DFIG Machine Design for Maximizing Power Output Based on Surrogate Optimization Algorithm

This paper presents a surrogate-model-based optimization of a doubly-fed induction generator (DFIG) machine winding design for maximizing power yield. Based on site-specific wind profile data and the machine's previous operational performance, the DFIG's stator and rotor windings are optimized to match the maximum efficiency with operating conditions for rewinding purposes. The particle swarm optimization-based surrogate optimization techniques are used in conjunction with the finite element method to optimize the machine design utilizing the limited available information for the site-specific wind profile and generator operating conditions. A response surface method in the surrogate model is developed to formulate the design objectives and constraints. Besides, the machine tests and efficiency calculations follow IEEE standard 112-B. Numerical and experimental results validate the effectiveness of the proposed technologies.


I. INTRODUCTION
W ITH THE increasing concern over global warming and the depletion of fossil fuels, the development of renewable energy technologies is becoming increasingly important. Among renewable sources of energy, wind energy plays a critical role in the establishment of an environmentally sustainable low carbon economy. Globally, the majority of installed medium and large-sized wind turbines employ doubly-fed induction generators (DFIGs) to provide the variable-speed operation needed to harvest wind energy. In general, the DFIGs available on the market are standard machines which may not be the best option for a specific site. These machines tend to be large and prone to faults, and thus they need to be repaired or rewound when they fail. Currently, the common practice in the machine repair industry is to return to the original machine design as close as possible. As a consequence, a unique opportunity of redesigning the failed machines is wasted [1]. Manuscript  This study takes a different view by optimizing the DFIG machine design at the rewinding stage, based on an in-depth understanding of the site-specific information (i.e., wind profile) and the machine's previous performance (i.e., maximum efficiency and operating points). In this case, only the stator and rotor windings are the refinement parameters as they are the components to be replaced during the rewinding process. For large DFIG machines, a small improvement in efficiency will lead to a significant energy saving and environment benefits [1]. In the European Union, a 3% increase in the energy efficiency of electrical machines would produce electricity savings of over $2 billion per year [2]. As a result, the economic implication is also significant when improving machine design for wind turbine applications.
In fact, the optimization of machine design is a multivariable and multimodal problem [3]. Thus, the multiphysics dimension of electrical machines should be taken into account. The conventional design optimization of DFIGs relies on analytical and empirical methods. With the development of computing techniques and numerical methods, the mainstream design optimization is now based on finite element method (FEM). Accurate FEM models can facilitate the exploration of alternative designs and reduce the resource and time in both design and repair stages. However, there are two challenges.
1) A high-fidelity FEM model is lengthy and complicated to process; it may take hundreds of iterations to arrive a solution. 2) Multiple variable and constraint functions are solved simultaneously while FEM is not suitable for local sensitivity calculations, which are crucial for general gradientbased optimization. In contrast, a surrogate modeling technique is widely used [4] to formulate an explicit relationship between the objective/constraint functions and design variables so as to reduce the FEM computational time. Furthermore, statistical methods are also useful to investigate the correlation between input parameters and numerical simulation outputs to identify significant parameters for an efficient optimal design. The obtained surrogate model can be easily used to evaluate the performance at trial design points.
It is well known that wind speed at a given installation site can vary significantly and are difficult to predict with accuracy. It is of particular importance to comprehend the distribution of wind speed frequency. At present, the Weibull model is popular because of its accuracy and easiness [5], [6]. However, the power losses in wind turbine generators are not considered when calculating the output power of wind turbines. This paper presents a new method based on FEM analysis, surrogate optimization and wind speed distribution for the optimal design of a DFIG in terms of maximum power output. A set of FEM models are first developed to estimate the power output of the DFIG under the characterized distribution of wind speeds at a specific site. Second, a surrogate model is constructed to approximate the input-output model and refine the machine design. Finally, the optimized design is verified by numerical and experimental tests.

II. WIND TURBINE MODEL
A schematic diagram of the DFIG wind turbine system is shown in Fig. 1. The wind turbine is connected to the DFIG rotor through a drivetrain system, which contains high and low speed shafts, bearings and gearboxes. The DFIG is constructed from a wound rotor induction machine where its stator is directly connected to the grid and its rotor is fed by bidirectional voltagesource converters, which are two four-quadrant IGBT pulse width modulation (PWM) converters [i.e., rotor side converter (RSC) and stator side converter (GSC)] connected back-to-back by a dc-link capacitor [7]. The crowbar is used to short-circuit the RSC in order to protect the RSC from overcurrents in the rotor circuit during transient disturbances [8]. The speed and torque can be regulated by controlling the RSC.

A. Wind Energy Estimation
It is possible to characterize a wind turbine power curve based on available wind energy and the rotor power coefficient, C p . In general, C p can be expressed as a function of the tip speed ratio λ, which is defined by where ω is the angular speed of the wind turbine rotor, R is the radius of the wind rotor, and U is the wind speed. For a wind turbine, C p is represented by a nonlinear curve in terms of λ and the pitch angle β [9] C p = (0.44 − 0.0167β) sin π(λ − 2) 13 − 0.3β − 0.00184(λ − 2)β.
(2) Therefore, the average wind machine power is found by where ρ is the air density, η is the drive train efficiency (generator power/rotor power). In practice, wind turbine is controlled to operate at a maximum power output.

B. DFIG Mathematical Model
The DFIG model in the d-q reference frame is given by ψ ds = (L ls + L m )i ds + L m i dr (8) ψ qs = (L ls + L m )i qs + L m i qr (9) ψ dr = (L lr + L m )i dr + L m i ds (10) ψ qr = (L lr + L m )i qr + L m i qs (11) where V ds and V qs are the d-and q-axis stator voltages; V dr and V qr are the d-and q-axis rotor voltages; i ds and i qs are the dand q-axis stator currents; i dr and i qr are the d-and q-axis rotor currents, respectively. Rs and R r are the phase resistances of the stator and rotor; L ls and L lr are the leakage inductances of the stator and rotor; L m is the magnetizing inductance; ω s and ω r are the synchronous speed and the rotor speed of the DFIG, respectively.

C. Finite Element Model
A 2-D model of DFIG is built by the FE software Infolytica MagNet, as shown in Fig. 2. Its ratings are listed in Table I. In this model, the skin effect was ignored. The finite element model of DFIG is given by [10]  where A Z is the axial components of the magnetic vector potential; J Z is the source current density; v is the material relativity; A Z 0 is the given values for the boundary τ . It is well known that the permeability of ferromagnetic materials is much greater than that of air so that the magnetic field lines are parallel to the boundary, i.e., Based on the numerical model, the iron loss can be calculated, making it is easier to focus on the winding design (material and size).

III. WIND SPEED DATA ANALYSIS
Actual wind speed data from a U.K. wind farm at Albemarle is analyzed by statistical methods to derive the wind speed probability density function. In the literature, the Pearson model, Rayleigh model and Weibull model are widely used to fit the distribution of wind speed frequency. However, it is found from a large number of measured data that the two-parameter Weibull distribution is a good representation of the wind speed [11] and thus adopted in this paper.

A. Weibull Model
Using the Weibull model, it is very accurate to analyze the wind speed data based on the shape factor k and the scale factor c. The Weibull probability distribution function is obtained by where v is the wind speed and the probability density is

B. Statistic Estimation Method (SEM)
In order to estimate the Weibull probability density function, the SEM is first used to analyze the wind speed data in terms of the wind average velocity v and the standard deviation σ. v can reflect the main tendency of data change and σ represents the extent of deviation from the mean [5]. v and σ are given by k and c can be expressed as However, it is quite complex to solve the Gamma function directly. Instead, these formulas are often solved by the empirical approach [12] As a result, the mean wind speed can be calculated by Following this procedure, the Weibull distribution for this wind farm is analyzed and presented in Fig. 3. A good fit can be found when compared with measured wind speed data.

IV. PROPOSED OPTIMIZATION ALGORITHM
The surrogate-based analysis and optimization is an effective tool for the design and optimization of computationally expensive models, and is widely used for airfoil shape optimization, mechanical structure and so forth. Typically, surrogate models can be comprehended as a nonlinear inverse problem, which is to determine a continuous function f of a set of design variables from a limited amount of available data f. But model estimation and error assessment are difficult. In this case, the predicted formula for the FEM output is f p (x) =f (x) + ε(x). In this paper, the Kriging model [13] is adopted to create a surrogate model and a nongradient heuristic search method [14] is used to analyze calculated data by the deterministic algorithm. The proposed optimization algorithm is shown as a flowchart in Fig. 4 and is further explained in four steps.

A. Step 1-Design of Experiment (DoE)
The DoE is a sampling plan in the design variable space [14], which is aimed at maximizing the amount of information acquired and minimizing the bias error. At certain sampling points, there is a clear tradeoff between the number of points selected and the amount of information that can be extracted from these points. Due to the computational resource, the number of sample points is limited severely. On the other hand, the balance between bias and variance errors shall be found during the construction of the surrogate model. Generally, the bias error can be reduced through a DOE by distributing the sample points uniformly in the design space.
One of the most popular DoE for uniform sample distribution is Latin hypercube sampling (LHS) [15]. For arranging p samples with n design variables by LHS, the range of each parameter will be divided into p bins, so that the total number of p n bins will be generated in the design space. The samples are randomly selected in the design space, and each sample will be set randomly inside a bin. Moreover, there is exactly one sample in each bin for all one dimensional projections of the p samples and bins. While LHS represents an improvement over unrestricted stratified sampling [16], it can provide sampling plans with very different performance. In this paper, the LHS approach is adopted. An LHS realization of 18 samples for stator windings with two design variables (n s = 2) and 20 samples for rotor windings with two design variables (n r = 2). Both rotor windings and stator windings have an impact on the machine performance, so an LHS realization of 56 samples for stator and rotor windings with four design variables (n s&r = 4).
Furthermore, the slot filling factor is one of the limit factors in winding design. Typically, it depends on the insulation thickness around the conductors and the slot, as well as the conductor shape. So this limitation shall be included into the LHS design.

B. Step 2-Numerical Simulation at Sampling Points
It is easily found that winding length and size impact on the winding resistance. In this paper, the multivariable optimization issue should consider four variables: stator winding turns, stator winding cross-sectional area, rotor winding turns, and rotor winding cross-sectional area. Among the LHS samples, if their output torque is similar, the sample with a higher efficiency is selected.

C. Step 3-Construction of the Surrogate Model
Normally, there are two surrogate model construction methods: parametric (e.g., polynomial regression, Kriging model) and nonparametric (e.g., projection pursuit regression, radial basis function) methods. The former assumes the relative global functional form between the response variable and the design variable is known while the latter constructs the whole model by using local models in different data regions.
In recent years, Kriging models are popular in dealing with computationally expensive engineering problems [17], [18], and is employed in this study. In the Kriging model, an approximation expression is given by where β is a constant, and z(t) is calculated by Gaussian distribution whose mean and variance are 0 and σ 2 , respectively. Ifŷ(t) is defined as the approximation model, and the mean-squared error of y(t) andŷ(t) are minimum, satisfying the unbiased condition. y(t) is estimated as where R −1 is the inverse of the correlation matrix R, r is the correlation vector, y is ns observed data vector, and q is the unit vector. The correlation matrix and correlation vector are . . , ns, k = 1, . . . , ns) (1) ), R(t, t (2) ), . . . , R(t, t (ns) ) T . (23) The parameters θ 1 , θ 2 , . . . , θ n are unknown, but they can be calculated by using the following equation.

D. Step 4-Particle Swarm Optimization (PSO) Algorithm
In the literature, PSO and genetic algorithm are the two prevalent evolutionary algorithms. They can be used in combination with analytical models [19] and especially the combination with FEA is gaining popularity [20], [21]. It is also suggested that PSO performs better in terms of simple implementation and high computational efficiency with few controlling parameters [22], [23] and thus used in this paper.
In an optimization case, the PSO can be described as an animal or particle moving from a certain position at random velocity in a search field. Within a population (called a swarm), each particle is treated as a point in a d-dimensional design space. Each particle keeps track of its position in the solution space that is associated with the fitness value, termed the personal best (pbest). Meanwhile, there exists a global best fitness value, called global best (gbest). It is achieved by the whole swarm. The operation of PSO is to gradually change the velocity of each particle toward its pbest and gbest positions at each time.
The new velocity and position follow where V k j and X k j are the velocity and location of the jth particle at iteration k; p k j is the pbest of particle j at the kth iteration; p k g is the gbest of the entire swarm at the kth iteration; C 1 and C 2 are acceleration factors; ϕ 1 and ϕ 2 are the uniformly distributed random numbers between 0 and 1; w is the inertia weight that controls the influence of previous velocity in the new velocity. Each particle will try to change its position by four variables, which include the position of the current particle, the velocity of current particle, the distance between the current position and pbest, and the distance between the current position and the gbest. The PSO algorithm is illustrated in Fig. 5.

V. NUMERICAL TESTS AND RESULTS
In this paper, the optimization objective is to design winding size in order to maximize the machine efficiency as (26), by taking into account the design constraints.
2-D variables are added to machine optimization: winding diameter D and winding turns N for rotor and stator, respectively. In total, there are six constraints: stator winding turns, stator winding diameters, rotor winding turns, rotor winding di-

Constraints
T ≥ 360 Nm According to the stator and rotor slot design (see Fig. 6), the stator and rotor slot areas are calculated as 193.25 and 223.28 mm 2 , respectively. Owing to the filling factor, the stator slot area of stator slot is available between 38.65 and 77.3 mm 2 while the rotor's available area is between 44.66 and 89.31 mm 2 .
The test results for the 2-D variables of the stator and rotor are obtained and plotted. Fig. 7 shows the stator winding optimization function for efficiency with the 2-D variables. Fig. 8 presents the rotor winding optimization for efficiency. As can IEEE TRANSACTIONS ON ENERGY CONVERSION Fig. 8. Efficiency contour for rotor winding optimization.  be seen that the local peak is optimized from the original point for both stator and rotor optimization so as to achieve maximum efficiency.
The proposed algorithm and the original plan are compared using the test function given in Tables II-V. It is clear that the proposed algorithm provides excellent results in terms of speed and accuracy. A winding design with four variables (number of turns and the diameter of both stator and rotor) is also  attempted. The optimization results are visually shown in Fig. 9 and summarized in Table VI.
Having established the impacts of the stator and rotor winding parameters on the efficiency of DFIG, the results based on the rated condition can provide guidance for selecting the appreciate stator and rotor windings. However, the DFIG operates mostly between 1000 and 1457 r/min and thus the winding design should consider a wider speed range.

VI. EXPERIMENTAL RESULTS AND ANALYSIS
In this paper, a 55-kW three-phase DFIG is simulated, prototyped and then tested. Its winding design with four variables (the number of turns and the diameter of both the stator and rotor) is optimized. In order to have the maximum power output at variable operating conditions, a power loss balance needs to achieve and the maximum efficiency point needs to move close to the effective operating condition [24]. Because this is a multimodal and multivariable optimization problem, it can generate many maximum efficiency samples. When the efficiency values are similar between these samples, the point with a higher torque is selected.
A test rig is set up for experimental validation, as shown in Fig. 10. A range of experiments are carried out on the machine including no load and load tests. Fig. 11 presents no-load test results from subtracting stator copper loss from the input power. This can be used to find the core loss, frictional and windage losses. A perfect linear curve indicates a good accuracy of the experimental setup and measuring equipment. Therefore, load tests are followed to test the machine from 100% to 50% load according to the standard procedures [25], [26]. Test results are presented in Figs. 12 and 13. Fig. 12 shows that the    Fig. 13 shows the improvement of approximately 1% in machine efficiency across an operational range than the original design.
Next, the prototype DFIG is used to calculate the power output under a field condition to match the specific site. This is achieved by applying the Weibull function of wind power at Albemarle. The detailed data are shown in Tables VII and VIII, and the wind turbine power curve is presented in Fig. 14. Because this wind turbine adopts the pitch-regulated control, the maximum power output is capped at 55 kW. For a given Weibull probability density function (sitespecific), a wind turbine power curve can be attained, and so is the annual energy production. Based on the wind speed data and machine performance, the annual wind energy output at the given wind profile can be found to be approximately 63 MWh, as shown in Table IX. Compared to the original design of the DFIG, the machine efficiency is improved by 1%. In that case, the annual yield increases by about 600 kWh.

VII. CONCLUSION
A surrogate-model-based optimization of a DFIG winding design for maximizing output power has been presented. The machine is matched with a specific site taking account of the actual wind profile and the machine's operational conditions. The particle swarm optimization-based surrogate modeling techniques are used in conjunction with the FEM to optimize the machine.
The key refinement parameters are the stator and rotor winding windings for they are renewed during the repair and rewinding procedure.
A 55-kW DFIG is simulated and experimentally tested to check the effectiveness of the proposed techniques. No-load and load tests have confirmed the numerical and analytical machine models. The further work will extend to design and test two large DFIGs which undergo repeated repairs.