Modeling and control of a permanent magnet synchronous generator dedicated to standalone wind energy conversion system

Abstract

The interest for the use of renewable energies has increased, because of the increasing concerns of the environmental problems. Among renewable energies, wind energy is now widely used. Wind turbines based on an asynchronous generator with a wound rotor present the inconvenience of requiring a system of rings and brooms and a multiplier, inferring significant costs of maintenance. To limit these inconveniences, certain manufacturers developed wind turbines based on synchronous machines with large number of pairs of poles coupled directly with the turbine, avoiding using the multiplier. If the generator is equipped with permanent magnets, the system of rings and brooms is eliminated. The control of the permanent magnet synchronous generator (PMSG) can be affected with the implementation of various techniques of control. This paper presented a new approach mainly based on the control strategy of power production system based on the PMSG. In fact, a mathematical model that simulates the Matlab chain was established with the introduction of control techniques, such as direct control of the torque (DTC) to control the load side converter (LSC), the control of the speed of the turbine and the DC-bus voltage ensured by PI regulators. To show the performance of the correctors used, some simulation results of the system were presented and analyzed.

Keywords

wind turbine / permanent magnet synchronous generator (PMSG) / converter / proportional-integral (PI) / control / direct control of the torque (DTC) / regulation

Introduction

It is well known that the electric power generation using wind energy source is receiving considerable attention, since it is inexhaustible, safe, environmental friendly and capable of supplying significant amount of power [ 1]. Wind power generation systems are growing rapidly in renewable energy applications [ 2].

Along with such a rapid growth, an enormous volume of research and development is being undertaken in the academia and industry on wind energy conversion systems (WECS). Different configurations of grid-connected WECS have been reviewed in a number of papers and books based on the types of generators and power electronic converters employed [ 3, 4]. However, there are, to the best knowledge of the authors, insufficient review and comparative studies on off-grid WECS [ 5]. Several control techniques are mentioned in the literature, such as, in variable-speed WECS or maximum power point tracking (MPPT) technique which has been used to adjust the speed of the generator. For induction motor (IM) fed by WECS, direct field oriented, indirect field oriented, and direct torque controls are used to ensure an efficient control.

In this work, direct control of the torque (DTC) strategy has been opted for, because is quite different from that of the field-oriented control (FOC) or vector control, which does not need complicated coordination transformations and decoupling calculation [ 6, 7]. Also it presents other advantages which can be summarized by a diminution of torque response time, even better than vector controller and absence of modulator block, as well as other controllers such as proportional-integral-derivative (PID) for flux and torque.

In the same context, the direct-drive permanent magnet wind turbine system has been a research hotspot in recent years, and the permanent magnet synchronous generator (PMSG) has been widely used with its good control precision, high efficiency, low maintenance and many other advantages [ 8].

This paper is directed towards modeling and behavioral Matlab simulation, after the description of the respective conversion chain based on PMSG considering the influence of the variation of the wind profile on the amplitude of the output voltage generator. It proposes to introduce different techniques of adequate controls, acting on the power electronic interface so as to exploit the system in order to have good energy efficiency and performance.

Overview of permanent magnet wind power generation system

The system studied is presented in Fig. 1. It includes a turbine connected to a permanent magnet synchronous generator (PMSG), a gearbox, a static converter side of the generator acting as a rectifier (GSC), a static converter load side playing the role of an inverter (LSC), and a load represented by a motor.

The turbine is normally coupled with the generator shaft through a gearbox whose gear ratio G is chosen in order to set the generator shaft speed within a desired speed range [ 9].

Modeling of system components

A wind turbine is a device that converts the kinetic energy of wind into mechanical energy available on a drive shaft and then into electrical energy via a generator, which is, in this case, a permanent magnet synchronous machine.

Model of wind turbine

The total kinetic power available for the turbine is given by

(1) P_{\omega }=\rho S{V}_{\omega }^3/\mathrm{2}=\rho \mathrm{\pi }{R}^{\mathrm{2}}{V}_{\omega }^3/\mathrm{2},

where r is the density of air (1.25 kg/m³), S is the area swept by the turbine (m²), R is the turbine radius (m), and V_w is wind speed (m/s).

The aerodynamic shaft power is given by

(2) P_{\mathrm{turb}}=C_p{P}_{\omega }=\rho \mathrm{\pi }{R}^{\mathrm{2}}{V}_{\omega }^{\mathrm{3}}{C}_p(\lambda ,\beta )/\mathrm{2},

where b is the orientation angle of the blades (^° ), l is the specific speed which is expressed in Eq.(3),

(3) \lambda ={\displaystyle \frac{{\omega }_{\mathrm{t}\mathrm{urb}}R}{V_{\omega }}},

where w_turb is the speed of the turbine (rad/s), C_p is the characteristics as a function of l for different values of the pitch angle b as illustrated in Fig.2

Figure 3 indicates the mechanical powers generated by the turbine as a function of rotor speeds for different wind speeds. The maximum power extraction within the allowable range can be achieved, if the controller can properly follow the optimum curve with the variation of wind speed [ 10].

Model of gearbox

The gearbox adjusts the slow speed of the turbine to the speed of the generator. This gearbox is mathematically modeled by Eqs. (4)–(7) [ 11].

(4) T_{\mathrm{mec}}={\displaystyle \frac{T_{\mathrm{tur}}}{G}},

where T_mec is mechanical torque (N·m), G is multiplier report,

(5) {\mathrm{\Omega }}_{\mathrm{mec}}=G{\mathrm{\Omega }}_{\mathrm{tur}},

where {\Omega }_{\mathrm{mec}} is generator speed (rad/s).

The fundamental equation of dynamics can be written as [ 12]

(6) J{\displaystyle \frac{\mathrm{d}{\mathrm{\Omega }}_{\mathrm{mec}}}{\mathrm{d}t}}=T_{\mathrm{tur}}-f{\mathrm{\Omega }}_{\mathrm{mec}},

(7) {{\displaystyle T}}_{\mathrm{tur}}={{\displaystyle T}}_{\mathrm{mec}}+{{\displaystyle T}}_{\mathrm{em}},

where f is the viscous friction coefficient (N·m·s/rad), T_tur is the total torque of wind (N·m), T_em is the electromagnetic torque of the generator (N·m), and J is inertia ((kg·m²).

Model of generator

The model of PMSG, expressed in d-q frame, is given by voltage system Eqs. (8)-(10) [ 13].

(8) \{\begin{array}{l}{V}_d=-R_s{I}_d-L_d{\displaystyle \frac{d{I}_d}{dt}}+\omega L_q{I}_q\\ V_q=-R_s{I}_q-L_q{\displaystyle \frac{d{I}_q}{dt}}+\omega L_d{I}_d+\omega {\phi }_f\end{array},

Figure 4 presents the model of PMSM in d-q axis [ 14].

The coupling electromagnetic equations are expressed in Eq. (9).

(9) \{\begin{array}{l}{\phi }_d=L_d{i}_d+{\phi }_f\\ {\phi }_q=L_q{i}_q\end{array},

where L_d is the stator inductance in d-axis (H), L_q is the stator inductance in q-axis (H), L_q and L_d are the supposed independent of q(H), and j_f is the magnet flux (Wb).

Equation (10) represents the expression of electromagnetic torque [ 13].

(10) {{\displaystyle C}}_{\mathrm{em}}=\mathrm{3}p[({{\displaystyle L}}_d-{{\displaystyle L}}_q){{\displaystyle i}}_d{{\displaystyle i}}_q+{{\displaystyle i}}_q{{\displaystyle \phi }}_f]/\mathrm{2.}

Model of power converter

Given that the two converters used in the realization of the proposed wind conversion chain have the same structure, only one needs to be modeled to facilitate the modeling and to reduce the time of simulation. The converter has been modeled by a set of ideal switches, which means zero resistance in the passing state, infinite resistance to the blocked state, and instantaneous reaction to control signals [ 15].

A “connection” function associated to each switch has been defined. It represents the ideal orders of commutation and takes the values [ 15, 16] of

(11) \begin{array}{l}{S}_{\mathit{ic}}=1\mathrm{when\; the\; switch\; isclosed},\\ S_{\mathit{ic}}=\mathrm{0}\mathrm{when\; the\; switch\; isopen},\mathrm{and}\\ S_{\mathit{ic}}\in \{\mathrm{1},\mathrm{2},\mathrm{3}\},\mathrm{with}\{\begin{array}{l}c\in \{\mathrm{1},\mathrm{2},\mathrm{3}\}\\ i\in \{\mathrm{1},\mathrm{2}\}\end{array}.\end{array}

The “c” indication corresponds to the cell of commutation, and the index “i” corresponds to the location of the switch of this cell.

For the three phases of the converter, the conversion functions m has been defined [ 15, 16].

(12) m=[m_{\mathrm{1}}{m}_{\mathrm{2}}]=\left[\begin{array}{lll}\mathrm{1}& \mathrm{0}& -\mathrm{1}\\ \mathrm{0}& \mathrm{1}& -\mathrm{1}\end{array}\right]\left[\begin{array}{l}{S}_{\mathrm{11}}\\ S_{\mathrm{12}}\\ S_{\mathrm{13}}\end{array}\right].

The condition of the converter conducting state is provided by a connection matrix consisting of three cells, where the interrupters are supposed complementary [ 13, 14].

S_{i\mathrm{1}}+S_{i\mathrm{2}}=\mathrm{1},\forall i\in \{\mathrm{1},\mathrm{2},\mathrm{3}\}.

The modeling of converters consists in expressing voltages in lines, according to DC bus voltage and the states of the switches [ 15].

The modulated voltages are obtained from DC bus voltage and conversion functions according to the expressions (13) [ 17].

(13) \{\begin{array}{l}{u}_{m\mathrm{13}}=m_{\mathrm{1}}u\\ u_{m\mathrm{23}}=m_{\mathrm{2}}u\end{array}.

The modulated simple voltages are drawn from the modulated composed voltages according to the expressions (14) [ 15].

(14) \{\begin{array}{l}{v}_{m-\mathrm{1}}={\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{u}_{m\mathrm{13}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{u}_{m\mathrm{23}}\\ v_{m-\mathrm{2}}=-{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{u}_{m\mathrm{13}}+{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{u}_{m\mathrm{23}}\end{array},

The modulated current is obtained from the filter currents and conversion functions [ 15], expressed as

(15) {{\displaystyle i}}_{m-red}={{\displaystyle m}}_{\mathrm{1}}{{\displaystyle i}}_{t\mathrm{1}}+{{\displaystyle m}}_{\mathrm{2}}{{\displaystyle i}}_{t\mathrm{2}}.

Modeling of DC bus

The capacitor current is drawn from a node where two modulated currents circulate in each converter [ 15].

(16) {{\displaystyle i}}_c={{\displaystyle i}}_{m-mac}+{{\displaystyle i}}_{m-red},

where i_m-mac is the current provided by the generator (A), and i_m-red is the current modulated by the PWM2 converter (A).

DC bus is modeled with the knowledge of the voltage of the capacitor terminals obtained by integrating the differential Eq. (17) [ 15].

(17) {\displaystyle \frac{\mathrm{d}u}{\mathrm{d}t}}={\displaystyle \frac{\mathrm{1}}{C}{i}_{\mathrm{c}}},

where

(18) u={\displaystyle \int {\displaystyle \frac{du}{dt}}}+u(t_{\mathrm{0}}),

u(t₀) is the voltage value at instant t₀.

Control strategies of wind power production system

The control strategy used in this paper includes the control of PMSM speed, the control of DC bus voltage, and application of the direct torque control on load side converter.

Control strategy of PMSG speed

To ensure a speed control of PMSG, optimum speed reference has been pulled from the MPPT technique, expressed by Eq. (19). The control strategy of the PMSG speed is illustrated in Fig. 5 [ 18].

The reference speed is

(19) {\Omega }_{\mathrm{ref}}={\displaystyle \frac{{\lambda }_{\mathrm{opt}}v}{R_t}}.

The coefficients of the PI controller obtained from the closed loop analysis are expressed by Eq. (20), where w_nd and t_sd are system dynamics and response time of the system, respectively [ 18– 20].

(20) \{\begin{array}{l}{K}_p=\mathrm{2}\xi {\omega }_{\mathrm{nd}}J\\ K_i={\omega }_{\mathrm{nd}}^{2}J\end{array},{\omega }_{\mathrm{nd}}={\displaystyle \frac{\mathrm{5.8}}{t_{\mathrm{sd}}}}.

Procedure for checking the voltage of DC-bus

The DC-bus voltage V_dc is sensed and compared with a reference value V_dc-ref, and the obtained error is used as an input for the PI controller.

The closed control loop of the capacitor voltage is demonstrated in Fig 6.

The current I_sd-ref is obtained from the control loop of the voltage, and the current I_sq-ref is set to zero to obtain a unity power factor [ 18].

There is

(21) \{\begin{array}{l}{K}_i=C_{\mathrm{dc}}{\omega }_{\mathrm{0}}^{\mathrm{2}}\\ K_p=\mathrm{2}{C}_{\mathrm{dc}}\xi {\omega }_{\mathrm{0}}\end{array},

where K_i is the integrator gain, K_p the proportional gain, x is the damping factor, and w₀ is the angular frequency (rad/s).

Direct torque control strategy for LSC

Direct torque control (DTC) is getting more and more popular nowadays ever since it has been proposed by Depenbrock and Takahashi [ 14, 21, 22]. The main features of DTC can be summarized as a direct control of flux and torque (by selecting the optimum inverter switching vectors), indirect control of stator currents and voltages, approximately sinusoidal stator fluxes and currents, high dynamic performance, no vector transformation as in vector-based control, no feedback current control and less dependence on motor parameters [ 23– 25].

Figure 7 represents the DTC technique applied for this case on the load side converter.

This technique is based on the direct control of the stator flux and torque in which, the supply voltage and stator current are sampled. In order to control the induction motor, the stator flux on the stationary reference axes a and b are calculated. The stator flux vector can be estimated using Eq. (22) [ 7].

(22) \{\begin{array}{l}{\phi }_{s\alpha }={\displaystyle {\int }_t^{\mathrm{0}}(V_{s\alpha }-R_s{i}_{s\alpha })dt}\\ {\phi }_{s\beta }={\displaystyle {\int }_t^{\mathrm{0}}(V_{s\beta }-R_s{i}_{s\beta })dt}\end{array},

where j_s is the stator flux vector and R_s is the stator resistance (Ω).

The stator-voltage space vector v_s is computed using the DC-link voltage v_dc and the inverter switch gating signals S_a, S_b, and S_c.

(23) \{\begin{array}{l}{v}_{sa}= \sqrt{{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}}\left(S_a-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}(S_b+S_c)\right)v_{\mathrm{dc}}, \\ v_{s\beta }= \sqrt{{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}}(S_b-S_c)v_{\mathrm{dc}} . \end{array},

where S_i = 1 when the phase i is connected to the supply positive polarity, S_i = 0 when the phase i is connected to the supply negative polarity.

The components of the stator-current space vector is derived from the measured currents i_a, i_b, and i_c are

(24) \{\begin{array}{l}{i}_{s\alpha }=\sqrt{{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}}{i}_{sa},\\ i_{s\beta }=\sqrt{{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}}(i_{sb}-i_{sc}).\end{array},

The electromagnetic torque T_em of the motor can be evaluated as

(25) T_{\mathrm{em}}=P({\phi }_{s\alpha }{i}_{s\beta }-{\phi }_{s\beta }{i}_{s\alpha }).

A general recap for the estimation of the torque and stator flux is illustrated in Fig. 8 [ 25].

The voltage vector plane is divided into six sectors so that each voltage vector divides each region into two equal parts. In each sector, four of the six non-zero voltage vectors may be used. Also, zero sectors are allowed. All the possibilities can be tabulated into a switching table (Table 1). The output of the torque hysteresis comparator is denoted as t, the output of the flux hysteresis comparator as f, and the flux linkage sector as q [ 14].

Simulation results and discussion

All simulations are performed in the Simulink interface of Matlab. The PMSG used in this paper is a 6 kW, whose nominal parameters are indicated in Appendix.

Figure 9 shows the variation of the wind speed according to the time up to t=10 s between generated random values ranging from v=8.4 m/s to v=9.5 m/s. This gives a reproduction of a real wind profile.

Figure 10 indicates the mechanical speed of the generator. It can be seen that the measured value of the speed is approximately 100 rad/s following the reference speed which demonstrate the efficiency of the control technique used. Figure 11 shows the power coefficient variation C_p which is kept around its maximum value C_p=0.29.

Figures 12 and 13 show the variations of the output voltage of the generator according to the profile of wind used in this simulation.

The result of the DC bus voltage control is represented in Figs. 14 and 15. Two settings values use U_dc-ref =300 V and U_dc-ref =470 V, where the curve indicates the robust PI controller in the pursuit of reference with a short time response time and an acceptable overrun (near to zero).

Figure 16 depicts the stator flux in the complex plan. It can be seen that the stator flux trajectory is almost circular and starts at point (0, 0) and rotates in the trigonometric direction to follow a circle of radius of 0.9 Wb fixed by the reference.

Figure 17 displays the supply voltages in the d, q plan, in which, for the supply voltage, the different vectors imposed by the vector command can be distinguished.

It can be visualized in Figs. 18 and 19 respectively that the velocity curve of induction machine which reaches the steady state speed of 1600 r/min and the zoom of stator currents i_sd and i_sq are well and sinusoidal phase of p/2.

Figure 20 exhibits the estimated flux. It can be noticed that the measured magnitude follows perfectly the reference and the flow is not affected by the variation of the load

From Fig. 21, it can be observed that the torque follows the value of the reference reflecting the efficiency of the control technique used.

Conclusions

In this paper, the simulation of a conversion chain based on PMSG is presented with a random wind profile similar to the real wind profile. Waveform of the output voltage obtained gives the idea of an adequate control technique in order to exploit the system for producing electrical energy for stand-alone WECS.

Three control techniques have been conducted for each part of the WECS, including wind speed control and control voltage of the DC-bus whose two techniques are based on the PI controller. The strategy controls are very robust for certain parameters such as generator speed, DC-bus voltage and torque, which give a very high power factor and small ripple in order to provide a fixed voltage fed to the inverter controlled by the direct torque control. The DTC has been chosen unlike traditional techniques, for simplicity, that is to say, not using nested loops or coordinate transformations or modulators. Moreover, it proves successful effectiveness of the used technique.

The simulation results obtained are satisfactory because they show good agreement between measured and reference quantities. The introductions of DTC in WECS are very promising. They achieved great performances, fast responses with no overshoot fluctuations in the steady state.

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