse saturation condition in core 1.

5.4.4 Model 4

In Model 4, dc biasing current is optimised while keeping fault resistance, dc winding turns

and no of turns in ac winding fixed. Here the range between which Id is varied is 10 to 20. The

values of other constant parameters are shown in the Table 5.9below

Table 5.9 Value of constant parameters for Model 4

Wc Wd Rf V0 length freq A Rn

15A 100 0.1ohms 50V 0.56m 50Hz 45.4e-4 5ohms

Fig.5.19 describes the nature of fault current variation with respect to Id

Fig.5.18. B vs time for core 2 Fig.5.17. B vs time for core 1

60

From the above figures it is evident that if dc biasing current decreases then fault current

also decreases implying less peak fault current. However, the change is very less significant

because dc biasing current decides mainly the operating zone in magnetic hysteresis curve for a

successful fault limiting operation. The optimum value of dc biasing current is obtained at

13.25A and at this point the value of optimum peak fault current is 102.8A and peak voltage is

232.2 V.Fig.5.20. describes the variations of fault current and dc no of turns at every iteration

during optimization process.

Table 5.10 Different cases in Model 4Optimisation

Cases Id(A)

Case 1 15

Case 2 18

Case 3 20

Idopti 13.25

Different cases of Table 5.10 can be compared by following responses

Fig.5.20. If, Id variations with iterations Fig.5.19 Ifvs Id

61

If dc biasing current is further decreased below the optimum point then peak fault current

decreases but net effective flux created in the core will drag the core on reverse saturation

direction.This will violate the constraint condition.

Fig.5.24 B vs time for Core 1 Fig.5.23 B vs time for core 2

Fig.5.22vdcvs time Fig.5.21 circuit current vs time

62

5.4.5 Model 5

The dc bias current, the no of ac turns and the no. of dc turns are varied together in model 5

during optimization process. However, the fault resistance is kept constant. The constant

parameters are listed in the Table 5.11below


Rf V0 length freq A Rn

0.1ohms 50V 0.56m 50Hz 45.4e-4 5ohms

No. of ac turns, no of dc turns and dc biasing current are varied between 20 to 50, 80 to 150

and 5 to 20 A respectively. At optimum condition the value of Id,wd,wc are 18.018A,80 and 50

respectively. The values of peak fault current and peak value of voltage induce across dc coil

obtained during optimization procedure are 29.31A and61.42Vrespectively.

Table 5.12 Different cases in Model 5 Optimisation

wc wd Id(A)

Case 1 25 100 14

Case 2 35 110 15

Case 3 45 120 18

opti3 50 80 18.018


63

It is observed from Fig. 5.27that optimum condition is on verge of saturation limit during

normal condition. Penalty value, scores and best individual during optimization process are

shown in Fig. 5.29below

Fig.5.28 B vs time for core2 Fig.5.27 B vs time for core1

Fig.5.26. vdcvs time Fig.5.25 circuit current vs time

64

5.4.6 Model 6

All four parameters are varied in Model 6 and the overall optimization of the design is

achieved. The ranges used for various parameters are shown inTable 5.13 . Also, different

constant parameters are listed in Table 5.14.

Table 5.13 Ranges of the parameters in Model 6

Parameters Lower range Upper Range

Id(A) 10 20

wd 80 120

wc 20 50

Rf(ohms) 0.1 1

Fig.5.29 a) Penalty Value vs generation, b) current best Individual, c)Best, Worst and Mean scores

65


freq V0 length A Rn

50Hz 50V 0.56m 45.4e-4 5ohms

The optimization point is obtained at Id = 18.083A, wd =80, wc =50 and Rf=0.998 ohms. The

optimised values of peak fault current and peak value of voltage induce across dc coils are

23.51A and 36.78V. Penalty value, selection function, scores, expectations of scores and

distances during each generation are described in the following figures

Fig.5.33 Expectation of raw scores Fig.5.32 Score vs Generations

Fig.5.31 Selection vs generation Fig.5.30 Penalty value vs generation

66

FromFig.5.30it is observed that mean penalty is 47.832 which is close to best penalty value.

So, a large no of populations are converging towards optimal solution. It can be also seen

fromFig. 5.34. Up to 20 generation the average distance between individuals are varying

between 5 and 20. But after wards the distance falls to nearly zero indicates that it starts to

converge towards a solution, limited by the tolerance of objective and constraint function.

Table 5.15describes four cases including the optimised one.

Table 5.15 Different cases in Model 6 Optimisation

Id(A) wd wc Rf(ohms)

Case 1 14 90 25 0.4

Case 2 16 100 35 0.6

Case 3 18 110 45 0.8

Opti4 18.083 80 50 0.998

Fig.5.34 Average Distance vs Generation

67


Fig.5.38 B vs time for core 2 Fig.5.37 B vs time for core1

Fig.5.36vdcvs time Fig.5.35 Circuit current vs time

68

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