Optimum Wire Busbar Design by Genetic Algorithm

This work shows genetic algorithm approach to the optimum wire busbar design. Parameters of the wire busbar are mapped into a chromosome like a string. This includes insulator string type, wire conductor type and number, bundle wire conductor distance, rigid spacer number, phase distance, wire conductor nominal stress, 3D truss L elements types and foundation type. GA uses natural selection on a population of such strings. The fitness function is the wire busbar price. GA performance is presented on one example with variation of input data – nominal current, short-circuit current and short-circuit duration.


INTRODUCTION
The high voltage outdoor busbars are used for transfer of the electrical energy between connected feeder and transformer bays. Busbars and the necessary connection to the equipment can be of wire or tube conductors. Each of these busbar versions has certain advantages and disadvantages. This article deals with wire busbars.
Photo of the existing wire busbars is presented in Fig.  1. The wire busbars consist of: wire conductors, insulator strings, spacers (for wire bundles), connectors (for incoming and outgoing bays), portals and foundations.
Aluminium wire conductors, steel reinforced aluminium wire conductors (ACSR) and aluminium alloy wire conductors (AAC) are usually used for wire (flexible) conductors. Wire conductor cross-section ranges 400÷800 mm 2 . The span length is usually 10÷60 m. For high nominal currents bundled wire conductors are used. They are composed of two wire conductors placed within the distance between bundled conductors, ranging 0.200÷0.400 m. Wire conductors at the required spacing hold rigid spacers. The distance between rigid spacers ranges 2÷30 m. Wire conductors are usually strained with an initial static stress in the order of 5÷20 N/mm 2 . This is less than 10% of the nominal rupture conductor strength. The total sag (the sum of the wire conductor sag and the insulator string sag) under initial static stress equals about 3% of the span length. The sag is about 30% greater at the maximum permissible conductor temperature (for example 80 °C).
Strained wire conductors are connected to the steel supporting structures by insulator strings. The insulator string consists of cap and pin porcelain or glass insulator elements. For safety reason, insulator strings for wire busbars are almost allways doubled (as in Fig. 1).
Steel supporting structures are typically portals, usually made of latticed or solid welded steel and erected on concrete foundations. In this paper 3D truss type latticed steel structures are used. Structure elements are isoscele Lbeam elements. They are divided to two kinds -chord and lacing.
The increase in static stress of wire conductors increases the force on connecting points and also the cost of steel structures and foundations. Wire conductors phase spacing increasing will also cause changes in cost of steel structures because of steel width, but steel structure elements can be lighter because short-circuit force between phase conductors is inversely proportional to square phase spacing.
Because of the complex interrelationships between the busbars parameters, a method based on the application of GAs is used. The paper presents an original GA approach and presents results derived from several test cases.
The goal of this investigation is to design the cheapest wire busbars including costs of wire conductors, insulators, spacers, connectors, portals, foundations and terrain, but satisfying all electrical, mechanical and civil requirements and conditions on construction site.
To the author's knowledge, there are no papers from other authors dealing with the optimization of wire busbars. The paper is a continuation research of already published papers by the same author [3][4][5][6] and follows the concept of presenting 3D truss latticed portals with L-beam elements. The basis for the development of the described GA method are general papers [7,8,9], papers about usage of GA in electrical application [10][11][12] and papers in the field of optimization of steel structures [13,14]. The program GAUSAB3D is developed in MATLAB, using Genetic Algorithm and Direct Search Toolbox.
Difference from past published papers is reduction of genetic variables by excluding dependent variables, including necessary connectors with belonging wire conductors for incoming and outgoing bays and cost for occupied terrain.

WIRE BUSBAR DESIGN
The mechanical effects of short-circuit currents on wire busbars are well documented [1,2]. The short-circuit currents passing through phase wire conductors make the phase wire conductors swing with lower phase clearances (a min ) and higher tension forces (tensile force, drop force and bundle contraction force). The solutions to this problem are a higher static tension force, a higher conductor unit mass or a higher phase distance. All variants have influence on overall wire busbars cost mostly because of steel supporting structures, foundation and terain cost. The wire busbar design must satisfy all criteria with the lowest overall price. The wire busbar design criteria are: • continuous current, The design goal is to optimize the cost of all components of the wire busbar system (C) (wire conductors, insulator strings, spacers, connectors, portals, foundations and terrain) that meet all technical requirements (electrical, mechanical and civil engineering).
Wire busbar system price consists of the following costs: wire conductors cost (C 1 ), insulator strings cost (C 2 ), spacers cost (C 3 ), connectors cost (C 4 ), portal (steel structure) cost (according to portal mass) (C 5 ), concrete foundations cost (according to foundation volume) (C 6 ) and terrain cost (according to area between foundations) (C 7 ).
In the first five costs cost of installation works (factor f work ) is included. Mark c denotes unit prices.
Input data for the optimization process is wire busbar components database, data depending on the particular case and group of optimization parameters.
The results of the optimization process are discrete (integer) and continuous (real) values.
Discrete values are wire conductor type, number of wire conductors, insulator string type, spacers type, number of spacers, bundle distance, connectors for incoming and outgoing bays, chord L-type beam, lace Ltype beam and foundation type. In program GAUSAB3D these values are generated as real but they are rounded (up and down) to the integer value.
Continuous values are maximum static conductor horizontal stress and phase distance (with 0,1 m step).
Calculation of short-circuit effects on wire busbars (forces at attachment points and conductors side swing) are calculated according to the method described in international standards [1,2]. Steel structures and foundations must withstand short-circuit forces. Distance between swing-out conductors must be greater or equal to allowed clearance that depends on wire busbar nominal voltage.
Calculations of steel structure are based on known relations between vector of external forces {F}, stiffness matrix [K] and vector of node shift {Q}.
Calculations of stress in steel structure elements are done by Eq. (10)  [ The results are node shift, internal forces in steel structure elements, forces on foundations and moments on foundations. Limitations of steel structure are stated earlier. Stress in tensioned elements must be less than allowable values for steel and stress in compressed elements multiplied by factor for buckling must be less than allowable values for steel.
Checking of foundation overturn is done by comparison of moments on foundation by vertical and longitudinal forces.
Calculation of foundation soil pressure is done by dividing sum of all vertical forces -weight of steel structure W ss , concrete foundation W cf and soil over foundation W sof by foundation foot area A x A y . Obtained value must be multiplied by factor depending on moments of longitudinal forces f mlf . Calculated foundation soil pressure must be lower than allowed soil pressure.

OPTIMIZATION BY GENETIC ALGORITHMS
Because of the mixed type of variables (integer and real) for the process of wire busbars optimization the method of GAs is selected. A Genetic Algorithm and Direct Search Toolbox for use with MATLAB aim to make the GAs accessible to the design engineer within the framework of a calculation/simulation package. This allows the retention of existing modelling and simulation tools for building objective functions and allows the user to make direct comparisons between genetic methods and traditional procedures.
The procedure method of GAs: 1. Generation of an initial population.  Other components of wire busbar are uniquely determined by above genetic variables.
Insulator string is uniquely determined by conductor and bundle distance.
Rigid spacer is uniquely determined by conductor and bundle distance.
Spacer with branch is uniquely determined by main conductor, bundle distance and branch conductor T-connector is uniquely determined by main conductor and branch conductor.
The tableau for the wire busbar design is shown in Tab. The structure of program GAUSAB3D is shown in Fig. 3. GAUSAB3D.m is own developed main program. It calls data base, reads input data, calls MATLAB GA procedure and prints results. Subroutine base3D.m reads data from data base. Subroutine price3D.m is fitness function and it calculates the cost for individual solution. Subroutine equipment.m determines dependent components (insulator string, spacer, spacer with branch and T-connector). Subroutine corona.m calculates electric surface field strength (corona effects). Subroutine chain calculates static forces at attachment points, horizontal stress and sag of wire busbars. Subroutine mesks.m calculates forces and conductor side swing under passing short-circuit current. Subroutine truss3D.m calculates deflection of steel structure, forces in L-type elements, forces and moments on foundations and buckling of L-type elements. Subroutine foundation3D.m calculates pressure on soil and checks overturn of steel structure and foundation.
Concrete foundation data are concrete density 2400 kg/m 3 , soil density 1800 kg/m 3 and allowable soil bearing pressure 150000 N/m 2 .
Main prices are steel price 2,67 EUR/kg, concrete price with steel armature 240,00 EUR/m 3 and terrain price 20,00 EUR/m 2 and cost of wire busbar installation 20 % (f work ).
Penalizing a fitness function for violating the criteria is equal to 100000,00 euros.
The wire busbars components database parameters are number of wire conductors 3, number of conductor phase distances 5, number of insulator strings 5, number of rigid spacers 12, number of rigid spacers with branch 12, number of T-type connectors 3, number of steel isosceles L type beams 7 and number of foundations 6.
Parameters of GA procedure are population size 200, maximum generations 50, stall generation limit 10, crossover fraction 0,5 and elite count 20.

Results
The results of 6 runs of programme GAUSAB3D are presented in Tab. 11. Case 1 is performed with data from chapter 4.1. Cases 2, 3, 4, 5 and 6 are performed with change in nominal current, short-circuit current and shortcircuit duration.

Analyses
Results from Tab. 11 show expected variation in prices for various combinations of nominal current, short-circuit current and short-circuit duration.
Increase of nominal current gives increase of wire conductor cross-section and overall cost.
Increase of short-circuit current gives increase in dimension of L-type steel elements and foundations, which results in increasing of overall cost.
Increase of short-circuit duration gives increase in dimension of L-type steel elements, foundations and distance between phase conductors, which results in increasing of overall cost.
Because of the stochastic nature of genetic algorithms user must try several runs with identical input data to obtain global optimum instead of local optimum. Recommended number of runs with identical input data equals 5.

CONCLUSION
The article describes the application of GA to the problem of designing the optimal wire busbar. Some of the design variables have discrete values and the others have continuous values. The reason for choice of GA method is the ability to calculate wire busbars and use GA within a MATLAB program with GA and Direct Search Toolbox. The fitness function (in our case is also objective function) is a sum of the wire busbars components prices. If the generated wire busbar does not meet the criteria set, then it is given high costs and thus eliminates the solution from the selection procedure.
Difference from past published papers is reduction of the number of genetic variables by excluding dependent variables, including necessary connectors with belonging wire conductors for incoming and outgoing bays and cost for occupied terrain.
Optimization of the busbar system by using developed program GAUSAB3D is demonstrated on test example. Figs. 6 ÷ 11 present the gradual approaching to the optimal solution.
Large population size (200) gives already at the initial generation possible solution for wire busbar. The price of the initial solution is gradually reduced to the cost of the optimal solution. The biggest impact on the cost of wire busbar have the costs of steel structure (C5), foundation (C 6 ) and terrain (C 7 ).
There is a lack of similar research due to the complexity of a program, the problem that includes electrical, mechanical and civil calculations.
The future development work can be done by improvement of the Genetic Algorithms efficiency and more complex steel structures. Genetic Algorithms efficiency can be improved by the choice of optimal GA parameters (population size, number of generations, selection procedure, crossover rate and mutation rate). The ultimate goal is to reduce the number of fitness evaluations (population size times number of generations) necessary to obtain the optimal solution.