23 Nov 2012 - Mathematica Code . ...... a random variable. will have a PDF which is the convolution of the PDFs .... interactive tool in Mathematica (Figure 43).
University of Technology, Sydney Faculty of Engineering and Information Technology Modelling of CT Saturation for Power System Protection Applications by Michael James Stanbury Student Number: 10739314 Project Number: A12-151 Major: Electrical Engineering Supervisor: Dr Youguang Guo Industry Co-supervisor: Mr John Dowsett (Ausgrid) A 12 Credit Point Project submitted in partial fulfilment of the requirement for the Degree of Bachelor of Engineering 23 November 2012
1
2
UTS: Engineering Subject: Capstone
Assignment Number: A12-151
Date Submitted: 23/11/2012
Assignment Title: Modelling of CT Saturation for Power System Protection Applications
Student Name(s) and Number(s) Michael Stanbury 10739314
Declaration of Originality: The work contained in this assignment, other than that specifically attributed to another source, is that of the author(s). It is recognised that, should this declaration be found to be false, disciplinary action could be taken and penalties imposed in accordance with University policy and rules. In the statement below, I have indicated the extent to which I have collaborated with others, whom I have named.
Signature
Statement of Collaboration:
3
4
Abstract Current transformers (CTs) are regularly used in electrical power systems to measure currents for protection purposes. In certain situations they may undergo saturation and fail to represent the current accurately. The traditional solution to this problem has been to use very large CTs which are unlikely to saturate. Ausgrid, an electricity distributor in New South Wales, is now procuring new fixed pattern switchgear for its zone substations. The smaller switchgear means that Ausgrid’s traditionally large 11kV CTs will not fit inside the enclosures. The aim of this work is to investigate whether smaller CTs can be used on Ausgrid’s network, without additional risk to the network. In this work I present a novel, physically based CT model which includes saturation, hysteresis, remanence and minor loop phenomena. This model is based on the Preisach theory of hysteresis, and its parameters are algorithmically deduced from experimentally measured B-H curves. I use this model to simulate several through-faults on City South transformer 4 on Ausgrid’s network. The resulting saturated waveforms were injected into a real differential protection relay to observe its behaviour under saturation. The results show that modern protection relays can appropriately handle a moderate amount of CT saturation. The tests also show the amount of saturation resulting from different CT core sizes. The performance of my model is compared to two others. Lastly, I developed a statistical model which predicts the probability that a CT will saturate for a given fault. This model will help Ausgrid to understand the complex influences which determine CT saturation, and calculate the probability of it occurring. This quantitative analysis will be important for Ausgrid to develop a new specification for 11kV protection CTs which are small enough to fit inside the new fixed pattern switchgear.
5
6
Acknowledgments I’d like to thank Youguang Guo for being my helpful and supportive supervisor, John Dowsett for generously offering his time and expertise, Tarik Hussein for giving me a solid foundation on which to build and also for his keen proof reading, Greg Bartolo for his dedication and eye for detail in the lab, and James Ruff for sanity checking my calculations and generously offering assistance with network data. I dedicate this work to my gorgeous wife Tracey for her unwavering support and compassionate encouragement. You inspire me each day.
7
Table of Contents Abstract .......................................................................................... 5 Acknowledgments ......................................................................... 7 Nomenclature .............................................................................. 10 Abbreviations ............................................................................................. 10 Symbols ...................................................................................................... 10
1.
Introduction ........................................................................ 11 1.1.
Background ................................................................................... 11
1.2.
Problem to be Solved .................................................................... 12
1.3.
Outline........................................................................................... 13
2.
Literature Review ............................................................... 14
3.
Model Development ............................................................ 15 3.1.
Theoretical Background ................................................................ 15
3.2.
Proposed CT Models..................................................................... 16
3.3.
The Ideal Saturation Model........................................................... 18
3.4.
The Static Preisach Hybrid Model ................................................ 28
3.5.
Static Tanh Model ......................................................................... 39
4.
Model Verification .............................................................. 42 4.1.
CT Error and the Sudden Saturation Hypothesis .......................... 42
4.2.
Model Verification with no DC Offset ......................................... 53
4.3.
Model Verification with DC Offset .............................................. 56
5.
8
Application of the Static Preisach Hybrid Model ........... 65 5.1.
Saturation Detection and Handling Test ....................................... 65
5.2.
Results of Saturation Detection and Handling Test ...................... 67
5.3.
Discussion of Saturation Detection and Handling Test ................ 81
6.
Statistical Model of Saturation ......................................... 84 6.1.
Effect of Cross Sectional Area on Saturation ............................... 95
6.2.
Effect of Low Remanence Gaps on Saturation ............................. 96
6.3.
Effect of Secondary Resistance on Saturation .............................. 98
6.4.
Verification of Statistical Model................................................. 100
6.5.
Validity of Assumptions ............................................................. 100
7.
Future Work ..................................................................... 103 7.1.
Effect of Primary Current on Saturation ..................................... 103
7.2.
The Distribution of Remanence on 11kV CTs ........................... 103
7.3.
Saturation Detection and Handling of Other Relays .................. 104
7.4.
Measurement of Secondary Resistance ...................................... 104
8.
Conclusion ......................................................................... 105
9.
References ......................................................................... 108
10. Appendices ........................................................................ 110 10.1.
Hand Wound Core Details .......................................................... 110
10.2.
Transient Equations of Various Fault Types .............................. 111
10.3.
Spreadsheet to Process B-H Curves............................................ 114
10.4.
Mathematica Code ...................................................................... 114
10.5.
Matlab Code................................................................................ 116
9
Nomenclature Abbreviations Abbreviation CT
Meaning
Abbreviation
Current Transformer
SPH model
Meaning Static
Preisach
Hybrid
model FORC
First
Order
Reversal
ODE
Ordinary
Curve PDF
Differential
Equation
Probability
Density
LLL
3 phase fault
LG
Single phase to ground fault
T
Tesla (unit of flux density)
Function (statistics) LL
2 phase fault
LLG
Double line to ground fault
Symbols Symbol
Meaning CT secondary EMF
Symbol
Meaning Primary and secondary currents through CT
Number of windings on the
Cross Sectional Area of the core
CT time
Total resistance in the secondary circuit
DC offset of the fault
Total
(-1
secondary circuit
)
inductance
in
the
Flux in the core
Core length
Magnetic Flux Density
The X on R ratio of the primary system
for
the
given
fault
location. Magnetic Field Intensity
Power system frequency in rad/s
The angle of the voltage sine
Remanence Factor = remanence
wave where the fault occurs
flux
density
divided
saturation flux density ( )
10
Instantaneous CT error
by
1. Introduction 1.1. Background Ausgrid currently faces a problem related to the sizing of its 11kV protection class current transformers (CTs). In the past, Ausgrid has used large switchgear in zone substations, but it is now procuring smaller, fixed pattern switchgear (Siemens NXPLUS C, Gas Insulated). The benefits of this smaller switchgear are less maintenance and less space being consumed. These benefits reduce the cost of land and the cost of running the network. One disadvantage is that the smaller enclosures will not be large enough to fit the CTs that Ausgrid has traditionally used. Figure 1 shows 11kV feeder CTs which only just fit into the switchgear.
Figure 1: Feeder protection CTs (600/300/5) in OLX switchgear (brown cylindrical objects at right). View from above switchgear, looking down into the top, switchgear cover plates have been removed. It can be seen that there is only just enough room for 3 CTs side by side in the enclosure. If Ausgrid makes the panels narrower, the 3 CTs of existing size will not fit in this arrangement.
The two main options are to reduce the size of the CTs, relocate them and/or use gapped CTs to reduce remanence. Ausgrid has decided to use smaller cores with low remanence gaps for the new switchgear (Ausgrid, 2012, p. 16), however there is still uncertainty and disagreement about this solution. In this thesis, I will investigate some of the issues related to reducing the size of the CTs and adding low remanence gaps. 11
Reducing the CT size increases the chance that cores will saturate when faults occur on the network. If the CT saturates, then the protection relays will not receive an accurate measurement of the fault current. This may mean that they fail to operate, or they may operate when they should not (a nuisance trip). A nuisance trip may cause a large number of customers to experience an unnecessary power outage. There are many factors which contribute to CT saturation including fault duration, primary system X/R ratio, the location of the fault on the AC waveform, and the amount of remanence in the core at the time of the fault (Turner, 2010). These issues are complicated by the fact that modern protection relays have the ability to detect CT saturation and may compensate for it, or restrain the trip threshold. Every relay uses a different algorithm to detect and deal with saturation, so there are no simple models to predict what will occur in this situation. In the long term the industry will likely move away from ferromagnetic core CTs, and towards magneto-optical current transducers. These use optical fibre to sense magnetic fields and do not saturate like traditional CTs, so they have the potential to solve this problem once and for all. Unfortunately, the market for these products is not yet mature. Some products exist, but there are no well-established standards for these devices to communicate with protection relays. There are also questions about whether these devices are robust enough to withstand 30 years of weather. Ausgrid’s position is to be fairly conservative with the adoption of new technologies for protection systems. If the technology fails, customers will lose power, and lives could be put at risk. For this reason, Ausgrid believes that traditional CTs will still be used in protection applications for approximately the next decade, and so a solution to this saturation problem is still needed.
1.2. Problem to be Solved The following 5 problems will be addressed in this thesis, and the answers will be presented in the conclusion (section 8). 1. Can Ausgrid use smaller CTs without risking saturation? 2. If saturation is inevitable, can saturation be suitably handled by modern relays? 3. If saturation is inevitable, how likely is it to occur for a particular fault?
12
4. How will a particular CT perform in transformer differential protection schemes (the most onerous case for CT saturation)? A model must be developed in order to simulate faults on the network. 5. Ausgrid needs a series of saturation waveforms to be produced which can be injected into a relay to test its saturation detection and handling algorithms. This will be used to evaluate relay products for procurement decisions.
1.3. Outline Here’s an outline of the document. In section 2 I will present a literature review of CT modelling strategies. In section 3 I present 2 CT models I’ve developed (the “Static Preisach Hybrid” and the “Static Tanh” models) and compare them to the Ideal Saturation model. In section 4 I present my “Sudden Saturation Hypothesis” which implies that the Ideal Saturation model should be highly accurate for real faults on the network, even though it’s not very accurate in lab tests with small input currents. If confirmed, this hypothesis could greatly simplify Ausgrid’s task of investigating CT saturation because complicated CT models will not be required. In section 4 I also present verification tests of the models. In section 5 I use my Static Preisach Hybrid model to generate fault waveforms for differential protection on a transformer at the City South zone substation. The Ausgrid relay test lab injected these currents into a relay to observe how it handles the saturation. These tests showed that the relay can appropriately handle a significant amount of saturation. In section 6 I present a novel analysis of the probability that a particular CT will saturate for a given fault. This statistical model provides Ausgrid with a completely new way to analyse the risk of saturation, and brings quantitative understanding to an issue which has previously been quite vague and poorly understood. Section 7 describes future work to be completed, and I summarise my contributions to the field in section 8.
13
2. Literature Review There is a great deal of literature on the modelling of ferromagnetic core transformers. Accurate simulation of a transformer under saturation is a complex process which is impossible to achieve without a computer. This is due to the complex non-linear characteristic of the core. There are many interdependent variables of magnetic field intensity, magnetic flux density, secondary voltage and secondary current. Zocholl and Smaha (1992) present a simplified model which describes 3 different circuits. One of them is the CT under low excitation, one under medium, and a third under high excitation. These models are useful for understanding the process, but are not useful for making accurate predictions of secondary current waveforms. Preisach (1935) devised the first theory which capably reproduced many of the phenomena of ferromagnetic cores including saturation, hysteresis and minor loops. This model assumes the core is composed of many ideal relay elements with different parameters. The sum of these ideal relays produces the core behaviour. This theory is widely used today. Mayergoyz (1986) showed there are some limitations to the Preisach theory including the restriction that B-H curves of different magnitudes must be congruent which is not strictly true of real ferromagnetic cores. However, the theory still produces quite accurate results for the saturated secondary current so it is widely used today. Bertotti (1992) has attempted to generalise the classical Preisach model to include dynamic effects such as the change in B-H curve with input frequency. These effects are modelled by limiting the rate at which magnetic domains can change state, depending on the surrounding magnetic field. The surrounding field is dictated by the state of other domains so they have dynamic interaction. The results show good agreement with various careful experiments, but this theory remains very difficult to implement in a robust and computationally efficient way. This makes the dynamic Preisach model important from a scientific standpoint, but too cumbersome for most engineering applications. Naidu (1990) describes a simple model which is capable of accurate secondary currents. This author presents an implementation of the classical Preisach theory and describes how it can be simulated with a computer. I will adapt this
14
method for application to current transformers. This paper also describes a way to calculate hysteron weighting values using an experimentally measured B-H curve from a real CT. Another method for predicting CT saturation is to perform low voltage tests at low frequency and extrapolate up to high currents at power system frequency. Changqing & Jianan (2011) present their work on power system CTs and show how low frequency measurements can be scaled up to apply to high currents. They claim that this method is well suited to on-site testing. Chandrasena et al. (2001) investigates the behaviour of gapped core CTs for power system protection applications. It claims to be one of the first papers to accurately simulate this phenomenon for protection purposes. Gapped core CTs could be an important part of the solution for Ausgrid as they have smaller remanence, which is the major contributor to CT saturation. Annakkage et al. (2000) presents a CT model based on the Jiles-Atherton theory of ferromagnetic hysteresis. They achieve good results, but the Jiles-Atherton model is particularly difficult to implement and it’s very difficult to experimentally determine the model parameters. Despite many decades of research, no ferromagnetic hysteresis model has been found to be both generally applicable and practical for engineering applications. Research in this field continues to this day. For this reason, I will focus my efforts on a model that is suitable for current transformers in power system protection applications.
3. Model Development 3.1. Theoretical Background The most difficult part of modelling a current transformer (CT) is the B-H characteristic. This is not a simple function but one that is:
Highly non-linear
Multi-valued
History dependent
15
Frequency dependent
Temperature dependent
B and H are vectors which may point in different directions.
Modelling all of these effects simultaneously is not practical. To develop my models I will ignore the last three effects, and focus on the first three. These first three are the effects captured by classical Preisach (1935) theory. I have also chosen to discretise the Preisach model as described by Pokrovskii (2001). This means my implementation will have a finite number of ideal relay elements called “hysterons”. This discretisation makes is easy to implement in a computer model and provides flexibility in representing many different core materials.
3.2. Proposed CT Models For this thesis I have developed several models to simulate CT behaviour under fault conditions on an electrical network. The first model provides a few simple algebraic formulae which predict whether a CT will enter saturation for a given fault. Another formula finds the time taken for saturation to occur. These formulas will be very useful to Ausgrid as they express CT saturation information in terms of readily available quantities like the fault level, X/R ratio, and burden resistance. These formulas will then be used to confirm the results of the second class of models. The second class of models I developed is a full dynamic simulation of the CT. To achieve this I developed several algorithms which implement the classical Preisach model in different ways. I then compared these algorithms and chose the most reliable, accurate and easy to use. This model was then used to simulate faults on Ausgrid’s network through a zone transformer at City South substation. In this thesis I will not use the traditional electrical equivalent circuit with a “magnetising impedance”. This model is simple for electrical engineers to understand, but it requires non-physical components like an ideal transformer and non-physical currents like the magnetising current. Instead, I will use physically based models which simulate the relationship between primary and secondary currents, mediated by a model of the B-H characteristic of a ferromagnetic core. This
16
is more suitable for the complex transient solutions required for differential protection schemes, and it more accurately demonstrates the physical processes within the core. All models presented here the following assumptions:
A constant cross sectional area around the core
Uniform flux density (B) at all parts of the core at all times
Uniform magnetic field intensity (H) around the core
The primary current is unaffected by the CT or burden; it is modelled as an ideal current source
All of these are standard assumptions for protection class CTs. They are good assumptions for this application because CT manufacturers construct their products to meet these assumptions. The following derivations use these symbols: Symbol
Meaning CT secondary EMF
Symbol
Meaning Primary and secondary currents through CT
Number of windings on the
Cross Sectional Area of the core
CT time
Total resistance in the secondary circuit
DC offset of the fault
Total
(-1
secondary circuit
)
inductance
in
the
Flux in the core
Core length
Magnetic Flux Density
The X on R ratio of the primary system
for
the
given
fault
location. Magnetic Field Intensity
Power system frequency in rad/s
17
3.3. The Ideal Saturation Model The first model I present was first described in (Wright, 1968). It is a piecewise linear model which assumes an ideal CT until the flux density reaches . At this point, the secondary current immediately falls to zero until the flux density falls below
and it becomes an ideal CT again (see Figure 2). This model
has no hysteresis, and is considered the simplest possible model of CT saturation. Despite its simplicity, it can reproduce the secondary current quite accurately. The BH curve however is very different to real materials, including Grain Oriented Silicon Steel which is commonly used for protection class CTs.
Figure 2: B-H curve of the Ideal Saturation model. It models an ideal transformer when not in saturation, and the secondary current instantly falls to zero when in saturation.
The Ideal Saturation model is currently Ausgrid’s most accurate model for simulating faults with a DC offset. Since DC offset is crucial for investigating CT saturation, the Ideal Saturation model will be the benchmark against which I compare my Static Preisach Model. In the following analysis I will derive several equations using the Ideal Saturation model which will predict whether a CT will saturate for a given fault, and the time at which saturation will occur. This model has been previously analysed by Ausgrid (Dowsett, 2011). My derived formulas will predict saturation using data which is commonly available to Ausgrid like the RMS fault level and the X/R ratio
18
of the fault. Due to the ease of use, these equations can be of great use to Ausgrid in determining whether a particular CT can possibly saturate, and if it can, the risk that it will do so. The main question for Ausgrid, and the one I will address, is whether this model’s simplicity makes it less accurate than more complicated models involving hysteresis. Note: In the following derivations, the secondary inductance
has been
included. In most cases this is small and may be neglected which will simplify the equations significantly. I have left the inductance in for the sake of older electromechanical relays which have an inductive burden. 3.3.1. Flux Density for a Given Fault We begin by deriving the CT’s flux density ( ) waveform for a given fault. Unless otherwise mentioned, all values are instantaneous (not RMS). Faraday’s Law of Induction says the CT’s secondary EMF will be:
Using the aforementioned assumption of uniform flux density:
We model the CT secondary as a resistance (
) and inductance ( ) due to
the CT winding, the leads from the CT to the relay, and the burden. From Ohm’s law in the CT secondary circuit:
Combining and rearranging these 3 equations gives a differential equation which can be used to find . For clarity, I will henceforth highlight functions of time in green, I.e.
is short for ( ). Constants will remain black.
Equation 1
Under the assumptions presented above, the magnetic field intensity
is: 19
We will now temporarily assume that the CT does not saturate. This assumption will be checked at the end. If the assumption is found to be true, the CT has not saturated. For a non-saturated CT, the secondary current is proportional to the primary current (as in an ideal transformer):
Substituting this into the equation for
to find that it is zero:
In section 3.3.3 we will quantify what values of
are small enough to be
considered approximately zero. In that section it will be seen that when zero (| |
) the CT is not saturated, and when
is close to
is large (| |
) the CT is saturated. In a CT on a real electrical network,
is either
very large or very small. It changes very quickly between large and small values when entering or leaving saturation. With our CT assumed to be non-saturated, we substitute our secondary current into Equation 1 and simplify: (
)
We will now integrate this equation with respect to time to find an equation for
over time in terms of the primary current
constant of integration
. Note: we will evaluate the
later on.
Equation 2
∫(
20
)
To perform this integral we must know
, which will be the fault current on
the network. Typical fault waveforms have a sinusoidal component and an exponentially decaying “DC offset” component. (The fault waveforms for various fault types are derived in the Appendix, section 10.1). In general, fault waveforms have the following form: Equation 3
√
[
(
)
]
Where:
is between -1 and 1. It describes the level of DC offset. 0 means no offset, 1 means full positive offset, and -1 is full negative offset. Full offset is the worst case because it makes the CT more likely to saturate. The value of D depends on the time on the 50Hz cycle that the fault occurs so this value is essentially random.
is the RMS, steady-state fault current. Ausgrid can easily obtain this value from a fault study.
is the X on R ratio of the primary system up to the point of the fault. This can easily be found with a fault study.
Now that we have our fault waveform
we can evaluate the integral in
Equation 2: √
[
(
)
(
)
(
)]
We can combine the cos and sin terms into a single sinusoidal term: Equation 4
√
[√
(
(
))
(
)]
21
We now calculate the constant of integration such that at
. We choose this value to be
(when the fault occurs), the flux density in the core ( ) equals the 1:
remanent flux
( )
By substituting
, and
into Equation 4, we can solve for : √
[
√
]
Having found , we can now substitute it into Equation 4 to obtain the final solution of the flux density waveform for a given fault, CT, remanence and burden. ( ) is the inverse cotangent function.)
(Note:
Equation 5
√
√
[
(
(
(
)
))
√
]
As an example of what Equation 5 looks like, Figure 3 shows a fault with some typical values from Ausgrid’s network, with the corresponding flux density.
1
Remanent flux can take a range of values, typically from -1.4 to +1.4 T. It depends on the history of the core before this fault occurs. Demagnetising the core sets this value to 0 T.
22
Figure 3: A 3 phase fault with worst case DC offset is shown at top (calculated with Equation 3). Below is the corresponding flux density in the CT core if there was no remanence (calculated with Equation 5). The saturation level of 1.7T is shown in dotted red. The flux density settles down to a value less than 1.7 T so the CT does not saturate.
3.3.2. Maximum Flux Density for a Given Fault Equation 5 is very useful for investigation of CT saturation when faults occur. For example, we will now use it to derive the maximum flux density
that the
core will experience for a given fault. If this is greater than the saturation flux density (around 1.7 T), then the core will saturate for the given fault. The maximum flux density in the core will occur when the exponential term in Equation 5 decays to zero, and the sinusoidal term is one. The maximum flux density
will be: Equation 6
√
[√
√
]
23
Note: in Equation 6 we have assumed
. If
, the equation would
need to be modified so the sinusoidal term equals -1. The resulting value would be the minimum (i.e. most negative) value of . In section 3.3.1 I mentioned that we would need to check the assumption that (i.e. that the core is not saturated). We will do this now. If Equation 6 is less than the saturation flux density (
) then
from and the
core is not saturated. Otherwise, the core will enter saturation and the flux density waveform found in Equation 5 is not valid. So using Equation 6 we can quickly and easily check whether a particular CT will saturate for a given fault. Note: So far we have assumed that
is a typical value of about 1.7 T for
Grain Oriented Silicon Steel. This is a rough value. In section 3.3.3 we will specify more rigorously based on the desired tolerance for CT error. For a more accurate prediction of saturation, this more rigorous value of the
should be compared with
of Equation 6. 3.3.3. Time Until Saturation In the previous section we discovered whether a CT will have any saturation
for a given fault. We now estimate the time this will occur. CTs do not saturate at the instant the fault begins. There is a time delay whilst the flux density ramps up (since B is proportional to the integral of the fault current, see Equation 2). It is not until the flux density ramps up to
that the CT will saturate. If this ramping time is long
enough, the fault may be cleared before the CT saturates. This is one way to avoid CT saturation. The following derivation will arrive at an equation which allows one to easily predict the time for a CT to saturate for a given fault. The bottom plot of Figure 3 showed the flux density oscillating and rising. We will create a curve which follows the peaks of these oscillations. When this curve rises to
, this will be a good estimate of the time of saturation. We begin by
modifying Equation 5 so that it follows the peaks of the oscillations. The peaks occur when the sinusoidal term is 1 (We will again assume term should be -1). The modified equation is:
24
. If
the sinusoidal
Equation 7
√
√
(
) √
[
]
A graph of this ‘Flux Density Peaks’ function is shown in Figure 4. We will now substitute
, and solve for
to find the time that the core reaches
saturation. Equation 8
(
⁄
)
√ √
(√ (
[
√
)
) ]
Equation 8 is very useful for specifying the size of CTs. Ausgrid has a policy that 11kV bus bar faults must be cleared in less than 1 second. In some circumstances, this could prevent saturation for a bus bar fault because the CT does not have time to do so. Also note that Equation 8 is conservative. It shows the earliest possible time that the CT could saturate. Most of the time it will be about 10 ms later because it is unlikely to occur precisely at a peak in the B waveform.
25
Figure 4: The curve ‘Flux Density Peaks’ is described by Equation 7, and the Flux Density by Equation 5. The Flux Density Peaks can be solved for time to find the instant when it crosses B sat. Note that the portion of the flux density curve which rises above Bsat is not realistic, but the time to saturation is correct because everything before that time is correct.
3.3.4. The “1 + X/R” Rule of Thumb Equation 7 embodies the conventional wisdom that the CT size must scale with (1+X/R). We can show this if we assume the secondary inductance is negligible, the DC offset is maximum (i.e.
), and that there is no remanence (
).
Under these assumptions, Equation 7 reduces to: √
[
]
Now if we assume the fault persists for a very long time so that
, the
equation now describes the maximum B that will appear in the core for this fault: Equation 9
√
The
[
]
term in Equation 9 shows that the higher the
ratio, the higher the
flux density in the core, and the bigger the core will need to be ( must increase) to accommodate the fault without saturating (so that
). This is the
conventional wisdom of CT sizing. However, the above analysis shows that one of the assumptions is not always true. It is not always the case that the fault will persist 26
for a long time. If we know the fault will be cleared in a certain time frame, then the rule may not be valid. Ausgrid ensures that 11kV bus bar faults are cleared in less than 1 second. If the exponential term in Equation 7 does not fall to zero after 1 second, then the
rule of thumb may result in an oversized CT. It’s clearly
worth knowing what the maximum fault duration will be, to see if this allows smaller CTs to be used. It is commonly known that reducing the secondary resistance reduce
will also
preventing saturation so this could be another avenue for preventing
saturation in smaller cores. This is perhaps one of the most practical ways to reduce the risk of saturation. We will investigate how secondary resistance affects saturation risk in section 6.3. Equation 9 suggests another way to prevent saturation. We can also reduce by increasing the number of turns on the CT (increasing
). Since
is
squared, a small increase in the number of turns will have a relatively large reduction in
. If Ausgrid wishes to make the core smaller, then increasing the number of
CT turns could be an effective way to prevent saturation. Of course, if
is
increased the CT currents will be smaller. It’s possible that modern relays are better able to accurately measure small currents than older electromechanical relays, so this could be an effective solution to the CT sizing issue. If Ausgrid wishes to reduce the size of its CTs, the above analysis may help to check which assumptions are correct, and arrive at a safe and reliable specification.
27
3.4. The Static Preisach Hybrid Model In this section I describe my CT model which I have named the “Static Preisach Hybrid” model (SPH model). My novel contribution is that I’ve developed it to be a hybrid of 2 separate modules. The first module is a discretised Preisach model to create First Order Reversal Curves (FORCs). The second module takes the current FORC and solves a set of ordinary differential equations to arrive at the output. Firstly I’ll give an overview of the model and its benefits, and then I’ll describe my Matlab implementation. 3.4.1. Overview of the Static Preisach Hybrid Model Although Preisach theory has been used since 1935, it is still very difficult to implement in a dynamic simulation. My approach in the SPH model is to discretise the Preisach plane, and use it to generate first order reversal curves (FORC) which are used as inputs to a continuous ordinary differential equation (ODE) solver. A FORC is simply a trajectory on the B-H curve. It is a single-valued function of B in terms of H. It goes from the current B-H location towards saturation in whichever direction H is heading. I find the current FORC trajectory by numerically evaluating Equation 10. The shape of a FORC depends on the initial B and H values, and the core material properties. My SPH model follows the FORC when moving along the B-H curve. Whenever H changes direction (e.g. from increasing to decreasing or vice versa), the ODE solver is stopped, and a new FORC is generated which starts from that turning point. The ODE solver then resumes with the new FORC which will give it a new trajectory on the B-H plane. I have named this technique the “Static Preisach Hybrid” model because it is a hybrid of 2 separate modules: 1) The Discretised Preisach Module is only used to generate FORCs when H changes direction. It is not involved in the dynamic calculations of secondary current. 2) The Equation Solver Module is an ODE solver which calculates the secondary current using the B-H trajectory of the FORC provided by the Discretised Preisach Module.
28
My hybrid approach has several key benefits:
The ODE solver is kept simple because it does not have to deal with the complex history-dependent behaviour of the B-H curve. It only deals with the FORCs which are smooth, single valued functions with no history dependence (though they are non-linear).
Because the ODE solver is simple, we can use built-in Matlab functions which provide very good results for this type of problem.
The solution is modular, so that the FORCs could easily be generated by a different process other than the Discretised Preisach Module. For example, lab measurements could measure the FORCs directly from a core, and these could be used by the ODE solver.
Once FORCs are generated by the Discretised Preisach Module, they can be smoothed to ensure accuracy and reliability of the ODE solver.
Since the SPH model is a hybrid of 2 modules, I’ll describe them separately. Firstly I’ll describe the Discretised Preisach Module, and then I’ll describe the Equation Solver Module.
3.4.2. The Discretised Preisach Module The discretised Preisach module is based on the Preisach theory of hysteresis (Preisach, 1935). This model assumes that frequency dependent effects are negligible. The Preisach theory models the magnetic domains as independent “hysterons” which are “particles of hysteresis”. Each hysteron has the B-H characteristic of an ideal relay, shown in Figure 5.
29
B +Bsat
α
β
H
-Bsat
Figure 5: The B-H characteristic of each “hysteron” according to the Preisach theory of hysteresis. Each hysteron is like a magnetic domain in the core which is modelled as an ideal relay. The flux density B can be either Bsat or –Bsat. The H values at which the relay changes state are α when decreasing and β when increasing. β is always greater than or equal to α.
Each hysteron has a different pair of switching values some statistical distribution. Each combination of factor. If there are many hysterons with particular
and and
and , according to also has a weighting
values, then the weighting
will be high. For rare values of
and
, the weight will be low. This forms a 2
dimensional weight function (
). Each hysteron also has a state: +1 if it’s in the
positive state, and -1 otherwise. This forms a 2 dimensional state diagram to describe the total state of all hysterons (
). The flux density
as a function of
is found
by the double integral over the weighted state (Naidu, 1990): Equation 10
( )
∫ ∫ (
The constant 2D function
(
)
(
)
) contains all the parameters needed to
characterise the core. The 2D function (
) specifies the current state of the core
and varies with time (Naidu, 1990). In my implementation I discretise the 2D functions to perform the integration. I’ve found that about 45000 hysterons produces good results. The weight function ( horizontal axes are
30
) of the core is shown in Figure 6 where the
and , and the vertical axis is the relative weight at each point.
Figure 6: Preisach weights for a Grain Oriented Silicon Steel core. This diagram completely specifies a particular ferromagnetic core. The cube root has been taken of the vertical axis to emphasise smaller weights. It can be seen that there is a large spike near α = 0, and β = 0. This tells us that the core has a small coercivity (i.e. a narrow B-H curve). It’s comparatively flat everywhere else, indicating that the flux density does not change very much when in saturation.
Figure 7 shows a series of images generated by my Static Preisach Hybrid model showing the B-H curve and the hysteron states evolving over time.
31
Figure 7: These figures illustrate how my SPH model produces hysteresis and saturation phenomena. The left images show B-H curves and the right images are corresponding Preisach state diagrams. On the left diagrams, the blue curve is measured experimentally and the red curve is the output of my model where the cross represents the current point in time. In the right images, blue areas represent hysterons in the negative state, red represents the positive state. Each row represents a different point in time. The core begins in the demagnetised state (top row). In the middle row, the current begins increasing and the flux density rises because some hysterons shift to the positive state. In the bottom row, the core has gone into positive saturation and is now decreasing so some hysterons are shifting to the negative (blue) state.
32
3.4.3. The Equation Solver Module This module uses the FORC generated by the Discretised Preisach Module as an input to a set of ordinary differential equations which are solved. The system treats the primary current
as an input signal, and the system of equations has 3
state variables:
The flux density
The secondary current
The magnetic field intensity
We will now derive the differential equations which relate these state variables to the inputs and to each other. Finding the flux density is made simple because the Discretised Preisach Module has generated a FORC which is used to find B in terms of H. The equation solver does not need to consider the multivalued or history-dependent nature of the core. It simply uses the generated FORC which is a single valued, predetermined function (until H changes direction and a new FORC is generated). Given this understanding, the equation solver uses this method to find the flux density: Equation 11
( ) Where the function
is simply the current generated FORC.
We now look at the second state variable,
. By rearranging Equation 1 we
find an implicit differential equation for the secondary current: Equation 12
Lastly, we look at the final state variable, the magnetic field intensity From the definition of
.
:
Equation 13
33
The above 3 equations are interrelated and must be solved simultaneously. This is done using Matlab’s implicit ordinary differential equation solver called ode15i( ). Because of the implicit input equations, care must be taken that the initial values are consistent. This is ensured using Matlab’s helper function decic( ). The Equation Solver Module solves for these 3 state variables over time until changes direction. At this point, the solver is stopped because whenever this occurs, the B-H trajectory should follow a different path. A new FORC is generated by the Discretised Preisach Module and the Equation Solver continues on the new path by following the new FORC.
3.4.4. Parameter Identification My SPH model has a very large number of parameters (thousands of values) because they describe the 2D function of the hysteron weights
(
). For this
reason, we must find the parameters from experimental measurements of a real ferromagnetic core. Here are the steps for parameter identification (see Matlab code Appendix 10.5, and spread sheet in Appendix 10.3): 1) Measure B-H curves of the desired CT at various input currents from non-saturated to deeply saturated. No burden is used (open circuit). Around 30 curves will produce good results. Low frequency currents should be used (< 1 Hz) to avoid frequency dependent effects. 2) Extract the “top side” of each B-H curve. This is just the portion where Figure 8).
34
is decreasing. These are called the “extrado” curves (see
Figure 8: Here are 30 B-H extrado curves shown side by side. The non-saturated curves are at the front and the highly saturated curves are at the back. This graph is of one of the data structures in my Matlab code for the SPH model.
3) Step 3 uses the extrado data to compute the weighting for each hysteron. It follows the “cell method” of Bernard, et al. (2000). If B-H curves are taken, we divide the Preisach plane into
(
)
cells. These cells represent each change in B for a change in H, and the combination of hysteron regions which contribute to each change. This produces
(
) equations which can be solved as a matrix
equation to produce the hysteron weights of each cell. An example is shown in Figure 9. See Bernard, et al. (2000) and my Matlab code for more details (note that Bernard, et al. swaps the definitions of
and
from those presented here).
35
Figure 9: Cell Weights found by solving the n(n+1) equations described in step 3). The cube root of the weights has been shown to emphasise small weights. These cell values will be interpolated to find the final weights of each hysteron.
4) The hysteron weights of each cell (found in step 3) are a low resolution representation of the hysteron weights, so the values are interpolated to obtain a smooth, high resolution representation of the weights function (
). (See Figure 6 for an example).
So the final weights function
(
) has been found which fully
characterises the core. It can now be used in Equation 10 to find the flux density each point in time, given a magnetic field intensity
at
. In my implementation, this is
used to generate First Order Reversal Curves (FORCs) on the B-H plane. 3.4.5. Features of the Static Preisach Hybrid Model Preisach theory is quite complicated but this is necessary to produce many of the complex phenomena seen in real ferromagnetic cores. In the following sections, I will briefly mention some complex phenomena and show that my SPH model is able
36
to reproduce them. Note that simple models like the Ideal Saturation model are unable to produce any of the following phenomena. 3.4.5.1.
Initial B-H Trajectory
The first complex phenomenon that my SPH model achieves is the initial BH trajectory. In Figure 10 we see that a demagnetised core has an initial trajectory in the middle which is different from the steady state behaviour. Preisach theory produces this behaviour naturally. Also note that the red curve (my SPH model) accurately follows the blue curve (experimental result), so this is good verification of my model.
Figure 10: The SPH model in red shows an initially demagnetised core undergoing sinusoidal excitation with an open circuit secondary. The curve starts at B=0, H=0 and moves up the initial trajectory. After reaching saturation it then moves into the steady state B-H curve. The blue curve was experimentally determined in the lab at low frequency (< 1Hz). There is close agreement between my SPH model and the experimental results.
The initial B-H trajectory is a property of real cores which is very important for transient behaviour. Accurate transients are important for modelling transformer differential protection schemes. 3.4.5.2.
Minor Loops
Ferromagnetic cores also exhibit the phenomenon of minor loops. If H changes direction at a B level near zero, it will travel back along a different path to
37
the one it took to arrive at that point. If it changes direction again, it will take yet another path, but once H reaches the value of the first direction change, the B-H curve should close the minor loop and continue on from where it left off. This is known as the “wiping out” property as described by Mayergoyz (1986). My implementation of Preisach theory neatly handles minor loops, as shown in Figure 11.
Figure 11: Minor loop behaviour is reproduced by my SPH model. This test simulates an open circuit secondary. The input current is 2 sinusoids of different frequencies to produce the extra direction changes necessary to form minor loops.
Ausgrid’s previous CT models did not have robust handling of minor loops, so this represents a discernible improvement in modelling power. Preisach theory naturally incorporates minor loops into its design because the hysterons change state in the same way that real magnetic domains do. 3.4.5.3.
First Order Reversal Curves (FORCs)
One of the challenges of modelling ferromagnetic cores is the production of first order reversal curves (FORCs). The SPH model produces a custom generated FORC whenever H changes direction. Figure 12 shows a B-H curve where a different FORC is generated for each cycle. Each FORC is subtly different, depending on the state of all the hysterons in the core at each point in time.
38
Figure 12: The input waveform is a fault with a large DC offset. As the DC offset decays with time, each cycle takes a different path on the B-H curve. For each unique path it takes, a custom generated FORC is produced. Each path has a slightly different shape which is dictated by the experimentally determined parameters.
The above features of the SPH model make it an accurate solution for CT saturation simulations. In section 0 I will use this model to simulate various faults on Ausgrid’s power network. The graphs in section 5.2 will show the output waveforms of my SPH model.
3.5. Static Tanh Model I also developed a second, simpler model called the Static Tanh Model. This was designed to be a compromise between the Ideal Saturation and the SPH models. It is more accurate than the Ideal Saturation model, but is much easier to use than the SPH. The Static Tanh model uses the same hybrid structure as the SPH model but instead of generating FORCs using Preisach theory, this model simply has one B-H trajectory which is used at all times. This model assumes hysteresis is negligible and it does not produce minor loops. It is like the Ideal Saturation model except that it has a smooth entry into saturation, not an instantaneous entry. My main reason for developing this model was to see whether hysteresis plays an important role in the behaviour of the CT. If it does not, then the Static Tanh and SPH models should show very similar results (comparison presented in section 4).
39
The B-H curve of the Static Tanh model is single valued function given by: Equation 14
( )
(
)
The tanh( ) function was chosen because it is easy to differentiate: Equation 15
(
)
A graph of the B-H characteristic is shown in Figure 13.
Figure 13: The B-H curve of the Static Tanh model.
Being much simpler than the SPH model, it’s faster to simulate and much easier to use. The model has only 2 parameters:
40
is the saturation flux density controls the slope of the B-H curve in the centre region
My implementation of the Static Tanh model uses Matlab’s ode15i function to solve the same differential equations as the SPH model. The only difference is the definition of ( ). Here are the 3 state variable equations for the Static Tanh Model: ( )
(
)
The only difference from the SPH model is that the B(H) function for the Static Tanh model is not a custom generated FORC, but is simply given by the function in Equation 14. Accordingly, when solving for the secondary current, we need to find
, and this is given by Equation 15. Apart from these differences in the
model, my implementation of the Static Tanh model is exactly the same as the Equation Solver Module for the SPH model (see section 3.4.3).
In the above sections I have presented 3 models; the Ideal Saturation, the Static Preisach Hybrid, and the Static Tanh. In the next section these models will be compared and verified with laboratory results.
41
4. Model Verification To verify the models introduced in section 3, I will first present theoretical derivations of CT error, and then compare the model results to lab experiments and also to each other. The experimental verification was performed by two methods. The first is with no burden (open circuit), and the second is with a resistive burden. Ultimately, the final verification of my models will require experiments at Ausgrid’s high voltage testing facility to observe what happens with a real network fault. This is expected to be performed in 2013 or 2014, and this thesis forms part of the preparation for these high voltage tests.
4.1. CT Error and the Sudden Saturation Hypothesis As part of the model verification, I now present my theoretical findings about CT error. I have performed a detailed theoretical analysis of CT error in saturation and non-saturation and the transition between them. I begin by introducing my hypothesis about CT saturation:
The Sudden Saturation Hypothesis: For small values of the product 𝑁 𝐼 (< 10 Amp-turns), the Ideal Saturation model is very inaccurate for simulating CT saturation because saturation occurs gradually. However, for large values of 𝑁 𝐼 (~5000 Amp-turns), the Ideal Saturation model is highly accurate because saturation occurs suddenly (over a few milliseconds).
Note that in a typical lab, only small values of
can be obtained, whereas
a real fault on the network produces a much larger
. The Sudden Saturation
Hypothesis implies that the Ideal Saturation model will be very accurate for simulating CT saturation for fault currents, however it will not be possible to verify this in a typical lab because of the low currents which can be obtained. Only high voltage testing will be able to verify it. In the following sections (4.1.1 and 4.1.2) I will provide a theoretical proof of my above hypothesis, and the experimental verification in sections 4.2 and 4.3 will provide further evidence that it is true.
42
4.1.1. CT Error When Not Saturated The Ideal Saturation model makes the assumption that when not saturated, . In reality,
continuously varies, but increases greatly when in saturation.
We must decide which values of
are small enough to be considered
“approximately zero”, in order to verify the Ideal Saturation model. The following derivation will allow us to quantify the accuracy of the Ideal Saturation model. Then we will see that if the Ideal Saturation model is accurate in a particular situation, then other more complex models will be even more accurate. So by verifying the Ideal Saturation model to a certain level of accuracy, we are also verifying the other models to be better than that level of accuracy. In order to properly quantify the values of
which can be considered
“approximately zero” (i.e. no CT error), we now define the instantaneous CT error . This is defined in terms of the difference between primary current and the secondary current (scaled by the turns ratio), as a proportion of the primary current. Equation 16
( ) ( )
( ) ( )
An error value of 0 means the secondary current precisely follows the primary (times the turns ratio). Large values of
mean the CT has some degree of
saturation and the secondary current is less than it should be, ideally. Note that this error is an instantaneous function of time and changes throughout the cycle. Solving Equation 16 for
( ):
Equation 17
( )
( )(
( ))
Now from the definition of the magnetic field intensity
under our general
assumptions (listed in section 3.2): Equation 18
( )
( )
( )
43
( ) and simplifying:
Substituting our equation for Equation 19
( )
( )
From Equation 19, it can be seen that CT error
is proportional to the instantaneous
for a given primary current. We can also observe that
is analogous to
the concept of magnetising current. If the instantaneous magnetising current is referred to the secondary of the CT, it is defined as: ( )
( )
( )
Equation 20
( )
( )
( )
Combining Equation 17, Equation 19 and Equation 20 we find an expression for the magnetising current in terms of H: Equation 21
( )
( )
So it can be seen from the above equation that magnetising current is proportional to
. This shows that
,
and
are all ways of measuring the
mismatch between primary and secondary current due to CT error. For this thesis I have chosen not to use magnetising current because it’s not a physical current. Instead, I have chosen to use H because it is a physical property of the system and does not imply the existence of an ideal CT. In any case, those familiar with the notation of magnetising current can use Equation 21 to convert We will now find a maximum value of
to
for which the error
. is suitably
small. When not in saturation, CT error is largest for large primary currents so we take the worst case where the primary current is at a peak, values of Equation 19:
44
. Taking maximum
Equation 22
Equation 22 is very useful for planning lab work on CTs as it helps to predict what sort of CT error will be encountered. More importantly, it will also help us greatly in modelling CT saturation. It means that for suitably large fault currents, it hardly matters which model we use. It doesn’t matter whether we accurately model the shape of the B-H curve or take into account temperature or frequency dependent effects. All that matters is that the portions of the curve which fit within will have a CT error less than
to
. This means that for applications with
large primary currents (like network faults) we can avoid much of the complications of modelling the core. Let’s take a practical example of how Equation 13 can be used. Let’s choose a suitably small error
= 1%. We choose this value so that if the error is less than
1% at all times, then the CT is considered to be working properly (i.e. not saturated). If we take a typical fault current of 4961 ARMS, and assume the CT’s core length to be 0.3778m, then we can find the maximum value of than or equal to
which will keep the error less
. In terms of the Ideal Saturation model, we can think of this as
the maximum value of
which can be considered “approximately zero” (as required
by the assumptions of the Ideal Saturation model). Inserting the above values into Equation 13: Equation 23
√
This is a very important result. It means that if the actual measured B-H curve of any core fits within the range -186 to 186 A/m, then we can model this CT with the ideal saturation model (or any more accurate model) and the result will have an accuracy better than 1% at all times. In fact the situation is better than this. The error will be at most 1% for a brief instant at the current peak, but will be lower for the rest of the cycle. In Figure 14 we represent the range
to
by the grey box. The
green and blue curves show the B-H characteristics for GOSS at 50Hz and
