IET Power Electronics
Special Issue: Flexible Operation and Control for Medium Voltage
Direct-Current (MVDC) Grid
Optimisation design of medium frequency
transformer for the offshore dc grid based on
multi-objective genetic algorithm
ISSN 1755-4535 Received on 27th April 2017 Revised 4th September 2017 Accepted on 13th September 2017 E-First on 30th November 2017 doi: 10.1049/iet-pel.2017.0309 www.ietdl.org
Liang Zhang
1,2, Dan Zhang
3, Henghua Shui
1, Yubo Yuan
4, Qian Pei
1,5, Jihong Zhu
21School of Power Electric Engineering, Nanjing Institute of Technology, No.1, Hong Jing Avenue, Jiangning Science Park, Jiang Ning, Nanjing,
Jiangsu Province, People's Republic of China
2Department of Computer Science and Technology, Tsinghua University, East main building room 9-310, Beijing, People's Republic of China 3Shanghai Jiao Tong University, Shanghai, People's Republic of China
4Jiangsu Electric Power Company Research Institute, Nanjing 211167, People's Republic of China 5State Grid Suqian Power Supply Company, Suqian 223800, People's Republic of China
E-mail: [email protected]
Abstract: A DC/DC converter is a vital component for offshore DC grids, in which a medium frequency transformer (MFT) is the
main element. Therefore, it is necessary to optimise the core material and the design method so that the loss of the MFT would be reduced. In this study, the multi-objective genetic algorithm is proposed, by which two MFTs with different core materials are both optimised. The magnetic flux density of the transformer core and the current density of the windings are introduced as the optimised variables. Also, the key equations for transformer design are put forward as the objective functions. Moreover, with constant iteration of the process of segmentation, parallel selection, mergence, recombination and variation, the Pareto optimal solution is determined. Furthermore, to verify the optimisation results, the temperature field and electric strength of the amorphous alloy MFT are calculated and simulated by finite-element analysis. Finally, a MFT prototype with an amorphous alloy core is built, and the experiment is carried out.
1 Introduction
The European Union and China both have ambitious plans for renewable energy exploitation. Only the offshore wind power would provide an amazing magnitude [1–3]. The new transmission grid is required, and also the DC transmission technology. Recently, a pure DC transmission grid is proposed for a large scale offshore wind farm [4–6], shown in Fig. 1, which exhibits good
features of low investment cost, low power transmission loss, no offshore operation platform and no reactive power compensation device. Therefore, it is considered as a promising technique for power transmission of an offshore wind farm, in which, large capacity DC/DC converters are the vital components. The DC/DC converter not only plays a role to boost the DC voltage in the wind turbine side to a required voltage level but also adjusts the voltage and current in real time to maintain the system stable.
A multi-level type DC/DC converter is the trend. In Fig. 2, a full bridge DC/DC converter is shown, which is composed of a three level converter, a medium frequency transformer (MFT) and an output rectifier. Obviously, the MFT is a key element. Its loss and efficiency directly affect the efficiency of the power transmission. Therefore, it is necessary to optimise the core material and design of the transformer. Since the magnetic flux density of the magnetic core Bm and the current density of the
winding j would directly affect the loss and efficiency, the optimisation of the MFT is an essence of a multi-objective and multi-variable optimisation problem [7].
There are many different optimisation methods for transformer design. The Monte-Carlo method was firstly put forward [8], while the computation for optimal solution is huge. Another optimisation method called the improved Powell method was proposed for the optimisation design of the transformer [9]. It was pointed out that when the original design scheme was used as the initial point, the convergence speed was fast. Especially, when the number of optimised variables was more, the advantage was more obvious. The simulated annealing method [10] was also used to optimise the power transformer, which took the initial cost as the objective function. However, the initial selection and management problems of the control parameter T needed further research in the practical application. On the other hand, the cyclic variable method [11] was introduced to the optimisation design for the transformer, which judged all possible combinations of various variables and calculated the objective functions of all combinations satisfying the condition. Meanwhile, the objective functions were sorted by size to select the optimal solution. However, the method would take longer time when the number of variables was more and the range
Fig. 1 Pure DC grid for offshore wind farm
Fig. 2 Full bridge DC/DC converter
was larger. Therefore, some efforts are made to solve the problem of low-computation efficiency. The quantum behaved particle swarm optimisation algorithm was proposed [12], which studied the fixed value control strategy of the expansion contraction factor and the linear control strategy in the quantum behaved particle swarm. The quantum behaved particle swarm optimisation algorithm improved the efficiency of the transformer optimisation. Nevertheless, it was very easy to be trapped in local optimal solution if an appropriate parameter control strategy was not selected. Recently, a genetic algorithm was applied to the optimisation design of a power transformer [13], which could effectively find the global minimum point and the approximate global minimum point. With this method, a 250 kVA three phase high-temperature superconducting transformer is built [14].
As such, the use of the genetic algorithm for MFT design is yet the main motivation to find an optimal solution. This paper is organised as follows. The loss calculation and the objective functions of MFT are introduced in Section 2, which includes the calculation formulae of the core loss and the winding loss. Besides, the multi-objective functions are also built in Section 2. In Section 3, the optimisation designs of the MFT based on the multi-objective genetic algorithm (MOGA) are presented. Meanwhile, the design results are compared with the empirical method. In Section 4, the verification of the temperature field and the electric strength by finite element analysis (FEA) was carried out. Also, a MFT prototype with an amorphous alloy core was manufactured.
2 Loss calculation and optimisation objective
functions
2.1 Loss calculation of MFT
2.1.1 Core loss calculation of MFT: Under the condition of
sinusoidal excitation, there are two approaches to calculate the core loss. One is the loss separation method. The other is Steinmetz experience formula.
The loss separation method was proposed by Bertotti, and the core loss is divided into magnetic hysteresis loss, eddy current loss and residual loss. However, the Bertotti loss separation method ignores the skin effect of the magnetic core. The formula for calculating the loss [15] is shown in (1).
Pc= ChBmαf + CceBm2f2+ CexBm1.5f1.5, (1)
where Ch, Cce, Cex are the hysteresis loss coefficient, eddy current
loss coefficient and residual loss coefficient, respectively. α is the magnetic flux density index.
However, parameters involved in the loss separation method are more. Meanwhile, for different core sizes, the parameters need to be refitted. Therefore, the loss separation method does not apply to the engineering calculation.
Steinmetz put forward the famous Steinmetz formula in 1982. He fits the sum of the core hysteresis loss, eddy current loss and residual loss by one formula, which greatly simplified the calculation model of the core loss. The Steinmetz experience formula is shown as
Pv= CmfαBmβ, (2)
where Pv is the loss power of per unit volume core, and its unit is
kW/m3. f is the working frequency and its unit is kHz. B
m is the
amplitude of the magnetic flux density, and its unit is Tesla (T).
Cm, α, β are core coefficients which are related to core materials
and working conditions. Generally, the range of possible values is that: 1 < α < 3, 2 < β < 3.
2.1.2 Winding loss calculation of MFT: As the transformer
works under the condition of medium frequency, the skin effect and proximity effect should be taken into account. Therefore, it is necessary to use AC resistance to calculate the winding loss. Dowell put forward the famous Dowell model [16] in 1966 and gave the calculation formula of AC resistance coefficient Kr
Kr= y M(y) + 23(m2− 1)D(y) (3)
Besides, the calculation formulas of y, δ, M(y) and D(y) are as
follows:
y =hδ ,c (4)
δ = 0.071√ f , (5)
M(y) = sinh(2y) + sin(y)cosh(2y) − cos(y), (6)
D(y) = sinh(y) − sin(y)cosh(y) + cos(y), (7) where hc, δ, f and m are conductor thickness, skin depth at 100∘C,
frequency and winding layers, respectively.
From the above formulae, the winding loss of the transformer could be expressed as
Pcopper= KrRdcI2= KrRdc(js)2, (8)
where I is the effective value of the winding current and j is the current density of the winding. s is the cross-sectional area of the conductor and Rdc is the DC resistance of the winding.
2.2 Optimisation objective functions for MFT
The current density determines the cross-sectional area of the winding and the transformer area product (AP) value. Therefore, the current density is the variable. This study considers the AP value of the transformer as the first objective function f1(X), the
core loss as the second objective function f2(X) and the winding
loss as the third objective function f3(X). Moreover, the magnetic
flux density of the core Bm and the current density of the winding j
are considered as the optimisation variables [17, 18]. The three-dimensional mathematical model is built as
min F(X) = min f1(X), f2(X), f3(X), T f1(X) =4KPt mf Bmj f2(X) = CmfαBmβV f3(X) = KrRdc(js)2 (9)
where Pt is the computing power of the transformer and Km is the
window space factor. s is the cross-sectional area of the conductor.
η is the efficiency.
3 Optimisation design of MFT
3.1 Method of MOGA
The genetic algorithm of the multi-objective optimisation problem has five common methods [19, 20] as the following:
i. Weight coefficient method. ii. Parallel selection method. iii. Permutation selection method. iv. Shared function method. v. Mixed method.
In this study, the parallel selection method is adopted for the multi-objective optimisation of the transformer. The basic idea of the parallel selection method is that: firstly, it divides the initial population into three sub-groups and the parallel selection operation is carried out for each sub-group. Secondly, individuals with higher fitness are selected to form a new sub-population. Then, they are merged into a complete group to have the operations
of recombination and variation. Then, the next generation of the complete group is produced. Furthermore, it ceaselessly carries on the process of segmentation, parallel selection, mergence,
recombination and variation. Finally, the Pareto optimal solution for the MFT could be obtained.
A schematic diagram of the parallel selection method for the multi-objective optimisation of the transformer is shown in Fig. 3.
3.2 Optimisation design based on MOGA
The material of the MFT core is the first element. The common materials are silicon steel sheets, ferrite and amorphous alloys. A silicon steel sheet has a high saturation magnetic induction strength, while its loss in high frequency is very large. When the temperature rises too high, the ferrite would lead to the loss of magnetism, which is not suitable for a high power transformer. Under the condition of medium frequency, the strip of amorphous alloy materials is thinner than that of the silicon steel sheet and ferrite. Besides, the core loss is also less than them.
Assume that the transformer parameters are as follows (Table 1).
For comparison, the optimal design based on MOGA is done for both the transformers of the silicon steel core and amorphous alloy core. According to (9), the corresponding optimisation objective functions are established.
The parameters of MOGA are shown in Table 2.
The program runs once to get a set of Pareto solutions. When the transformer core loss and winding loss is equal, the efficiency of the transformer is the most [21]. The optimal solution is selected from the results of all the run programs. A three-dimensional Pareto solution for the optimisation of the amorphous alloy core transformer is shown in Fig. 4.
The coordinates of these results and the corresponding Bm and j are shown in Table 3. The star point is the optimal solution and the coordinate is 36.1 W (core loss), 36.0 W (winding loss) and 1068.5 cm4 (AP value).
The three-dimensional Pareto solution for the optimisation of the silicon steel sheet core transformer is shown in Fig. 5.
Fig. 3 Schematic diagram of parallel selection method for MOGA
Table 1 Parameters of the transformer
Rated
power, kW Rated inputvoltage, V Rated outputvoltage, V frequency, kHzRated
15 200 600 1
Table 2 Parameters of MOGA
Size
population 100
recombination probability 0.7
variation probability 0.1
generation gap 0.9
maximum genetic algebra 30
Fig. 4 Three-dimensional Pareto solution of MFT with amorphous alloy
core
Table 3 Coordinates of three-dimensional Pareto solution
of MFT with an amorphous alloy core
Bm, T J, A/mm2 Core loss, W Winding loss, W AP value, W
0.760 2.05 31.7 33.0 1203.5 0.812 2.34 35.5 43.1 986.4 0.952 2.11 46.8 35.0 933.4 0.522 2.41 16.5 45.7 1489.4 0.582 2.31 19.9 42.0 1393.4 0.878 2.25 40.7 39.8 949.1 0.928 2.02 44.8 32.1 1000.6 0.602 2.48 21.1 48.4 1255.4 0.722 2.25 29.0 39.8 1153.7 0.631 2.13 22.9 35.7 1393.7 0.999 2.48 51.0 48.4 756.3 0.654 2.00 24.4 31.4 1433.5 0.702 2.12 27.6 35.4 1259.3 0.972 2.41 48.6 45.7 800.4 0.788 2.33 33.7 42.7 1020.8 0.820 2.14 36.1 36.0 1068.5 0.541 2.28 17.5 40.9 1519.4 0.532 2.25 17.0 39.8 1566.6 0.746 2.42 30.6 46.0 1038.6 0.783 2.12 33.4 35.3 1128.9 0.878 2.20 40.7 38.0 971.1 0.852 2.13 38.6 35.5 1035.6 0.823 2.25 36.4 39.8 1012.1 0.965 2.34 48.0 43.0 830.3 0.802 2.39 34.8 44.7 979.8 0.653 2.32 24.3 42.3 1237.1 0.987 2.38 49.9 44.3 799.5
Fig. 5 Three-dimensional Pareto solution of MFT with silicon steel
The coordinates of these results and the corresponding Bm and j are shown in Table 4. The star point is the optimal solution and its coordinate is 114.6 W (core loss), 114.9 W (winding loss) and 2136.0 cm4 (AP value). The optimised models of the two kinds of
transformers built in FEA are shown in Fig. 6.
To make the highest efficiency, the transformer iron loss and copper loss should be almost the same. Based on this view, a group of the most appropriate Pareto optimal solution is achieved to realise the copper loss minimised. The optimisation results are shown in Table 5.
3.3 Comparison between MOGA and empirical method
Moreover, it is also compared with the transformers with two kinds of cores which are designed by the empirical method. Optimised results of the two kinds of transformers with MOGA and empirical methods are shown in Table 6.
It shows that the MOGA method has the advantages of low core loss, low winding loss and high efficiency. Also, it could be found that the transformer of the amorphous alloy core is the most efficient and the lowest loss.
4 Verification
With the increase of temperature, the loss of the transformer will increase. Therefore, the temperature of the transformer should be verified before it is built. However, the existing calibration formula is only suitable for small power transformers with temperature within 60°C [22]. Therefore, it has to be calculated by the finite element method to simulate the temperature field of the optimised transformer.
The Ansys and Ansoft Maxwell are combined to have a thermal analysis on the transformer. Besides, the optimisation model of the transformer has been established in the Ansoft Maxwell and then import it into Ansys for the analysis of the temperature field. Due to the symmetry of the model, 1/8 of the amorphous alloy transformer model is taken for thermal analysis. The temperature field of the amorphous alloy transformer is shown in Fig. 7. The comparison results of the characteristic parameters of the amorphous alloy material and the actual simulation parameters are shown in Table 7. It could be found that the maximum temperature of the transformer is lower than the allowable working temperature. That is to say, the temperature rise of the optimised amorphous alloy transformer meets the requirements.
Insulation performance is another issue for the design of the MFT. Before the transformer is built, it is necessary to check the electrical strength to improve the reliability.
A two-dimensional simulation of an electric field cloud chart is made to analyse the transformer and the simulation adopts the voltage as the exciting source. It is assumed that if the insulation of
Fig. 6 Contours of the two kinds of transformers
(a) Model diagram of amorphous alloy core transformer, (b) Model diagram of silicon
steel sheet core transformer
Table 4 Coordinates of three-dimensional Pareto solution
of MFT with a silicon steel sheet core
Bm, T J, A/mm2 Core loss, W Winding loss, W AP value, W
0.417 2.00 113.1 105.2 2248.4 0.520 2.18 168.4 125.0 1654.0 0.492 2.40 152.6 151.5 1586.5 0.799 2.44 364.9 157.2 959.8 0.414 2.30 111.6 139.1 1969.7 0.400 2.50 105.1 163.7 1877.9 0.430 2.44 119.7 156.9 1784.8 0.551 2.22 186.9 129.6 1532.8 0.688 2.41 278.7 152.8 1130.8 0.775 2.50 345.3 164.0 968.9 0.502 2.12 158.0 118.2 1761.8 0.511 2.32 163.2 141.6 1581.6 0.618 2.35 229.7 145.2 1291.1 0.420 2.09 114.6 114.9 2136.0 0.646 2.44 248.8 156.6 1189.5 0.767 2.04 338.9 109.5 1198.3 0.722 2.39 303.9 150.2 1086.6 0.587 2.03 209.4 108.8 1570.4 0.465 2.23 137.7 130.8 1808.2 0.603 2.15 219.8 121.6 1446.3 0.562 2.38 193.6 149.0 1401.8 0.420 2.28 114.6 136.7 1958.0 0.740 2.43 317.7 155.3 1042.7 0.770 2.29 341.3 137.9 1063.3
Table 5 Results of the proposed method
Silicon steel
sheet Amorphous alloy
design flux density, T 0.42 0.82
winding current density, A/mm2 2.09 2.14
core loss, W 114.6 36.1
winding loss, W 114.9 36.0
efficiency, % 98.47 99.52
Table 6 Optimised results of the two kinds of transformers
Silicon steel sheet Amorphous alloy Empirical
method Proposedmethod Empiricalmethod Proposedmethod
design flux density, T 0.5 0.42 0.9 0.82 winding current density, A/mm2 2.2 2.09 2.4 2.14 core loss, W 156.9 114.6 42.5 36.1 winding loss, W 127.3 114.9 45.3 36.0 efficiency, % 98.11 98.47 99.41 99.52
Fig. 7 Temperature field of the amorphous alloy transformer
the transformer is not broken down at the peak of the voltage, and then it would never be broken down under the other voltage value. Therefore, the instantaneous values of the primary side voltage and secondary side are set 200 2 V and 600 2 V, respectively. The load voltage of the core is set 0 V. The electric field cloud chart of the transformer and the circle section is a magnified drawing of the local strong electric field region shown in Fig. 8, where the maximum electric field strength is 0.73 kV/mm, while the insulation strength of the insulation material is 25–40 kV/mm. Consequently, the electrical strength is enough to meet the insulation requirements.
According to the optimisation results, the MFT prototype with an amorphous alloy core was manufactured, shown in Fig. 9.
The experiment was carried out and the result is shown in Fig. 10.
5 Conclusions
A DC/DC converter is a vital component for offshore DC grids, in which the MFT is the main element. Therefore, it is necessary to optimise the core material and the design method so that the loss of the MFT would be reduced. In this study, MOGA is used to optimise the transformer in the offshore DC grids, and the
optimised characteristics of the transformer are compared with that of the traditional empirical method. The amorphous alloy material is selected for the MFT, which significantly reduces the loss of the transformer. The temperature field and the electric strength of the amorphous alloy MFT are verified by FEA. According to the verification results, the amorphous alloy MFT prototype is manufactured. It was found that the MOGA gives a significant support for the optimisation design of the MFT.
6 Acknowledgments
The authors are grateful for the support by the National Natural Science Foundation of China (51607084), the National Key Research and Development Program of China (2017YFB0903504), the Natural Science Foundation of Jiangsu (BK20130740), the Jiangsu Key Research and Development Program (BE2014876, BE2017169), and 12th Six talent peaks project (2015-ZNDW-008) in Jiangsu Province.
7 References
[1] Vrana, T., Torres-Olguin, R., Liu, B.: ‘The north Sea super grid-a technical perspective’. 9th IET Int. Conf. on AC and DC Power Transmission (ACDC, 2010), 2010
[2] Torres-Olguin, R., Garces, A., Molinas, M.: ‘Integration of offshore wind farm using a hybrid HVDC transmission composed by the PWM current-source converter and line-commutated converter’, IEEE Trans. Energy Convers., 2013, 28, (1), pp. 125–134
[3] Feltes, J., Gemmell, B., Retzmann, D.: ‘From smart grid to super grid: solutions with HVDC and FACTS for grid access of renewable energy sources’. 2011 IEEE Power and Energy Society General Meeting, 2011, pp. 1–6
[4] Ortiz, G., Biela, J., Bortis, D.: ‘1 megawatt, 20 kHz, isolated, bidirectional 12 kV to 1.2 kV DC-DC converter for renewable energy applications’. 2010 Int. IEEE Power Electronics Conf. (IPEC), 2010, pp. 3212–3219
[5] Nishikata, S., Fujio, T.: ‘A new interconnecting method for wind turbine/ generators in a wind farm and basic performances of the integrated system’, IEEE Trans. Ind. Electron., 2010, 57, (2), pp. 468–475
[6] Lundberg, S.: ‘Wind farm configuration and energy efficiency studies: series DC versus AC layouts’ (Chalmers University of Technology, 2006) [7] Yang, H.N., Zhang, Y. S., Liu, G.: ‘Multi-objective optimization of electronic
transformer based on an improved NSGA-II algorithm’, Sci. Technol. Eng., 2015, 15, (19), pp. 139–145
[8] Andersen, O. W.: ‘Optimized design of electric power equipment’, IEEE Comput. Appl. Power, 1991, 4, (1), pp. 11–15
[9] Wang, M.: ‘Research on optimization design of amorphous alloy distribution transformer’ (South China University of Technology, 2014)
[10] Padma, S., Bhuvaneswari, R., Subramanian, S.: ‘Optimal design of power transformer using simulated annealing technique’, IEEE Trans. Ind. Technol., 2006, 55, (6), pp. 1015–1019
[11] Zheng, L.: ‘Optimum design of oil-immersed transformer with Tridimensional wound core’ (South China University of Technology, Guangzhou, 2013) [12] Pan, Z., Zhang, Z., Pan, X.: ‘Optimal design of power transformers using
quantum-behaved particle swarm optimization’, J. Electr. Eng., 2013, 28, (11), pp. 42–47
[13] Fan, S., Wang, G., Xie, W.: ‘Application study of genetic algorithm in optimal design of power transformer’, Proc. Chin. Soc. Electr. Eng., 1996, 16, (5), pp. 346–348353
[14] Daneshmand, S.V., Heydari, H.: ‘A diversified multiobjective simulated annealing and genetic algorithm for optimizing a three-phase HTS transformer’, IEEE Trans. Appl. Supercond., 2016, 26, (2), pp. 1–10 [15] Boglietti, A., Cavagnino, A., Lazzari, M.: ‘Predicting iron losses in soft
magnetic materials with arbitrary voltage supply: an engineering approach’, IEEE Trans. Magn., 2003, 39, (2), pp. 981–989
[16] Dowell, P.L.: ‘Effects of eddy currents in transformer windings’, Proc. Inst. Electr. Eng., 1966, 113, (8), pp. 1387–1394
[17] Du, Y., Baek, S., Bhattacharya, S.: ‘High-voltage high-frequency transformer design for a 7.2 kV to 120 V/240 V 20 kVA solid state transformer’. IECON
Table 7 Temperature verification of the amorphous alloy
transformer
Core material Amorphous alloy
simulated maximum temperature, °C 96
curie temperature, °C 392
allowable working temperature, °C < 130
Fig. 8 Electric field cloud chart of the transformer
Fig. 9 Prototype of the MFT
Fig. 10 Experiment for the prototype
2010-36th Annual Conf. on IEEE Industrial Electronics Society, 2010, pp. 493–498
[18] Wang, G.: ‘Optimal design of high-power electronic transformer’ (North China Electric Power University, 2012)
[19] Adly, A.A., Abd-El-Hafiz, S.K.: ‘A performance-oriented power trans-former design methodology using multi-objective evolutionary optimization’, J. Adv. Res., 2015, 6, (3), pp. 417–423
[20] Lei, Y.: ‘MATLAB genetic algorithm toolbox and application’ (Xi'an Electronic and Science University Press, 2005), pp. 45–48
[21] Montoya, R., Mallela, A., Balda, J.C.: ‘An evaluation of selected solid-state transformer topologies for electric distribution systems’. 2015 IEEE Applied Power Electronics Conf. and Exposition (APEC), 2015, pp. 1022–1029 [22] Billings, K.H.: ‘Switchmode power supply handbook’ (McGraw-Hill Inc.,
USA, 1989)
