Finite Elements in Analysis and Design 43 (2007) 659–665www.elsevier.com/locate/finel

Design of distribution transformers based on a knowledge-based system and2D finite elements

C. Hernández, M.A. Arjona∗

Departamento de Ingeniería Eléctrica–Electrónica, Instituto Tecnológico de la Laguna, Carrara 371, Col. Torreón Residencial, 27268 Torreón, Coahuila, México

Received 8 September 2006; received in revised form 29 January 2007; accepted 29 January 2007Available online 13 March 2007

Abstract

This paper presents the development of a hybrid system for the design of distribution transformers. It is based on a knowledge-based system(KBS) and two-dimensional (2D) finite-element (FE) models, which are used to compute the electromagnetic fields within the transformer.A KBS is a software product based on established artificial intelligence techniques that has been successfully applied in many areas, whereas theFE method is a sound numerical technique for solving partial differential equations. Both methods are combined to offer transformer designersa computational tool for designing distribution transformers. First order elements are employed in two FE models, an axi-symmetric one isused to compute leakage reactance and eddy losses in transformer windings and a 2D cartesian model is used to calculate the iron loss andmagnetizing current. The KBS mimics the designers experience through the representation of heuristic, procedural and graphical knowledge.The proposed design system is applied to the design of a 1500 kVA, 13.2 kV–220V, delta-star, oil-filled distribution transformer.� 2007 Elsevier B.V. All rights reserved.

Keywords: Expert systems; Intelligent system; Transformer; Finite elements; Design

1. Introduction

Electrical transmission systems of increasing size are cru-cially dependent on power distribution transformers, whoseefficiencies are significantly greater than a century ago.Transformers are electrical devices based on electromagneticinduction; they change or transform voltage and current lev-els between two circuits (primary and secondary). However,the power transferred between the circuits is almost unalteredbecause the losses which occur in the process are small [1,2].Distribution transformers are normally considered to be thosethat provide the transformation from 13.2 kV to lower voltagesdown to the level of the final distribution network. They arethe most numerous and varied types of transformer used onthe electricity supply network. Consequently, there are moremanufacturers who make smaller transformers than largerones. However, their design is normally based on traditional

∗ Corresponding author. Tel.: +52 871 7051331x115;fax: +52 871 7051326.

E-mail address: marjona@ieee.org (M.A. Arjona).

0168-874X/$ - see front matter � 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2007.01.004

calculations and empirical knowledge, which is in the hands ofexperienced design engineers.

It is well known that designing transformers can be a time-consuming task because of its dependence on a high numberof design variables. But with the appearance of more power-ful personal computers, it is now possible to incorporate artifi-cial intelligence, numerical and programming methods into thedesign process. Decision trees, neural networks and geneticalgorithms have been successfully applied to reduce iron lossesin distribution transformers, where measurements at the coreconstruction stage are taken to feed the artificial intelligencemethods [3]. Test measurements are needed prior to the designstage—this may disadvantage new designs.

Geromel and Souza [4] applied artificial neural networksto the design of power transformers; they used an existingdatabase with experimental data that allow them to train andbuild neural networks at different design stages. Knowledge-based systems (KBSs) have also been applied to the design ofsmall core inductors and transformers for low electrical powersupplies [5–7]. Optimization has been applied to the design oftransformers, where a simplified mathematical model of the

660 C. Hernández, M.A. Arjona / Finite Elements in Analysis and Design 43 (2007) 659–665

transformer is usually made (neglecting, for instance, magneticcore nonlinearity and winding eddy currents due to the radialflux density component [8,9]).

The finite element (FE) method offers a solution to the non-linear Maxwell equations that govern the transformer’s electro-magnetic behavior [10,11]. It also allows more accurate compu-tation of some transformer quantities that are otherwise proneto error when analytical or experimental relationships are used.A hybrid approach, that integrates FE and a KBS for designingdistribution transformers, has not been published in the liter-ature. Although a KBS does not yield an optimum design asglobal optimization methods do, it generates a design that sat-isfies the specifications. Using FE in design optimization hasthe drawback of requiring a lot of computing power, especiallyif global methods are used [12]. Consequently a design toolthat can combine both accuracy and speed would definitely beof interest to manufacturers of distribution transformers.

In this paper, a novel approach for transformer design is pre-sented. It is based on the use of two-dimensional (2D) FE tosolve the nonlinear electromagnetic model of a transformer. Italso includes a KBS that mimics the heuristic knowledge ofa design engineer. With this hybrid scheme, it is possible toyield more accurate designs than using traditional techniques.A description and validation of this design methodology is pre-sented. A computer aided design package was developed in thewindows environment.

2. Transformer design

The general problem of design may be defined as that ofdetermining the most suitable form of the transformer. A satis-factory design is not obtained simply by the solution of a set ofequations; it requires several iterations. The transformer prin-ciple is based on the law of electromagnetic induction, wheretwo or more coils are magnetically coupled through a commonhigh permeability magnetic core. The core of the transformercan be of shell or column type. Fig. 1 shows side and top viewsof a three-phase transformer with three limbs, the low voltage(LV) and high voltage (HV) windings are also indicated.

The sinusoidal induced voltage, V, in any transformer wind-ing is given by [13,14]

V = 4.44Nf BmAc, (1)

Core

LV Windings

HV Windings Gap

Fig. 1. Side and top views of a three-phase transformer.

where N represents the winding turns, f the frequency (Hz),Bm is the amplitude of the sinusoidal flux density (T) and Acdenotes the net cross section area (m2) of the core. Anotheruseful relationship is the turns ratio, a, which is defined as

a = Np

Ns= Vp

Vs= Is

Ip, (2)

where I is the rms phase current and V is the rms phase volt-age, subscripts p and s denote primary and secondary wind-ings, respectively. Eq. (2) is also known as the ampere-turnbalance. Assuming or using a recommended core diameter andthe desired working flux density, the turns of the primary andsecondary windings can be determined with Eqs. (1) and (2).Next, the conductor bare areas are calculated assuming a cur-rent density, a value of 2–4 A/mm2 is usually recommendedfor copper conductors [14].

The transformer has different voltage levels; therefore it isnecessary to determine the electric clearances of windings,coils, core and yoke to avoid insulation failure due to highelectric field stresses. These distances are determined usingrecommended values according to the transformer voltages.Dimensions of ventilation ducts between groups of coils arealso taken into account at this stage, these will be dependenton the type of transformer (oil-filled or dry). The total numberof calculated turns must be accommodated in the transformerwindow area, thus an initial distribution of windings must becomputed. A transformer can have disk, helical, helical mul-tilayer or crossover windings, which must be considered inthe winding distribution [13]. The electrical clearance betweenhigh and low voltage windings is determined using distancesrecommended by transformer manufacturers.

A distribution transformer must satisfy a series of design con-straints to avoid penalty costs from the customer. These con-straints are: (a) efficiency, (b) percentage of leakage impedance,and (c) no-load current. In addition, the designer must guaran-tee the specified electrical power of the unit.

There are different sources of power loss in the transformercomponents. Hysteresis and eddy core losses are due to themagnetization under no-load condition. They depend on thefrequency and the working flux density. This core loss is cal-culated using experimental data provided by the steel manu-facturer. In addition, a curve is available that gives the specificmagnetizing power as a function of the amplitude of the fluxdensity. Stray losses are due to the leakage fluxes that induceeddy currents in frames and tank; in this study they are assumedto be negligible.

The winding itself has three types of loss: loss due to dcresistance, loss due to induced currents arising from alternatingflux leakages and circulating loss. In this paper, the conductorstrands are assumed to be properly transposed so circulatingcurrents can be considered negligible. The conductor eddy losscalculation is presented in the next section, and it is based onFE calculations. The dc resistance loss of a single winding iscalculated with the following equation:

Pr = J 2�mCu × 109

�dCu, (3)

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where J stands for the current density (A/mm2), � denotes theconductor resistivity (� mm), mCu is the copper weight (kg)and �dCu is the density of copper (kg/m3). The copper weightcan be computed with

mCu = �dCuNlms, (4)

where lm is the length of the mean turn (m) and s is the cross-sectional area of a conductor (m2). It must be recalled thattotal joule loss is three times the joule loss of a phase (low andhigh voltage windings). The transformer efficiency can then berepresented by the following expression:

� =[

1 − PFe + PCu

S cos � + PFe + PCu

]× 100, (5)

where S stands for the transformer rated power (VA), PCu rep-resents the total winding losses (W), � denotes the power factorangle and PFe the iron loss (W).

The total winding loss is

PCu = Pr + Peddy, (6)

where Pr and Peddy denote total joule and eddy current windinglosses, respectively.

The winding temperature is determined as a function of theradial and axial dimensions of a coil, current density and mmflinear distribution [15]. Based on the maximum voltages, tankdimensions are found. The amount of loss to be dissipated willdefine the total dissipating area, hence the radiators requiredsurface. Although a transformer is a simple device, its numer-ous, highly interelated and heterogeneous design parametersmake a closed-form solution to the design problem impossible.For example, if wire resistances are too great, the transformermay exhibit an undesirable temperature rise. When wire sizesare increased to reduce resistance, the windings may no longerfit in the core window.

3. FE calculation

The FE method can handle the solution of Maxwell equa-tions, which are partial differential equations. Closed formsolutions of the electromagnetic field inside the transformer aredifficult to obtain. Furthermore the magnetic nonlinearity pro-hibits the use of analytical methods. For a design engineer itis desirable to predict leakage fields, reactances, the flux distri-bution in the iron core and tank-wall losses. Thus, even thoughthe transformer may seem a simple device, its practical real-ization is complex. The coils are of cylindrical form, as are thelimbs. However the axial length, the yokes and the assembly ofthree phases converts the total assembly to three-dimensions.Fortunately, useful results can be obtained from axi-symmetricand 2D models. In this paper, first order FE and a commercialsolver are used throughout [16].

3.1. Leakage reactance

An axi-symmetric FE model is used to compute the trans-former leakage reactance. A single limb of the three-phase

Fig. 2. FE models: (a) axi-symmetric and (b) cartesian.

transformer is considered in isolation. Although there are somedistribution transformers that have square coils and limbs, themajority are circular. Therefore, the core and coils have rota-tional symmetry about the axis of the limb. Of course, othertransformer elements like frames, magnetic shunts and tank donot have this symmetry feature. However, this model representsa practical solution for transformer designers. An axi-symmetricmodel is represented in Fig. 2a. The tank wall is made of mildsteel and its depth of penetration at supply frequencies is muchless than its thickness so a surface impedance representationis a good approximation to its electromagnetic behavior. Thephase windings (high and low voltage) are excited with spec-ified source currents that are uniformly distributed over theirrespective areas. Nonlinearity is taken into account by repre-senting the BH data by a power equation in the model. Thismodel can be used to predict the reactance value and leakageflux density components. The short-circuit or load test simula-tion allows the field quantities to be obtained. The electromag-netic model is given by the equation [17]

�

�z

�

r

�(rA)

�z+ �

�r

�

r

�(rA)

�r+ j��

r(rA) = Js, (7)

where � represents the reluctivity, � is the angular frequency,� is the electric conductivity, A is the complex magnetic vec-tor potential, Js is the source complex current density and r isradius. The flux density, B, which has radial and axial compo-nents, is given by

B = ∇ × A. (8)

The solution of Eq. (7) gives the � component of the magneticvector potential, rA�, at each node of the FE mesh. The mag-netic vector potential can be expressed in terms of the magneticflux linkage, , in a closed path as

=∮

A · dl. (9)

Consequently, the flux linkage of a transformer winding isfound by integrating the potential in the peripheral direction:

= 2(rA�). (10)

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The impedance of a particular winding is calculated as

Z = Nd

dt

/I. (11)

The reactance in a two-winding transformer is found by addingthe individual referred winding quantities. In the transformerimpedance computation, the phase winding resistance must beadded to the reactance.

3.2. Winding eddy losses

In the calculation of eddy losses due to stray flux, it is nec-essary to study an individual wire or strand, which could bepart of a transposed cable. It is assumed to have a rectangularcross section. The magnetic field across this strand will havea direction relative to its orientation. The losses are calculatedseparately for each field component and then added together.It is then possible to derive a closed form solution for the elec-tromagnetic governing equation, which is given by [2]

�2Hy

�x2 = ���Hy

�t, (12)

where Hy is the external magnetic field intensity in the verticaldirection, x the horizontal length and t denotes time. After solv-ing (12), the eddy winding loss for low frequency is given by

Peddy = 2f 2Vw

6�(b2B2

y + c2B2x ), (13)

where Peddy is the eddy current loss in a winding, b is theradial strand thickness, c the axial strand thickness, Bx and By

corresponds to the average peak flux density in the radial andaxial directions respectively and Vw is the winding volume.

The calculation of the winding eddy-loss involves solving theaxi-symmetric FE model and taking the average of the squaredflux density components at each winding node to obtain the fluxdensity components. The radial and axial flux components aregiven by Eq. (8). It should be point out, while there is an approx-imate analytical expression for calculating the axial flux density,there is no such expression for the radial component; howeverthey both lack of the accuracy obtained with the FE model.

3.3. Iron loss and magnetizing current

A designer needs to predict the iron loss and magnetizingcurrent at the design stage to avoid penalties, which will affectthe manufacturer’s profit; in practice both quantities can be ob-tained from the no-load test. In addition, the losses are neededfor calculating the efficiency (Eq. (5)). It is known that corelosses are due to hysteresis and eddy currents in the core lam-inations. Iron losses depend on the grade of the silicon grain-oriented steel and its numerical value depends on the amplitudeof flux density. Here, a 2D FE cartesian model is employedto get the iron flux density modulus, which is then used toobtain the specific loss and apparent power at each FE by usingtest data from the core manufacturer; the computed values arethen modified by a factor to reflect the effect of gaps and loss

due to core construction. In the two-dimensional model, thethree-dimensional length is determined by comparing the FEand test results for a built transformer. The flux distributionunder this operating condition is shown in Fig. 2b. The FEmodel is voltage fed; therefore the FE equations must be solvedsimultaneously with the electrical external circuit equations.This model uses complex numbers to represent sinusoidal volt-ages, currents and fields and its governing equations are[

S + j�N −Pj�Q R + j�L

] [AIw

]=

[0

Ve

], (14)

where S is the FE stiffness matrix, P and Q are winding weight-ing matrices, R is the external resistance, L external inductance,Iw a vector of winding currents, Ve denotes the external poten-tial sources and A represents a vector of complex potentials.

The total iron loss in the transformer is computed with theequation

PFe = Vc�c

m∑e=1

Pe, (15)

where Pe is the elemental specific iron loss (W/kg), �c is thedensity of oriented steel (kg/m3), Vc is the core volume (m3)and m is the number of elements in the FE core region.

The no-load magnetizing current, Im, can be calculated withthe following equation:

Im = Vc�c√3VL

m∑e=1

Se, (16)

where Se is the elemental specific magnetizing apparent power(VA/kg), and VL is the line voltage (V).

4. KBS-FE for designing distribution transformers

The integration of a KBS with the classical design proceduresfor transformers and the FE method represents a challenge. Theresulting hybrid system gives manufacturers a design tool wherethe knowledge of their experienced engineers can be recordedand used by novice designers. As was mentioned above, theknowledge takes the form of algebraic equations, Eqs. (1)–(6),and jointly with the FE models, they reproduce the steady-statetransformer performance. In addition, heuristic knowledge inthe form of charts, e.g. electrical clearances, saturation and corelosses are also considered. Conductor information, i.e. wiregauge, bare and insulated diameters are stored in a database.A description of the system is given below.

4.1. Knowledge-based system

A KBS is a complex software product that uses knowledge,facts and reasoning techniques to solve problems that normallyrequire the abilities of a human expert. It is made in a pro-gramming tool specially designed to describe a heuristic pro-cess. Such a tool is called an “expert system shell”. CLIPS(C-Language Integrated Production System) was the expert sys-tem shell used for this project because it can be embedded

C. Hernández, M.A. Arjona / Finite Elements in Analysis and Design 43 (2007) 659–665 663

easily into the classical design and FE solver, which were codedin C + + language [18].

There are three main parts in a KBS: a user interface, aknowledge base and an inference engine. The user interfaceprovides mainly windows, displays and objects. The knowl-edge base consists of rules, methods, data structures, databasesand external programs. In most KBS, rules consist of a condi-tion and an action part. The latter is executed when the spec-ified condition is satisfied. Rules are written in the languageprovided by the expert system shell. This language can rep-resent symbolic and conceptual knowledge, which is difficultto deal with in other computer languages. Methods are simi-lar to sub-routines in conventional programming. However, in-stead of being called, they are associated with the change ofvariables. The inference engine processes the knowledge avail-able in the knowledge base. Decisions can be performed in aforward manner, which means that they are made when suffi-cient data is available. Alternatively, a backward chaining canbe performed when the reasoning is goal driven, which impliessearching from an assumption to find information to supportit. Backward and forward chaining can be combined to yield avery flexible and powerful processing strategy.

4.2. Knowledge representation

Knowledge contained by the system was obtained fromexperts in the field of transformer design and literature

Fig. 3. Window snapshot with input data, winding results, performance, tank and core dimensions of the designed transformer.

[1,2,14,15,19]. The methods employed were formal and in-formal interviews, literature analysis and practical design op-erations. In order to represent this knowledge, programmingheuristics, sequences of numerical commands and data wererequired. The physical composition of transformers and themethods for computing their properties are represented as a hi-erarchy of objects. Associated with each object there exist sev-eral variables known as attributes. The relationships betweendifferent objects are defined and operations are carried out onthese objects by the inference engine.

Expert knowledge is represented by English-like sentencesknown as rules. They are modular pieces of knowledge in theform of IF condition THEN action. Rules use the availabledata to reason towards the final goal of finding an acceptabletransformer for a set of design specifications. A rule examplecan be of the form: Given the number of axial LV parallel stripsand the proportion between axial and radial strip thickness, ifthe proportion is less than 1.5 decrease by 1 the number ofaxial LV parallel strips. Otherwise, if the proportion is greaterthan 2.5 increase by 1 the number of axial LV parallel copperstrips.

4.3. KBS-FE execution

The design strategy employed is similar to that used by hu-man designers: generate and test design alternatives for a givendesign specification. The design process is started via the user

664 C. Hernández, M.A. Arjona / Finite Elements in Analysis and Design 43 (2007) 659–665

Fig. 4. KBS-FE architecture for designing transformers.

Table 1Comparison of KBS-FE and manufacturer design results

Designs Efficiency No-load Load Impedance Flux Magnetizing(%) loss (%) loss (%) (%) density (T) current (%)

Manufacturer 98.68 0.21 0.85 4.89 1.65 –KBS-FE 98.73 0.18 0.85 4.18 1.65 0.511

interface shown in Fig. 3 (input data window), where the userprovides basic information such as rating, primary and sec-ondary voltages, frequency, tapping values, connections, typeof magnetic material and power factor. Initial values must alsobe given for flux density, core diameter, core height, LV andHV winding current densities and parallel strips. Finally, thedesigner must give the desired values for impedance, efficiencyand no-load current. The KBS-FE tool then generates a firstdesign alternative for the transformer based on the calculationsshown in Fig. 4, which is then checked to see if it satisfies theconstraints imposed by the designer. If the generated designviolates input constraints, the KBS-FE suggests correspondingmodifications to produce what is considered a new alternativedesign that still needs to be evaluated. This process is repeatedgenerating a set of new design alternatives, that are queued in aFIFO (first in first out) order and coordinated by the ‘list man-ager’ (Fig. 4), which keeps track both of alternatives that satisfyall requirements and alternatives pending evaluation. The sys-tem execution stops when all alternatives generated have beenevaluated.

5. Results

In order to evaluate the performance of the developed KBS, acomparison between the results obtained from the standard de-sign procedure of a local manufacturer and the results obtainedby the KBS-FE approach is presented. The transformer spec-ifications are: 1500 kVA, 13200/220V, delta-star, 60 Hz, 0.8power factor, 1.65 T, efficiency greater than 98%, impedanceof 5.5%, magnetizing current less than 1%. Fig. 3 also shows awindow with the HV and LV winding results, i.e. dimensionsof copper strips, losses, resistance, winding diameters, windingheight and current densities. In addition, the values obtainedfor the performance and the dimensions of the core and tankfor what the system considers the best design alternative areshown in the window at the bottom of Fig. 3. Finally, a compar-ison of the final design alternative with that obtained by a localmanufacturer is presented in Table 1, where a good agreementcan be seen.

At present, the KBS-FE can handle the design of three-phaseoil-filled distribution transformers up to 10 MVA, with primary

C. Hernández, M.A. Arjona / Finite Elements in Analysis and Design 43 (2007) 659–665 665

voltages up to 34.5 kV and secondary voltages from 127 to600V. The design tool determines the following transformerelements: stepped core, three-limb core, concentric windings,off-load tap-changer, HV and LV bushings, tank, conservatorand radiators.

6. Conclusions

A hybrid approach that combines FE models and a KBSfor designing distribution transformers has been developed.The KBS is based on the use of expert systems, which is awell-established artificial intelligence technique within manyengineering areas. To improve accuracy in the design, two FEmagnetodynamic models were used. An axi-symmetric modelallows computation of the leakage reactance and eddy currentwinding loss, whereas a 2D cartesian model computes the ironloss and magnetizing current that are dependent on the mag-netic field distribution within the core. The KBS mimics thedesigner’s experience thorough a knowledge base, which rep-resents different kinds of knowledge, heuristic, graphical, pro-cedural and analytical. The design process is governed by aset of heuristic rules, which are transparent to the engineer.Using 2D FE models achieves an acceptable accuracy and fastcomputations when compared to full three-dimensional models.Although the final design is not the optimum one, it satisfiesthe specified constraints. The KBS-FE system was tested bydesigning a 1500 kVA, 13.2/0.22 kV, oil-filled, delta-star dis-tribution transformer. A close agreement with a manufacturerdesign result was found.

Acknowledgments

The authors would like to thank to CoSNET, DGEST andInstituto Tecnológico de la Laguna for financial support andto the local transformer manufacturer IMESA for all technicaldiscussions.

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