During the design process, electromagnetic and structural parameters must be selected while meeting specified performance indicators, aiming for minimal size and weight. Simultaneously, through raw material price analysis, the most cost-effective solution at current prices should be chosen to minimize the cost of major consumables. The solution to the objective function in transformer optimization design exhibits complex characteristics with multiple extrema, and the objective function and constraint functions are implicit forms of the design variables.
References [1-4] take amorphous alloy dry-type transformers as the optimization design object, take the main material cost as the optimization objective function, and optimize the design of amorphous dry-type transformers by particle swarm optimization (PSO) and genetic algorithm (GA), respectively, with obvious optimization effect.
Reference [5] uses a multi-objective genetic algorithm to optimize the design of a medium-frequency transformer, and conducts experiments with magnetic flux density and winding current density as optimization variables to obtain optimization results.
Reference [6] proposes a combination of nonlinear programming and genetic algorithm for the optimal design of transformers, which has good robustness.
This paper focuses on the structural parameter design of traction transformers for coal mining machines. With the goal of minimizing the consumption of major transformer materials such as windings and cores, the paper uses mathematical modeling and an adaptive weighted particle swarm optimization algorithm to optimize the design of mining traction transformers. The effectiveness and feasibility of the algorithm are verified by taking the optimization design of a 3.3 kV/190 kVA transformer as an example.
1
Transformer mathematical model
The core of the traction transformer used in coal mining machines has a rectangular cross-section and generally adopts a three-phase, five-column structure, as shown in Figure 1. The three middle columns of the core house the high-voltage and low-voltage windings, while the two outer columns are unwound. The design aims for a large effective cross-section, which allows for a reduction in the number of coil turns, saving materials and reducing energy loss. Considering the maximum magnetic flux density of silicon steel sheets, the transformer core window width and height are used as parameters in the transformer optimization design process, with material cost as the objective function, and no-load loss, load loss, unloaded loss, voltage impedance, and temperature rise as constraints.
(1) Core weight:
In the formula: GFe is the weight of the iron core; GZ is the weight of the iron core column; Ge is the weight of the iron yoke; d is the width of the cross-section of the iron core column; γ is the specific gravity of the cold-rolled silicon steel sheet, γ=7.65×10-4.
The effective cross-sectional area of the iron core column is its geometric cross-sectional area multiplied by the lamination factor, which is usually related to the thickness of the silicon steel sheet, the thickness of the insulating varnish film on the surface, the flatness of the silicon steel sheet, and the degree of compression.
(2) The cross-section of the high and low voltage side windings is a rectangle with rounded corners. The winding weight is:
In the formula: GCu is the total copper weight of the winding; GCuh and GCul are the copper weights of the high-voltage and low-voltage windings, respectively; Lh and Ll are the lengths of the high-voltage and low-voltage windings, respectively; Nh and Nl are the number of turns of the high-voltage and low-voltage windings, respectively; R and r are the chamfer radii of the high-voltage and low-voltage windings, respectively; xh, xl, yh, yl, Akh, and Akl are the length, width, and cross-sectional area of the individual conductors of the high-voltage and low-voltage windings, respectively.
(3) Transformer losses:
In the formula: P0, Pk, and Pkn are the no-load loss, actual load loss, and rated load loss, respectively; KFe is the unit core no-load loss; Kn is the load rate; and KCu is the load loss per unit weight of copper under rated load.
(4) Transformer impedance[7]:
In the formula: Uk, Ukr, and Ukx are the percentage values of the short-circuit impedance, resistive component, and reactance component of the transformer, respectively; SN is the rated capacity of the transformer; f is the frequency; I is the rated current; N is the total number of turns of the transformer; ∑D is the equivalent area of leakage flux; ρ is the Rogowski coefficient; K is the additional reactance coefficient; et is the potential per turn; and Hk is the average height of the high-voltage and low-voltage windings.
(5) Temperature rise:
The temperature rise values of the high and low voltage windings can be determined based on the transformer loss value and physical dimensions.
In the formula: τ0, τ1, and τ2 are the temperature rises of the iron core and the inner and outer windings, respectively; q0, q1, and q2 are the unit heat loads of the iron core and the inner and outer windings, respectively.
In the formula: P0, P1, and P2 are the losses of the iron core and the inner and outer windings, respectively; S1 and S2 are the heat dissipation surface areas of the inner and outer windings, respectively; S0', S1', and S2' are the heat dissipation surface areas of the iron core and the inner and outer windings that are shielded, respectively; kα0, kα1, and kα2 are the heat dissipation coefficients of the heat dissipation surface of the iron core and the inner and outer windings that are shielded, respectively.
(6) Transformer cost:
In the formula: FFe and FCu are the core cost and winding material cost (yuan), respectively; CFe and CCu are the unit prices of silicon steel sheet and copper wire (yuan/kg), respectively.
2
Transformer optimization design
The optimized design of traction transformers for coal mining machines can be achieved through intelligent optimization algorithms. This paper uses the particle swarm optimization algorithm to find the optimal objective function f(x) in the solution space X that satisfies the constraints.
According to transformer design standards:
In the formula: f(x) and gj(X) are the objective function and constraint conditions, respectively; j is the constraint condition number; X1 and X2 are the core and winding parameters in the optimization solution space, respectively.
The optimization objective is to minimize the cost of the main materials.
In the formula: f(X1) is the core cost; f(X2) is the winding cost; F(X) is the total cost; X is a set of data variables, including x1 and x2 are the core column length and width respectively, x3 and x4 are the core inner window height and window width respectively, x5 to x10 are the high and low voltage winding conduction width, thickness and length respectively, x11 to x15 are the high and low voltage winding spacing, winding to core distance and transformer to shell distance respectively, and x16 is the magnetic flux density.
The constraints are represented by g(X)≤0, and mainly include: (1) magnetic saturation constraint of the main circuit of the transformer core; (2) total loss less than the allowable value; (3) temperature rise of high and low voltage windings less than the limit value; (4) current density of high and low voltage windings less than the limit value; (5) short-circuit reactance not less than the allowable value; (6) no-load current less than the national standard; (7) efficiency greater than the required value; (8) the spacing between high and low voltage windings, the distance from the winding to the core, and the distance from the transformer to the outer shell meet the insulation requirements. The mathematical expression is shown in equation (10).
In the formula: B, P0, Pk, TH, TL, JH, JL, Uk, I0, η, hh, hw, and ht represent the core magnetic flux density, no-load loss, load loss, average temperature rise of the high-voltage winding, average temperature rise of the low-voltage winding, conductor current density of the high-voltage winding, conductor current density of the low-voltage winding, impedance voltage drop, no-load current, transformer efficiency, core height, core length, and core width, respectively; “￣” represents the upper limit of the corresponding value, and “_” represents the lower limit.
3
Transformer Optimization Design Based on Improved Particle Swarm Optimization Algorithm
The flight behavior rules of particles in the PSO algorithm are similar to those of birds. The solution space is iteratively searched by imitating the foraging process of birds [8], and it is applied to the optimization design of traction transformers for coal mining machines. In order to reduce optimization variables, according to the insulation and heat dissipation design, the spacing between high and low voltage windings and the insulation distance between windings and ground are both adopted as required under air-cooling conditions. For optimization problems of dimension D, the particle flight in the particle swarm algorithm is set to N, and the position of each particle is represented as:
The iterative equations for PSO, where the particle's historical optimal position Pi and the population's optimal position Pg are:
An inertial weighting coefficient w is introduced to achieve effective control and adjustment of the particle's flight velocity. The expressions for the particle's velocity and position are as follows:
In the formula: T is the total number of iterations; t is the current iteration number; i represents the particle number, i=1,2,3,…,N is the total number of particles; w is the inertia weight; vi is the iteration speed of the i-th particle; Xi is the i-th particle, and the corresponding actual variable value is Yi. Each particle corresponds to a transformer parameter scheme; pi is the historical best position of the i-th particle; pg is the historical best position of the particle swarm; c1 and c2 are learning factors; r1 and r2 are random numbers in the interval [0,1].
To balance the global search capability and local improvement capability of the PSO algorithm, an adaptive weight coefficient formula is adopted, the expression of which is:
In the formula: wmax and wmin are the maximum and minimum values of inertia weight, respectively; f, favg, and fmin are the current objective function value of the particle, the average objective value of all particles, and the minimum objective value, respectively.
The PSO algorithm with adaptive weights was used to optimize the parameters of the traction transformer for a 3.3 kV/190 kVA coal mining machine. The optimization results of the transformer's economic indicators are shown in Table 1. The unit prices of the main materials are: iron core 8.1 yuan/kg, high voltage and low voltage winding conductors 73 yuan/kg.
To demonstrate the algorithm's efficiency and global convergence, the evolutionary process is plotted as shown in Figure 2 (the target value is plotted every 5 generations for easy observation).
As can be seen from the figure, when the adaptive particle swarm optimization algorithm is used, the swarm produces the optimal solution when iterates to the 185th generation. The total material cost of the mining traction transformer is reduced by 3.75% compared with the traditional manual design, thus achieving the goal of reducing the material cost of the transformer.
4
Conclusion
An optimization problem for traction transformers used in coal mining machines was modeled and analyzed, and solved using a particle swarm optimization algorithm. This method is highly efficient, reduces the cost of major consumable materials, and can generate significant economic benefits. This optimization design method can also be applied to other dry-type transformers, demonstrating strong flexibility, versatility, and practicality.
