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Electric Power Systems Research 80 (2010) 1111–1120

Contents lists available at ScienceDirect

Electric Power Systems Research

journa l homepage: www.e lsev ier .com/ locate /epsr

mproved modelling of power transformer winding using bacterial swarminglgorithm and frequency response analysis

. Shintemirov, W.J. Tang, W.H. Tang, Q.H. Wu ∗

epartment of Electrical Engineering and Electronics The University of Liverpool, Brownlow Hill, Liverpool L69 3GJ, UK

r t i c l e i n f o

rticle history:eceived 10 April 2008eceived in revised form 19 June 2009ccepted 2 March 2010

a b s t r a c t

The paper discusses an improved modelling of transformer windings based on bacterial swarming algo-rithm (BSA) and frequency response analysis (FRA). With the purpose to accurately identify transformerwindings parameters a model-based identification approach is introduced using a well-known lumpedparameter model. It includes search space estimation using analytical calculations, which is used for the

vailable online 13 April 2010

eywords:ransformer windingathematical model

acterial swarming algorithm

subsequent model parameters identification with a novel BSA. The newly introduced BSA, being devel-oped upon a bacterial foraging behavior, is described in detail. Simulations and discussions are presentedto explore the potential of the proposed approach using simulated and experimentally measured FRAresponses taken from two transformers. The BSA identification results are compared with those usinggenetic algorithm. It is shown that the proposed BSA delivers satisfactory parameter identification and

be us

requency response analysisarameter identification

improved modelling can

. Introduction

Power transformer is a major apparatus in a power system, andts correct functioning is vital to system operation. It is thereforeery necessary to closely monitor their in-service behavior, in ordero avoid catastrophic failures and costly outages and improve the

anagement of maintenance and servicing.Among various techniques applied to power transformer condi-

ion monitoring, frequency response analysis (FRA) is suitable foreliable winding displacement and deformation assessment andonitoring. It has been established upon the fact that frequency

esponse shape of a transformer winding in high frequenciesepends on changes of its internal distances and profiles, whichre concerned with its deviation or geometrical deformation [1].

However, the interpretation of FRA data is mainly conductedanually by trained experts. Measured FRA traces are comparedith the references taken from the same winding during previous

ests or from the corresponding winding of a “sister” transformer,r from other phases of the same transformer. The shifts in reso-ant frequencies and magnitude of FRA traces are believed to be

ndicators of a potential winding deformation. However, the ques-

ion of potential deformation location in a winding is still requiredo be investigated [2].

A range of research activities have been undertaken to uti-ize FRA in the development of suitable mathematical models of

∗ Corresponding author. Tel.: +44 151 7944535; fax: +44 151 7944540.E-mail address: qhwu@liv.ac.uk (Q.H. Wu).

378-7796/$ – see front matter © 2010 Published by Elsevier B.V.oi:10.1016/j.epsr.2010.03.001

ed for FRA results interpretation.© 2010 Published by Elsevier B.V.

transformer windings. Considering the simplified equivalent modelof transformer winding, various experimental research was per-formed with the purpose to observe the model behaviors in thefrequency domain [3,4]. A winding equivalent model and an iden-tification method of transformer equivalent circuit were proposedin [5,6], where equivalent circuits of transformer winding for thelow, medium and high frequency ranges were discussed and itsfrequency responses were compared with experimental data inorder to identify the models’ parameters. These models representthe overall windings by combinations of single lumped elements:inductances, resistances and capacitances. This allows estimat-ing only the overall winding parameters in a particular frequencyrange, which makes these models unsuitable for deformation anal-ysis of each winding section.

The calculation of internal parameters plays an important part inaccurate simulations of transformer winding frequency behavior.Modelling of a real winding in order to obtain frequency responses,being close to experimental ones, is an extremely complex tasksince a detailed transformer model must consider each turn or sec-tion of a winding separately. The reason is the fluctuation of realwinding parameters such as inductances and resistances per turnlength as well as interturn capacitances. The insulation propertydeviation should also be taken into account, which is frequencydependent.

In [7] efficient procedures to calculate turn self inductances,mutual inductances and capacitances were proposed whichdemanded additional experimental tests and knowledge of geo-metric and physical characteristics of a transformer. A transferfunction approach is used in [2] to study the discriminating changes

1112 A. Shintemirov et al. / Electric Power Systems Research 80 (2010) 1111–1120

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Fig. 1. Equivalent circuit of singl

ntroduced into a winding physical model. In [8–11] analyticalxpressions are used to estimate parameters of an equivalent modelased on the geometry of windings. The well-known finite-elementethod was applied in [12,13] for more precise calculation of wind-

ng parameters for an equivalent circuit model. These techniqueshow higher degree of accuracy compared with experiment mea-urements.

However, in industry measurements it is not always possible toonduct additional tests for precise measurements of transformereometry or insulation parameter estimation. Recently, evolution-ry algorithms were utilized to overcome such difficulties, that offerway to identify model parameters using limited measurement

ata. Regarding transformer winding modelling, at first, two simi-ar simplified winding model parameter identification approachessing particle swarm optimizer (PSO) [14] and genetic algorithmsGAs) [15] were proposed in [16,17] respectively. However, onlyimplified one-winding lumped parameter models were consid-red in these work for parameter identification. The questions ofnitial estimation of the model parameters to establish search spaceor the evolutionary algorithms were not discussed either.

A variety of well-established biologically inspired computa-ional methodologies has emerged in the past a few years, suchs GAs, PSO, evolutionary programming (EP) [18], bacterial forag-ng algorithm (BFA) [19], etc. However, data processing in theselgorithms may be time consuming, especially when a large num-er of multi-dimensional variables need to be optimized. Thus, it

eads to a slow convergence rate and reluctant application in manyroblems involving a large number of parameters to be optimized,rimarily because of the huge computational burden imposed. Aey advance in this field will therefore be met by a significant reduc-

ion in the computational time-costs whilst further improving thefficiency of global research capabilities of these algorithms [20,21].

In this paper an improved modelling of transformer windings isresented by using a model-based identification approach to derivehe parameters of transformer windings models. The approach,

se power transformer windings.

firstly introduced in [16,17], is further modified with a novel bac-terial swarming algorithm (BSA) [21,22], presented and utilized toundertake parameter identification. The newly introduced BSA isbased on BFA, which incorporates ideas from the modelling of bac-terial foraging patterns [22]. Simulation studies and discussionsare presented to explore the potentials of the proposed modellingapproach.

The remainder of this paper is organized as follows: Section 2describes a lumped parameter mathematical model of transformerwindings utilized in this study, then in Section 3 the analyti-cal expressions for model parameter estimation are presented.BSA and a model-based identification approach are introduced inSection 4. Subsequently, results of parameter identification usingsimulated and experimentally measures frequency responses fromtwo transformers are presented and discussed in Section 5. Finally,conclusions are given in Section 6.

2. Lumped parameter model of transformer windings

Since resonances of an FRA trace are related to the valuesof capacitances and inductances within a transformer winding,lumped-parameter equivalent circuit models have been widely uti-lized to analyze frequency domain behaviors. Each section of anequivalent circuit usually represents one or a few discs in the caseof a disc type windings as well as one or a few turns for helicaltype windings [9–11]. Therefore, despite of the model simplicity incomparison with those based on traveling wave theory [20,23,24]and multiconductor transmission line theory [25–28], it retains aphysical veracity and can be useful for frequency response simula-tion in a restricted frequency diapason up to 1 MHz. Fig. 1 shows a

typical equivalent circuit of single-phase power transformer wind-ings [8–12], where the following notations for section parametersof a high voltage (HV) and a low voltage (LV) windings are in use:

KHV and KLV series capacitances of HV and LV winding sections

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HV and CLV ground capacitances of HV and LV winding sectionsHVLV capacitances between HV and LV winding sectionsHVLV conductance between HV and LV winding sectionHV and gLV series conductances of HV and LV winding sectionsHV and GLV ground conductances of HV and LV winding sectionsHV and LLV inductances of HV and LV winding sectionsHV and RLV resistances of HV and LV winding sectionskm mutual inductance between the kth andmth winding sec-

tionsnumber of sections in one winding

The mathematical description of the model in frequency domains usually given in a matrix form of nodal equations applying therst and second Kirchhoff’s laws [8–11]:

Y U = A I + QU0,

Z I = −ATU + PU0,(1)

here vectors U and I represent node voltages and branch currentsespectively and voltage U0 denotes an input sinusoidal signal.

Each element of the admittance matrix Y is a combination ofdmittances, corresponding to g − K and G − C parallel branchesespectively as shown in Fig. 1. Using matrix notations for theapacitance and conductance matrices, Y = jωC + G, whereω is anngular frequency and j denotes the imaginary operator. The branchmpedance matrix Z = jω L + R consists of the self- and mutual sec-ional inductances L and M, being combined into the matrix L, andquivalent section resistances R, building the matrix R [8].

The incidence matrix A consists of −1, 0 and 1 and serves toink nodal voltages with branch currents. Matrices Q and P areormed as a component vector of Y and A matrices correspondinglyn accordance with terminal connections of the winding model suchs external voltage source imposition, node grounding, internodalonnections, etc. [9–11,29].

From Eq. (1), the branch currents and nodal voltages vectors cane expressed as [8,9,11,29]:

U = (Y + AZ−1AT )−1

(AZ−1P + Q)U0,

I = Z−1(−ATU + PU0).(2)

An output signal can be chosen arbitrarily from the variable vec-or U or I in order to obtain the transfer functionH(jω) of a windingn frequency domain, i.e. the ratio of nodal voltages and inductiveranch currents with respect to the applied input voltage.

. Estimation of model parameters

Model parameters are usually estimated using physicalimensions of a winding. In practice some simplification andpproximations of winding geometrical structures are acceptedhich allow to apply analytical formulae [8,9]. On the otherand, geometry simplifications can be avoided using finite-elementethod [12,13] for parameter calculation. In both cases the

requency dependent behavior of resistive elements should beccounted as well as frequency dependent insulation properties.n this study, the analytical expressions for initial estimation of the

odel parameters are presented.

.1. Capacitance and conductance

One of the common methods to calculate the ground and inter-inding capacitances C between windings, tank (or core) is to use

he expression for cylindrical capacitance having an axial height ofhe model section [9,10,13].

On the other hand, the evaluation of series capacitances Kepends on winding types. For instance, for a disc type winding theeries capacitance is determined by the amount of stored energy

ms Research 80 (2010) 1111–1120 1113

in the disc, which can be estimated by assuming the equal voltagedrop across each disc and the existence of equipotential surfacesin the interdisc space as stated in [9,10,13,30]. Transformer insula-tion materials at capacitance calculation are usually represented byeffective dielectric permittivities calculated as proposed in [9,13]using dielectric material reference sources or additional test results.

It is known that the insulation conductivity is frequency depen-dent due to dielectric losses characterized by tan ı. Therefore,expressions for series and ground conductances g and G can beobtained using the well-known formulae [27,28]:

G = C ω tan ıg,g = K ω tan ıs,

(3)

where tan ıg and tan ıs are the effective loss tangents of the insula-tion between winding and ground, and the intersection insulationrespectively.

3.2. Inductance and resistance

Research reported in [8,29] assumes that in the high frequencyregion above 10 kHz the core effect is not significant and can beneglected. Hence, the self and mutual inductances of model werecalculated using air-core case expressions. This method showed ahigh degree of accuracy in comparison with the results performedon several experimental transformers with the core removed andsubstituted by a hollow metal cylinder.

However, according to [13] the core effect has to be taken intoaccount during inductance calculation in order to accurately modelpractical transformers. This can be achieved using the analyticalexpressions derived by Wilcox [31,32] as described below, whichare adopted in this study.

In Fig. 2, two coils representing the kth and mth sections of atransformer winding are illustrated. These coils have Nk and Nmturns respectively wounded concentrically on a magnetic core ofradius b at intersection distance z. Each coil is characterized by theaverage a, internal a1 and external a2 radii respectively, and cross-section weightw and height h. The mutual impedance between thecoils is given as [32]:

Zkm = sLkm + Z1(km) + Z2(km), (4)

where s denotes the Laplace transform operator, Lkm correspondsto the mutual inductance upon the air-core assumption, Z1(km) rep-resents the impedance due to the flux confined to core and Z2(km)is the impedance owing to leakage flux upon introducing the core.The second and third terms of Eq. (4) are thoroughly defined in [32].

Regarding the first term, Lkm, it is proposed to use the followingapproximate formula [32]:

Lkm ∼= �0NkNm√akam

2�

[(1 − �2

2

)K(�) − E(�)

], (5)

where �0 is a free space permeability, K(�) and E(�) are completeelliptic integrals of the first and the second kinds respectively, and

� =√

4akamz2 + (ak + am)2

. (6)

However, a more precise calculation of the mutual inductancescan be achieved by the following expression [31]:

Lkm = 2�0NkNmakam

∞∫I1(ˇam)K1(ˇak) cos(ˇz)dˇ, (7)

0

where K1 and I1 are the modified Bessel functions.In both Eqs. (5) and (7) in the case of self impedance calculation,

e.g. Zkk, z = 0.2235(h+w) [32].

1114 A. Shintemirov et al. / Electric Power Systems Research 80 (2010) 1111–1120

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The calculation of winding resistance is the one of the majorhallenges due to eddy current effect in winding conductor andore. There are a lot of methods being proposed and utilized8,11,12] for the resistance calculation, among them, the Dowell’spproach [33] is one of the most referenced:

R= Rdc�

(sinh(2�) + sin(2�)cosh(2�) − cos(2�)

+ 2(p2 − 1)3

· sinh(�) − sin(�)cosh(�)+ cos(�)

),

(8)

here Rdc is the DC resistance of one-winding section, p is theumber of layers in the section and

=(�

4

)3/4 d3/2

ı t1/2, (9)

n which d is the conductor equivalent diameter and t is the dis-ance between the centers of two adjacent conductors. The skinenetration depth can be found as follows:

=√

2�0�r�ω

, (10)

here � and �r are the conductor conductivity and relative per-eability, respectively.

. Model-based identification approach with bacterialwarming algorithm

.1. Bacterial foraging algorithm

In the past a few years, the development of evolutionary algo-ithms received great attention in the computational intelligence

ions on a core.

community worldwide. The BFA [19] is one of the emerging opti-mization methods, utilizing an optimization model for E. colibacterial foraging to mimicry the self-adaptability of bacteria inthe group searching activities.

In general, an E. coli bacterium has a control system that enablesit to search for food molecules (nutrient) and try to avoid noxioussubstances. This activity of individual meretriciously flagellatedbacteria, which is called “chemotaxis”, can be described in termsof run intervals during which the microbe swims approximatelyin a straight line interspersed with tumbles so that the organismundergoes a random reorientation, and it alternates between thesetwo modes of operation in its entire lifetime.

E. coli bacteria also demonstrate a particularly interestinggroup behavior—“cell–cell communication”, which is a process thatallows bacteria to search for similar cells in their close surroundingsusing secreted chemical signaling molecules called autoinducers.Other bacteria release the same autoinducers in response. One-cell organisms in effect become multi-cellular organisms and canrespond together [34].

A BFA optimization model operates by a population (set) ofbacteria executing processes of chemotaxis, reproduction and dis-persion. A chemotactic process in BFA consists of Tumble andRun steps for each bacterium in the population, and a stepfitness is evaluated at each step of the process. The total fit-ness of each bacterium is calculated as the sum of the stepfitness during its life, which is obtained after all chemotactic

steps.

In the reproduction process, all bacteria are sorted in reverseorder according to their fitness. Only the first half of population sur-vive and a surviving bacterium splits into two identical ones, whichoccupy the same positions in the environment at 1st step. Thus,

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he population of bacteria remains constant in each chemotacticrocess.

The dispersion process prevents the optimization process to berapped around local optima or initial positions. In BFA, the disper-ion event happens after a fixed number of reproduction processes.

bacterium is chosen, according to a preset probability ped, toe dispersed and moved to another position within the environ-ent. BFA adopts a strategy of “individual-based attraction and

epellent” for “cell–cell communication”, which largely increasests computational complexity [19].

BFA is claimed to have a satisfactory performance in optimiza-ion problems and has been applied to power flow optimization35] and stabilizers design [36] for power systems. However, thelgorithm has limitations, e.g. it is unable to obtain satisfactoryesults in a certain range of optimization problems, in particular,igh dimensional and multi-modal problems, as some parts of the

oraging process are artificially set in such a way that the charac-eristics of the problem (landscape) were ignored [21].

Comparative studies have demonstrated that a recent develop-ent upon the bacterial foraging study, BSA, outperforms BFA in the

forementioned problems. The numerical comparison study [21]as shown that BSA has demonstrated a superior performance inomparison with some popularly used algorithms, such as PSO andast Evolutionary Programming. Therefore, this algorithm is usedn this study to perform intelligent optimization with the purposeo identify transformer winding model parameters and is describedn detail in the next subsection.

.2. Bacterial swarming algorithm

The main difference between BSA and BFA is the absence of theeproduction process in BSA. All bacteria are kept in the populationith only their positions updated in the search domain accord-

ng to their fitness values. The BSA model executes a combinationf both chemotactic and “group-based attraction and dispersion”rocesses, which is described below in details.

In the chemotactic process, a unit walk with random directionepresents a Tumble and a unit walk with the same direction ofhe last step indicates a Run. The process consists of one step ofumble and followed byNs steps of Run, depending on the variationf environment.

In the process of Tumble, the position of the ith bacterium can beepresented as

i(j + 1, r) = �i(j, r) + Ci(j)∠i(j), (11)

here �i(j, r) indicates the position of the ith bacterium at the jth

hemotactic step in the rth iteration loop;(j) is the direction anglef the jth step for bacterium i, and it is a random angle generatedithin a range of [0, 2�] [21]. Ci(j) is the length vector of a unit walk

or the ith bacterium at the jth chemotactic step in the rth iterationoop, which is defined as follows [21]:

i(j) ={CinitB : if TumbleD1r1B, : if Run

, (12)

here Cinit is the step size for a unit walk, D1 is a constant, r1 is aandom number, r1 ∈ [0,1], and B is the length vector of the bound-ries of the search domain, depending on a particular optimizationroblem.

The fitness of the ith bacterium at the jth chemotactic step isepresented by Ji(j, r). If Ji(j + 1, r) is better than Ji(j, r), then the

rocess of Run follows, which can be represented by:

ˆ l+1i

(j + 1, r) = �̂li(j + 1, r) + Ci(j)∠i(j) (13)

here �̂li

denotes the position of the ith bacterium in the lth step ofun, 1 ≤ l ≤ Ns, 1 ≤ j ≤ Nc, 1 ≤ r ≤ Nr. The notation Ns corresponds

ms Research 80 (2010) 1111–1120 1115

to the swim length limits when it is on a gradient,Nc is the numberof chemotactic steps per bacteria lifetime and Nr is the numberof iteration steps. This process continues until Jl+1

i(j + 1, r) is not

better than Jli(j + 1, r) [21].

The “group-based attraction and dispersion” process is adoptedto mimicry “cell–cell communication”. The individual with mostenergy (best fitness value) gained inNc chemotactic steps is definedas the best cell and its position �p(r) is kept for updating thepositions of other bacteria in the next selection process. A certainpercentage of bacteria are involved in an attraction action, whichare selected according to a probability pa. Based on their currentpositions and the global best position, the positions of bacteriabeing attracted are recalculated as follows [21]:

�i(1, r + 1) = �i(Nc, r) + r2D2(�p(r) − �i(Nc, r)), (14)

whereD2 is a constant and r2 a random number, r2 ∈ [0,1]. The restbacteria are dispersed to positions around the best individual witha randomly chosen mutation step and a mutation angle, using thefollowing equation:

�i(1, r + 1) = �p(r) + r3B∠ (15)

where r3 is a random number, r3 ∈ [0,1] and is a random anglechosen from [0,2�].

In summary, a BSA process can be briefly expressed in the formof a sequence of the following operations:

(i) Random generation of the initial population.ii) Performing the chemotactic process for fitness evaluation of

each bacterium in the population.iii) Performing the “group-based attraction and dispersion” pro-

cess.iv) Repeating steps ii–iv until a termination criterion is met.(v) Presentation of the best bacterium in the population as the BSA

output.

4.3. Model-based identification approach

The model-based learning approach is based on searching of theoptimal model parameters by minimizing the difference, i.e. fitness,between reference frequency responses and simulated model out-puts. It is achieved by measuring the errors between the originalresponses and the model outputs. Therefore, for each individual(bacterium) of a population in BSA, its total fitness value is given asfollows:

minS∑j=1

||H0(ωj) −H(ωj)|| ∗wj, (16)

where H0(ωj) and H(ωj) ∈R1 are the reference and simulated withthe identified parameters frequency responses at frequencyωj, j =1, . . . , S, where S is the number of frequency points involved in BSAlearning process and wj is the relative weight of the jth point.

Due to iterative nature of evolutionary algorithms, processinga large number of data points can greatly slow down a learningprocess. In the case of FRA, frequency responses are characterizedmainly by resonant and antiresonance frequencies and correspond-ing magnitude values. Therefore, as proposed in [17], the dimensionof processed FRA data can be reduced by selection of points of reso-nance and antiresonance and its vicinities for more speedy analysis,which are weighted accordingly.

The following steps are performed for parameter identificationof the transformer windings model:

• Measurement FRA data or predefined model parameters are usedto obtain the reference frequency responses.

1116 A. Shintemirov et al. / Electric Power Systems Research 80 (2010) 1111–1120

Table 1Comparison of the reference and identified parameters of the transformer winding model.

Parameter Reference value(reprinted from [29])

Estimated value(expressions in Section 3)

Identified value Deviation from the reference (%)

BSA GA Estimated BSA GA

L1, �H 186.0534 194.3 190.1 177.5 4.43 2.17 4.59M12, �H 141.4607 145.7 147.8 164.0 2.99 4.48 15.93M13, �H 91.0820 91.8 93.9 95.4 0.79 3.09 4.74M14, �H 65.3182 63.8 65.5 79.3 2.32 0.28 21.4M15, �H 48.9240 48.1 48.8 55.8 1.68 0.25 14.05M16, �H 37.5984 38.1 38.4 44.2 1.33 2.13 17.55M17, �H 29.4244 29.9 30.9 32.7 1.61 5.02 11.13M18, �H 23.3602 22.9 22.8 18.9 1.97 2.39 19.09M19, �H 18.7726 18.2 17.9 20.7 3.05 4.64 10.26M110, �H 15.2490 15.3 15.4 16.4 0.33 0.99 7.55M111, �H 12.5083 12.9 13.2 12.4 3.13 5.53 0.87M112, �H 10.3530 10.3 10.5 8.2 0.51 1.42 20.79M113, �H 8.6410 8.2 8.1 7.0 5.10 6.26 18.99M114, �H 7.2688 7.2 7.2 6.1 0.94 0.94 16.08M115, �H 6.1593 6.4 6.5 9.1 3.91 2.28 41.55M116, �H 5.2552 5.4 5.3 4.3 2.75 0.85 18.17Cg, pF 5.0224 5.085 4.8017 4.2384 1.25 4.39 15.61K, pF 85.686 127.41 79.92 87.55 48.69 6.72 2.18Rdc, – 0.0151 0.0544 0.0204 – – –

•

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some inter-coil distance the mutual inductance begins to increaseslightly, which may be explained by the presence of the (2/�) termin Eq. (5), being in dependence on inter-coil distance z. On the otherhand, Eq. (7) gives more appropriate results as illustrated in Fig. 3.

tan ıs – 0.03tan ıg – 0.03

Reference response points of resonance and antiresonance andits vicinities are selected and weighted accordingly to create areference dataset, being employed as training targets for BSAlearning.Each point of the training dataset is weighted according to itsdegree of importance of being accurately repeated in simulatedfrequency responses.Assuming approximate geometrical and material parameters ofthe tested winding are known, parameter values of an utilizedwinding model are initially estimated using analytical formulaein order to establish the possible search space for each parameterof the model.BSA learning is performed, in each step of which the predefinedtraining dataset is compared with the corresponding values of thesimulated frequency responses at the same frequency points. Thesimulated frequency responses are generated using the modelparameters obtained during BSA learning process.

. Simulation result and comparison

.1. BSA accuracy analysis using numerical simulation

.1.1. Reference response simulationIn order to analyze the identification accuracy of BSA, the mod-

lling results of a transformer winding presented in [29] are chosenue to its high degree of simulation accuracy in comparison withxperimental measurements. The test object is a disc-type windingonsisting of 60 discs with 9 turns in each discs. The ratio of an out-ut neutral current flowing to ground and an input signal injected

nto terminal end of the winding is used to produce frequencyesponse measurements.

Using listed model parameters in [29], the frequency responsesf the test transformer winding have been calculated using the pre-iously described lumped-parameter winding model, simplified forne-winding case, and 71 selected points are defined as a reference

ataset in the this study. FRA simulations are performed assuminggrounded winding through an impedance Zout. Since the modeloes not allow to directly obtain the total neutral current, it is moreonvenient to express it as a product of the grounded node voltagen+1 and Zout.

0.0153 0.0670 – – –0 0 – – –

5.1.2. Initial estimation of model parametersWith the purpose to establish the search space for transformer

winding parameter identification with BSA, the analytical expres-sions, described in Section 3, are employed to provide initialestimates of winding parameters. The parameter estimation is per-formed using a double disc basis as a unit section of the lumpedcircuit model. This allows to compare the estimated and subse-quently identified with BSA parameters with the reference ones,calculated in [29] using the same winding model partition.

It is known that the mutual inductance between winding sec-tions decreases with an increase of the intersection distance.However, numerical estimation of the inductance using approxi-mate formulae, Eq. (5), reveals some discrepancies. As shown inFig. 3, the mutual inductance goes down rapidly with the inter-coildistance raising. However, the decrease is not monotonic since at

Fig. 3. Mutual inductances of the 1st section of the tested winding.

A. Shintemirov et al. / Electric Power Systems Research 80 (2010) 1111–1120 1117

Table 2BSA parameters.

Parameter Notation Value

No. of bacteria in the population p 50No. of chemotactic steps per bacteria lifetime Nc 5Swim length limit when bacteria is on a gradient Ns 6No. of iteration steps Nr 5Initial step length Cinit 10e−4

Parameter for calculating step length D1 0.5

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The GA parameters are chosen based on various preliminarytrials and listed in Table 3.

Fig. 8 illustrates the fitness convergence of one of the best runs ofBSA and GA, where the total number of iterations for BSA is defined

Attraction factor D2 1Probability for attraction pa 0.8Search space variation from estimated values ± (10–50)%

Table 1 lists the reference and estimated using the analyticalxpression in Section 3 parameters of the analyzed winding model.he analysis of the table shows that inductance estimates given byq. (7) are very close to the reference values with the deviation notxceeded 5.1%. It should be noted that, since the test winding doesot have a magnetic core, the remaining parts of Eq. (4), i.e. Z1 and2 are not used [9].

.1.3. Parameter identification with BSAAs shown in Fig. 3, the mutual inductances between distant

ections are negligibly small in comparison with self-inductance.herefore, it is decided to consider only the self-inductances and thelosest 15 mutual inductances, similar to the approach proposed in17]. In addition, DC resistance Rdc is of interest to calculate therequency variable section resistance R with aid of Eq. (8), as wells the series and ground capacitances and loss tangents K, tan ıs

nd C, tan ıg respectively. Regarding the latter parameter, tan ıg,ased on preliminary BSA runs it is assumed that better results cane obtained without consideration of ground loss, i.e. tan ıg = 0.hereby, in total 20 parameters of the model as shown in Table 1 areo be investigated with BSA learning. These parameters are coded asnite-length strings (bacteria), representing potential solutions ofhe parameter identification problem. Accepted search space vari-tions for parameter identification with BSA learning are limited toe within ±10% for the inductive and ±50% ranges for the rest ofarameters from the corresponding estimated values.

The BSA learning parameters are selected on the basis of therevious study on bacterial foraging optimization [21,22] andumerous trials with various BSA parameters. The parameters are

isted in Table 2.Fig. 4 illustrates the comparisons of the analytically estimated

nd the identified with BSA magnitude frequency responses,hereas in Fig. 5 the corresponding phase frequency responses are

iven.Table 1 summarizes the reference parameter and the identified

arameter values with BSA. It should be noted that the frequencyependent reference values of section resistances and conduc-ances are given in a form of data vectors [29] and are not includedn Table 1. The table contains the results of one successful run withSA and its deviation from the reference in percents, the analysis ofhich shows negligible difference between the identified param-

ters, therefore, confirming the convergence stability of the BSAith respect to the investigated problem.

Considering inductance parameters, BSA provides its accuratedentification with maximum deviation of 6.26% from the corre-ponding reference values. This does not essentially differ from thenalytically estimated values of inductances with the deviation of.1% from the corresponding reference values.

However, the major improvement of BSA identification is andjustment of series capacitance K to 6.72% deviation from the ref-rence, while the initial estimation was not successful and showed8.69% difference. The large deviation of the initial estimation of Krom the reference caused failure to repeat all resonance frequen-

Fig. 4. Comparison of the transfer function magnitude frequency responses: iden-tified with BSA, estimated and reference.

cies when using the model with estimated parameters. This resultsin clear shifts to the left of the resonant points with regard to thereference frequency responses in Figs. 4 and 5.

Nevertheless, despite of slight deviation from the reference val-ues, the utilization of the estimated parameters as a search basisfor BSA parameter identification essentially improves the modelperformance as illustrated in Figs. 4 and 5.

5.2. Comparison with GA

In order to compare the performance between BSA and an evolu-tionary algorithm widely utilized for the parameter identificationpurposes, GA is employed to conduct parameter identification ofthe equivalent lumped parameter model using the same referenceresponses and fitness function (16). Due to the stochastic nature ofboth the algorithms, BSA and GA, initial populations of individu-als (bacteria) are generated in random order using the same searchspace limits, specified in Section 5.1.3.

Fig. 5. Comparison of the transfer function phase frequency responses: identifiedwith BSA, estimated and reference.

1118 A. Shintemirov et al. / Electric Power Systems Research 80 (2010) 1111–1120

Table 3GA parameters.

Parameter Value

Population size 80Selection Algorithm tournament [37]Crossover Algorithm scattered [37]Crossover Fraction 0.8Mutation Algorithm adapt feasible [37]No. of Elite Individuals 2Search space variation from estimated values ± (10–50)%

Ft

aosatFbbateu

i

Fw

ig. 6. Comparison of the transfer function magnitude frequency responses: iden-ified with GA, estimated and reference.

s the product of number of bacteria in the population, numberf chemotactic steps per bacteria lifetime, the number of iterationteps and the number of swim length steps when a bacterium is ongradient. The number of GA total iterations denotes the product of

he GA population size and the number of generations considered.rom Fig. 8 it is clear that BSA converges faster than GA, which coulde due to a different search principles of the algorithms. However,oth the algorithms are able to reach fitness minimum value withinlmost the same number of iterations. With regard to the compu-ation demand, the parameter identification process takes almost

qual computation time (about 100 s) using both the algorithmssing the same Intel Duo 2 Core computer.

Frequency responses obtained using the model parametersdentified with GA are given in Figs. 6 and 7. Visual comparison

ig. 7. Comparison of the transfer function phase frequency responses: identifiedith GA, estimated and reference.

Fig. 8. Fitness functions convergence.

of Figs. 4 and 5 with Figs. 6 and 7 respectively shows that thesimulated magnitude frequency response with BSA identificationis closer to the reference. On the other hand, GA identificationgives closer resemblance of the phase frequency response with thereference.

Table 1 presents the identified parameters with GA and its devi-ation from the reference. As seen from the table, the deviation ofthe GA identified parameters becomes greater than those obtainedwith BSA. For instance, the deviation of the GA identified mutualinductance M115 reaches 41.55% in comparison with only 2.28%deviation by the BSA identification. Moreover, GA gives a worseestimate of ground capacitance Cwith 15.61% deviation comparingwith the one identified with BSA (4.39% deviation). On the contrary,GA performs better in identification of series capacitance K withonly 2.18% deviation against 6.72% deviation with BSA.

The above difference in the identified results can be explainedby the fact that both the algorithms optimize the combinationsof inductances and capacitances constituting the mathematicalmodel, which define the model resonance frequencies. Thus, acapacitance reduction jointly with an increase of an inductancecould provide the same resonance frequency as the capacitanceincrease with the corresponding decrease of the inductance.Besides, the algorithms are generally guided by the fitness func-tion, which computes only the total deviation of the model outputsfrom the reference. Therefore, due to different learning principles,BSA and GA identify diverse parameters, despite of achieving a closeresemblance with the reference (Fig. 8).

In summary, considering more accurate parameter identifica-tion using BSA in comparison with the reference values, it can beassumed that BSA is more appropriate for the given optimizationcase.

5.3. Parameter identification using experimental FRA results

In practice, each phase of a power transformer includes LV andHV windings, which are connected to each other via interwindingcapacitances and mutual inductances as shown in Fig. 1. This inter-winding coupling as well as a terminal connection mode of onewinding, i.e. open or short circuited, grounded, etc., affects mea-sured frequency responses of the other winding. Therefore, in order

to verify the proposed approach with BSA, it is applied to iden-tify transformer winding parameters on the basis of experimentallymeasured frequency responses. A single-phase experiment trans-former without a core, consisted of a 30-double disc HV windingand a 23-turn helical LV winding, was used to measure HV winding

A. Shintemirov et al. / Electric Power Systems Research 80 (2010) 1111–1120 1119

Fq

im

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Fr

issues need to be further investigated in order to achieve a more

ig. 9. Comparison of the analytically estimated input admittance magnitude fre-uency response with the experiment measurements.

nput admittance frequency responses using standard FRA equip-ent [9,10].The transformer parameters are estimated on a double disc basis

s a unit section of the two winding lumped circuit model basedn the geometrical dimensions provided in [10]. In general, it cane assumed that the LV winding does not essentially affect inputdmittance responses taken from the HV side of a transformer.herefore, the parameters of the LV winding can be analytically cal-ulated and used directly in the model. As discussed earlier, Eq. (7)ives more accurate inductance estimation, therefore only a nar-ow search range of ±10% from the analytically estimated valuess considered for BSA parameter identification of the HV windingnductive parameters.

Figs. 9 and 10 show the comparisons of the magnitude andhase frequency responses, being analytically estimated and exper-

mentally measured. As seen from the figures, there is an observed

hift in resonant frequencies between the simulated responsessing estimated model parameters and the measured ones, whichrimarily concerns series capacitance KHV, DC resistance RdcHVnd insulation characteristics tan ıs and tan ıg respectively. In

ig. 10. Comparison of the analytically estimated input admittance phase frequencyesponse with the experiment measurements.

Fig. 11. Comparison of the BSA identified input admittance magnitude frequencyresponse with the experiment measurements.

addition, the gradual increase of the measured magnitude fre-quency response at higher frequencies corresponds to the presenceof bushing capacitance and capacitance of the measurementleads, which have to be taken into account in a form of anadditional ground capacitance parallel to CHV1. Therefore, theabove parameters are optimized using the proposed parameteridentification approach with BSA and the results are shown inFigs. 11 and 12.

It can be observed from the figures that the simulated with BSAidentified parameters and measured responses are close to eachother as far as the general shape and resonant frequencies are con-cerned. However, the developed model provides more dampingat the higher resonant frequencies with respect to the measuredresponses. This could be due to an essential discretization of thelumped model and an existence of additional frequency dependantloss mechanisms that are not considered in the model. All these

precise parameter identification of the transformer model.

Fig. 12. Comparison of the BSA identified input admittance phase frequencyresponse with the experiment measurements.

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Systems I 55 (8) (2008) 2433–2442.

120 A. Shintemirov et al. / Electric Powe

. Conclusion

In this paper a model-based identification approach is for-ulated to determine the parameters of a well-known lumped

arameter model of transformer winding with BSA learning. Initialearch space for identification of the model parameters is estab-ished based on parameter estimated values, being calculated usingnalytical expressions. The analysis of the BSA performance usingimulated reference frequency responses and a comparison withA has shown that that BSA is more accurate for the consideredase of the model-based parameter identification of transformerindings. There is a slight difference between the identified andreset parameters, which is negligible in a practical sense.

The model parameter identification using experimental inputdmittance frequency responses shows that the proposed approachan be utilized for the experimental FRA results interpretation aim-ng at winding fault diagnosis.

In the current study, only single-phase model transformersithout a laminated core are considered. However, further studyeeds to be undertaken to verify the model by simulation studiesf real transformers with core involved. As a variant, the core effectan be accounted in the inductance calculation as proposed in [32].

cknowledgment

The first author would like to thank the Center for Internationalrograms for granting Kazakhstan Presidential Bolashak Scholar-hip to support his PhD research in the University of Liverpool, UK.e is also indebted to Dr. N. Abeywickrama and Professor S. M.ubanski from the Chalmers University of Technology, Göteborg,weden, for providing experimental data.

eferences

[1] S.A. Ryder, Diagnosing transformer faults using frequency responce analysis,IEEE Electrical Insulation Magazine 19 (2) (2003) 16–22.

[2] S. K. Sahoo, L. Satish, Discriminating changes introduced in the model forthe winding of a transformer based on measurements, Electric Power SystemResearch, doi:10.1016/j.epsr.2006.07.007.

[3] E.P. Dick, C.C. Erven, Transformer diagnostic testing by frequency responceanalysis, IEEE Transactions on Power Apparatus and Systems 1 (6) (1978)2144–2153.

[4] L. Xiaowei, S. Qiang, Test research on power transformer winding deforma-tion by FRA method, in: Proceedings of 2001 IEEE International Symposium onElectrical Insulating Materials, 2001, pp. 837–840.

[5] S.M. Islam, K.M. Coates, G. Ledwich, Identification of high frequency equivalentcircuit using matlab from frequency domain data, in: Proceedings of the 32ndIAS Annual Meeting. Conference Record of the 1997 IEEE Industry ApplicationsConference, New Orleans, LA, 1997, pp. 357–364.

[6] S.M. Islam, Detection of shorted turns and winding movements in large powertransformers using frequency responce analysis, in: Proceedings of 2000 IEEEPower Engineering Society Winter Meeting, 2000, pp. 2233–2238.

[7] F. de Leon, A. Semlyen, Efficient calculation of elementary parameters of trans-formers, IEEE Transactions on Power Delivery 7 (1) (1992) 420–428.

[8] E. Rahimpour, J. Christian, K. Feser, Transfer function method to diagnose axialdisplacement and radial deformation of transformer windings, IEEE Transac-tions on Power Delivery 18 (2) (2003) 493–505.

[9] N. Abeywickrama, Y.V. Serdyuk, S.M. Gubanski, Exploring possibilities for char-acterisation of power transfromer insulation by frequency response analysis(FRA), IEEE Transactions on Power Delivery 21 (3) (2006) 1375–1382.

10] N. Abeywickrama, Effect of dielectric and magnetic material characteristics onfrequency response of power transformers, PhD Thesis, Chalmers University ofTechnology, Department of Materials and Manufacturing Technology, 2007.

11] M. Florkowski, J. Furgal, Detection of transformer winding deformations basedon the transfer function—measurements and simulations, Measurement Sci-ence and Technology 14 (11) (2003) 1986–1992.

[

[

ms Research 80 (2010) 1111–1120

12] E. Bjerkan, H.H.P. Dick, C. Erven, High frequency FEM-based power transformermodeling: investigation of internal stresses due to network-initiated overvolt-ages, Electric Power System Research 77 (11) (2007) 1483–1489.

13] E. Bjerkan, High frequency modelling of power transformers: stresses and diag-nostics, PhD Thesis, Norwegian University of Science and Technology, 2005.

14] J. Kennedy, R.C. Eberhart, Swarm Intelligence, Morgan Kaufmann Publishers,2001.

15] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learn-ing, Addison-Wesley Longman, Inc, 1989.

16] W.H. Tang, S. He, Q.H. Wu, Z.J. Richardson, Winding deformation identificationusing a particle swarm optimiser with passive congregation for power trans-formers, International Journal of Innovations in Energy Systems and Power 1(1) (2006) 8.

17] V. Rashtchi, E. Rahimpour, E. Rezapour, Using a genetic algorithm for parame-ter identification of transformer R-L-C-M model, Electrical Engineering 88 (5)(2006) 417–422.

18] D.B. Fogel, Evolutionary Computation, IEEE Press, New York, 1995.19] K.M. Passino, Biomimicry of bacterial foraging strategy for distributed optimi-

sation and control, IEEE Control Systems Magazine 22 (3) (2002) 52–67.20] A. Shintemirov, W.H. Tang, Z. Lu, Q.H. Wu, Simplified transformer wind-

ing modelling and parameter identification using particle swarm optimiserwith passive congregation, in: Applications of Evolutinary Computing, vol.4448 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2007, pp.145–152.

21] W.J. Tang, Q.H. Wu, J.R. Saunders, A bacterial swarming algorithm for globaloptimization, in: Proceedings of the IEEE Congress for Evolutionary Computa-tion (CEC 2007), Singapore, 2007.

22] W.J. Tang, Q.H. Wu, J.R. Saunders, A novel model for bacterial foraging invarying environments, in: Computational Science and Its Applications, vol.3980 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2006, pp.556–565.

23] A. Shintemirov, W.H. Tang, Q.H. Wu, Modeling of a power transformer windingfor deformation detection based on frequency response analysis, 5-506–5-510,in: Proceedings of the 26th Chinese Control Conference (CCC2007), Zhangjiajie,Hunan, China, 2007.

24] A. Shintemirov, W.H. Tang, Q.H. Wu, A hybrid winding model of disc-type powertransformers for frequency response analysis, IEEE Transactions on PowerDelivery 24 (2) (2009) 730–739.

25] J.A.S.B. Jayasinghe, Z.D. Wang, P.N. Jarman, A.W. Darwin, Investigations on sen-sivity of FRA technique in diagnostic of transformer winding deformations, in:IEEE International Symposium on Electrical Insulation, Indianapolis, USA, 2004,pp. 496–499.

26] Y. Shibuya, S. Fujita, N. Nosokawa, Analysis of very fast transient overvoltagein transformer windings, in: IEE Proceedings–Generation, Transmission andDistribution, Vol. 144, 1997, pp. 461–468.

27] H. Sun, G. Liang, X. Zhang, X. Cui, Modeling of transformer windings under veryfast transient overvoltages, IEEE Transactions on Electromagnetic Compatibil-ity 48 (4) (2006) 621–627.

28] M. Popov, L. van der Sluis, R.P.P. Smeets, J.L. Roldan, Analysis of very fast tran-sients in layer-type transformer windings, IEEE Transactions on Power Delivery22 (1) (2007) 238–247.

29] E. Rahimpour, J. Christian, K. Feser, H. Mohseni, Modellierung der Transforma-torwicklung zur Berechnung der Übertragungsfunktion für die Diagnose vonTransformatoren, Elektrie 54 (1) (2000) 18–30.

30] R.M.D. Vecchio, B. Poulin, R. Ahuja, Calculation and measurement of windingdisk capacitances with wound-in-shields, IEEE Transactions on Power Delivery13 (2) (1998) 503–509.

31] D.J. Wilcox, M. Conlon, W. Hurley, Calculation of self and mutual impedancesfor coils on ferromagnetic cores, in: IEE Proceedings–Science, Measurementand Technology 135 (7), 1988, pp. 470–476.

32] D.J. Wilcox, W.G. Hurley, M. Conlon, Calculation of self and mutual impedancesbetween sections of transformer windings, in: IEE Proceedings–Generation,Transmission and Distribution 136 (5), 1989, pp. 308–314.

33] P.J. Dowell, Effects of eddy currents in transformer windings, in: IEE Proceed-ings 113 (8), 1966, pp. 1387–1394.

34] K.M. Passino, Biomimicry for Optimization, Control, and Automation, Springer,2004.

35] W.J. Tang, M.S. Li, Q.H. Wu, J.R. Saunders, Bacterial foraging algorithm for opti-mal power flow in dynamic environments, IEEE Transactions on Circuits and

36] T.K. Das, G.K. Venayagamoorthy, U.O. Aliyu, Bio-inspired algorithms for thedesign of multiple optimal power system stabilizers: SPPSO and BFA, IEEETransactions on Industry Applications 44 (5) (2008) 1445–1457.

37] Genetic Algorithm and Direct Search Toolbox User’s Guide for use with MAT-LAB, 2nd edition, The MathWorks, Inc., 2005.