Three-phase Transformer optimization design based on NSGA2 algorithm

To solve the problems of lack of diversity and low quality of solution set in the solving process of traditional transformer optimization algorithm, a multi-objective optimization algorithm based on NSGA-II is proposed. Firstly, a multi-objective optimization model of power transformer winding was established with manufacturing cost, additional coil loss and transformer short-circuit impedance as optimization objectives. Secondly, NSGA-II algorithm was used to optimize the model design. The optimization results showed that, compared with traditional MOPSO, MODE and MOEA. NSGA-II algorithm had better fitness values for each target. Finally, the reliability of the scheme is proved in a 110 kV/63000 kVA prototype, which has better economy compared with the traditional scheme.

design magnetic density, no-load loss of distribution transformer can be effectively reduced.Subsequently, many scholars have carried out transformer optimization design according to the above ideas [8][9][10][11].The above transformer optimization model based on analytical method is more reliable when CMOPS solves the model with high confidence and the algorithm with high accuracy.
In view of the shortcomings of traditional design schemes, this paper proposes a transformer multiobjective optimization design model which considers the use of core, leakage reactance and materials comprehensively, and uses an improved multi-objective algorithm to solve the equation.Finally, finite element method and engineering test are used to prove the feasibility of the scheme.

Multi-objective optimization model
Based on the well-known principle of electromagnetic induction, the induced voltage U can be calculated via electromagnetic induction formula: 4.44 (1) where, f is the current frequency, B is the core peak flux density, S is the net core cross-sectional area, N is the turns in primary winding.
According to Formula (1), the relationship between induced voltage U and the core cross-sectional area is linear in an ideal state.Core is the main magnetic circuit and energy transfer parts, it's materials and geometric cross-sectional dimensions directly deter-mine the power transfer, leakage inductance and manufacturing cost of the transformer.The optimization equation of the core will be firstly discussed under the background of practical engineering application.

Core section optimization equation
The larger the core section is in the core transformer, the higher the economic index is.Therefore, it is of great significance to maximize the section area under given constraints.Figure 1 shows a schematic diagram of a core section with circumcircle radius D(mm) and series n.In Figure 1, since the section of each layer is symmetric about the X and Y axis at the same time, the dimension of the section of level i can be located by the coordinates (x i ,y i ) of its matrix vertex, for which the optimization objective Equation (2) can be obtained: max 2 ∑ 2 (2) In addition, in order to facilitate the clamping of the clamp and effectively resist the short-circuit mechanical force in the transformer operation condition, the last stage of the laminate size must have a reserved inch for the clamp.Combined with the actual processing requirements, the optimization objective constraint condition Equation (3) can be obtained: where, to ensure the effective fixation of the clamp, the vertex of any level-1 rectangle should be constrained to be within two concentric circles of diameter D/2±γ, where γ is the relaxed coefficient.δ x is the X-axis dimension demand of the heat dissipation oil channel, and δ y is the Y-axis dimension demand of the heat dissipation oil channel.MOD is the remainder operation, which restricts the length and thickness of each level in the core section to a multiple of a specific set.a i is the thickness of the ith silicon steel sheet group, and A is the set composed of a i .In order to facilitate processing and reduce the workload of silicon steel sheet stacking in workshops, the thickness of all levels of general transformers should be the value in the specified set.Similarly, b i is the dimension value of its length, and B is its corresponding set.

Short circuit equation
The short-circuit impedance of a transformer is of paramount importance in the field of electrical engineering.It refers to the impedance that a transformer exhibits when a short-circuit occurs across its terminals.The short circuit currency, loss and impact are directly determined by the short circuit impedance.Based on Rogowski method [12], the leakage impedance can be written as: where V T is the volt per turn, H eq is the equivalent height of the transformer, μ 0 is the permeability under vacuum.∑ ATD and H eq can be computed via the following equation: where , , are te radial dimensions of primary winding, clearance and sec-ondary winding, respectively; , , are the corresponding diameter size, respectively.The optimization objective is to minimize the value of the calculated impedance and the target impedance X st and can be described as:

Transformer load loss optimization equation
Transformer loss is composed of no-load loss and load loss.It directly determines the additional cost generated by the operation of the transformer under the rated working condition and is one of the most important design parameters in the transformer design process.Transformer no-load loss is composed of eddy current loss and hysteresis loss.They are usually not separated in the calculation process.When the applied voltage is a sine wave, then: (7) where, is the core peak flux density, is the thickness of silicon steel sheet, n is the Loss coefficient of silicon steel sheet, K 1 and K 2 is the coefficient of correction which is determined by Processing method and heat treatment method.
The load loss of the transformer is mainly composed of the winding resistance loss , the additional loss of the winding and stray loss .This whole loss P can be approximately calculated by the following empirical formula: where, m is phase number, r is the resistance of the winding, is the width of the wire in windings, J is the current density in the winding, is the unit resistance value of coil material at rated temperature.
is the length of the tank, is the height of the tank, is the thickness of the tank cover, is the thickness of the tank bottom cover, is the Rated capacity of transformer, S is the Actual operating capacity of transformer.

Overall optimization objective
Combined the context in Chapter 2, the multi-objective optimization equation can be written as: where, the corresponding constraint boundary and parameter description can be found in above Equations ( 2) ~ ( 8).The multi-objective optimization equation will be optimized in the next chapter., return , break.selection and evolution to find a set of solutions that are optimal in terms of multiple objectives.It maintains a population of candidate solutions and applies various genetic operators such as selection, crossover, and mutation to generate new offspring.These offspring compete with the parent solutions to form the next generation.This algorithm proven to be effective in solving real-world optimization problems with multiple conflicting objectives.It allows decision-makers to explore a diverse set of solutions and make informed trade-offs between different objectives.The algorithm has been widely implemented and applied in various domains, including engineering, finance, and scheduling, where multiple objectives need to be considered simultaneously.The algorithm is consisted by the following parts: non-dominated sorting algorithms, crowding comparison, and elite retention strategy.The corresponding algorithm flow in generation t is given in Table 1.
As the given Algorithm flow in Table 1, the next generation population retention strategy is similar to the genetic algorithm with elite retention strategy.In single objective optimization, the performance of population can be directly decided via its corresponding value and the retention strategy is to keep the population with lower fitness.However, the multi-objective optimization is pareto dominance relation based, the population with lower dominated rank will be retained.For the individuals with the same dominance level, the selection will be based on crowding degree.More detailed about reservation strategy, congestion sorting process and fast non-dominated sorting can be found in [10].The development environment is python 3.9.11, the toolkits of Numpy 1.21.0 and Scipy 1.24.0 are used in this work.The computer performance is: CPU 8750H and RAM 16GB.

Optimization case
To effectively measure the performance of MAWGA algorithm in transformer optimization design.The optimized design is now carried out for a three-phase, three-column transformer core with a voltage and capacity of 63,000 kVA / 110 kV.Its design technical requirements are shown in Table 2.The main design parameters, parameter limits and requirements are given in Table 2.The mentioned multi-objective algorithm will be tested under above requirements.

Algorithms performances
Classical multi-objective optimization algorithms are mainly given by MODE (Multi-Objective Differential Evolution algorithm), MOEA (Multi-Objective Evolutionary Algorithm based on decomposition) and MOPSO (Multi-Objective Particle Swarm Optimization algorithm) algorithms.To prove the performance of presented NSGA-II algorithms, the optimization Equation ( 8) is also solved via MODE, MOEA and MOPSO.Set the population as 1000, the number of iterations is 1000.Due to the complexity of the multi-objective problem, the mean values of each optimization object in pareto set were recorded to judge the performance of algorithm.Figures 2-4 show the recorded values of core sectional area, absolute impedance deviation and load loss within 1000 iterations.Table 3 shows the mean value performance of each algorithm's pareto solution in the final iteration.Due to advantages of evolutionary strategy, the elite individual has greater probability to retain and been utilized to generate the next generation population.As shown in Figures 2-4, the NSGA-II has better performance in each optimization objectives compared with MODE, MOEA and MOPSO.Combined with Table 3 and Figures 2-4, it is reasonable to believe that NSGA-II has better global search ability and can obtain better Pareto front solutions.

Short circuit impedance
In order to effectively analyze the transformer short-circuit impedance and magnetic leakage distribution under this scheme, the finite element analysis model in Figures 5-6 is established, and the transformer mesh is divided under the 20 mm mesh setting.It can be calculated that the impedance of the transformer is 17.3%, which meets the required interval value of 17%±2%.A large number of engineering practice has shown that the finite element method calculation error is usually not more than 0.05%, the design parameters calculated by NSGA-II can highly meet the requirements of short circuit impedance in Table 2.

Engineering test
To effectively measure the actual impedance of the transformer, the short circuit test of the transformer is carried out, and its load loss is measured.In addition, the no-load loss is also measured via no-load test.Figure 7 shows the internal assembly diagram of the prototype designed by NSGA-II.In Table 4, the electrical performance of the prototype optimized by presented NSGA-II algorithm is compared with the original scheme: In the table, the optimized transformer prototype has better economy and various performances, and it is reasonable to believe that the improved NSGA-II algorithm can be effectively used in the field of transformer optimization design.

Conclusions
This paper proposed a NSGA-II based methodology to carry out the multi-objective optimization of transformer.Several achievements have been obtained: (i) A multi-objective equation considers the short circuit impedance, core cross section area and no-load loss was built to measure the effect of
10. else: # Remains N-| | objects need to be found in 11. ∅U , K= N-| | #Add the satisfied population to the next iteration 12. Dis = Dis ( ) #calculated the degree of congestion of population in 13.Choose K objects from the with lower degree of congestion.14.End 3. NSGA-II algorithm 3.1.Non-dominated sorting algorithms NSGA-II (Non-dominated Sorting Genetic Algorithm-II) is a widely used multi-objective optimization algorithm.It is an extension of the original NSGA algorithm and is designed to solve problems with multiple conflicting objectives.The key idea behind NSGA-II is to simulate the process of natural MEIE-2023 Journal of Physics: Conference Series 2591 (2023) 012024

Figure 2 .
Figure 2. Wave of core sectional area.
on transformer permeances.(ii) the NSGA-II algorithm were utilized to optimize the transformer and the result showed the NSGA-II has better performance compared with MODE, MOEA and MOPSO in each optimization objective.(iii) the test of engineering prototype showed the transformer optimized via NSGA-II has better performance compared with original.The NSGA-II can be utilized in transformer optimization.

Table 1 .
Generation t of NSGA-II.
Input: Pt ← parent population, N← Number of population, Dis ← the function of Crowding degree calculation

Table 3 .
Mean value in Pareto solution of each algorithm.