Robust Variable-Pitch Control Design of PMSG Via Perturbation Observer

Wind turbine employs pitch angle control to maintain captured power at its rated value when the wind speed is higher than rated value. This work adopts a perturbation observer based sliding-mode control (POSMC) strategy to realize robust variable-pitch control of permanent magnet synchronous generator (PMSG). POSMC combines system nonlinearities, parametric uncertainties, unmodelled dynamics, and time-varying external disturbances into a perturbation, which aims to estimate the perturbation via a perturbation observer without an accurate system model. Subsequently, sliding mode control (SMC) is designed to completely compensate perturbation estimation in real-time for the sake of achieving a global consistent control performance and improving system robustness under complicated environments. Simulation results indicate that, compared with vector control (VC), feedback linearization control (FLC), and nonlinear adaptive control (NAC), POSMC has the best control performance in ramp wind and random wind and the highest robustness in terms of parameter uncertainty. Specially, the integral absolute error index of !m of POSMC is only 11.69%, 12.10% and 15.14% of that of VC, FLC and NAC in random wind speed.


Introduction
Energy is an essential material basis and support for human survival and social-economic development [1]. However, extensive consumption of limited fossil energy sources such as coal, oil, and natural [2] lead to severe environmental pollution, greenhouse effect, and increased global warming which are the common challenges in the world [3]. Hence, develop various renewable energy, e.g., wind [4], solar [5], geothermal [6], tides [7], waves [8], etc., and improve its efficiency have become a global consensus [9]. Specially, wind energy is one of promising alternative energy with the merits of pollution-free, cheap, widespread, and unlimited supply [10]. According to statistics of Renewables 2020 Global Status Report, the total growth rate of wind power capacity is around 228.79% in the globe over the past decade, which leads to a total of 651 GW installation up to 2019 [11].
And the operating regions of PMSG can be divided into three parts, as shown in Fig. 1. Specially, in Region 2, wind speed is between cut-in wind speed and rated value. The task of WT is to control turbine speed at the optimal value for extracting the maximum output power [13]. In Region 3, wind speed is between rated value and cut-out wind speed. The variable pitch controller adjusts the blade pitch angle to maintain output power at its rated value as the capacities of generator and converter are limited [14].
Various variable pitch control techniques have been reported in publications during the past decades so far. Conventional vector control (VC) using proportional-integral (PI) and proportional-integral-derivative (PID) control are widely adopted in industrial processes owing to its strengths of simple configuration and convenient implementation [15]. Nevertheless, it is an approximated linearization model based on an equilibrium point in which performance will inevitably degrade when the operating point is changed [16]. The linear quadratic Gaussian (LQG) is another common method in pitch angle control which can provide high robustness in terms of the phase and gain margins [17]. Nevertheless, wind energy conversion systems (WECS) are highly nonlinear due to the randomicity, intermittence, and seasons of wind energy such that this linear controller only has poor performance [18]. Hence, a series of advanced control strategies for pitch angle control, e.g., nonlinear control, fuzzy control, robust control, and selfadaptive control are presented to overcome the defects of VC and LQG. Adaptive PID [19] control and fuzzy self-tuning PID control [20] tune PID parameters on-line, which can suppress a variety of nonlinear, time-varying factors. But the inherent drawback of PID is still retained in the above frameworks, which cannot obtain consistent control performance [21]. Moreover, Senjyu et al. [22] proposes a generalized predictive control (GPC) for wind generators in all operating regions which can effectively mitigate the adverse effects of changed operating points. Van et al. [23] develops a low-cost fuzzy logic controller for variable-speed WT without consideration of expensive wind speed measurements. And Wang et al. [24] designs a two-degree-of-freedom motion mechanism with feedback linearization control (FLC) for the large WT with improved robustness and stability. Moreover, multi-layer perceptron and radial basis function neural networks are investigated in work [25] to prevent WT from overloading or shutting down during high wind speed. Recently, perturbation observer has been applied widely in nonlinear system control, such as WECS [26], photovoltaic systems [27], VSC-HVDC systems [28] and so on, which can on-line estimate unknown nonlinearities, parametric uncertainties, and time-varying external disturbances for nonlinear system without the requirement of detailed system model [29].
In this work, a perturbation observer based sliding-mode control (POSMC) is adopted for PMSG to limit the turbine output power and generator speed in Region 3. Firstly, the perturbation observer generates the new perturbation via combining the system nonlinearities, parametric uncertainties, unmodelled  Figure 1: Three operating regions of PMSG dynamics, and time-varying external disturbances. Then sliding mode control is utilized to completely compensate the perturbation estimation in real-time. The proposed POSMC retains the strong robustness of sliding-mode control (SMC), and only require the measurements of d-q axis current and mechanical rotation speed. Three case studies are studied by Matlab/Simulink, e.g., ramp wind speed, random wind speed, and parameter uncertainty. Simulation results validate that, compared with VC control, feedback linearization control (FLC) and nonlinear adaptive control (NAC), POSMC can achieve satisfactory robust control performance under various operating conditions.
The rest of this article is organized as follows: Section 2 gives the model of PMSG system; Section 3 introduces the theory of POSMC; Section 4 develops the detailed design of POSMC for variable-pitch PMSG. In Section 5, case studies results are discussed and analyzed. And the last Section summarizes this work and draws conclusions.

Modelling of PMSG System
A representative topology of PMSG system is described in Fig. 2. Firstly, mechanical power is transformed into electrical power via WT. Then, the electrical power is injected into power grid through back-to-back voltage source converters, filters, and the transformer. Specially, the major task of the machine-side converter (MSC) is to capture mechanical power and provide the required stator voltage, while active power and reactive power are regulated by the grid-side converter (GSC) [30]. In addition, MSC and GSC can realize a fully decoupled control by the DC link.

WT Model
The tip speed ratio of WT is denoted as [15] ¼ where R represents the blade radius, x m is the mechanical rotation speed, V is the wind speed.
According to aerodynamic theory, the electrical power extracted by the WT can be described as where q is the air density. And the power coefficient C P ðb; Þ can be represented by pitch angle b and

Generator System Model
The voltage and torque equations of PMSG can be denoted as where V d and V q represent d-q axis stator voltages, i d and i q are d-q axis stator currents, L d and L q are d-q axis inductances, x e ¼ px m is electrical rotate-speed, T e is the electromagnetic torque, p is the number of pole pairs, and K e is the permanent magnetic flux.

Drive Train Model
The dynamics model of mechanical shaft system can be represented by where J tot denote the total inertia of the drive train, T m is the mechanical torque, D is the viscous damping coefficient. Note that T e has a much faster response than T m , thus let _ T e ¼ 0 in this paper to realize simplified calculation.

Pitch Angel Control Model
Pitch angel control actuator could regulate the blade pitch based on the required value. And the firstorder linear model of pitch angel control actuator without considering the delay characteristics can be given as [14] where b r is the required pitch angle, and s b is the time constant of actuator.
3 Perturbation Observer Based Sliding-Mode Control 3.1 Design of SMSPO An uncertain nonlinear system is denoted as . . . ; x n ½ T 2 R n , u 2 R, y 2 R denote state vector, control input, and system output, respectively. a x ð Þ : R n 7 !R and b x ð Þ : R n 7 !R represent unknown smooth functions, and d t ð Þ : R þ 7 !R is time-varying external disturbance. The perturbation of system (11) is described as where b 0 is constant control gain.
Suppose y = x 1 is the sole measurable state, a (n+1)th order SMSPO is designed to estimate the system states and perturbation, obtains wherex represents the estimated value of x,x¼x Àx denotes the estimation error, positive constants a i (i = 1,2,…,n) are observer gains, and positive constants k i (i = 1,2,…,n) are sliding surface gains.

Design of Sliding-Mode Controller
An estimated sliding surface is defined aŝ where the estimated sliding surface gains q i ¼ C iÀ1 nÀ1 nÀi c (i = 1,2,…,n) place all the poles of estimated sliding surface on the left half À c of complex plane with c > 0.
Finally, POSMC of system is given as where & and ' are controller gains, o c is the thickness layer boundary of controller. x where x 2 R 4 , u 2 R 3 , and y 2 R 3 are state vector, control input, and system output, respectively. R s is the stator resistance, V d and V q are d-q axis stator voltages, and x e ¼px m is the electrical generator rotation speed.

Pitch Angle Control
Differentiate control output y 1 ¼ x m until it appears explicitly, as where Eq. (24) can be rewritten into matrix form, yields where Note that det B 1 ðxÞ ½ 6 ¼ 0 when V 6 ¼ 0 and b 6 ¼ 0, thus B 1 ðxÞ is nonsingular in all the feasible zone. In other words, above input-output linearization is valid.
Define perturbation w 1 ðÁÞ to describe the nonlinearities and uncertainties of F 1 ðxÞ and B 1 ðxÞ, yields where B 1 ð0Þ¼b 10 is constant control gain.
Define tracking error Then, a third order sliding-mode state and perturbation observer (SMSPO) is adopted to estimate w 1 ðÁÞ, as where positive constants a 11 , a 12 , a 13 , k 11 , k 12 , and k 13 are observer gains.
The estimated sliding surface of system (29) is chosen aŝ Finally, the POSMC of system (29) is designed as where q 1 is estimated sliding surface gains, & 1 and ' 1 are controller gains.

Generator Control
Differentiate control output y ¼ y 2 ; y 3 ½ T ¼ i d ; i q Â Ã T until it appears explicitly, as where is nonsingular in all the feasible zone. In other words, above input-output linearization is valid.
Define perturbation w 2 ðÁÞ and w 3 ðÁÞ to describe the nonlinearities and uncertainties of F 2 ðxÞ, F 3 ðxÞ, and B 2 ðxÞ, yields Then, two second order sliding-mode perturbation observers (SMPOs) are adopted to estimate w 2 Á ð Þ and w 3 Á ð Þ, as where positive constants a 21 , a 22 , a 31 , a 32 , k 21 , k 22 , k 31 and k 32 are observer gains.
At this end, the overall block diagram of POSMC is shown in Fig. 3.

Analysis and Discussion
As described in Assumption 2, the perturbation and its derivative are locally bounded. And the deduction of these bounds is given as Hence, the validity of the developed perturbation observer is demonstrated.

Case Studies
Three cases, e.g., ramp wind speed, random wind speed, and parameter uncertainty, are undertaken to assess the performance of POSMC compared with that of VC [15], FLC [24], and NAC [14]. The simulation is implemented based on MATLAB R2019a by a desktop computer with an Intel® Core™ i5 CPU at 3.4 GHz and 16 GB of RAM. And the parameters of PMSG and POSMC are listed in Tabs. 1 and 2, respectively.

Ramp Wind Speed
A ramp wind signal changing from 18 m/s to 14 m/s is exerted to WECS, as shown in Fig. 4. The simulation outcomes of four controllers under ramp wind are shown in Fig. 5. One can easily find that VC has the longest convergence time and the biggest tracking error of x m due to its control gains are obtained by local linearization of specific system operating point. FLC has obvious oscillation of x m , P m , and Q m during running period which needs full state measurements. Meanwhile, NAC has the highest overshoot of i md and i mq compared with other three methods. And POSMC can obtain the satisfactory control performance with the fast convergence speed and the small tracking error. Specially, the convergence time of VC, FLC, NAC, and POSMC in terms of x m are 4.16 s, 5.71 s, 5.80 s, and 16.46 s, respectively. And the maximum overshoot of P m of VC, FLC, NAC and POSMC is 8.70% and 3.55%, 1.25% and 0.15%, respectively. Meanwhile, the errors between the estimations and actual values of the designed observers are shown in Fig. 6 which provides satisfactory estimation.

Parameter Uncertainty
In this case, the variation of field flux K e from 1 (p.u.) at t = 4s to 0.9 (p.u.) at t = 9s is implemented to system for evaluating the robustness of four controllers. And wind speed remains at 18 m/s during all the simulation time. Fig. 10 shows the simulation outcomes of four controllers under above scenario. It is clear that POSMC can restore perturbed system with the fastest speed. Though VC and FLC have the lower oscillation of i md because of the simple mechanism, they have the worst control performance in other seven output variables. And the maximum overshoot of x m of VC, FLC and NAC is 5.16% and 1.24% and 0.36%, respectively. While POSMC can realize nearly smooth tracking performance of all output variables with the strongest robustness.

Statistical Analysis
Integral absolute error index IAE x ¼ R T 0 jx À x Ã jdt is widely used in the quantitative analysis of control errors, which describes the error accumulation of system output compared to its reference value over a period of time T. Tab. 3 gives IAEx m , IAEi d , and IAEi q of four controllers in three cases. VC obtains the smallest IAEi d and IAEi q in most cases thanks to its simple structure, but it has the longest recovery time and most obvious oscillations of x m due to the linear frame. And POSMC can obtain the smallest IAEx m in all the four controllers under three cases. Moreover, the overall control costs R T 0 ðjb r j þ jV d j þ jV q jÞdt of four controllers are shown in Fig. 11. Although VC have the lower control costs than POSMC in ramp wind, it has poor control performance. While POSMC has the smallest control costs than VC, FLC, and NAC in random wind due to the integration of nonlinear real-time perturbation estimation and robust SMC structure. Specially, the overall control costs of POSMC are 99.58%, 99.52% and 96.73% of that of VC, FLC, and NAC in random wind.

Conclusions
In this paper, POSMC is applied in variable-pitch control of PMSG to limit generator's output power at its rated value when the wind speed is higher than rated value. The main novelties/contributions can be concluded as follows: (i) POSMC combines nonlinearities, parametric uncertainties, unmodelled dynamics, and time-varying external disturbances into a new perturbation estimating via the perturbation observer. Subsequently, sliding mode control is designed to completely make up for the perturbation estimation in real-time for the sake of realizing a global consistent control performance and improving the robustness of the system under various operation conditions.  Figure 11: The overall control costs of four control approaches (ii) Compared with VC, POSMC is designed based on nonlinear architecture which is not affected by the changed system operating points.
(iii) Compared with FLC, POSMC only requires the measurements of d-q axis current and mechanical rotation speed x m rather than full-state measurements which are easy to implement.
(iv) Compared with NAC, POSMC has the better control performance in ramp wind and random wind, the higher robustness in terms of parameter uncertainty, the smaller IAE indexes and overall control costs. Specially, the IAEx m of POSMC is only 1.45%, 15.14% and 1.79% of that of NAC in ramp wind, random wind and parameter uncertainty, respectively. And the overall control costs of POSMC are 96.86% and 96.73% of that of NAC in ramp wind and random wind, respectively.
Future studies will be focused on carrying out the HIL experiment of variable-pitch PMSG to further prove the implementation feasibility of POSMC.
Funding Statement: The authors thankfully acknowledge the support of the Noise problem of electric vehicle brushless DC motor starting (S202010641109).

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.