Speed sensorless model predictive control method for a direct-drive wind energy conversion system

Abstract

Considering the problems of the internal and external disturbances of wind speed in the direct-drive wind energy conversion system based on a permanent magnet synchronous generator, a novel model predictive control based on the extended state observer method without the accurate mathematical system model is proposed in this paper. First, a model predictive control method is employed as the feedback controller, while the mathematical model of the control system can be adjusted online via the rolling optimization strategy. Second, an extended state observer is introduced to estimate the state variables and lumped disturbances, that is, the internal disturbances including nonlinear characteristic, multi-variety coupling effect, uncertainties of system parameters, and external disturbances including variations of wind speeds and uncertainties of the natural environment. Third, the effect of lumped disturbances can be attenuated by the estimated disturbance value via a feedforward channel. In addition, in order to achieve the real-time speed control performance of the permanent magnet synchronous generator, a speed sensorless algorithm based on a flux observer is proposed to solve the problem of unsuitability of mechanical speed sensor. Finally, the simulation results with several wind speed types show that the proposed sensorless model predictive control with the extended state observer strategy is an effective way to improve the performance of anti-disturbance and ability of tracking maximum wind energy of the wind power control system.

Keywords

Model predictive control extended state observer speed sensorless permanent magnet synchronous generator flux observer

Introduction

As a renewable, pollution-free natural resource, the wind energy has shown significant prospects of social development. To a certain extent, it can replace the increasingly scarce fossil fuels and solve the problems of energy shortage and environmental pollution in today’s world. With the development of social economy and civilization, people put forward significant requirements for wind power generation systems. Permanent magnet synchronous generator (PMSG) is a gearless wind energy conversion system (WECS) with the advantages of direct energy transmission and low noise level. Moreover, this kind of generator also has reliable operation and low rotor loss, since PMSG can work at low speed without brush. Therefore, it is of practical significance to perform research on PMSG-based WECSs. Due to the multivariable coupling and nonlinear features of a PMSG-based WECS, it is difficult for the wind power system to obtain excellent dynamic performance while the WECS is subjected to lumped disturbances, that is, internal disturbances and external disturbances including uncertainties of system parameters, modeling error, uncertainties of wind speeds, and natural environmental factors.

Moreover, the generator usually is operated in the variable-speed mode below rated wind speed, since the significant control purpose of the WECS with PMSG is power efficiency maximization in the low-speed region. The optimum tip speed ratio method is a great way to achieve maximum power tracking performance. Due to the above reasons, many researchers have developed a large number of control strategies based on optimum tip speed ratio methods for obtaining maximum power tracking performance of PMSG-based WECSs in recent decades. A novel proportional–integral–derivative (PID) controller for a direct-drive wind turbine generator is proposed to track maximum power in Hong et al.¹ An adaptive duty cycle method with gradient approximation is introduced to enhance the power capture performance. Pole placement based on a linearized system model in the state space domain is proposed to maximize the aerodynamic efficiency of a horizontal axis wind turbine with a PMSG.² The simulation results show that the proposed method can obtain good performance with a nonlinear model subjected to random wind speeds. A composite maximum power point tracking (MPPT) controller for the small signal model of a grid-connected PMSG-based wind turbine system is proposed to enhance the smoothness and rapidity of capturing power in Nasiri et al.,³ including the tip speed ratio method and optimal torque control. These tip speed ratio methods based on traditional feedback controllers can capture the maximum wind power with its dependence on the accurate mathematical model of the wind turbine system. The industrial applications of these methods are limited, since it is difficult to establish the accurate dynamic model of the actual system.

Compared with the traditional feedback control methods, since the model predictive control (MPC) method is independent of the mathematical model of the generator and also due to its advantages of rolling optimization and online feedback, it has been widely used in industrial cranes, series elastic actuators, wind turbines, and other industries in recent years.^4–8 However, the internal and external disturbances of the industrial systems called lumped disturbances, such as modeling error, parameter variation, and external uncertainties, can not only cause inaccurate feedback correction to the predictive model of the MPC, but also degrade the stability and control performance of the industrial systems. Some popular disturbance observer methods are introduced to attenuate the above total disturbances, such as linear disturbance observer,⁹ equivalent-input-disturbance approach,¹⁰ neural network observer,¹¹,¹² and Luenberger observer.¹³ An extended state observer (ESO) with a high order than the system is another observer method employed to obtain excellent dynamic performance. This kind of disturbance and state variable estimation method is first put forward and completely expressed by Han.¹⁴ A linear structure ESO–based control scheme is proposed by Gao¹⁵ with the advantages of anti-disturbance performance. Due to the features of the concise framework, convenient form for parameter tuning, analysis, and implementation, these kinds of linear ESO–based control methods have been broadly applied in dealing with dynamic uncertainties, disturbances, and nonlinear factors of the industrial systems.^16–20

In order to realize real-time speed control of a PMSG, it is necessary to measure the speed of the generator as the feedback signal. Compared with the current or voltage sensor, the cost of speed sensor detection accuracy is much more expensive. Even worse, the mechanical speed sensor may not be suitable for a harsh wind power operating environment, since research pays more and more attention to the study of speed sensorless algorithms. There are three kinds of speed sensorless algorithms, that is, open-loop algorithm based on the motor mathematical model,²¹ closed-loop algorithm based on various observer models,²² and an algorithm based on nonideal characteristics.²³ These kinds of speed sensorless technologies have been introduced into the PMSG-based wind turbine systems because of their practical advantages. A wind turbine control and an analogue speed estimator based on a Kalman filter are introduced to estimate the shaft speed for capturing the maximum power of wind turbines in Antarctica.²⁴ In Abdelrahem et al.,²⁵ a sensorless field-oriented control (FOC) scheme is proposed for a PMSG in a variable-speed wind turbine. The experimental results show that the power capture performance is excellent under many operating conditions and parameter variations of the PMSG. A novel sensorless method based on the output voltage and current of a rectifier is proposed to improve maximum power extraction of a small-scale WECS using a PMSG, without the wind speed information and turbine characteristics.²⁶ The simulation and experimental results show that the extraction performance and efficiency are higher than those of the conventional perturb and observe method.

Considering the limitations of MPC for PMSG-based WECSs, the effectiveness of MPC will be decreased when the WECSs are subjected to serious total disturbances, such as nonlinearity, strong coupling, multi-varieties, and uncertainties of natural wind, since a second-order ESO due to the advantage of its anti-disturbance ability is employed to enhance the extraction performance of the whole WECS. Moreover, the ability of the whole control system is improved by the feedforward compensation part. Nevertheless, the measurement of stator current and voltage is relatively easy, and the technology of avoiding magnetoelectric field disturbance is relatively perfect. Therefore, the speed sensorless algorithm based on the flux observer technology is proposed for the wind power control system.

The rest of this paper is organized as follows. Section “Modeling of the PMSG-based WECSs” gives description of the system dynamic model. A novel structure of the speed sensorless MPC method for a PMSG-based WECS is introduced in section “Composite MPC based on the speed sensorless method.” And the design process and stability analysis of the proposed speed observer strategy are also discussed in this section. In section “System simulation verification,” numerical simulation results based on several wind speed types show the effectiveness of the proposed control strategy. Finally, the conclusions end the paper.

Modeling of the PMSG-based WECSs

The wind turbine model

According to the principle of aerodynamics, the mechanical energy of the wind turbine can be converted from wind power through equation (1)²⁰

P_{m} = \frac{1}{2} ρ π R^{2} C_{P} (λ, β) v^{3}

(1)

where the mechanical power $P_{m}$ is from the wind turbine; the air density $ρ$ is a constant parameter; $R$ represents the blade radius; $v$ indicates the natural wind speed; $λ$ and $β$ are the tip speed ratio and blade pitch angle, respectively; the parameter $C_{P}$ represents the power factor and is a nonlinear function of the wind turbine parameters $β$ and $λ$ , so its size depends mainly on the type and operating condition of the blade.

Thus, the driving torque of the blade with wind speed $v$ can be defined as follows

T_{m} = \frac{ρ π C_{P} (λ, β) R^{3} v^{2}}{2 λ}

(2)

The tip speed ratio of the system can be obtained by equation (3)

λ = \frac{ω R}{v}

(3)

where $ω$ is defined as the angular velocity.

According to Li et al.,²⁰ the wind power coefficient $C_{P}$ can be obtained by the following equation

C_{P} (λ, β) = 0.22 (\frac{116}{λ_{i}} - 0.4 β - 5) e^{- \frac{12.5}{λ_{i}}} + 0.0068 λ

(4)

These parameters satisfy the equation as $(1 / λ_{i}) = (1 / (λ + 0.08 β)) - (0.035 / (β^{3} + 1))$ .

It can be seen from the mechanical power equation (1) that the maximum power can be obtained when the parameter $C_{P}$ reaches a maximum value at a given wind speed.

The relationship between $λ$ and $C_{P}$ of equation (4) can be simulated as the curve shown in Figure 1 via MATLAB. From Figure 1, it is easy to see that the smaller the pitch angle $β$ , the greater the power coefficient of the wind turbine $C_{P}$ in a certain range. And for a certain pitch angle, there is an optimal tip speed ratio $λ_{opt}$ , which makes the $C_{P}$ the of the wind turbine to reach the maximum. Therefore, under the rated speed of the generator, the optimal value $λ_{opt}$ can be obtained by choosing the blade speed $ω$ and wind speed $v$ , that is, the power coefficient $C_{P}$ is the largest and the maximum output power of the generator can be obtained under this operating condition. It is well known that the best tip speed ratio $λ_{opt} = 8$ can also be achieved when $β = 0 °$ from Figure 1. The direct-drive WECS studied in this paper only considers maximum power tracking performance of the motor side, so the uncertainties of the grid side are equal to those of the motor side.

Figure 1.

Relationship between $λ$ and $C_{P}$ .

It is obvious from equation (3) that the optimal speed $ω_{opt}$ can be obtained by specifying the wind speed $v$ , and thus the optimal blade tip speed ratio satisfies $λ = λ_{opt}$ , that is, when the wind turbine runs at the optimal blade tip speed ratio $λ_{opt}$ , the maximum output power of the generator can be obtained. The optimum power of wind turbines $P_{\max}$ can be written using the following equation from equations (1) and (3)

P_{\max} = K_{opt} ω_{opt}^{3}

(5)

Equation (5) indicates the nonlinear relationship between optimal power and angular velocity. The parameters $K_{opt}$ and $λ_{opt}$ satisfy the relationship given using the following equations

\begin{matrix} K_{opt} = \frac{0.5 ρ π ω_{opt}^{3} C_{P_{\max}} R^{5}}{λ_{opt}^{3}} \\ λ_{opt} = \frac{ω_{opt} R}{v} \end{matrix}

The mathematical model of the PMSG

The PMSG is the core device of the direct-drive PMSG, and the correct establishment of the mathematical model plays a significant role in the research process. In order to develop a simplified and practical generator model, the following assumptions are made before the modeling:

Three-phase electricity is symmetrical;

The cogging torque effect of the PMSG is not considered;

The saturation effect of the generator core can be neglected;

The magnetic path is assumed as linear, regardless of the effect of eddy current and hysteresis loss.

The mathematical model of the PMSG in the α–β coordinate system can be expressed as follows

[\begin{matrix} u_{α} \\ u_{β} \end{matrix}] = [\begin{matrix} R_{s} & 0 \\ 0 & R_{s} \end{matrix}] [\begin{matrix} i_{α} \\ i_{β} \end{matrix}] + [\begin{matrix} {ψ'}_{α} \\ {ψ'}_{β} \end{matrix}]

(6)

where $[\begin{matrix} ψ_{α} \\ ψ_{β} \end{matrix}] = [\begin{matrix} L & 0 \\ 0 & L \end{matrix}] [\begin{matrix} i_{α} \\ i_{β} \end{matrix}] + ψ_{f} [\begin{matrix} \cos (n_{p} θ) \\ \sin (n_{p} θ) \end{matrix}]$ , $u_{α}$ and $u_{β}$ are the voltage components in the $α$ and $β$ axes; $i_{α}$ and $i_{β}$ are the current components in the $α$ and $β$ axes; the parameter $n_{p}$ refers to the number of generator pole pairs; the mechanical angle of the rotor is described by $θ$ ; $ψ_{α}$ and $ψ_{β}$ are the stator flux linkages in the $α$ and $β$ axes; $R_{s}$ and $ψ_{f}$ are the armature resistance and permanent magnet flux of the generator, respectively; and $L$ is the equivalent armature inductance under the rotating d–q coordinate.

The electromagnetic torque in the α–β coordinate system can be described as follows

T_{e} = \frac{3}{2} n_{p} ψ_{f} [i_{β} \cos (n_{p} θ) - i_{α} \sin (n_{p} θ)]

(7)

Through the Park transformation, the mathematical model of the PMSG in the α–β coordinate system can be converted into a mathematical model in the d–q coordinate system as follows

[\begin{matrix} u_{d} \\ u_{q} \end{matrix}] = [\begin{matrix} R_{s} & - n_{p} ω L \\ n_{p} ω L & R_{s} \end{matrix}] [\begin{matrix} i_{d} \\ i_{q} \end{matrix}] + [\begin{matrix} L & 0 \\ 0 & L \end{matrix}] [\begin{matrix} {i'}_{d} \\ {i'}_{q} \end{matrix}] + n_{p} ω ψ_{f} [\begin{matrix} 0 \\ 1 \end{matrix}]

(8)

where $u_{q}$ and $u_{d}$ are the voltage components in the d and q axes, respectively, and $i_{d}$ and $i_{q}$ are the current components in the orthogonal axes. The electromagnetic torque in the d–q coordinate system can be rewritten as follows

T_{e} = \frac{3}{2} n_{p} ψ_{f} i_{q}

(9)

The dynamic equation of the whole wind turbine system can further be expressed as follows

J \frac{d ω}{dt} = T_{m} - T_{e} - B ω

(10)

where the parameters $J$ and $B$ are the moment of inertia and viscous friction coefficient of the PMSG, respectively.

Composite MPC based on the speed sensorless method

Design of a composite model predictive controller

Model predictive controller for a PMSG-based WECS

Let the load torque $T_{m} = 0$ in equation (10). The frequency-domain model of the wind conversion system is rewritten using the Laplace transform as follows

G (s) = \frac{ω_{r} (s)}{T_{e} (s)} = \frac{K}{Js + B}

(11)

where the auxiliary parameter $K = n ψ$ .

Considering that MPC is a kind of computer algorithm based on the model, it is necessary to use the zero-order holder to discretize the PMSG model and the Z transfer function can be obtained as follows

G (z) = Z [\frac{1 - e^{- Ts}}{s} \cdot \frac{K}{Js + B}] = \frac{a z^{- 1}}{1 + b z^{- 1}}

(12)

where the symbol $z$ is the Z-transform, the time coefficient T represents the sampling time, and the label s is a complex variable in Laplace transform. The other coefficients of equation (12) satisfy the following equation: $a = K (1 - e^{- TB / J}) / B$ and $b = - e^{- TB / J}$ .

The present sampling step is defined as k, so $ω (k)$ is the current actual speed of the generator at time $kT$ ; furthermore, the parameter $i_{q} (k)$ represents the current of the q axis at time $kT$ . Moreover, the difference equation can be obtained using the following equation

{\begin{matrix} ω (k) = a i_{q} (k - 1) - b ω (k - 1) \\ ω (k + 1) = a i_{q} (k) - b ω (k) \end{matrix}

(13)

The prediction model for the speed of PMSG can be obtained by subtracting the two equations in equation (13), since it can be written as follows

ω_{p} (k + 1) = a Δ i_{q} (k) + (1 - b) ω (k) + b ω (k - 1)

(14)

where $ω_{p} (k + 1)$ is the predicted speed of the generator at time $(k + 1) T$ . The difference component $Δ i_{q} (k)$ means the increment of current in the $q$ axis at time $kT$ , and this current variable is also the output of the speed loop controller for the WECS-based PMSG.

The speed error $e (k)$ is introduced as the part of the feedback loop to enhance the stability of the system. The compensation amount is $e (k) = ω (k) - ω_{p} (k)$ and indicates the difference error between the predicted rotational speed and the actual rotational speed at the time of $kT$ . The closed-loop predicted output speed of the system at $kT$ is obtained using the following equation

ω_{pe} (k + 1) = ω_{p} (k + 1) + e (k)

(15)

Based on the dynamic characteristics of the WECS, the reference trajectory generally takes the form of a first-order exponential change to prevent drastic changes in input and output parts. Hence, the expression can further be given as follows

y_{r} (k + 1) = α ω (k + 1) + (1 - α) ω_{r} (k)

(16)

where $α = e^{- T_{s} / τ}$ with the time parameter $τ$ being the time constant of the reference output.

It is well known that the optimal controller output is determined by the system performance index, so the quadratic performance index is employed in this paper considering the practicability and significance of this kind of index function. Frequently, the output error and the weight of the control amount are also used, so the predicted output $ω_{pe} (k + 1)$ can not only coincide with the expected output of the system, but also effectively reduce the drastic changes of the control increment. The quadratic performance index function is expressed as follows

I_{q} (k) = \min {W_{1} {[ω_{pe} (k + 1) - y_{r} (k + 1)]}^{2} + W_{2} {[Δ i_{q} (k)]}^{2}}

(17)

where $W_{1}$ and $W_{2}$ are the weighting coefficients of the output error and the control increment, respectively. Obviously, the parameters $ω (k)$ , $ω (k - 1)$ , and $y_{r} (k + 1)$ can be definite at time $kT$ . According to equations (14)–(17), an optimal control function can further be obtained as shown in equation (18)

I_{q} (k) = \min {W_{1} {[\begin{matrix} a Δ i_{q} (k) + (1 - b) ω (k) \\ + b ω (k - 1) - y_{r} (k + 1) \end{matrix}]}^{2} + W_{2} {[Δ i_{q} (k)]}^{2}}

(18)

To minimize the optimization control function $I_{q} (k)$ , the derivative of $I_{q} (k)$ needs to be solved with $Δ i_{q} (k)$ . Hence, the following equation can be obtained with the first-order differential equation $(d I_{q} (k) / d Δ i_{q} (k)) = 0$

Δ i_{q} (k) = - \frac{W_{1} a}{W_{2} a^{2} + W_{2}} [(1 - b) ω (k) + b ω (k - 1) + e (k) - y_{r} (k + 1)]

(19)

Design of ESO

In the actual operating environment, there are internal and external disturbances in the wind power system, such as wind speed variation, uncertainties of the natural condition, and modeling error. The model predictive controller performs an online calibration system model based on the feedback information output by the PMSG. The total disturbances will cause unfavorable effects on the feedback part of the model predictive controller, which will inevitably lead to a decrease in the control accuracy of the MPC. Moreover, they can affect the control accuracy of the entire WECS. Hence, the ESO is introduced to estimate the internal and external disturbances of the system. These disturbances can further be compensated for and eliminated by the feedforward channel. And the first-order state space model can be expressed as follows

{\begin{matrix} x' (t) = f (x, t) + d (t) + b_{1} u (t) \\ y (t) = x (t) \end{matrix}

(20)

where $f (x, t)$ is an unknown function and relevant to the system state variables and internal disturbance, the parameter $d (t)$ refers to the external disturbance, the label $b_{1}$ represents the control gains relating to the system model, and the control input and output signals can be defined as $u (t)$ and $y (t)$ , respectively.

Defining $x_{1} = y$ and $x_{2} = l_{d} (t) = f (x, t) + d (t)$ as the total disturbances including the internal and external uncertainties, the state equation of equation (20) can further be expressed as

{\begin{matrix} {x'}_{1} = x_{2} + b_{1} u (t) \\ {x'}_{2} = {l'}_{d} (t) \end{matrix}

(21)

The ESO can be designed as follows

{\begin{matrix} {z'}_{1} = z_{2} - β_{1} (z_{1} - x_{1}) + b_{1} u (t) \\ {z'}_{2} = - β_{2} (z_{1} - x_{1}) \end{matrix}

(22)

where $z_{1}$ is the estimated value of state variable $x_{1}$ and $z_{2}$ is the estimated value of total system disturbance $x_{2}$ , which are defined as the extended state variables, respectively. The observer bandwidth–dependent parameters $β_{1}$ and $β_{2}$ can be adjusted for improving the performance of the whole system.

The characteristic equation of equation (22) can be given using equation (23) to satisfy the convergence and stability of the ESO

λ (s) = s^{2} + β_{1} s + β_{2}

(23)

Selecting the ideal characteristic equation $E_{c} (s) = (s + ω_{o})^{2}$ , there is the following equation

{\begin{matrix} β_{1} = 2 ω_{o} \\ β_{2} = ω_{o}^{2} \end{matrix}

(24)

where $ω_{o}$ is defined as the observer bandwidth. Therefore, the ESO based on PMSG can be designed as follows

{\begin{matrix} {z'}_{1} = z_{2} - 2 ω_{o} (z_{1} - ω) + b_{1} u (t) \\ {z'}_{2} = - ω_{o}^{2} (z_{1} - ω) \end{matrix}

(25)

Hence, the feedback control part can further be designed as follows

{\begin{matrix} u_{0} (t) = k (ω^{*} - z_{1}) \\ u (t) = u_{0} (t) - \frac{z_{2}}{b_{1}} \end{matrix}

(26)

Design of the speed sensorless method

The d–q axis usually has the same speed as the generator rotor. In other words, the conversion from the α–β coordinate system to the d–q coordinate system requires rotor speed information to determine the position of the d and the q axis. Unfortunately, the Clarke transformation of the current in the A–B–C coordinate system requires only the rotor position signal. Therefore, the speed sensor proposed in this paper is based on the mathematical model of PMSG in the α–β coordinate system.

According to equations (1)–(6), the state space equation of PMSG can be written as follows

{\begin{matrix} x' = Ax + U \\ y = Cx \end{matrix}

(27)

where

x = [\begin{matrix} i_{α} \\ i_{β} \\ ω \end{matrix}], U = [\begin{matrix} u_{α} / L \\ u_{β} / L \\ 0 \end{matrix}], C = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], A = [\begin{matrix} - R_{s} / L & 0 & k_{fs} / L \\ 0 & - R_{s} / L & - k_{fc} / L \\ 3 k_{fs} / 2 J & - 3 k_{fc} / 2 J & (k_{t} ω - B) / J \end{matrix}]

with

{\begin{matrix} k_{fc} = n_{p} ψ_{f} \cos (n_{p} θ) \\ k_{fs} = n_{p} ψ_{f} \sin (n_{p} θ) \\ k_{t} = ρ π R^{5} C_{p} (λ, β) / 2 λ^{3} \end{matrix}

Define the matrix $Q_{o}$ as

\begin{matrix} Q_{o} = {[\begin{matrix} C & CA & C A^{2} \end{matrix}]}^{T} \\ = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ - R_{s} / L & 0 & k_{fs} / L \\ 0 & - R_{s} / L & - k_{fc} / L \\ \frac{R_{s}^{2}}{L^{2}} + \frac{3 k_{fs}^{2}}{2 JL} & - \frac{3 k_{fs} k_{fc}}{2 JL} & k_{fs} k_{tb 1} \\ - \frac{3 k_{fc} k_{fs}}{2 JL} & \frac{R_{s}^{2}}{L^{2}} + \frac{3 k_{fc}^{2}}{2 JL} & k_{fc} k_{tb 2} \end{matrix}] \end{matrix}

(28)

where ${\begin{matrix} k_{tb 1} = - \frac{R_{s}}{L^{2}} + \frac{k_{t} ω - B}{JL} \\ k_{tb 2} = \frac{R_{s}}{L^{2}} + \frac{k_{t} ω - B}{JL} \end{matrix}$

Since $k_{fc}$ and $k_{fs}$ are not zero at the same time in the actual process, the equation of state of the velocity observer can be described as

ω' = \frac{1}{J} (T_{tur} - \frac{3}{2} n_{p} ψ_{f} I_{α β} - B ω)

(29)

Substituting $I_{α β} = i_{β} \cos (n_{p} θ) - i_{α} \sin (n_{p} θ)$ , the above formula can be converted into the following equation

\hat{ω}' = \frac{1}{J} (T_{tur} - \frac{3}{2} n_{p} ψ_{f} {\hat{I}}_{α β} - B \hat{ω}) + h_{ω_{1}} {\bar{i}}_{α} + h_{ω_{2}} {\bar{i}}_{β}

(30)

where ${\hat{I}}_{α β} = {\hat{i}}_{β} \cos (n_{p} θ) - {\hat{i}}_{α} \sin (n_{p} θ)$ . The parameters $h_{ω_{1}}$ and $h_{ω_{2}}$ are the error feedback gains. The other parameters satisfy the equations ${\bar{i}}_{α} = i_{α} - {\hat{i}}_{α} and {\bar{i}}_{β} = i_{β} - {\hat{i}}_{β}$ .

In order to obtain the maximum power of current wind speed, the inner control loop is employed in the FOC method. In addition, it is well known that $i_{d}^{*}$ is defined to equal zero in order to decouple the currents of the d–q axis. Although it is not necessary for the actual current i_d to be equal to zero, the effect can be attenuated by introducing ESO. The block diagram of the proposed composite control based on FOC for a WECS with PMSG based on the combined speed sensorless and MPC method is shown in Figure 2. For the difference of speed loop and current loop control requirements, the model predictive controller is selected as the speed loop controller, and the current loop uses the traditional proportional–integral (PI) regulators. A composite MPC scheme for the maximum power output of PMSG-based wind turbines is designed. In the feedforward channel of the outer speed loop, the internal and external disturbances of the system are estimated by the ESO, and the observer value of the disturbances is transmitted to the model predictive controller through the feedforward channel. Then, according to the actual output speed of PMSG and the observed disturbance, the model of the system is optimized and corrected online, so as to eliminate the control errors caused by internal and external disturbances. During this process, the PMSG actual speed value is fed back to the composite controller via the speed observer.

Figure 2.

Wind turbine control structure block diagram based on the speed sensorless composite model predictive control method.

System simulation verification

The parameters of the PMSG used in the simulation are as follows: the rated speed is $n_{N} = 3000 r / \min$ ; the viscous friction coefficient is $B = 8.29 \times 10^{- 5} kg m^{2} / s$ ; the rated power is $1.5 kW$ ; the rotary inertia is $J = 2.5 \times 10^{- 4} kg / m^{2}$ ; the armature winding resistance satisfies $R = 2.875 Ω$ ; the number of pole pairs is $n_{p} = 2$ ; the permanent magnet flux linkage is $ψ_{f} = 0.175 Wb$ ; and the armature winding inductance is $L = 8.5 \times 10^{- 3} H$ .

The parameters of the wind turbine used in the simulation are as follows: the power coefficient is $C_{P} = 0.48$ ; the air density satisfies $ρ = 1.25 kg / m^{3}$ ; the optimum blade tip speed ratio is defined as $λ_{opt} = 8$ ; according to the “IEC 61400-1 design requirements” and the rated speed of the PMSG, the speed of $v = 6 m / s$ is chosen as the basic wind speed; the radius of the wind turbine is $R = 1.5 m$ ; the parameters of the ESO used in the simulation are given as $b_{1} = 71$ and $p = 150$ ; and the parameters of the speed sensorless method used in the simulation are $h_{ω 1} = 200$ and $h_{ω_{2}} = 200$ .

Figures 3 and 4 show the comparison of the estimated speed of the speed observer with the actual speed of PMSG. It is not difficult to observe that the speed observer in this paper has an excellent speed tracking effect and will not have a great impact on the stability and convergence speed of the system. This strategy can reduce the cost of physical sensors in the actual operation of wind farms and also has no significant impact on the anti-disturbance ability for the WECSs. Therefore, it can replace the speed sensor to a certain extent, also reduce the cost of the system, and achieve high economic benefits on the premise of guaranteeing the control accuracy of the wind power system.

Figure 3.

Velocity observation under basic wind.

Figure 4.

Velocity observation under gradient wind.

Figures 5 –8 show the comparisons of the generator speed tracking under basic wind and random wind. In order to simulate the disturbance of the wind power system during the actual operation, when t = 1, a sudden change is applied to the driving torque of the system. The generator speed tracking result at this time is shown in Figure 6. Figure 7 shows the ESO’s observation of disturbances in this case. It can be seen that pure MPC and composite model predictive controller can achieve good tracking effect when the system runs smoothly. However, when the system is disturbed or the wind speed changes abruptly, the tracking performance of the composite model predictive controller is faster and more stable than the traditional MPC. The fluctuation amplitude is smaller, that is, the anti-disturbance effect is better than that with pure MPC. The ESO can observe the total disturbances in time and compensate for them through the feedforward channel when the system is subject to internal and external disturbances, so that the accuracy of the generator speed observation is higher.

Figure 5.

Comparison of generator speed tracking under basic wind.

Figure 6.

Comparison of generator speed tracking when disturbed under basic wind.

Figure 7.

ESO observations when disturbed under basic wind.

Figure 8.

Comparison of generator speed tracking under random wind.

Conclusion

First, the wind speed, wind turbine, and PMSG are modeled separately in this paper to enhance the dynamic performance of the wind power system in the actual operation. Second, based on the in-depth study of model predictive controller and ESO, a composite MPC scheme is proposed to design the control system for capturing the maximum power. According to the different control requirements of speed loop and current loop, the model predictive controller is chosen as the main controller of the speed loop and the ESO is introduced into the feedforward channel to observe and compensate for the internal and external disturbances of the system. In addition, a speed sensorless control scheme based on the flux observer is proposed to estimate the speed of the generator. Hence, the whole composite MPC scheme is designed to track maximum power of a wind turbine without a mechanical sensor. Finally, the simulation results obtained using MATLAB show that the identification accuracy of generator speed with the proposed sensorless method is acceptable. Compared with the system using only the model predictive controller, the wind power system under the composite MPC strategy based on ESO has better tracking performance and anti-disturbance properties, which greatly improves the wind energy conversion capability and actual operating cost of the wind power system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was supported in part by the National Natural Science Foundation of China (Grant no. 61773335), Natural Science Foundation of Jiangsu Province (Grant nos. BK20171289 and BK20160476), Six Talent Peaks Foundation of Jiangsu Provincial (Grant no. KTHY2018038), the Natural Science Foundation of Yangzhou City for Outstanding Young Scholars (Grant no. YZ2017099), State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant no. LAPS19003).

ORCID iD

Shengquan Li

References

Hong

Chiou

. MPPT for PM wind generator using gradient approximation. Energ Convers Manage 2009; 50(1): 82–89.

Gonzalez

Morales

. Wind turbine speed control by state feedback. Cienc Ing 2017; 38(2): 123–130.

Nasiri

Milimonfared

Fathi

. Modeling, analysis and comparison of TSR and OTC methods for MPPT and power smoothing in permanent magnet synchronous generator-based wind turbines. Energ Convers Manage 2014; 86: 892–900.

Sun

Fang

Chen

, et al. Nonlinear stabilizing control for ship-mounted cranes with ship roll and heave movements: design, analysis, and experiments. IEEE T Syst Man Cy A 2018; 48: 1781–1793.

Sun

Fang

, et al. Nonlinear antiswing control for crane systems with double-pendulum swing effects and uncertain parameters: design and experiments. IEEE T Autom Sci Eng 2018; 15: 1413–1422.

Sun

Yang

Chen

, et al. Adaptive anti-swing and positioning control for 4-DOF rotary cranes subject to uncertain/unknown parameters with hardware experiments. IEEE T Syst Man Cy A 2019; 49: 1309–1321.

Yan

Zhang

Narayan

, et al. Generalized dynamic predictive control for nonparametric uncertain systems with application to series elastic actuators. IEEE T Ind Inform 2018; 14(11): 4829–4840.

Lin

Chen

Liu

, et al. Coordinated mechanical loads and power optimization of wind energy conversion system with variable-weight model predictive control strategy. Appl Energ 2019; 236: 307–317.

Sun

Lee

. Enhanced decentralized PI control for fluidized bed combustor via advanced disturbance observer. Control Eng Pract 2015; 42: 128–139.

10.

Fang

Zheng

Qiu

, et al. A control system for the ball mill grinding process based on model predictive control and equivalent-input-disturbance approach. J Adv Comput Intell Intell Inf 2016; 20(7): 1152–1158.

11.

Sun

Chen

, et al. An energy-optimal solution for transportation control of cranes with double pendulum dynamics: design and experiments. Mech Syst Signal Pr 2018; 102: 87–101.

12.

Yang

Sun

Chen

, et al. Neural network-based adaptive antiswing control of an underactuated ship-mounted crane with roll motions and input dead-zones. IEEE T Neur Net Lear. Epub ahead of print 6 May 2019. DOI: 10.1109/TNNLS.2019.2910580.

13.

Yan

Zhang

Liu

, et al. Disturbance rejection for nonlinear uncertain systems with output measurement error: application to a helicopter model. IEEE T Ind Inform. Epub ahead of print 15 April 2019. DOI: 10.1109/TII.2019.2910841.

14.

Han

. Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE T Ind Electron 2014; 61(11): 6285–6293.

15.

Gao

. Scaling and bandwidth parameterization based controller tuning. In: Proceedings of the 2003 American control conference, Denver, CT, 4–6 June 2003, pp. 4989–4996. New York: IEEE.

16.

Zhang

Yan

Wen

, et al. A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: algorithms and experimental tests. IEEE T Ind Electron 2018; 65(11): 8828–8839.

17.

Wang

Yang

. Extended state observer-based sliding mode control for PWM-based DC-DC buck power converter systems with mismatched disturbances. IET Control Theory A 2015; 9(4): 579–586.

18.

Zheng

Gao

. Predictive active disturbance rejection control for processes with time delay. ISA T 2014; 53(4): 873–881.

19.

Fang

Yuan

Liu

. Active-disturbance-rejection-control and fractional-order-proportional-integral-derivative hybrid control for hydroturbine speed governor system. Meas Control 2018; 51(5–6): 192–201.

20.

Zhang

, et al. On the rejection of internal and external disturbances in a wind energy conversion system with direct-driven PMSG. ISA T 2016; 61(1): 95–103.

21.

Al Kouzbary

Abu Osman

Wahab

AKA

. Sensorless control system for assistive robotic ankle-foot. Int J Adv Robot Syst 2018; 15(3): 1729881418775854.

22.

Zhao

Jiao

Chen

. Sensorless control of a linear permanent-magnet motor based on an improved disturbance observe. IEEE T Ind Electron 2018; 65(12): 9291–9300.

23.

Hong

Chen

. Maximum power point tracking-based control algorithm for PMSG wind generation system without mechanical sensors. Energ Convers Manage 2013; 69(12): 58–67.

24.

Liano

McMahon

. Modeling, control and sensorless speed estimation of micro-wind turbine for deployment in Antarctica. IET Renew Power Gen 2018; 12(3): 342–350.

25.

Abdelrahem

Hackl

Kennel

. Implementation and experimental investigation of a sensorless field-oriented control scheme for permanent-magnet synchronous generators. Electr Eng 2018; 100(2): 849–856.

26.

Putri

Pujiantara

Priyadi

, et al. Maximum power extraction improvement using sensorless controlled based on adaptive perturb and observer algorithm for PMSG wind turbine application. IET Electr Power App 2018; 12(4): 455–462.