Application of finite-time control Lyapunov function in low-power PMSG wind energy conversion systems for sensorless MPPT

https://doi.org/10.1016/j.ijepes.2018.09.039Get rights and content

Highlights

Abstract

This paper discusses the problem of observer-based finite-time control (FTC) of a grid-connected wind turbine and a low-power permanent magnet synchronous generator (PMSG). An adaptive nonlinear observer is employed to estimate the mechanical variables. Maximum power point tracking (MPPT) is obtained using the estimation of rotor speed and torque from the adaptive observer, and excluding the wind speed sensor. Improvement of the MPPT technique through the designed FTC is investigated. The proposed controller stabilizes the WECS and tracks the reference trajectories in a short pre-known time alternative to common nonlinear controllers with large settling time. The suggested controller is also robust against uncertainties in WECS parameters. Parameters’ variations are compensated by robust control design. Finite time stability and robustness of the proposed WECS controller is mathematically proved. Moreover, the advanced performance of the suggested FTC is demonstrated by simulation and is compared to a conventional asymptotic convergent controller (ACC). The proposed FTC provides fast and robust rotor speed regulation and thus enhances the sensorless MPPT. The proposed FTC improves the WECS performance for tracking of ramp references and robustness against parameter uncertainties. Furthermore, advanced control of the grid-side converter yields improved resiliency and reliability.

Introduction

Nowadays, distributed energy resources (DER) are putative as a technology of utility generation, for their capabilities in providing; backup power, demand and peak power reduction, improved power quality, and ancillary services to the grid [1]. Wind energy, as a DER, is noticeably taking part in electricity generation for its renewable nature, reduced environmental impacts, high installed capacity, its applications in smart grids, and islanded operation capability [1].

The wind energy is converted into mechanical power through a wind turbine (WT) and consequently into electrical energy with an induction or a synchronous generator. Usually, power electronic devices are used to interconnect the synchronous generators to the grid [1], [2]. Fundamentally, wind energy interfacing systems can be categorized as direct connection, partially-rated power electronics, and full-scale power electronics interface (FSPE) [2].

In a FSPE system, generation system is a conventional synchronous generator or a PMSG to interface the WT to the grid. WT’s output power should be converted to a utility-compatible variable voltage and variable frequency output [2] in order to be serviceable. Although FSPE system causes extra losses in this design, variable speed mode operation ability of FSPE topology provides more energy to be captured, and therefore enables maximum power point tracking (MPPT) [1]. PMSG has numerous advantages to improve power extraction from WTs such as high power density, exclusion of the gearbox, and reduced need for maintenance. For a PMSG with FSPE system, control of the back-to-back voltage source converter (VSC) provides some advantages including MPPT, low harmonics in injected line side current (IEEE 519), wide range of operation, bidirectional power flow, and islanded operation [1]. A detailed methodology based on analytical characteristics of the PMSG-based WECS is developed to determine the critical steady state operating points and assess their feasibility in [3].

Naturally, the wind speed dynamic is inherently variable; consisting average value, gusts, noises and ramps. So, rotor speed must be updated with wind speed changes in order to get MPPT. There are two categories of MPPT methods; The first category named as ‘WT characteristics-based MPPT methods’ requires online wind and rotor speed measurement [4], [5]. Since the turbine blades are large in diameter, the average wind speed through the blades is measured instead of the exact value, which reduces the MPPT accuracy. A linear Kalman filter can be utilized to approximate the load and torque by measuring the rotor speed [5]. The drawback of this method is using linear approximation of the system which is completely nonlinear in reality [6]. The second category such as “extremum seeking” or “perturb and observe” are appropriate for WTs with small inertia [7], [8] in which the WT characteristics is not necessary [9].

In order to establish MPPT, the wind speed is measured by means of an anemometer installed at nacelle, which could not measure the accurate effective speed at the blades [4]. Essence of mechanical sensors for wind speed, rotor position and speed, and mechanical output torque is a limiting factor in most control and MPPT techniques. It is advantageous to have a controller independent of mechanical sensors which also follows reduction in costs and maintenance requirements of low-power wind turbines [10]. An observer-based controller would also be beneficial in the presence of mechanical sensors in order to improve the reliability of the WECS and enable the sensor fault detection [7], since the controller can get the information from either measurement or observation. Although the rotor speed and mechanical torque can easily be measured in practical applications, the observation for the two parameters can be used to increase the reliability of the control loops in the case of sensor failure as well as fault detection. The observer-based controller can switch to use the observed information, whenever a fault is detected from the mechanical sensors.

High frequency signal injection methods [11], [12] has been used to determines the rotor position in low speeds. The back electromotive forces of PMSG at very low wind speeds are insufficient to accurately observe the stator flux estimation. Moreover, the dead-time and forward voltage drops of the switching devices of the VSC substantially impact its performance accuracy at low wind speeds. Furthermore, nonlinear characteristics of the switching devices of the VSC deteriorate the precision of flux and torque estimation at very low wind speeds. Therefore, specific estimation methods are required at low speeds, such as high frequency signal injection [11], [12]. Nonetheless, this method is undesirable fallacious while operating in medium and high speed, in which observer-based estimation methods provide high precision. A changeover algorithm can be employed to switch between the estimation methods regarding the speed [12]. The excitation method in [13] detects the rotor position using stator currents and voltages with no signal injection. The induced stator voltage or back EMF is also used to observe the rotor position by Kalman filters in [14], [15] which is suitable for medium and high operating speed but not for low speeds with low back EMF. PLL is also used to determine the position and speed of the rotor in [16]. The impact of PLL dynamics on sub-synchronous interactions and stability of the grid-connected PMSG-based WECS with PI controllers was investigated using modal analysis in [17]. An observer based on quasi-sliding modes is used for estimation of rotor position and stator flux linkage in [18] with direct torque control through a proportional controller. Direct torque control scheme was also used along with space vector modulation to track the flux and torque references in [19]. In this scheme, the references were achieved through online optimization, and MPPT was established using an observer to calculate the reference electrical torque [19]. Sliding mode model-reference scheme was used for adaptive observation of the rotor speed with a fuzzy WECS controller [20]. The neural network model reference method requires an extensive data for offline training process of the neural network [21]. Therefore, wind speed estimation using artificial neural network with online training was investigated for MPPT [22]. Moreover, an adaptive nonlinear observer is developed in [23] which is appropriate for PMSG and does not have the problems mentioned above. The nonlinear adaptive observer in [23] can be used to estimate the rotor speed and flux as the system states and the load torque as a variable parameter, by measuring the stator currents and voltages at the stationary reference frame [24].

Most nonlinear observers involve continuous-time measurements of the outputs [25]. For real-worlds practices and digital implementations of the observer systems, the sampled data nonlinear observer designs [25], [26], [27], [28] has been developed, to estimate the mechanical variables of a permanent magnet synchronous machine through sampled output measurements. The observers were designed based on Lyapunov stability theorem and provide exponential convergence of the estimated variables to real values. Moreover, a robust observer based on sampled and delayed output measurements was proposed in [27], considering the effect of external disturbances. The maximum allowable time-delays and sampling period was determined to ensure exponential convergence of the observed states in [27]. The output-feedback control of the direct-drive WECS using sampled-data state-observer was developed by [28] to improve the performance and reduce the costs of the state-feedback control methods. The backstepping method was employed to design the nonlinear sampled controller based on a Lyapunov-Krasovskii candidate function. The practical implementation of the output-feedback nonlinear controller was discussed realizing the discrete-time sampled controller of the WECS [28].

Linear control methods are the most common schemes for the WECS such as the PI controllers [29], [30], [31], [32], [33], the fuzzy PID [34], and the self-tuned adaptive PI control [35]. Linear control schemes are fundamental to stability of WECS and requires detailed analysis for stability for connection of WECS to grid. PI control gains are tuned based on the stability and transient performance of the linearized model of WECS at the specified operating point, considering the desired bandwidth of the control loops. The instability caused by the intrinsic positive zero and pole in linearized transfer function of the VSC model is the key problem in designing linear controller for the PMSG-WECS, particularly when connected to a weak grid [36].

The PI and linear control schemes are designed based on the operating point of the WECS, in which the system is linearized. Thus, the performance of linear controllers is degraded in the presence of external disturbances and WECS parameter variations. Therefore, the PI [29], [30], [31], [32], [33], [34], [35] and linear controllers [18], [37], [38] provide limited robustness against large external disturbances and parameter uncertainties which are the foremost drawback of these controllers.

Intelligent and fuzzy methods employs extensive data gathering and experts’ knowledge to develop the WECS controller. Designing the intelligent controllers entails a general training data to extract the rules for a fuzzy control system or train an artificial neural network. Performance of the intelligent controllers [39], [40], [41] are impacted by the learning and training conditions, especially the training epochs.

Nonlinear controller design using the sliding mode technique [42], [43] provides improved robustness against parameter uncertainties of WECS. Moreover, the super-twisting sliding mode [43] reduces the negative impact of chattering using integral terms.

Although a faster convergence can be achieved by increasing the bandwidth of the control loops, utilizing a low-order linear controller limits the response characteristics of the control loops of the WECS, considering the stability margins (i.e. gains and phase margins) as a trade-off between the robustness and transient performance of the control loops of WECS. The linear and PI controllers [29], [30], [31], [32], [33], [34], [35], [37], [38], and the nonlinear Lyapunov [24] and sliding mode controllers [42], [43] result in asymptotic convergence of the state trajectories to the reference values.

Despite the nonlinear and linear controllers with asymptotic convergence, the finite-time control schemes yield finite-time stability. The finite-time controllers provide inherent advantages of fast transient responses, high steady state accuracy, and finite-time convergence of the system errors to the equilibrium. Utilizing finite-time controllers, the state trajectories converge to reference values in a predefined finite time, providing improved and fast transient performance as well as steady state accuracy, in comparison with conventional asymptotic convergent controllers (ACCs) and linear controllers. FTCs as nonlinear controllers realize improved robustness against disturbances and uncertainties [44].

In this paper, a robust finite-time control method for the PMSG-based WECS is proposed. The proposed nonlinear regulators of the machine side and the grid side converters are developed based on the back-stepping design technique and the control Lyapunov function method. The proposed controller provides advanced tracking performance considering the transient settling time and the steady state accuracy. Additionally, an adaptive observer is utilized for mechanical-sensorless MPPT and improving the reliability of WECS in case of sensor failures. The adaptive observer provides estimates of rotor speed, load torque, and the rotor flux. The multi-loop output-feedback FTC contains rotor speed, voltage, grid current, and power flow control loops. The objectives of the proposed FTC is to stabilize the system, to improve MPPT by lowering the transient convergence time in comparison with conventional ACCs, to improve the steady state performance in tracking ramp wind speeds, and to provide robustness against parameter changes and variation.

The remaining parts of the paper are organized as follows: Section 2 summarizes the sensorless MPPT technique. The PMSG-based WECS is modeled in Section 3. In Section 4, the controller is designed using control Lyapunov function method and its finite-time convergence is mathematically proved. Robustness of the proposed FTC is discussed in Section 5. The Adaptive state and parameter observer is introduced in Section 6. Simulation results are included in Section 7. Section 8 concludes the paper.

Section snippets

Sensorless maximum power point tracking

In a variable speed PMSG-based WECS, an optimum rotor speed reference tracking could lead to MPPT. An optimization system is assumed to calculate the optimum rotor speed in order to deliver the maximum power that could be captured from the WECS. The optimum speed-power curve MPPT technique [24] is utilized and improved by the proposed robust FTC. The captured power from a WT is:Pw=0.5cpρAvwind3where Pw, vwind, A, ρ and cp are wind power, wind speed, swept area of the turbine blades, air

PMSG-based WECS modeling

The WT with PMSG is connected to the grid with an AC/DC/AC converter in order to achieve MPPT, soft start-up, unity power factor in grid side and bidirectional current flow mode.

The whole system is modeled in state space form in dq reference frame with rotor position as (3) including WT, PMSG and the VSCs [37]. Since there is no gearbox and the coupling is direct, generator mechanical output torque is equal to the turbine shaft torque.ddtωisqisd=-FJKMJ0-KMLsRsLspω-pisq0RsLsωisqisd-1Ls0u1u2+1JTg0

Finite time controller design via control Lyapunov function

The WECS including a synchronous permanent magnet aero-generator, the proposed FTC with the nonlinear adaptive observer and the regulating loops are depicted in Fig. 3. The function and explanations of each block are presented in the following sections. The ac side of the machine-side converter (VSC1) is connected to the PMSG and controls the rotor speed for MPPT. The reference rotor speed is calculated through the optimal speed-power curve of the MPPT algorithm. Besides, the ac side of

Robustness against bounded uncertainty and external disturbances

Uncertainties and disturbances, which are not involved in the model of the system under study, are inevitable. These uncertainties include modeling error, mechanical vibration, faults, variations of WECS parameters, time dependency of the parameters, and external disturbances. Accordingly, robustness against uncertainty is reasonably significant in practice. Robustness of the closed-loop WECS with the proposed controller is proved in the following.

Suppose an uncertain parameter (i.e. Π)

System state and parameter estimation with adaptive observation

The proposed FTC is based on the dq reference frame model of the WT and PMSG. The dq transformation of the utilized model is based on rotor position. It is supposed that the rotor position is not available since it needs mechanical sensor. Because of this, the PMSG model in the αβ frame is presently used which does not need the rotor position for modeling and controller designs. The PMSG model in the αβ stationary coordinates in matrix form is [50]:Ẋ1=A1X1+G1+0,1/JTTgẊ2=A2X2+G2y1=isαy2=isβ

Numerical simulation

The WT, PMSG, VSCs, and the control loops of Figs. 3 and 4 are simulated in Matlab®/Simulink R2017a using the Simscape toolbox blocks. The ODE4 is selected with type of fixed step as the solver. Matrix operations of the DSP system toolbox are exploited to evaluate the presented adaptive observer using matrix calculations. The switching frequency is 1350 Hz, and the sampling time is 1μs.

Five cases are studied to analyze the proposed FTC performance as:

  • I.

    Transient performance of the proposed FTC is

Conclusion

This paper addressed the recent studies in connecting a PMSG-based WECS to the grid and designed a novel robust finite-time controller (FTC) based on control Lyapunov function method. The proposed FTC improves the speed-tracking transient performance and the robustness of the low-power WECS. Using the designed FTC, the error trajectories of the WECS converge to zero in a short pre-known finite time in comparison with the conventional ACCs with asymptotic convergence. The proposed FTC improves

Acknowledgement

This paper is published as part of a postdoctoral research project supported by Research Affairs Office of University of Tabriz, Tabriz, Iran under Grant No. S-27-8.

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