Simulation of grid connection and maximum power point tracking control of brushless doubly-fed generator in wind power system

Abstract

In this paper, based on the analysis of the mathematical model in a common synchronous reference frame of the brushless doubly-fed generator (BDFG), the grid connection strategy and maximum energy extraction control were both analyzed. Besides, the transient simulation of no-load model and generation model of the BDFG have been developed on the MATLAB/Simulink platform. The test results during cutting-in grid confirmed the good dynamic performance of grid synchronization and effective power control approach for the BDFG-based variable speed wind turbines.

Keywords

brushless doubly-fed generator (BDFG) / modeling / grid connection control / back-to-back pulse-width modulation (PWM) converter / wind power generation

Introduction

The brushless doubly-fed machine (BDFM), also known as a self-cascaded machine, is composed of two three-phase windings in the stator of different pole numbers called power winding (PW) and control winding (CW) and a special rotor winding. Typically the two stator supplies are of different frequencies, one a fixed frequency supply connected to the grid via switch, and the other a variable frequency supply derived from a power electronic frequency converter (inverter), as illustrated in Fig. 1. The natural synchronous speed of the machine is equal to

(1){\omega }_r={\displaystyle \frac{{\omega }_p\pm {\omega }_c}{p_p+p_c}},

where {\omega }_p and {\omega }_c are the electrical angular velocities of the PW and CW voltages.

The variable speed constant frequency (VSCF) wind turbine generator system is more important to improve the efficiency by capturing the maximum wind energy and using the high quality, efficient and controllable power, where the major challenge is the independent control of the active and reactive powers exchanged between the BDFG and the grid.

The different control strategies that have been used until now in the BDFM are the scalar current control [1], the direct torque control [2], the L₂ robust control method [3], the H_∞ control [4], the fuzzy power control [5], the sliding mode power control [6], and the rotor flux oriented control [7]. A new vector controller using a dynamic model with a unified reference frame based on the PW flux was investigated for the BDFM by Poza et al. [8], and a simplified controller oriented with the PW stator flux with a complete mathematical derivation frame was presented by Shao et al. [9] with some experimental results presented on both speed and reactive power regulating.

The grid connection condition of the DFIG is that the frequency, amplitude and phase angle of the stator voltage are the same as those of the grid voltage [10-13], which indeed have numerous advantages and flexibilities over the DFIG-based wind system for different speeds. The synchronization between the stator and grid voltages is maintained to overcome the defects of angular velocity constant when using the synchronous generator.

A no-load mode simulation of cascade brushless doubly-fed generator (BDFG) was built [14] according to the d-q mathematical model based on the double synchronous reference frame. This paper presented a novel research focusing on grid-connection BDFG analysis in detail and relying on the vector control algorithm via regulating the power after connection as well as independent control of active and reactive power of PW. The maximum power point tracking (MPPT) was implemented for optimal energy capture by the wind turbine. All the control algorithms were confirmed by simulation study and the system performance were evaluated in detail.

Control mechanism of the maximal wind energy capturing

From the Betz theory, the power captured from wind energy by a horizontal axis wind turbine can be expressed as [3,4]

(2)P={\displaystyle \frac{\pi }{\mathrm{2}}}{C}_p{R}^{\mathrm{2}}\rho {\nu }^{\mathrm{3}},

where \rho is the air density; R, the turbine radius; and\mathit{\nu }, the wind velocity. The power coefficient C_p is a function of the tip speed ratio

(\lambda ={\omega }_{t}R/\nu )

as well as the blade pitch angle\beta; and {\omega }_t is the angular speed of the wind turbine.

(3)C_p(\lambda ,\beta )=\mathrm{0.5176}\left({\displaystyle \frac{\mathrm{116}}{{\lambda }_i}}-\mathrm{0.4}\beta -\mathrm{5}\right)e^{-\frac{\mathrm{21}}{{\lambda }_i}}+\mathrm{0.0068}\lambda ,

where

(4){\displaystyle \frac{\mathrm{1}}{{\lambda }_i}}={\displaystyle \frac{\mathrm{1}}{\lambda +\mathrm{0.08}\beta }}-{\displaystyle \frac{\mathrm{0.035}}{{\beta }^{\mathrm{3}}+\mathrm{1}}}.

Clearly, the turbine speed has to be changed along with the wind speed so that the optimal tip speed ratio is maintained for the maximum power captured and the generator active power matches the output power of the turbine.

Figure 2 shows the curve of the power coefficient versus λ for a constant value of the patch angle β. It is observed from Fig. 2 that there is a value of λ for which C_p is maximized, maximizing thus the power for a given wind speed.

Figure 3 depicts the power-speed characteristics of the wind turbine. It is observed from Fig. 3 that the peak power of each wind speed occurs at the point where C_p is maximized. To maximize the power generated, it is, therefore, desirable for the generator to have a power characteristic that follows the maximum C_{p-\max }line.

To extract the maximum power generated, the λ must be maintained at the optimal command rotor speed{\lambda }_{\mathrm{opt}}. The estimate of wind turbine speed can be obtained by

(5){\omega }_t={\displaystyle \frac{{\lambda }_{\mathrm{opt}}\nu }{R}},

where the relation of the wind turbine speed and the generator speed is

(6){\omega }_m=G{\omega }_t.

The block diagram of the turbine model with the control of the speed is represented in Fig. 4.

The action of the speed corrector must achieve the control of the mechanical speed {\omega }_mwith its reference {\omega }_m^{\ast }. The electromagnetic torque reference is defined as

(7)T^{*}=\left(k_p+k_i{\displaystyle \int }\right)({\omega }_m^{*}-{\omega }_m).

Mathematical model of BDFG

The model of the BDFG in the PW flux frame is expressed as [8,9,15]

(8)\{\begin{array}{l}{V}_{dp}=R_p{i}_{pd}+{\displaystyle \frac{d{\psi }_{pd}}{dt}}-{\omega }_p{\psi }_{pq},\hfill \\ V_{qp}=R_p{i}_{pq}+{\displaystyle \frac{d{\psi }_{pq}}{dt}}+{\omega }_p{\psi }_{pd},\hfill \end{array}

(9)\{\begin{array}{l}{V}_{dc}=R_c{i}_{dc}+{\displaystyle \frac{d{\psi }_{dc}}{dt}}-\left[{\omega }_p-\left(P_p+P_c\right){\omega }_r\right]{\psi }_{qc},\hfill \\ V_{qc}=R_c{i}_{qc}+{\displaystyle \frac{d{\psi }_{qc}}{dt}}+\left[{\omega }_p-\left(P_p+P_c\right){\omega }_r\right]{\psi }_{dc},\hfill \end{array}

(10)\{\begin{array}{l}{V}_{dr}=R_r{i}_{dr}+{\displaystyle \frac{d{\psi }_{dr}}{dt}}-\left({\omega }_p-P_p{\omega }_r\right){\psi }_{qr},\hfill \\ V_{qr}=R_r{i}_{qr}+{\displaystyle \frac{d{\psi }_{qr}}{dt}}+\left({\omega }_p-P_p{\omega }_r\right){\psi }_{dr}.\hfill \end{array}

The flux equations are given as

(11)\{\begin{array}{l}{\psi }_{dp}=L_p{i}_{dp}+M_p{i}_{dr},\hfill \\ {\psi }_{qp}=L_p{i}_{qp}+M_p{i}_{qr},\hfill \end{array}

(12)\{\begin{array}{l}{\psi }_{dc}=L_c{i}_{dc}+M_c{i}_{dr},\hfill \\ {\psi }_{qc}=L_c{i}_{qc}+M_c{i}_{qr},\hfill \end{array}

(13)\{\begin{array}{l}{\psi }_{dr}=L_r{i}_{dr}+M_c{i}_{dc}+M_p{i}_{dp},\hfill \\ {\psi }_{qr}=L_r{i}_{qr}+M_c{i}_{qc}+M_p{i}_{qp}.\hfill \end{array}

The electromagnetic torque is expressed as [16,17]

(14)T_e={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{p}_p{M}_p(i_{qp}{i}_{dr}-i_{dp}{i}_{qr})-{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{p}_c{M}_c(i_{qc}{i}_{dr}-i_{dc}{i}_{qr}).

The active and reactive powers of the PW are defined as

(15)P_p={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}(V_{dp}{i}_{dp}+V_{qp}{i}_{qp}),

(16)Q_p={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}(V_{qp}{i}_{dp}-V_{dp}{i}_{dp}).

Power decoupled control

Control of the BDFG with a PW field oriented

If the d-axis of the PW synchronous reference frame is aligned with the PW air gap flux, the PW

R_p

is neglected. Then, the relation between the PW voltage and its flux is

(17)\{\begin{array}{l}{V}_{dp}=0,\hfill \\ V_{qp}=V_p={\omega }_p{\psi }_p,\hfill \end{array}

(18)\{\begin{array}{l}{\psi }_{dp}=L_p{i}_{dp}+M_p{i}_{dr},\hfill \\ 0=L_p{i}_{qp}+M_p{i}_{qr}.\hfill \end{array}

From Eq. (18), the equations linking the rotor currents to the PW currents are deduced as

(19)\{\begin{array}{l}{i}_{dr}={\displaystyle \frac{{\psi }_p}{M_p}}-{\displaystyle \frac{L_{dp}}{M_p}}{i}_{dp},\hfill \\ i_{qr}=-{\displaystyle \frac{L_{qp}}{M_p}}{i}_{qp}.\hfill \end{array}

PW flux estimator

The derivation in the stationary reference frame (\alpha -\betareference frame) form the PW voltage Eq. (8) is given as

(20)\{\begin{array}{l}{\psi }_{\alpha p}={\displaystyle \int \left(V_{\alpha p}-R_p{i}_{\alpha p}\right)}dt,\hfill \\ {\psi }_{\beta p}={\displaystyle \int \left(V_{\beta p}-R_p{i}_{\beta p}\right)}dt.\hfill \end{array}

The PW flux angle can be expressed as

(21){\theta }_p=\arctan {\displaystyle \frac{{\psi }_{\beta p}}{{\psi }_{\alpha p}}}.

Control of PW current

Suppose that the BDFG is running in a steady-state, then the dynamic model can be transferred to the state model [2,18].

(22)\{\begin{array}{l}{V}_{dp}=R_p{i}_{dp}-{\omega }_p{L}_p{i}_{qp}-{\omega }_p{M}_p{i}_{qr},\hfill \\ V_{qp}=R_p{i}_{qp}+{\omega }_p{L}_p{i}_{dp}+{\omega }_p{M}_p{i}_{dr},\hfill \end{array}

(23)\{\begin{array}{l}{\displaystyle \frac{s_2}{s_1}}{V}_{dc}={\displaystyle \frac{s_2}{s_1}}{R}_c{i}_{dc}-{\omega }_p{L}_c{i}_{qc}-{\omega }_p{M}_c{i}_{qr},\hfill \\ {\displaystyle \frac{s_2}{s_1}}{V}_{qc}={\displaystyle \frac{s_2}{s_1}}{R}_c{i}_{qc}+{\omega }_p{L}_c{i}_{dc}+{\omega }_p{M}_c{i}_{dr},\hfill \end{array}

(24)\{\begin{array}{l}0={\displaystyle \frac{1}{s_1}}{R}_r{i}_{dr}-{\omega }_p{L}_r{i}_r-{\omega }_p{M}_c{i}_{qc}-{\omega }_p{M}_p{i}_{qp},\hfill \\ 0={\displaystyle \frac{1}{s_1}}{R}_r{i}_{qr}+{\omega }_p{L}_r{i}_{dr}+{\omega }_p{M}_c{i}_{dc}+{\omega }_p{M}_p{i}_{dp},\hfill \end{array}

where s_{\mathrm{1}}ands_{\mathrm{2}} are the slips, which are defined as

(25)s_{\mathrm{1}}={\displaystyle \frac{{\omega }_p-p_p{\omega }_p}{{\omega }_p}},s_{\mathrm{2}}={\displaystyle \frac{{\omega }_c-p_p{\omega }_p}{{\omega }_c}}.

Equations (26) and (27) can be obtained by combining Eq. (22) with Eq. (24) and considering Eqs. (17) and (19), and neglecting the PW resistance.

(26)i_{dc}={\displaystyle \frac{(L_r{L}_p-M_p^{\mathrm{2}})i_{dp}}{M_p{M}_c}}-{\displaystyle \frac{{\psi }_p{L}_r}{M_p{\omega }_p{M}_c}}+{\displaystyle \frac{R_r{L}_p}{M_p{M}_c{\omega }_p{s}_{\mathrm{1}}}}{i}_{qp},

(27)i_{qc}={\displaystyle \frac{(L_r{L}_p-M_p^{\mathrm{2}})i_{qp}}{M_p{M}_c}}+{\displaystyle \frac{R_r{\psi }_p}{M_p{M}_c{\omega }_p{s}_{\mathrm{1}}}}-{\displaystyle \frac{R_r{L}_p}{M_p{M}_c{\omega }_p{s}_{\mathrm{1}}}}{i}_{dp}.

Equations (26) and (27) represent the relationship of the power courant and control wind courant.

The first term of Eqs. (26) and (27) defines the direct coupling between i_c and i_p, while the second term performs as a constant, and the third term reflects the cross coupling.

Control of CW current

Combining Eqs. (9), (12), (19) and (24), the CW voltage can be derived as

(28)V_{dc}=R_c{i}_{dc}+{\displaystyle \frac{L_c{L}_p{L}_r-L_c{L}_p^{\mathrm{2}}-L_p{L}_c^2}{L_p{L}_r-L_p^2}}{\displaystyle \frac{d{i}_{dc}}{dt}}-{\displaystyle \frac{L_p^{\mathrm{2}}{M}_c{R}_r}{s_{\mathrm{1}}{\omega }_p{M}_p(L_p{L}_r-M_p^{\mathrm{2}})}}{\displaystyle \frac{d{i}_{qp}}{dt}}-[{\omega }_p-(P_p+P_c){\omega }_r]\left(L_c{i}_{qc}-{\displaystyle \frac{M_c{L}_p}{M_p}}{i}_{qp}\right),

(29)V_{qc}=R_c{i}_{qc}+{\displaystyle \frac{L_c{L}_p{L}_r-L_c{L}_p^{\mathrm{2}}-L_p{L}_c^2}{L_p{L}_r-L_p^{\mathrm{2}}}}{\displaystyle \frac{d{i}_{dc}}{dt}}+{\displaystyle \frac{L_p^{\mathrm{2}}{M}_c{R}_r}{s_{\mathrm{1}}{\omega }_p{M}_p(L_p{L}_r-M_p^2)}}{\displaystyle \frac{d{i}_{qp}}{dt}}+[{\omega }_p-(P_p+P_c){\omega }_r]\left(L_c{i}_{qc}+M_c{\displaystyle \frac{\left|{\psi }_p\right|-L_p{i}_{dp}}{M_p}}\right).

The first term

R_c{i}_{dc}+{\displaystyle \frac{L_c{L}_p{L}_r-L_c{L}_p^{\mathrm{2}}-L_p{L}_c^2}{L_p{L}_r-L_p^{\mathrm{2}}}}{\displaystyle \frac{d{i}_{dc}}{dt}}

shows the direct relation between V_{qc} and i_{qc}.

The second term

{\displaystyle \frac{L_p^{\mathrm{2}}{M}_c{R}_r}{s_{\mathrm{1}}{\omega }_p{M}_p(L_p{L}_r-M_p^{\mathrm{2}})}}{\displaystyle \frac{d{i}_{qp}}{dt}}

represents the cross coupling which can be neglected in the steady-state.

The third term

[{\omega }_p-(P_p+P_c){\omega }_r]\left(L_c{i}_{qc}-{\displaystyle \frac{M_c{L}_p}{M_p}}{i}_{qp}\right)

shows another cross coupling which can be neglected compared with the direct coupling term.

A similar derivation can be applied to the analysis of Eq. (28). Therefore, V_c and i_c can be a first order relation.

Grid synchronization of BDFG

Before grid connection, the generator operates with the no-load model. That is to say, the PW current is zero. The switch between the grid and the PW is opened before grid connection, under the constraint conditions of

(30)i_{dp}=\mathrm{0},i_{qp}=\mathrm{0.}

Based on Eqs. (30) and (8) to (13), which is the no-load mathematical model for the BDFG,

(31)\{\begin{array}{l}{V}_{dp}={\displaystyle \frac{d{\psi }_{pd}}{dt}}-{\omega }_p{\psi }_{pq},\hfill \\ V_{qp}={\displaystyle \frac{d{\psi }_{pq}}{dt}}+{\omega }_p{\psi }_{pd},\hfill \end{array}

(32)\{\begin{array}{l}{V}_{dc}=R_c{i}_{dc}+{\displaystyle \frac{d{y}_{dc}}{dt}}-\left[{\omega }_p-\left(P_p+P_c\right){\omega }_r\right]{\psi }_{qc},\hfill \\ V_{qc}=R_c{i}_{qc}+{\displaystyle \frac{d{\psi }_{qc}}{dt}}+\left[{\omega }_p-\left(P_p+P_c\right){\omega }_r\right]{\psi }_{dc},\hfill \end{array}

(33)\{\begin{array}{l}{V}_{dr}=R_r{i}_{dr}+{\displaystyle \frac{d{\psi }_{dr}}{dt}}-\left({\omega }_p-P_p{\omega }_r\right){\psi }_{qr},\hfill \\ V_{qr}=R_r{i}_{qr}+{\displaystyle \frac{d{\psi }_{qr}}{dt}}+\left({\omega }_p-P_p{\omega }_r\right){\psi }_{dr},\hfill \end{array}

(34)\{\begin{array}{l}{\psi }_{dp}=M_p{i}_{dr},\hfill \\ {\psi }_{qp}=M_p{i}_{qr},\hfill \end{array}

(35)\{\begin{array}{l}{\psi }_{dc}=L_c{i}_{dc}+M_c{i}_{dr},\hfill \\ {\psi }_{qc}=L_c{i}_{qc}+M_c{i}_{qr},\hfill \end{array}

(36)\{\begin{array}{l}{\psi }_{dr}=L_r{i}_{dr}+M_c{i}_{dc},\hfill \\ {\psi }_{qr}=L_r{i}_{qr}+M_c{i}_{qc}.\hfill \end{array}

The power control of the BDFG is not needed in grid synchronization. The control principle of the PC current in grid synchronization is based on the no-load mathematical model for the BDFG and considering the PW field oriented. Considering Eq. (30), Eq. (18) yields

(37)\{\begin{array}{l}{\psi }_{dp}=M_p{i}_{dr},\hfill \\ i_{qr}=0.\hfill \end{array}

Substituting Eqs. (18) and (37) into Eq. (35), Eq. (35) can be obtained.

(38)\{\begin{array}{l}{i}_{dc}^{*}={\displaystyle \frac{y_{dc}}{L_c}}-{\displaystyle \frac{M_c{V}_{qp}}{{\omega }_p{M}_p{L}_c}},\hfill \\ i_{qc}^{*}={\displaystyle \frac{{\psi }_{qc}}{L_c}}.\hfill \end{array}

The CW flux estimator used the voltage Eq. (8), whose derivation in the stationary reference frame (\alpha -\betareference frame) is

(39)\{\begin{array}{l}{\psi }_{\alpha c}={\displaystyle \int \left(V_{\alpha c}-R_c{i}_{\alpha c}\right)}dt,\hfill \\ {\psi }_{\beta c}={\displaystyle \int \left(V_{\beta c}-R_c{i}_{\beta c}\right)}dt.\hfill \end{array}

The CW flux must be transformed from the local(dq) using

(40)\left[\begin{array}{c}{\psi }_{dc}\\ {\psi }_{qc}\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{\psi }_{\alpha c}\\ {\psi }_{\beta c}\end{array}\right].

A PLL structure is used to the CW grid voltages angle estimation as presented in Fig. 5.

The control diagram is demonstrated in Fig. 6, which can be obtained according to the power decoupling and the no-load grid connection mathematical model. The no-load model and the generation model of the BDFG are established by using the method of separate modeling and time-sharing working. The control of the CW-side converter adopts current inner loop before grid connection, power outer loop and current inner loop after grid connection. The inner loop is the same and the outer loop should be switched.

A PLL circuit is used to compensate for the phase shift between the stator EMF and the grid voltage, as displayed in Fig. 5.

A soft connection of the generator to the grid is obtained when the phase, frequency and amplitude of the PW and grid voltages are equal before the switches S₁, S₂, S₃ are closed path, as shown in Fig. 6. The BDFG is switched to power control. Next, the power reference is adjusted to extract the maximum power based on the wind speed by closing S₄.

Simulation results

The simulation under MATLAB^®/Simulink^® has been conducted with an ode 3, fixed-step solver with a step size of 2e-5s. The sample machine used in this simulation model is 3Y-3Y connected and its stator winding is 6-2 pole. The main parameters of the BDFG simulation model are listed in Table 1.

The frequency of the BDFG PW voltage is not affected by the angular velocity. The grid connection is implemented at 0.2 s, and the BDFG is switched to the MPPT control at 1 s.

Before grid connection, the BDFG operates with no-load, that is to say, the PW current is zero. If the PW voltage is the same as the grid voltage, the case in which BDFG is connected from the grid before the switches S1 S2 S3 of Fig. 6 is closed. To evaluate the dynamic performance of the system, a step change was made in wind speed, as exhibited in Fig. 7.

The power and currents of the PW are all zero because the PW is open-circuited before grid-connection, as shown in Figs. 8 and 9.

Figure 10 shows the waveforms of the PW voltage and grid voltages of the grid- connection at 0.2 s. The two curves are in close agreement in phase, frequency, and magnitude. A real advantage of this control is that the connection to the grid may be realized at any rotational speed which is implemented via regulating the CW exciting current.

In Fig. 11, the PW flux d-q axis components are shown for conditions described above. The PW q-axis flux component is back to zero, showing the correct orientation of the reference frame.

The active and reactive powers and their references are reported in Fig. 12, which represents a good track of their references and verifies the decoupling between the active and reactive power.

Figure 13 shows that the output DC voltage is regulated and it follows the reference voltage command of 600 V.

When the active power is increasing, the DC voltage is also trying to increase. The DC-link capacitor voltage is constant by balancing the real power at the BDFG machine-side and grid-side converter, and to compensate the BDFG reactive power as much as possible.

Figure 14 illustrates, in detail, the dynamic variation of the MPPT in wind speed change (7, 5, 11 ms) in the rotor speed- power characteristics of the BDFG-turbine. It can be noted it is in accord with the optimal value. These results realize the maximum wind energy tracking control. Figure 15 shows that, when the wind profile changes, C_p can quickly reach around the optimal value. The power coefficient is kept around its optimum when C_pmax = 0.48 occurs at a λ_opt = 8.1

Consequently, between the active and reactive powers, this leads to a good control of the power flow between the grid and the machine at all time and the maximum power point tracking can be realized. In the end, it can be observed from the simulation result that the control of system BDFG-wind turbine has good performance.

Conclusions

A grid connection of the BDFG to the utility gird using power flux oriented vector control scheme has been proposed in detail. During grid connection, the active and reactive power is controlled to realize the MPPT. A complete simulation model of the VSCF BDFG wind power generation is built based on Matlab/Simulink and the simulation results have proved that the grid connection control is effective. A better dynamic response is very good and non-coupled. Besides, the maximum wind energy capture is realized. Finally, this theoretical study with simulation is encouraging although more research should be conducted on the grid connection strategy of brushless doubly-fed wind power generation system.

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