IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010
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Optimal Design of Doubly Fed Induction Generators Using Field Reconstruction Method Wei Wang, Morgan Kiani, and Babak Fahimi Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76019 USA Doubly fed induction generator is attracting attention among options in distributed wind energy harvest. Traditional design and analysis of doubly fed induction generator (DFIG) dominantly rely on lump-parameters and finite-element models. Although finite-element analysis (FEA) method is available, the computational time limits its application in iterative optimal design practices. This paper is based on the use of field reconstruction method (FRM) which can greatly reduce the computation cost, whereas maintaining acceptable accuracy. In order to conduct efficiency optimization, the procedure to calculate flux density and core losses are described. Finally, an optimal design method of DFIG towards maximum annual energy production in a given area with available wind speed information is presented. Index Terms—Doubly fed induction generator (DFIG), field reconstruction method, optimal design, wind profile.
I. INTRODUCTION
II. FIELD RECONTRUCTION MODEL OF DFIG
OUBLY FED INDUCTION GENERATOR (DFIG) is used in more than 70% of the commercial available high-power wind turbines due to its superior advantages including variable operation speed, low converter cost, and the flexibility of power control [1]. DFIG is a wound rotor induction machine. Although there are significant number of papers on optimal design of induction machine, it is still valuable to discuss the optimal design of DFIG in the context of wind energy harvesting application. Traditional design methods are based on the evolutionary method and the lumped parameters model. It also relies heavily on the experience of engineers. Although finite-element analysis (FEA) method is available, its high computation cost has limited its application in iterative optimal design of induction machine. In fact, either super computer or parallel computation has to be used if the FEA optimal design is involved. Several authors have proposed optimal design methods of electric machines using FEA coupled with alternative optimization algorithms with relatively lower computational effort. References [2]–[4] evaluated the performance by analyzing the model extracted from the FEA. Reference [5] does the parallel computation via internet instead of traditional parallel hardware as vector computer. In this paper, the field reconstruction (FR) method [6] is applied to the optimal design of DFIG. FR method can greatly reduce the computation time, while maintaining a good accuracy compared with FEA. The machine performances are evaluated directly by field data not the lumped parameters analytical model. Also, the over all efficiency over one year operation is optimized; with the input mechanical power converted from wind profile whereas the traditional method usually focus on rated power efficiency.
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Manuscript received December 22, 2009; revised February 10, 2010; accepted February 12, 2010. Current version published July 21, 2010. Corresponding author: W. Wang (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2043934
A. Field Reconstruction (FR) Method FEA is the mainstream tool in the design and analysis of electric machines. However, an optimal and iterative design of electrical machines, using FEA, typically requires much more computation cost than lump-parameters model. In the absence of magnetic saturation, FR method [6] can offer the distribution of the flux density in the middle of air gap with acceptable accuracy using superposition of two preexisting magneto-static FE field solutions. In this paper, the FR method is extended to the whole machine instead of air gap. It is understood that distributed wind generators that are grid connected should not operate under saturated conditions to avoid emission of undesirable harmonics. Although FEA is used to obtain basis functions, the magnitude of stator and rotor excitation is kept to a level that does not cause any magnetic saturation. Therefore, the basis functions do not include any magnetic nonlinearity. With the linearity assumption, the normal and tangential flux densities in DFIG can be viewed as the superposition of the stator and rotor contributions (1) (2) refer to In (1) and (2), normal component of flux density generated by stator and rotor denote the 3 phases currents. tangential components of flux density generated by stator and can be defined as rotor excitations, respectively. (3) (4) are the basis functions only In which, related with the geometry parameters of DFIG. The radius and refer to polar position relative to stator axis. Due to angle the symmetry of the stator, the flux component of stator can be expressed as
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Fig. 1. Normal flux density in the middle of air gap calculated by FEA and FR. (a) FR method and FEA result. (b) The error between FEA and FR result.
Fig. 2. Tangential flux density in the middle air gap calculated by FR and FEA. (a) FEA and FR result. (b) The error between FEA and FR result.
(6) where is the number of pole pairs. Since the structure of the rotor is similar to that of stator, the rotor flux component is given by (7) (8) (9) B. Basis Functions Extraction Magneto static FEA of DFIG model in commercial package Maxwell is used to compute the basis functions . By setting the A, while other phases of stator and rotor are and can be kept open, basis functions obtained using a single FEA field solution. Similarly, the basis functions of rotor flux density can be calculated. One 4 poles, 3 phase DFIG is studied in this paper. The machine parameters are described in Appendix. Figs. 1 and 2 show the normal and tangential flux densities in the middle of air gap calculated by A, FR and FEA methods when the excitation is 3 A, A at stator phase a, b, c, respectively. The rotor excitation is the same with that of the stator. Our results show that FRM is typically two orders of magnitude faster than FEA and can have a high impact on computational time and resources. The differences between the two methods are virtually nonexistent. III. PERFORMANCE EVALUATION A. Loss Calculation In this paper, we have extended the FR method to compute the local distribution of the flux within and throughout rotor and
Fig. 3. Stator core loss by FEA and FR method.
stator laminations. Starting from the knowledge of the flux distribution in the machine lamination, traditional core loss model [7] of the ferromagnetic materials is applied to calculate the core loss. Since the normal and tangential component of magnetic field is already computed by FR method, the calculation of core loss distribution is straight forward according to (10) (10) refer to the hysteresis loss, eddy current in which, loss, and excess loss coefficients, respectively, which only reis the magnitude of flux lated with material properties and density. As the rotor core loss is negligible compared with stator core loss, it is not considered in the proposed optimization method. Also, due to the symmetrical structure of stator, only one tooth area’s core loss over one electrical cycle is calculated then multiplied by the total tooth number to obtain the total core loss. Fig. 3 shows the stator core loss of the studied model calculated by FEA and FR method. The final error is less than 0.5%. FEA needs to calculate flux density waveform at every point over one period thus caused the transient in Fig. 3. This process is time consuming. The copper loss is calculated based on the lumped parameters model extracted from the field solution and winding resistances.
WANG et al.: OPTIMAL DESIGN OF DOUBLY FED INDUCTION GENERATORS USING FIELD RECONSTRUCTION METHOD
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In order to determine the stator and rotor current, DFIG is modsynchronous reference frame with stator flux in eled in axis [8]. It is also assumed that the magnetizing current is supplied by stator only. B. Torque Calculation Maxwell Stress Tensor (MST) method [9] is used to calculate the force distribution in the middle of air gap and the torque on rotor. They are thereby (11) (12) (13) and are the normal and tangential flux density where is the stack length and is the component, respectively. radius of the integral contour. IV. OPTIMIZATION METHOD A. Optimization Procedure Starting with the initial machine geometry parameters, the basis functions are extracted from two static FEA solutions. Once the FR model is built, the loss under different conditions can be calculated as described previously. Different objective functions and boundary conditions can be built according to different optimization requirements. In this paper, the primary goal is to minimize the total loss (maximize the overall efficiency) in a one year operation period. The objective function is the sum of the loss over one year. In order to solve the constrained nonlinear multivariable optimization problem, sequential quadratic programming (SQP) methods, which is part of commercial package Matlab optimization toolbox, is used. Specifically, the solver fmincon is applied in this optimization. Each time the new geometric parameters are generated by fmincon, the FR model is updated from the FEA solutions of the new machine model. The objective function is evaluated based on the new FR model. Fig. 4 described the optimization procedure. B. Coupling Commercial Package Maxwell With Matlab Commercial package Maxwell is a convenient tool for FE analysis of the electromagnetic devices. Matlab is a powerful for computation and programming. By using the Maxwell scripts, the combined simulation between Matlab and Maxwell can be created. This enables automatic updating of basis functions each time the new machine parameters generated. V. OPTIMAL DESIGN OF DFIG FOR MAXIMUM OVERALL EFFICIENCY
Fig. 4. Optimization procedure.
Fig. 5. Wind turbine P.U. output power probability distribution.
in which, is the wind speed, is average wind speed. However, in the practical design, it is better to take advantage of specific site wind speed distribution data if it is accessible. In actual operational condition, due to pitch control and limitation of wind turbine, the output power distribution is totally different from the probability density described by (14). Once, the turbine reaches its rated speed, the output power is controlled to remain constant. Assume the wind farm has an average wind speed of 13 m/s with 13 m/s rated speed turbine installed, combing the wind speed distribution function and normalized turbine output power data [10], the output power distribution is calculated and plotted in Fig. 5. The optimization goal in this paper is to minimize the total annual energy loss under such input mechanical power condition.
A. Wind Speed Distribution and Power Analysis
B. Objective Functions
One drawback of the wind energy is that the available power relies heavily on the weather condition. There would be large fluctuation in wind energy due to the uncertainty of wind speed. To facilitate the optimal design, the Rayleigh distribution, which has reasonable approximation of real wind speed condition, is applied in this paper. The probability density function is given by
Typically, the optimization design of electric machines is aimed to maximize the efficiency at the rated operational condition. However, due to the uncertainty of wind speed, DFIG often works off the rated condition. It is valuable to evaluate its efficiency over one year operation period. Thus, the objective function is defined in (15)
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TABLE II DFIG PARAMETERS
Fig. 6. Design variables definition. TABLE I OPTIMAL DESIGN RESULTS
single iteration whereas FR methods only costs 3 min with same time step of 0.1 millisecond. The computation time advantage is 45:1. Totally, the proposed optimization method takes about 5 h (100 iterations) to finish the optimization. VI. CONCLUSION An optimal design method of DFIG based on FR and FEA, aimed at maximum efficiency over one year operation period, is proposed in this paper. Using the proposed method, optimal design can be achieved while the computation cost would be significantly reduced. The optimal design result of DFIG shows the advantage of this method.
Fig. 7. Efficiency at different power level.
where is the power loss at different input power, the probability of input power.
is
APPENDIX The DFIG model parameters are listed in Table II.
C. Optimal Design The wind farm is assumed to have average wind speed of 13 m/s which is same as the rated speed of wind turbine. This assumption is different from [3] because it is unreasonable to install high rated wind speed turbine in low-speed wind farm. The wind turbine output power is the actual experimental data from a 13 m/s wind turbine [10]. The design variables are selected as stator tooth width and the depth of stator slot, as described in Fig. 6. Although we only defined two variables for the purpose of validating the proposed optimization method, more variables can be defined if the need arises. D. Results The optimal design results are compared in Table I. Although the overall efficiency is only improved by 1.1%, the loss is reduced by 17%. The efficiency at variable power level is compared in Fig. 7. The optimal model showed lower efficiency at low-power level and higher efficiency at high-power level. This confirms the power probability distribution illustrated in Fig. 5. The efficiency is high at the point where the maximum energy is produced. On a platform of 2.99 GHz Intel Core 2 Duo machine, it takes FEA transient analysis 135 min to calculate the total loss for
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