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Transformers for Frequency Response Analysis

Article in IEEE Transactions on Power Delivery April 2009

DOI: 10.1109/TPWRD.2008.2007028

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3 authors, including:

Almas Shintemirov

W.H. Tang

Nazarbayev University

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730

Transformers for Frequency Response Analysis

A. Shintemirov, W. H. Tang, Member, IEEE, and Q. H. Wu, Senior Member, IEEE

transformer winding for frequency response analysis (FRA) based

on traveling wave and multiconductor transmission line (MTL)

theories. Each disc of a winding is described by traveling wave

equations, which are connected to each other in a form of MTL

matrix model. This significantly reduces the order of the model

with respect to previously established MTL models of transformer

winding. The model is applied to frequency response simulation of

two single-phase transformers. The simulations are compared with

the experimental data and calculated results using lumped parameter and MTL models reported in other publications. It is shown

that the model can be used for FRA result interpretation in an extended range of frequencies up to several mega Hertz and resonance analysis under very fast transient overvoltages (VFTOs).

Index TermsFrequency response analysis, multiconductor

transmission lines, transfer functions, transformer windings.

Magnetic permeability of free space.

Insulation thickness between adjacent turns.

Insulation thickness between adjacent discs.

HV winding minimum radius.

LV winding maximum radius (core radius if LV

winding is absent).

Equivalent circular radius of disc conductor.

Conductivity of disc conductor.

,

Winding Parameters Per Unit Conductor Length:

Average interturn capacitance of the

NOMENCLATURE

,

,

,

,

,

disc.

disc.

and

discs.

discs.

Time and space dependant voltage and current.

and

discs.

Impedance of the

disc.

Angular frequency.

Admittance of the

disc.

Imaginary unit.

discs.

disc.

disc.

and

disc.

disc.

disc.

and

Manuscript received December 12, 2007; revised July 14, 2008. First published March 04, 2009; current version published March 25, 2009. This work

was supported by the Center for International Programs under a Kazakhstan

Presidential Bolashak Scholarship and by a Grant from JSC Science Fund

for research at the University of Liverpool, Liverpool, U.K. Paper no. TPWRD00805-2007.

The authors are with the Department of Electrical Engineering and

Electronics, University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail:

qhwu@liv.ac.uk).

Digital Object Identifier 10.1109/TPWRD.2008.2007028

I. INTRODUCTION

condition monitoring, only FRA is most suitable for reliable winding distortion and deformation assessment. It is based

upon a fact that winding deviation or geometrical deformation

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SHINTEMIROV et al.: A HYBRID WINDING MODEL OF DISC-TYPE POWER TRANSFORMERS FOR FRA

winding, which are related to internal capacitances and inductances. Sufficiently different values of combinations of these internal parameters reply essentially altered frequency responses,

which can be observed by applying variable frequency signals

to the reference point of winding and measuring output response

signals [1], [2].

Despite of extensive FRA practice, transformer winding condition assessment is usually conducted by trained experts. The

obtained FRA traces are compared with the references taken

from the same winding during previous tests or from the corresponding winding of a sister transformer, or from other phases

of the same transformer. The shifts in resonant frequencies and

magnitude of FRA traces may indicate a potential winding deformation.

A range of research activities have been undertaken to utilize

FRA in the development of suitable lumped parameter mathematical models of transformer windings, where each section

of a winding represents a group of turns or discs. In [3][6]

analytical expressions were used to estimate parameters of an

equivalent lumped parameter models based on the geometry of

winding. Well-known finite-element method was applied in [7]

for more precise calculation of winding parameters for an equivalent circuit model. These work showed high degree of accuracy

compared with experiment measurements. However, the precise

simulation of high-frequency behavior of winding above 1 MHz

can be achieved only with small sectioning of the models which

leads to its essential complexity [8].

On the other hand, the theories of distributed parameter systems and traveling wave in transmission lines offer an appropriate mathematical foundation to model, in a compact mathematical form, the propagation processes of a voltage signal

injected into a transformer winding [9]. This can be used for

interpretation of FRA measurements in an extended range of

frequencies up to 10 MHz. The advantages of such high frequency measurements for FRA were discussed in [10], [11],

which show that FRA measurements are more sensitive to minor

winding movements in the high frequency range.

In practice, distributed parameter modeling of transformer

windings is based upon the application of MTL theory [12].

The theory has been employed to machine winding analysis

[13] and, subsequently, transformer winding mathematical description [14][19] used for partial discharge location purposes

and VFTOs studies. However, these models do not take into account constructional features of transformer windings. Generally, each turn of a winding is represented as a single transmission line [14][16], [19], which makes MTL models complex to

operate in case of analysis of a winding with a large number of

turns.

In this paper a novel approach to model disc-type transformer

winding frequency behavior at FRA testing is presented. Firstly,

injected source signal propagation along a single winding disc

is considered using the analytical approach and results of the

transformer winding analysis obtained by Rdenberg [20].

Then, the derived traveling wave equations are applied for all

discs of the winding and connected to each other in a matrix

form of MTL model. This significantly reduces the number of

equations in the model with respect to the previously estab-

731

application of the model to simulate frequency responses of

an input impedance and winding resonances under VFTOs

is presented. Simulation comparisons are given with the previously reported experimental and calculated results being

obtained using lumped parameter and MTL models to verify

the proposed model.

II. EQUIVALENT DISTRIBUTED PARAMETER CIRCUIT OF

TRANSFORMER WINDING

In this paper a continuous disc-type winding is analyzed,

since it represents the essential part of HV transformer windings. Each disc of the winding consists of a number of turns

wounded in a radial direction. In practice, continuous disc-type

windings are wound to provide the same direction of a flowing

current along all discs of a winding. In some transformers, the

reinforcement of an insulation at the ends of a winding results

to a reduced number of turns for a several discs closest to the

winding ends. Thus, the conductor length of the extreme discs

can differ from those of the middle discs. However, in this

study for the sake of simplicity an equal conductor length of all

winding discs is accepted.

In order to obtain a general equivalent circuit of the winding,

each disc is considered as an equivalent distributed parameter

circuit with turns being stretched out in radial direction of

the winding, similar to an equivalent circuit of transformer

winding proposed with respect to the axial direction [21].

Entering interdisc connections and taking into account above

simplifications, the equivalent distributed parameter circuit of

disc-type transformer winding at FRA testing can be composed

as in Fig. 1.

As clear from the equivalent circuit the disc numeration is

marked from the top till down and the direction of the space

coordinate is denoted from the left terminal of each disc towards the right end. Each disc has its own parameters signified

by a corresponding disc number and the discs are connected to

,

each other by a curve line. Thus, notations of

and

,

denote the voltages and currents at an input

for the

terminal and an output end of the winding, i.e., at

for the

disc. Note that the term ground in

first disc,

the definitions of the above capacitance and conductance

means that the parameters can be considered as being between

the winding and the core or the tank, or as interwinding in case

of LV winding presence.

III. SIGNAL PROPAGATION ALONG A WINDING DISC

The fundamentals of the theory of transients in coils were formulated by K.W. Wagner. He considered a coil as a transmission line with additional interturn capacitance acting between

the turns [22]. The similar theoretical results were obtained by

Rdenberg [20] wherein he elaborated and extended the traveling wave theory to lossless transformer winding analysis. His

results were adopted in order to include losses, associated with

insulation and conductor resistance, and subsequently develop

mathematical descriptions of signal propagation along a single

uniform winding disc.

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732

one is

turn to the

(3)

and to the

turn is

(4)

The sum of (3) and (4) gives the total interturn current per

as follows:

disc conductor length

(5)

where

is the voltage difference between adjacent turns;

denotes the difference of the

s between successive turns.

Since only interturn relations are considered, variable turn

,

length of a disc conductor is accepted as a unit length:

and the second difference of voltage can be rewritten in differential form [20]

(6)

Substituting (6) into the sum of (2) and (5), the space derivative

turn is obtained as

of the total currents decrease in the

follows:

Fig. 1. Equivalent circuit of a disc-type transformer winding.

The analysis of signal propagation is based upon the Telegraphers equations for lossy transmission lines [21], which are

expressed using the notations in Fig. 1 as follows:

(7)

or

(8)

(1)

and

denote combined ground and interdisc cawhere

pacitances and conductances, respectively, for simplicity of the

analysis.

Upon these fundamental dependencies, in the following subsections the derivation of current and voltage propagation equations is presented.

If the

turn, having self-inductance per unit length, were

separated from the rest of the disc, the voltage induced by the

current in it would be expressed as

(9)

per unit length, induce a voltage in the

turn [20]

The charging current

of the

turn, flowing to ground due to correlength

sponding capacitance and insulation conductance, depends on

of the

turn and is described as follows:

the voltage

(2)

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(10)

SHINTEMIROV et al.: A HYBRID WINDING MODEL OF DISC-TYPE POWER TRANSFORMERS FOR FRA

are produced in the

turn by

voltages proportional to

each succeeding turn and, hence, it can be collected together

using (9) and (10) which lead to:

(11)

where as defined above is the self-inductance of a disc being

given per unit length and derived by total inductive effects between all turns or simply is the self-inductance of the entire disc

divided by its total length [20].

Considering the interturn relationships, the second difference

of current has a form of second-order space derivative:

733

In order to develop a mathematical model of a disc-type transformer winding, the above obtained (15) and (16) are employed

to describe signal propagation along each disc of the winding.

Considering the equivalent circuit in Fig. 1, the equations include interdisc capacitances and conductances, which are subsequently combined into a matrix form similar to that of a MTL

model.

The mathematical equations describing the first disc of a

winding are as follows:

(17)

(18)

(12)

where

which relates to an influence of the immediately adjacent turns,

because the induced effects of other turns in a disc have already

been included in (11).

turn is equal to the decrease

The total voltage in the

which, using (10) and (11), becomes

(13)

(19)

disc, where

are similar

The expressions for the

to (17) and (18) with minor modifications.

The combination of the equations corresponding to all the

discs in a matrix form gives the following matrix equation:

and the voltage space derivative along the disc with aid of (12),

transforms into

(20)

(14)

where

With the purpose of further processing, the last element of

(14) is to be neglected within practical accuracy, since the mutual inductance of adjacent turns is much less than self-inductance , which already includes mutual inductances itself.

..

.

..

..

..

..

.

(21)

In order to convert the above expressions (8) and (14) into

frequency domain, a Laplace transform is employed. Assuming

zero initial conditions

..

.

..

..

..

..

.

(22)

(23)

then (8) and the simplified equation (14) are transformed into

frequency domain as follows:

(15)

of each disc and

interdisc admittances per unit conductor length are defined as

follows:

for

(24)

(16)

for

and

are the Laplace transforms of the

where

and current

correspondingly.

voltage

Thus, the derived (15) and (16) describe the injected signal

propagation along a uniform winding disc in frequency domain.

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(25)

734

and

(26)

voltage signal. In order to derive the expressions for transfer

functions, (31) is rearranged in the form as follows:

as follows:

(33)

where [12]

(27)

or

(34)

(28)

where

(35)

(29)

[16], [19]:

Consequently, the following equations are derived:

..

.

..

.

(36)

(30)

The solution of matrix (28) for the voltages and currents at the

and

is obtained

ends of transformer winding when

in the form that follows [12]:

(31)

where is a

chain parameter matrix.

Since the admittance matrix Y is nonsymmetrical, i.e.,,

, where

denotes the transpose of Y, (30) are difficult to decouple in order to obtain an analytical solution using

the similarity transformations as in the case of multiconductor

transmission line equations [12].

On the other hand, since the turn length varies depending

on turn number in a winding disc, which affects the calculation

can be calculated in an iterative numerical

of Y, the matrix

form using a matrix exponential function [12] as follows:

where

is a

Finally, by inverting matrix

follows:

..

.

matrix.

, (36) can be rewritten as

..

.

(37)

is of

order [13].

where

In this research, input impedance response and transfer functions of voltages from various points of the winding are simulated and compared with the measurement results and simulations from other models. Input impedance of the winding can

be found assuming a grounded winding. Thus, the following terminal condition is applied to the model:

(38)

(32)

where is the turn length of the

turn and is the number

of turns in a disc.

it is possible

By calculating the chain parameter matrix

to derive expressions for the transformer winding transfer functions, which are presented in next section.

to the

gives the equation for input impedance of

input current

the winding, as follows:

(39)

V. TRANSFER FUNCTIONS OF TRANSFORMER WINDING FOR

FREQUENCY RESPONSE ANALYSIS

In practice, frequency responses of transformer winding are

obtained in a form of transfer functions of selected input or

winding, the transfer functions are calculated as the ratio of a

voltage

from the end terminal of an arbitrary disc to

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SHINTEMIROV et al.: A HYBRID WINDING MODEL OF DISC-TYPE POWER TRANSFORMERS FOR FRA

735

[19]

(40)

Using the above obtained (39) and (40), the

frequency responses of the winding are simulated with

. At the same time, calculations of the

the substitution

model parameters are performed using analytical formulae, as

illustrated in the following section.

VI. ESTIMATION OF MODEL ELECTRICAL PARAMETERS

Adequate estimation of the winding model parameters depends on winding geometrical dimensions and properties of

winding conductor and insulation. The presence of cylindrical

pressboard barriers and spacers between HV and LV windings,

spacers between winding discs along with insulation oil can

be accounted by an effective permittivity using the approach

described in [5], [7].

Winding capacitances are estimated in a similar manner to

those obtained for MTL models upon the simplified geometry of

a winding, known effective permittivity of a winding insulation

and adjusted per unit of disc conductor length [14]. Average

interturn and interdisc capacitances and , respectively, are

calculated using the expression for parallel plate capacitances

as follows:

(41)

An average ground capacitance of a disc-type HV winding

can be interpreted as a cylindrical capacitance between a

winding and a tank, considering an external turn of a winding

disc. On the contrary, for the internal turn of the disc, a ground

capacitance is calculated as the capacitance between HV and

LV windings or a core

(42)

calculated using the expressions derived by Wilcox [23], [24]

and adjusted to per unit of disc conductor length base.

Taking into account skin effect at high frequencies, the conductor resistance can be found using the following expressions

[25]:

(43)

The frequency-dependent conductances ,

,

are

of the correestimated using the effective loss tangents

sponding insulation mediums [14], [15], [18], [19].

VII. SIMULATION RESULTS AND COMPARISON

A. Model Verification With FRA

In order to verify the model, it is applied to simulate frequency responses for a single-phase experiment transformer

Fig. 2. Simulated with the proposed model and measured responses of open

circuit impedance of the experimental transformer.

winding and a 23-turn helical LV winding, with the geometrical

dimensions provided in [6]. The simulated input impedance

response with aid of (39), as well as experimentally measured

input impedance from the HV side of the transformer [5], are

shown in Fig. 2.

It can be observed that the simulated and measured responses

are close to each other as far as the general shape and resonant frequencies are concerned. However, the most clear deviations of 10.8%, 5.57%, and 5.58% in the first three resonant frequencies, respectively, may be due to approximations

of winding conductor and insulation properties accepted at the

stage of parameter estimation. In addition, the developed model

provides less damping at the resonant frequencies with respect

to the measured response. This could be due to the existence

of additional loss mechanisms [5], i.e., losses in the tank or in

the conducting cylinder used to represent equipotential surface

of the core, that are not considered in the proposed model, or to

approximations in the expressions used for the model parameter

calculation.

On the other hand, considering the input impedance response

of the same experimental transformer, presented in [5] and calculated using a lumped parameter model, it is clear that the responses from both the models are almost identical to each other

in terms of resonant frequencies and magnitudes. Although the

developed model does not provide higher accuracy in comparison with known lumped parameter models in the frequency

range up to 1 MHz, the developed model is capable of high frequency simulation with frequencies above 1 MHz, as discussed

in the following subsection.

B. Winding Resonance Simulation

In practice, a FRA result interpretation includes visual crosscomparison of measured frequency responses aiming to notice

newly appeared suspicious deviations of the investigated trace

comparing with various etalon responses. The appearance of

clear shifts in resonance frequencies or new resonant points

736

Fig. 3. Amplitudes of transfer function between the terminal of turn 20 and the input [(a) and (b) reprinted from [19]].

Fig. 4. Amplitudes of transfer function between the terminal of turn 40 and the input [(a) and (b) reprinted from [19]].

may characterize faulty conditions of windings. This comparative approach was extended by simulating frequency responses

with lumped parameter models to analyze the effect of different

winding faulty modes on deviation of resonance frequencies

in comparison with the reference simulations corresponding to

normal state of a winding [3][7]. However, the considered frequency range was limited by the reasons stated in Section I regarding the models applied. Recent work [10] showed the potential for FRA result interpretation in an extended range of frequencies up to 10 MHz, which indicates the importance of continuing the study using suitable transformer winding models.

Therefore, it is necessary to investigate the proposed model correctness in a high frequency range, and a large attention should

be paid, at first, to the analysis of resonance frequencies as the

main sources of information about the object and secondly the

ability of a model to reflect the general shape of a winding response.

Unfortunately, the reference FRA data in the frequency range

of interest were not available for the analysis. However, considering that the derivation of the transfer function expressions,

presented in Section V, is similar to those used for VFTOs

for the comparative study using data from resonance analysis

under VFTOs of an experimental transformer winding. The

winding is composed of 18 discs with 10 turns per disc and its

basic parameters are given in [19].

One way to analyze the resonance phenomena under very fast

transients overvoltages is to compare amplitude-frequency responses of winding transfer functions measured from different

points in the winding. Thus, potentially vulnerable locations in

a winding, where resonance could occur, are determined [19].

Figs. 35 illustrate measured (a) and calculated with the MTL

model frequency responses (b) of transfer functions from turns

20, 40, and 60 of the experimental transformer winding respectively, having been published in [19]. For comparison purposes,

frequency responses of the transfer functions taken from the end

terminals of the discs 2, 4 and 6, that correspond to turns 20, 40

and 60 of the winding respectively, are calculated using (40) and

arranged in the same figures as (c). The resonant frequencies of

the studied winding discs are listed in Table I.

As seen from Figs. 35, on the whole, the simulated frequency responses using both the MTL and developed models

SHINTEMIROV et al.: A HYBRID WINDING MODEL OF DISC-TYPE POWER TRANSFORMERS FOR FRA

737

Fig. 5. Amplitudes of transfer function between the terminal of turn 60 and the input [(a) and (b) reprinted from [19]].

TABLE I

RESONANT FREQUENCIES IN DISCS 2, 4, AND 6 OF THE EXPERIMENTAL

TRANSFORMER WINDING

are close to measurements and correctly emphasize main resonant points of the winding. The analysis of Table I reveals that

the proposed model determines the main resonant frequencies

close to measurement ones in a range up to 3 MHz. In addition,

a high-frequency resonance component at around 6.5 MHz frequency is more accurately located by the proposed model with

respect to the MTL one, which is of the particular interest for

resonance analysis [19].

However, there are deviations in resonant amplitudes and frequencies of the model responses with respect to the measured

ones. It can be referred to an approximate choice of some of

the experimental transformer parameters being not listed in [19]

and to approximations in the expressions used for the model parameter calculation. In addition, as shown in [1], [10], [11] for

the case of FRA measurements, there is an effect of ground connections and length of measurement leads on high-frequency regions of the measured responses, which also needs to be taken

into account during simulation.

Comparing the two applied models, the MTL model can replicate more accurately the shape of the experimental responses,

which employs a more detailed representation of the winding

by the MTL model, where each turn of the winding is described

by individual equations. However, this leads to a sharp increase

of the order of the MTL model [19] with respect to the order of

complexity of the MTL model. For instance, with respect to the

employed model winding, the matrix is of the order 19 for the

developed model, whereas for the MTL model [19] the order of

the corresponding matrix is 181, which is almost 9.5 times more

and drastically increases computational time. Using the same

computer, the calculation of the 1000 frequency points takes 26

and 333 s in MATLAB for the proposed and MTL models respectively.

It should also be mentioned that the authors various preliminary attempts to employ a lumped parameter model for resonance simulation of the analyzed transformer winding have

failed to provide high frequency resonance components at frequencies above 1 MHz due to essential discretization of the

lumped model.

VIII. CONCLUSION

A novel mathematical model of disc-type transformer

winding for FRA study based on the combination of traveling

wave and MTL theories is presented. During the derivation

of the model, each disc is represented by the traveling wave

equations describing voltage signal propagation, which are then

connected to each other in a form of MTL model. It allows to

significantly reduce the order of the model determined only

by the number of discs in a modeled winding, whereas known

MTL models of transformer windings are mostly of the order

of total turn number.

Using the derived transfer functions, numerical computation

of frequency responses of winding input impedance and resonances under very fast transient overvoltages is undertaken. The

simulations are compared with the experimental and simulation results obtained using lumped parameter and MTL models,

being reported in other publications.

In the current study, only single-phase transformers without a

laminated core are considered. In general, a further study needs

to be undertaken to verify the model by the simulation of real

transformers with core involved, which will affect calculation of

the model parameters, such as inductances and ground capacitances. In this regard, different terminal connections for three-

738

phase multiwinding transformers should also be applied to investigate its effect on the model accuracy for FRA result interpretation.

The deviation of the model responses primarily concerns an

inexact estimation of winding parameters, especially for correct

calculation of its frequency dependant behavior. One way to

overcome this difficulty is to apply evolutionary techniques for

winding parameters identification, such as Genetic Algorithms

[26] and Particle Swarm Optimizer [27], etc.

In general, it is deduced that the developed model correctly

reflects the interactions between capacitances and inductive elements in a disc-type power transformer winding in a wide frequency range up to several MHz. Thus, it could be used for FRA

result interpretation at higher frequencies above 1 MHz, where

lumped parameter models are not capable. The model is also

shown to be useful for resonance analysis under VFTOs with

purpose to reduce computational complexity with respect to the

traditional MTL models.

ACKNOWLEDGMENT

The first author, A. Shintemirov, would like to thank the

Center for International Programs for awarding the Kazakhstan

Presidential Bolashak Scholarship and the JSC Science Fund

for a financial grant within the frame of the Sharyktau

competition to support his Ph.D. research at the University of

Liverpool, Liverpool, U.K. He is also indebted to N. Abeywickrama and Prof. S. M. Gubanski from the Chalmers University

of Technology, Gteborg, Sweden, for providing experimental

data, and to Mr. H. Sun from the North China Electric Power

University, Baoding, China, for valuable discussions on MTL

models.

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electrical engineering at Pavlodar State University, Pavlodar, Kazakhstan, where he received the

M.Eng. and Cand.Tech.Sci. (Ph.D.) degrees in 2001

and 2004, respectively. He is currently pursuing

the Ph.D. degree in the Department of Electrical

Engineering and Electronics, The University of

Liverpool, Liverpool, U.K.

His research interests include transformer winding

modeling and evolutionary algorithm applications for

transformer condition monitoring.

SHINTEMIROV et al.: A HYBRID WINDING MODEL OF DISC-TYPE POWER TRANSFORMERS FOR FRA

degrees in electrical engineering from Huazhong

University of Science and Technology, Wuhan,

China, in 1996 and 2000, respectively, and the Ph.D.

degree in electrical engineering from The University

of Liverpool, Liverpool, U.K., in 2004.

He was a Postdoctoral Research Assistant at The

University of Liverpool from 2004 to 2006. Since

2006, he has held a Lectureship in Power Engineering in the Department of Electrical Engineering

and Electronics, The University of Liverpool. His research interests are transformer condition monitoring, power system operation,

evolutionary computation, multiple criteria decision analysis, and intelligent

decision support systems.

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Authorized

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in electrical engineering from Huazhong University

of Science and Technology (HUST), Huazhong,

China, in 1981 and the Ph.D. degree in electrical

engineering from The Queens University of Belfast

(QUB), Belfast, U.K., in 1987.

From 1981 to 1984, he was Lecturer of Electrical

Engineering at QUB. He was a Research Fellow and

Senior Research Fellow at QUB from 1987 to 1991

and a Lecturer and Senior Lecturer in the Department of Mathematical Sciences, Loughborough University, Leicestershire, U.K., from 1991 to 1995. Since 1995, he has held the

Chair of Electrical Engineering in the Department of Electrical Engineering and

Electronics, The University of Liverpool, Liverpool, U.K., acting as the Head of

the Intelligence Engineering and Automation Group. His research interests include adaptive control, mathematical morphology, computational intelligence,

information management, condition monitoring, and power system control and

operation.

Dr. Wu is a Chartered Engineer and a Fellow of IEE.

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