Application Of Covariance Matrix Adaptation Accounting Essay


In power sector, design of electrical equipment is essential to achieve an optimum balance between two apparently contradictory demands, performance and cost. Optimal transformer design is a complex multivariable, nonlinear, multi-objective and mixed variable problem. This paper discusses the application of covariant matrix adoption evolution strategy (CMA-ES) for designing the distribution transformer, considering various objectives such as minimum purchase cost, minimum mass, minimum loss, and minimum total life-time cost. Two new design variables such as voltage per turn and type of magnetic material are proposed in addition to the usual transformer design variables. The effectiveness of the proposed methodologies have been examined with a sample 400KVA, 20/0.4 KV transformer. Simulation results obtained by CMA-ES are compared with the conventional design procedure and mixed integer non linear programming combined with branch-bound (MINLP-BB) technique. The computational results and comparison demonstrate that the evolutionary technique, CMA-ES applied for this transformer design optimization problem offer best result, faster convergence and consistency. Four different case studies, implementing three sets of transformer design vectors are handled in order to prove the effectiveness of proposed modifications in the design variables (MDV). Superiority of the proposed MDV is proved in terms of cost savings, material savings and loss reduction for the respective objective function.

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All power utilities are much worried about high failure rate of distribution transformers. Transformers can be expected to operate 20-30 years. But losses in the form of heat reduce the transformer life by causing damage to the insulation. Finding ways to decrease the losses of transformer is an important factor towards reducing transformer failure rate, costs and CO2 emissions. So designing/buying a transformer based on profitability i.e., lesser purchase cost, without considering the future losses is therefore meaningless/ uneconomical. A transformer with low total life time cost (TLTC) would be expected to have longer life, lower failure rate as well as low losses. Selecting energy efficient distribution transformers (SEEDT) project also concluded that electricity distribution companies and commercial and industrial users should use the TLTC method for making transformer purchasing decisions. TLTC is the total life cycle cost which considers the future operating costs of a unit over its lifetime, brought back into present day cost and then added to its total purchase price. Hence the decision is whether to choose a transformer with low purchase cost (i.e.,) high TLTC or high purchase cost (i.e.,) low TLTC. Energy efficient transformers are more expensive, but use less energy resulting in lesser TLTC than inefficient transformers. Effective transformer design (TD), based on minimizing total life time cost (TLTC) can drop off the failure rate by reducing losses, amount of power generation needed to accommodate the losses, operating cost, emission of green house gases and environmental cost.

From the overview of research papers in TD, efforts are focusing on prediction of specific transformer characteristics, techniques adopted for global transformer design optimization, transformer post design performance and modeling, and recent trends on transformer technology. In a nutshell, TD optimization problem remains an active area [1]. TD Optimization can be minimization of no-load loss [2-4], minimization of load loss [5], maximization of efficiency [6, 7], maximization of rated power [8, 9], minimization of weight [9, 10] or minimization of cost [6, 7, 9-15, 17-29]. Transformer manufacturers (TM) use the optimization techniques to optimize the objective taking into account the constraints imposed by international standards, transformer specifications and customer needs. The techniques that are employed for TD optimization must be able to meet the design requirements, while remaining cost-effective and flexible and there is a difficulty in achieving a balance between cost and performance. Hence the research associated with design optimization is more restricted involving different optimization methods [16]. More research in TD using computers was pioneered by [13, 25, 20, 27, 30-37]. Comparison of nonlinear programming techniques for optimum design of transformers was presented by Sridhar et al [19]. Jabr [9] used deterministic method of geometric programming and provided solution for TD optimization problem of both low and high frequency transformers, minimizing the total weight of the transformer. But this method lacks in flexibility and finds difficulty in combining with cost estimation algorithm due to its large number of coefficients in the polynomial. It also requires development of mathematical model for each specific transformer type in advance.

Mixed integer programming finite element methodology (MIP_FEM) [11, 12], numerical field computation [17], bacterial foraging algorithm (BFT) [6, 7], and simulated annealing technique [14] have been adopted for minimization of main material cost (MMC) of transformers. However, all these methods have got their own drawbacks. MIP-FEM is sensitive to the selection of value range of design variables and fails to find the global optimum. Numerical field analysis technique has a disadvantage, due to the complexity of required mesh size, especially in 3D configurations. The well known drawbacks of simulated annealing technique are finding difficulty in extending itself to the multi objective case and long searching-time to find the optimum. The performance of basic BFT is satisfactory for small problems with moderate dimension and searching space. But when the search space and complexity grow exponentially in scalable problems, basic BFT would not be suitable, as cells move randomly, and swarming is not achieved by cell-to-cell attraction and repelling effects.

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Hybrid numerical analytical technique [18] and heuristic solution [21], where the user optimizes its design based on trial and error were carried out for transformer manufacturing cost (MC) minimization problem. But in these two design objectives, MMC and MC minimization losses have not been considered in the design aspect.

TD problem formulated as a mixed-integer non linear programming problem, by introducing an integrated design optimization methodology, based on evolutionary algorithms and numerical electromagnetic computations in Ref. [23] and another widely used numerical field analysis technique, FEM-BE which is combination of boundary and finite elements in Ref. [24] have addressed minimization of TLTC of transformer. These papers overcome the above said weakness, by considering losses. However, the magnetic material is not optimized whereas fixed as MOH-0.27 [23]. In order to take in to account, conflicting transformer design objectives, (example: minimizing purchase cost, minimizing TLTC), single core material has not been fixed in this paper. Moreover, when the type of MM is preferred as decision variable, optimum material to be used, for the respective design objective (example: lower loss or lower purchase price), can be easily identified from the variety of core materials available, satisfying various other TD constraints.

Apart from the deterministic methods, soft computing techniques such as genetic algorithms for MC minimization [22], incorporating type of MM as one of the design variables, transformer cost plus running cost minimization [26, 28], optimal placement of distribution transformers [29], and neural network techniques for prediction and minimization of power losses [17], no load loss minimization [2, 3] have been extensively used. Recently global optimization technique using genetic algorithm has been successfully applied to various engineering optimization fields. But in order to enhance the capability of searching for global optimum or quasi optimum to optimal transformer design problem, within a reasonable computation time, an efficient optimization technique CMA-ES, which can overcome the aforementioned problem to some extent, is applied in this paper.

Covariance Matrix Adapted Evolution strategy (CMA-ES) is a stochastic method for real parameter optimization of non-linear, non-convex functions. An important property of CMA-ES is its invariance against the linear transformations in the continuous search space, when compared to other evolutionary algorithms. This algorithm outperforms all other similar classes of learning algorithms on the benchmark multimodal functions [39, 42-44].

The main contributions of the paper are: (a) application of CMA-ES for TD optimization for the first time, assuring accuracy, consistency and computational speed (b) incorporation of type of magnetic material (TMM) as one of the design variables for representing 10 different materials (c) inclusion of variable, voltage per turn in place of LV turns. (d) optimization of four different objectives such as minimization of purchase cost, minimization of TLTC, minimization of mass, and minimization of loss suggesting the designer a set of optimal transformers instead of single solution, so that he can choose which of them best fits the requirement of the customer and application under consideration (e) comparison of simulation results with recent report [23] and conventional design procedure of industrial practice [41].

This paper is organized as follows: Section II describes the design of distribution transformers, section III presents CMA-ES technique, section IV explains the CMA-ES technique for TD optimization, section V includes computational results and section VI concludes the paper.

Design of distribution transformer


The design procedure is presented for three phase oil immersed shell type, wound core distribution transformer. The specified information consists of desired performance variables, core variables, conductor variables, cost variables and various other input variables required for the transformer design, employing analytical formulae to calculate the transformer parameters.

Performance variables:

These parameters list include transformer rating i.e. name plate details, design requirements on allowable no load losses (PSNLL), and load losses (PSLL) based on IEC 60076-1 standards, allowable standard short circuit impedance (UkS), full load minimum allowed efficiency, maximum allowed temperature rise and voltage regulation.

Core variables:

The core data include mainly the core stacking factor (Kc), mass density (gc), magnetization curve, variation of specific core loss (pc) for different values of maximum magnetic flux densities at 50Hz frequency for 10 different magnetic materials such as M3-0.27, M4-0.27, MOH-0.23, MOH-0.27, 23ZDKH90, 27ZDKH95, 23ZH90, 23ZH95, 23ZDMH85, 27ZDMH. Specific Core loss data has been taken from Nippon steel catalogue [], with handling factor of about 40%.

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The expression for core loss (PNLL) is calculated in watts using [41],

PNLL = GFe x pc


Weight of the core GFe in kilogram is

GFe = 2 x (GSC + GLC) ------(2)

Weight of the small core GSC is

GSC = LMTsc * Wcl * Tcl * Kc * gc ------(3)

Weight of the large core GLC is

GLC = LMTlc * Wcl * Tcl * Kc * gc ------(4)


LMTsc is the length of mean turn of small core,

LMTlc is the length of mean turn of large core,

Wcl is the width of core leg,

Tcl is the thickness of core leg.

Conductor variables:

This list includes the resistivity of the conductor material, i.e., copper at the maximum specified temperature, type of internal and external winding, typical practical values for insulation of conductor, distance and insulation between windings and core, mass density of the LV and HV conductor, distance between two adjacent cores, maximum ambient and winding temperature, direction space factor for turns and layer, HV taps. Here, copper sheet is used for LV conductor and copper wire is used for HV conductor.

Cost variables:

This includes unit price of various main materials in euro per kilogram namely, 10 MMs (CFe-1) to (CFe-10), conductor (CCu), sheet steel (Css), mineral oil (Coil), insulating paper (Cins), duct strips (Cds), and corrugated panel (Ccor).

Other variables:

It includes insulation outside HV winding, insulation between layers of HV winding, insulation between LV and HV winding, insulation between LV winding and core, insulation between layers of LV winding, number of ducts, width of the LV and HV duct strips, thickness of LV and HV duct strips, mass density of sheet steel, mineral oil, insulating paper, and duct strips, width, height, and weight per unit area of the corrugated panel etc.

B. Mathematical model

Mathematically, multivariable, nonlinear, multi-objective, mixed variable and mixed programmed transformer design optimization problem can be stated in the form as,

Find x={x_(1,) x_(2,).x_n },which ------ (5)

Minimize z_k (x), for k=1,2,,l ------(6)

Subject to g_j (x)=c_j for j=1,2,,m ------(7) x_i =0, for i=1,2,,n ------(8)

x_imin = x_i =x_imax, for i=1,2,,n ------(9)


zk, kth objective function and gj, jth inequality constraint are functions of n decision variables, x={x_(1,) x_(2,).x_n },

'c_j' for j=1,2,,m are constants?

'x_imin' for i=1,2,,n is the lower limit on x, and

'x_imax' for i=1,2,,n is the upper limit on x.

Minimum purchase cost design-Objective function: z1

The objective (z1) minimizes the cost of transformer materials like core, copper, insulating paper, duct strips, oil, corrugated panel, and sheet steel using [41]:


C_TM={C_(FE ) ?.G?_(FE )+? C?_(CU ) ?.G?_(CU )+ C_(INS ) ?.G?_(INS )+ C_(DS ) ?.G?_DS+ C_(OIL ) ?.G?_(OIL )+ C_(CORR ) ?.G?_(CORR ) + ? C?_(SS ) ?.G?_(SS ) } -----(10)


G_FE,G_CU,G_INS,? G?_DS,G_OIL,G_CORR,G_(SS ) are weight of magnetic materials (1-10), copper, insulating paper, duct strips, oil, corrugated panel, and sheet steel in kg respectively.

Minimum total life time cost design-Objective function: z2

The cost optimal design of transformer has to minimize the sum of transformer cost and running cost. In this objective, transformers selling price and losses are considered as transformer price. So material saving and energy saving are the two most important aspects in minimization of TLTC. The objective (z2) minimizes TLTC of transformer using [41]:


TLTC={?(?(C_TM+C_rem+C?_(lab )))/((1-S_m ) )+ A?.P?_(NLL )+B?.P?_(LL )?_ } -----(11)


CTM, TLTC, A, B, Crem, Clab, Sm, PNLL and PLL are cost of transformer materials in euro, total life time cost in euro, no load loss cost rate in euro per watts, load loss cost rate in euro per watts, remaining materials cost in euro, labour cost in euro, sales margin, designed no-load loss in watts and designed load loss in watts respectively.

Minimum Mass design-Objective function: z3

The objective (z3) minimizes the total mass of transformer materials like core, copper, insulating paper, duct strips, oil, corrugated panel, and sheet steel using:


?Tweight={ G?_FE+G_CU+G_INS+G_DS+G_OIL+G_CORR+G_SS} -----(12)

Minimum loss design-Objective function: z4

The objective (z4) minimizes the total loss of the transformer using:


Totalloss=?{P?_(NLL )+? P?_(LL ) } -----(13)

Design vector x: Design variables are collectively called as design vector. MDV are clearly described in the following three sets of design vectors [dv1, dv2, and dv3], implemented for the TD case study.

?dv?_1={x_(1,) x_(2,).x_6 } -----(14)

?dv?_2={x_(2,) x_(3,).x_7 } ----- (15)

?dv?_3={x_(2,) x_(3,).x_8 } -----(16)


x_1= LV turns

x_2= Width of the core leg in mm

x_3= Core window height in mm

x_4= Maximum magnetic flux density in tesla

x_5= Current density in LV winding in A/mm2

x_6= Current density in HV winding in A/mm2

x_7= Voltage per turn in volts

x_8= TMM

Constraints of performance indices:

Transformer performance must meet certain standards and rules, including,

No-load loss constraint:

P_(NLL )= ? P?_( SNLL ) (1+e_1 ) - ----(17)

Load loss constraint:

P_(LL )= ? P?_( SLL ) (1+e_1 ) - ----(18)

Total loss constraint:

P_(NLL )+ P_(LL )= ? P?_( SNLL )+ P_(SLL ) (1+e_2 ) -----(19)

Impedance voltage constraint:

U_(ks ) (1-e_2 ) = ?U_(k )= ?_( ) U_(ks ) (1+e_2 ) -----(20)


e1 and e2 are tolerances, whose values are specified in IEC 60076-1.

Constraints of process technology indices:

These constraints are based on the processing conditions of transformer factories and transformer manufacturer specifications.

Ratio of width of the core leg to the height of the core window should be less than or equal

to unity.

x_2 = x_3 -----(21)

2-multiplied core leg thickness must be between a minimum of half core leg width and a

maximum of 90% core leg width.

0.5x_2 =2T_cl =0.9x_2 -----(22)

Constraints of material performance:

Heat transfer constraint: Total heat produced by total loss of transformer must be smaller than the total heat dissipated by convection (Hconvec) through cooling arrangement.

? P?_(NLL )+ P_(LL )= ? H?_(convec ) -----(23)

III. CMA-ES technique

CMA-ES is a class of continuous Evolutionary Algorithm (EA) that generates new population members by sampling from a probability distribution that is constructed during the optimization process. Owing to the learning process, this algorithm performs the search independent of the coordinate system, reliably adapts topologies of arbitrary functions and significantly improves convergence rate especially on non-separable and/or badly scaled objective functions. It includes adaptation of covariance matrix, C of a normal search (mutation) distribution, N and overall standard deviation (global step size), ??. The covariance matrix C is adapted by the evolution path and vector difference between the best individuals in the current and previous generation. A standard CMA-ES with weighted intermediate recombination, step size adaptation, and a combination of rank - update and rank-one update is considered in this paper [39]. The various processes

Generation of population

At each iterations of algorithm, a set of search points ? X?_1^((g+1)), ? X?_?^((g+1)) are generated by sampling the distribution, N(m, C) using mean value m Rn covariance matrix C Rnxn and step size s^(( g ) ). The new individuals are sampled at generation g+1 as:

X_k^((g+1) )=N(m^(( g ) ),(s^(( g ) ) )^2 C^(( g ) ) ), for k=1,,? . (24)

where, R denotes the linear transformation, n N. X_k^((g+1)) is the kth offspring at generation g + 1. ?? = 2, population size, sample size, number of offspring.

Selection and recombination

The ?? sampled points are ranked in order of ascending fitness and best are selected. The new mean m^(( g+1 ) ) of all current population vectors is a weighted average of selected vectors from the samples? X?_1^((g+1)), ? X?_?^((g+1))with weight parameter, wi. and is updated using weighted recombination of the selected points as:

m^(( g+1 ) )= ?_(i=1)^?w_i X_(i:? )^((g+1) ) ? (25)

where, = ??, is the parent population size and by setting w_i = 1/, eqn (25) calculates the mean value of selected points. X_(i:? )^((g+1) ) denotes the ith ranked best individual of the ? sampling points.

Adapting the Covariance matrix

i) Estimating rank- update of C:

? C?^((g+1) )=(1- c_cov ) C^((g) )+c_cov {?_(i=1)^w_i ( (X_(i:? )^((g+1) )-m^((g) ))/s^((g) ) ) ( (X_(i:? )^((g+1) )-m^((g) ))/s^((g) ) )^T } (26)

ii) Utilizing the evolution path: Cumulation

Evolution path p_c^g is computed with initial values p_c^((0))=0,s^((0))=0.25(X_u-X_l ),? C?^((0))=I as:

? p?_c^((g+1) )=(1-c_c ) p_c^((g) )+v((c_c (2-) c_c)_eff)( (m^((g+1) )-m^((g) ))/s^((g) ) ) (27)

The rank-one update of the covariance matrix via the evolution path is given by:

C^((g+1) )= (1- c_cov ) C^((g) )+c_cov p_c^((g+1)) p_c^((g+1)^T ) (28)

iii) Combining rank--update & cumulation:

The final CMA update formula for the covariance matrix,? C?^((g+1) )combines (26) and (28), with _cov=1, weighting between rank- and rank-one update:

? C?^((g+1) )=(1-c_cov ) C^((g) )+c_cov/_cov p_c^((g+1) ) p_c^((g+1)^T )+c_cov (1-1/_cov )?_(i=1)^?w_i ( (X_(i:? )^((g+1) )-m^((g) ))/s^((g) ) ) ?

( (X_(i:? )^((g+1) )-m^((g) ))/s^((g) ) )^T (29)

?where C?^((g) )= B^((g) ) ??(D?^((g) ))?^2 ??(B?^((g) ))?^T and _eff=1/( ?_(i=1)^w_i^2 ) is variance effective selection mass.

c_cov={ 1/_cov 2/?(n+v(2))?^2 +(1-1/_cov ) min??(1,((2_cov-1))/((n+2)^2+_cov ) ) ?} is the learning rate of C^((g) ).

c_c= 4/((n+4)) , learning ratio determines the cumulation step for the evolution path. X_u and X_lare upper and lower bounds of decision variables.

Step- size control

To control step size s^((g) ) correlation of mean trajectory by ? p?_s^((g)) is utilized.

p_s^((g+1) )=(1- c_s ) p_s^((g) )+{v((c_s (2-) c_s)_eff)B^((g) ) ? D?^((g)^(-1) ) ? B?^((g)^T ) ( (m^((g+1) )-m^((g) ))/s^((g) ) )} (30)

The step size s^((g) ) is adapted with conjugate evolution path ? p?_s^((g)) as:

s^((g+1) )=s^((g) ) exp?(c_s/d_s ((? ||p?_s^((g+1) ) ||)/(E||N(0,I)||)-1)) (31)

where p_s^((0) )=0 initially. c_s=10/((n+20)) is the backward time horizon of the evolution path.

d_s=max?(1,(3_cov)/((n+10) )+c_s) is a damping parameter for the step size. B^((g)) is an orthogonal basis of eigenvectors and the diagonal elements? of D?^((g)) are the square roots of the eigen values of ? C?^((g)).

IV CMA-ES based TD optimization

The step by step CMA-ES implementation for transformer design optimization is given below. Set the transformer design objective function Z and design vector dv. Assume proper values for the transformer performance variables, core variables, conductor variables, cost and other variables, lower and upper bounds of the transformer design variables and constraints.

Step 1: Input C^((0)),? m?^((0)). Set CMA-ES parameters ?,,_cov,d_s,_cov c_s,?c_c,w?_i ?,c?_cov.

Step 2: Initializep_s^((0) )=0 ,p_c^((0) )=0. Choose step size s^((0))=0.25(X_u-X_l ),and generation count gmax.

Step 3: Initialize generation g = 0;

Step 4: Termination criterion: g=gmax , If termination met, then go to step 13.

Step 6: Generate ? candidate solutions using gaussian sampling of multi-variate normal distribution with covariance matrix, overall standard deviation from eqn (24), and the information of lower and upper bounds of transformer design variables from step 0.

Step 7: Determine the transformer objective function values and rank the ? design vectors in the order of ascending fitness. Select best search points.

Step 8: Update the mean value m^(( g+1 ) ) using eqn (25).

Step 9: Update the search points using eqn (24).

Step 10: Update the covariance matrix ? C?^((g+1) ) by eqns (27) and (29).

Step 11: Update the global step size s^((g+1) ), using eqns (30), and (31)

Step 12: Increment generation count, g = g + 1;

Step 13: Stop the optimization process.

Computational Results

To demonstrate the effectiveness of the proposed MDV, a design example of 400KVA, 50Hz, 20/0.4 KV, 3 phase, shell type, wound core transformer with vector group = Dyn11 has been considered. The upper and lower bounds of the design variables, no load loss cost rate, load loss cost rate, unit price of transformer materials, Crem, Clab, Sm, tolerances e1 = 15%, e2 = 10% are derived from [23]. As reference transformer, the transformer with loss category AB according to CENELEC [40] is selected, which means that PSNLL = 750W and PSLL = 4600W, and Uks = 4%. Coding for transformer design optimization is developed using MATLAB 7.4 on Intel core, i3 processor Laptop, operating at 3.2 GHZ, with 3 GB RAM. Suitable modifications are incorporated for handling TD constraints in the coding of CMAES algorithm [45]. The population size and maximum number of function evaluations are fixed at 100 and 10,000 respectively.

Case 1:

The CMA-ES technique minimizes objective functions z1, and z2, subject to the constraints from eqns. (17) to (23), by finding optimum values for dv1 as in [23]. Table I compares the solution of z1, and z2 by CMA-ES technique with MINLP-BB [23] and conventional design [41].

Table I: Comparison of optimization results of dv1




vector Minimum Purchase cost design

(z1) Minimum Total life time cost design


Conventional design [41] MINLP-BB[23] CMA-ES Conventional design [41] MINLP-BB[23] CMA-ES

x1 19 19 18 17 20 17

x2 230 230 212 230 231 242

x3 245 261 241 245 299 242

x4 1.8 1.8 1.8 1.6 1.6 1.6

x5 3 3.4 3.6 3 3 3

x6 3 3.3 3.6 3 3 3

CTM 4182 3954 3863 4667 4428 4617

PLL 4479 4890 5012 4246 4613 4265

NLL 813 818 854 710 719 696

TLTC 27563 28301 28774 26821 27467 26680

As can be seen from Table I, while optimizing the objective functions z1 and z2, three techniques converged to three different solutions. In particular, the CMA-ES technique converges to the best result comparatively. For objective function z1, CMA-ES technique has given cost savings of about 2.36% and 8.26%, when compared with MINLP-BB method and conventional design. Similarly for objective function z2, CMA-ES technique provides cost savings of 2.95% and 0.528%, when compared with MINLP-BB method and conventional design.

Case 2:

The CMA-ES technique optimizes the design vector dv2 by minimizing objective functions z1 to z4, subject to the constraints from eqns. (17) to (23), and results are tabulated in Table II for the same design example. This case discusses the effectiveness of the addition of proposed TD variable, voltage per turn x7 to the existing design vector [23] for TD optimization problem. Since variables voltage per turn and LV turns (x1) are inter-dependent, x1 is not taken into account as optimization variable. To show the performance of the variables, x2 to x7, this case study is examined. Variable x7 is varied at discrete levels in the interval [0 15], with step 0.01. The type of magnetic material used for building the core is fixed as MOH-0.27 from case 1 [23].

Table - II: Optimization results of dv2

Design vector/


vector Minimum Purchase cost design (z1) Minimum TLTC design

(z2) Minimum Mass design (z3) Minimum Total loss design (z4)

(with dv1) (with dv2) (with dv1) (with dv2) (with dv1) (with dv2) (with dv1) (with dv2)

x1 18 NA 17 NA 19 NA 16 NA

x2 212 224 242 245 248 244 232 220

x3 241 227 242 245 270 275 232 220

x4 1.8 1.8 1.6 1.6 1.8 1.8 1.6 1.6

x5 3.6 3.6 3 3.0 3.6 3.6 3.0 3.0

x6 3.6 3.6 3 3.0 3.0 3.0 3.0 3.0

x7 NA 13.22 NA 13.20 NA 11.85 NA 14.93

CTM 3863 3817 4617 4543 3993 3970 4754 4837

PLL 5012 4886 4265 4297 4771 4781 4092 3958

PNLL 856 854 685 662 818 804 737 767

TLTC 28790 28394 26588 26371 28060 27936 26786 26820

Tweight 1271 1260 1460 1428 1238 1229 1533 1589

Totalloss 5868 5740 4950 4959 5589 5585 4829 4725

Objective values obtained after the addition of variable x7 are lesser than the results of the case before adding x7, for all the objectives. From Table II, it is evident that the optimization results of dv2 for all the objective functions, z1 to z4 have yielded performance improvement of 1.21%, 0.823%, 0.732% and 2.20%, respectively on comparison with the results of dv1.

Case 3:

In this case, CMA-ES technique optimizes the design vector dv3, for the objective functions z1 to z4 subject to the constraints from eqns (17) to (23) and the simulation results are depicted in Table III. In the TD optimization problem, soundness of the TD variable, type of magnetic material x8 is analyzed, by adding it with the design vector dv2. The magnetic material is not fixed as a single material in this case, whereas it is also optimized during the run. The proposed variable x8 is used, such that the no-load losses and cost due to no load losses for different MM can be evaluated during the design phase, and the appropriate core material, out of 10 materials considered (M3-0.27, M4-0.27, MOH-0.23, MOH-0.27, 23ZDKH90, 27ZDKH95, 23ZH90, 23ZH95, 23ZDMH85, and 27ZDMH) can be optimized suitably, for each design objective. Variable x8 is defined as an integer, such that it varies from 0 to 10 for representing the 10 magnetic materials.

+Table- III: Optimization results of dv3

Design vector/


vector Minimum Purchase cost design (z1) Minimum TLTC design (z2) Minimum Mass design (z3) Minimum Total loss design (z4)

(with dv1) with dv2) (with dv3) (with dv1) with dv2) (with dv3) (with dv1) with dv2) (with dv3) (with dv1) with dv2) (with dv3)

x1 18 NA NA 17 NA NA 19 NA NA 16 NA NA

x2 212 224 224 242 245 245 248 244 232 232 220 219

x3 241 227 227 242 245 245 270 275 285 232 220 219

x4 1.8 1.8 1.8 1.6 1.6 1.7 1.8 1.8 1.8 1.6 1.6 1.75

x5 3.6 3.6 3.6 3 3.0 3 3.6 3.6 3.0 3.0 3.0 3.0

x6 3.6 3.6 3.6 3 3.0 3 3.0 3.0 3.45 3.0 3.0 3.0

x7 NA 13.22 13.22 NA 13.20 13.20 NA 11.85 11.27 NA 14.93 14.90

x8 NA NA 4 NA NA 5 NA NA 10 NA NA 9

CTM 3863 3817 3817 4617 4543 4629 3993 3970 4820 4754 4837 5678

PLL 5012 4886 4886 4265 4297 4215 4771 4781 4981 4092 3958 3828

PNLL 856 854 854 685 662 613 818 804 563 737 767 663

TLTC 28790 28394 28394 26588 26371 25882 28060 27936 27648 26786 26820 26836

Tweight 1271 1260 1260 1460 1428 1360 1238 1229 1216 1533 1589 1469

Totalloss 5868 5740 5740 4950 4959 4828 5589 5585 5544 4829 4725 4491

From Table III, it is explicit that the optimization results of dv3 by CMA-ES technique are superior for the objective functions z2, z3, and z4 than the solutions of dv2. The variable x8 can impart appreciable changes in the objective values, only when the core loss evaluations are used by the objective. Since z1 design does not rely on losses computation, type of magnetic material optimized in case 3 (No: 4) is same as that of the material fixed in case 2, i.e., MOH 0.27. Hence in the z1 design, the purchase cost optimized with dv2 and dv3 remains same. For z2 incorporating dv3, CMA-ES has provided least TLTC with cost savings of about 1.89%, 2.73% on comparison with CMAES-dv2, CMAES-dv1 respectively.

Similarly for z3 incorporating dv3, CMA-ES technique has given material savings of about 1.07%, 1.81% with CMAES-dv2, CMAES-dv1 respectively. In the same way, Table III explains for the objective z4, wherein the total loss of the transformer evaluated is very less comparatively. With dv3, percentage loss reduction is found to be 5.21% and 7.53% on comparison with the cases dv2 and dv1 correspondingly.

From the simulation results, it is evident that CMA-ES technique and the proposed MDV are able to find the optimal solution for TD optimization problem, irrespective of the objective function.

Case 4:

Due to the stochastic nature of CMAES algorithm, 25 independent runs are performed to prove the consistency in obtaining optimal solutions and statistical results are reported in this case study, for all the four objective functions z1 to z4 with the design vector dv3. Their performances are compared with respect to their solution accuracy, and mean computation time to reach the optimum in the Table IV below.

Table IV: Performance of CMA-ES for TD optimization problem

Objective function Best value Worst value Mean variance Standard deviation variance Mean computation time (sec)

z1 25882 25882 25882 0 86

z2 3817 3826 3819.8 3.82 113

z3 1216 1215.8 1215.6 0.113 92

z4 4491 4491 4491 0 105

For all the objective functions employed, the numerical results of CMA-ES technique are found more satisfactory, in terms of performance, consistency, faster convergence for TD optimization problem.


In this paper CMA-ES optimization technique was employed for the optimum design of three phase distribution transformer, considering various TD objectives such as minimum purchase cost, minimum mass, minimum loss, and minimum total life-time cost. The work proposed aiming at minimizing the objective using MDV, taking into account the constraints imposed by international standards, transformer specifications and customer needs. The validity of the CMA-ES algorithm for TD optimization problem was illustrated by its application to a 400KVA distribution transformer. Case studies 1-3 have clearly investigated the significance of MDV for all the TD objective functions and have proven that MDV is reasonably efficient for the TD optimization problem. The proposed MDV are not only capable of producing manufacturable optimum design, but also render considerable cost savings, material savings and loss reduction. Case study 4 has clearly demonstrated the effectiveness of CMA-ES with respect to its global searching, excellent solution precision, consistency in obtaining solutions, faster convergence, computational speed and bright application prospect in the fields of transformer design. It is evident that for the functions z1 and z2 incorporating MDV dv3, CMA-ES is able to give least purchase cost and TLTC with reasonable cost savings of about 3.59% and 6.12%, when compared with MINLP-BB method respectively. From the analysis of results, the soundness of the proposed methodology is undoubtedly pointed out with the design example and their optimization results are found satisfactory for all the objective functions.