Theoretical and Applied Contributions to Robust Stability Analysis of Complex Systems
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Jun Liu, Feihang Zhou, Chencong Zhao, Zhuoran Wang, "A PIType Sliding Mode Controller Design for PMSGBased Wind Turbine", Complexity, vol. 2019, Article ID 2538206, 12 pages, 2019. https://doi.org/10.1155/2019/2538206
A PIType Sliding Mode Controller Design for PMSGBased Wind Turbine
Abstract
Because the PMSG (permanent magnet synchronous generator)based WECS (wind energy conversion system) has some uncertainties, the conventional control strategy with poor robustness is sometimes difficult to meet the performance requirements of control. In order to ensure efficient and stability of the system, this paper proposed a novel PI (proportionalintegral)type SMC (sliding mode control) strategy for PMSGbased WECS uncertainties and presented the detailed analysis and design process. Compared with the conventional control method, the PItype SMC proposed in this paper not only can make the closedloop system globally stable, but also has a better robustness and slightly reduced the current ripple and distortion. Finally, the simulation results verify the correctness and effectiveness of this algorithm.
1. Introduction
Environment problems such air pollution and global warming caused by fossil fuels have drawn the world’s attention to exploration and utilization of renewable energy sources, in recent years [1–5]. At present, wind energy is the fastest growing renewable energy source and is most prevalent in coastal regions spanning temperate and boreal climates [6–8]. There is great potential for wind power development in China, USA, Denmark, and other countries, due to their high average wind velocities [9]. Therefore, the research on wind power generation technology has a significant value nowadays. Compared with the constantspeed constantfrequency wind turbine, the variablespeed constantfrequency wind turbines can obtain the maximum energy conversion due to its rotational speed could vary with wind speed to ensure that the system has the OTSR (optimal tip speed ratio) and maximum wind energy utilization coefficient. The variablespeed constantfrequency wind energy conversion system (WECS) consists of DFIG (doubly fed induction generator)based WECS and PMSG (permanent magnet synchronous generator)based WECS. The PMSGbased WECS was selected to study in this paper, due to the fact that PMSG has many superior characteristics such as wider speed control range, higher reliability, and more efficient performance compared with DIFG [10].
In fact, the practical systems have many uncertainties. The uncertainties of PMSG or PMSM (permanent magnet synchronous motor) consist of the unmodeled converter dynamics and the parameters perturbations [11–14]. A robust control scheme for PMSM uncertainties based on an adaptive DOB (disturbance observer) was introduced in [11]. The results indicated that the controller obtained a good control performance. Reference [12] proposed a robust nonlinear predictive control strategy for PMSM uncertainties. The control system obtained a high speed tracking precision. In [13], a PFC (predictive functional control) + ESO (extended state observer) method was studied. After that, the effectiveness of this new method was verified. Reference [14] proposed a novel decoupled PI current control method for the PMSGbased WECS. This method can successfully achieve improved the transient performances and the nominal performance recovery under the model uncertainty.
SMC (sliding mode control) first proposed in the early 1950s has a good robustness and powerful ability to reject the plant uncertainties and disturbances [15, 16]. References [15, 16] summarized the development of SMC and examined key technical research issues and future perspectives. Although the SMC has been widely and extensively employed in some industrial applications for a long term, it is still the researching focus for scholars and worth studying in depth. A DPC (direct power control) based on SMC for the gridconnected WECS was presented in [17]. When the grid voltage is unbalanced, the controller also can regulate the instantaneous active and reactive powers directly in stator stationary reference frame. Reference [18] enhanced the exponential reaching law to improve the control efficiency and performance of SMC used in PMSGbased WECS.
An adaptive secondorder SMC strategy was explored in [19]. This method can effectively deals with the presence of model uncertainties, intrinsic nonlinear behavior of WECS, and random wind. A SMC method for mismatched uncertainties via a nonlinear DOB was developed in [20]. Meanwhile, the ISMC (integral slidingmodel control) with a good steadystate precision was mentioned in [20, 21]. The PItype SMC is derived by the ISMC and FBL (feedback linearization) approach and has been widely used in renewable energy conversion systems and the electric motor drives [21–23]. Furthermore, compared with the ISMC, the proportional and integral parameters of the PItype SMC are able to be adjusted to better meet the control performance indexes such as the celerity and accuracy.
Based on the above background, this paper presents a novel PItype SMC for the PMSGbased WECS. Although the inspiration for this paper comes from literature [20–23], the whole designing thought and procedure of the proposed controller are completely different from the methods in [20–23]. Furthermore, [20, 23] just only referred to the control of PMSG without the consideration of gridconnected control. And the study subjects in [22, 23] and this paper are different. At the same time, the suppression of flexible drivetrain torsional vibration also has been regarded as a control target in this paper. Then, we set up a detailed 2 MW WECS simulation test platform based on MATLAB/SIMULINK/SimPowerSysterms to verify the effectiveness and correctness of the proposed method. A large number of existing packaging modules in SimPowerSysterms are used in the simulation test platform that is relatively close to the real physical system. Finally, the simulation results indicate the proposed strategy has good control performance.
2. PMSGBased WECS Model
The simplified PMSGbased WECS mainly consists of a wind turbine, a flexible drivetrain, a PMSG, fully rated converters, and its control level shown in Figure 1. In order to capture the maximum wind energy, the maximum power point tracking (MPPT) strategy is adopted to control the machineside converter (MSC). In the loop control level, the gridside converter (GSC) control is to regulate the reactive power and keep DClink voltage U_{dc} stable at 1800V. Meanwhile, in order to suppress the torsional vibration of flexible drivetrain, the damping compensation torque T_{Damp} was introduced.
2.1. PMSG and MSC Dynamic Model
The PMSG and MSC mathematical model is [5, 20–27]where U_{sd} and U_{sq} are the daxis and qaxis stator armature voltages. i_{sd} and i_{sq} are the daxis and qaxis components of stator currents. L_{sd} and L_{sq} are the daxis and qaxis stator inductances. ω is the rotor speed. represents magnetic pole logarithms. is the stator resistance. ψ represents the permanent magnet chain. And the disturbance vector and represent model uncertainties including the external disturbances and the PWM offset. and were generally assumed to be bounded by D_{sd} and D_{sd}. where D_{sd} and D_{sq} are the boundaries of and .
The PMSG torque is given by
2.2. GSC Dynamic Model
The GSC dynamic model is given bywhere and are the control voltages. and are the components of gridside currents. is the power grid frequency. L_{c} and R_{c} are the filter inductance and resistance. and express the daxis and qaxis power grid potential components (usually, = 0). The and can be gotten by the voltage phaselocked loop. and are also the uncertainties and meet and are the boundaries of and .
3. PIType Sliding Mode Controller Design
3.1. Control Objectives
If the state vector x and the reference state vector x_{_ref} arethe control objectives of WECS can be expressed aswhereis the error vector. In the above formula, usually we havewhere K_{P} and K_{I} are the PI (proportionalintegral) parameters. The optimal torque T_{opt} in (9) meets [1, 26–29]where R is the wind wheel radius, ρ is the air density, is defined as the maximum wind energy conversion coefficient, λ_{opt} is the OTSR (optimal tip speed ratio), and the damping compensation torque T_{Damp} is given by [29–38]where and H(s) is the transfer function of bandpass filter shown in [29–38].
3.2. Design of ProportionalIntegral (PI)Type Sliding Mode Controller
The PItype sliding surfacecan be defined aswheres is Laplace variable. and (i=1,2,3,4). Obviously, we also can getif the Lyapunov function is defined aswhereEquations (19)(22) can be gotten by taking the derivative of (17).
LetEquation (27) can be gotten.where . Therefore, the control laws are given by (28) (31) and the control structure diagram is shown in Figure 2.From (17) and (27), it is clear that the Lyapunov energy function V is greater than or equal to 0 and the derivative of V is less than or equal to 0. Hence, the whole system is asymptotically stable based on Lyapunov stability theorem.
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3.3. Tracking Accuracy Analysis
In order to analyze the tracking accuracy, we need to introduce Theorem 1 here.
Theorem 1. For any dynamic system with state vector and assuming is a Lyapunov function of system, if or where , then .
Proof. There are three cases.
① If , then , and further . Due to , thus . This means that there are not continuous and stable equilibrium points such as stable limit cycle in this system.
② Assuming , due to , then, if , then , and furthermore . Obviously, this is in contradiction with . Thus .
③ Given , according to Lyapunov stability criterion, the system is stable. So, there must be t_{1} meeting . Otherwise . This is clearly in contradiction with system stability. If we redefined t_{1} as 0, we can get by ②.
When , we have the same conclusion in the same way.
Based on Theorem 1, we haveFrom (12) and (33), we can getThe above formula shows that the steadystate error of the system is always 0 regardless of the input. Therefore, the proposed strategy can obtain a good tracking accuracy.
4. Algorithm Analysis and Verification
In this section, a detailed simulation test platform is constructed to verify the correctness and effectiveness of the new algorithm, shown in Figure 3. The simulation test platform is based on the MATLAB/Simulink environment and the packaged modules in SimPowerSysterms are used in this simulation test platform, including the wind turbine, PMSG, and VSCs. Meanwhile, the two mass block spring damping model mentioned in [21–31] is adopted to describe the dynamics of flexible drive chain. Therefore, the simulation test platform is relatively close to the actual physical system, due to the adoption of many packaged modules. The controller of WECS consists of MSC controller and GSC controller which are mentioned in Figure 3. The system parameters are shown in Table 1. In general, the actual wind speed is timevarying and random; however we also believe the wind speed is constant for a short period of time. Hence, we assume that the wind speed meets v = 12 + 0.4 × rand(t). The DClink currents i_{1} and i_{2} are shown in Figure 4 and the other response curves of WECS are shown in Figure 5.

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From Figure 4, we know that the DClink currents i_{1} and i_{2} are bidirectional. This means that the MSC or GSC is able to switch between rectifier and inverter state. However, because of the mean values of i_{1} and i_{2} are positive, therefore the MSC is basically working as a rectifier and the GSC is mostly in the state of inverter. This is in accordance with the actual accident situation and can verify the accuracy of the model in a sense.
Figure 5 shows the response curves of wind energy utilization coefficient C_{P}, DClink voltage U_{dc}, daxis, and qaxis components of stator current and grid current. The disturbance component of the wind speed is the main causes of C_{P}, U_{dc}, current, and power fluctuations. Figure 5(a) depicts the wind energy utilization coefficient C_{P} is always in the vicinity of the maximum value (or 0.4763) under the proposed control strategy. This implies the WECS is operating in MPPT mode to capture the maximum wind energy under the rated wind speed. It is clear that the DClink voltage U_{dc} is very close to its reference (or 1800V) and the deviation does not exceed 0.8V by Figure 5(b). Figures 5(b)–5(f) show that the proposed control can reduce the current ripple of stator currents and grid currents compared with the conventional control, because the uncertainty of the unmodeled dynamics is taken into account in the design of the proposed controller. In one word, the proposed control strategy has a better robustness than the conventional control strategy.
The electromagnetic torque and sliding mode surfaces curves are shown in Figures 6 and 7. It is clear that the electromagnetic torque fluctuation under the proposed control method is smaller, compared with the conventional control method. Meanwhile, Figure 7 shows all steadystate values of sliding mode surfaces are tend to zero. Thus, the correctness of the analysis in Section 3.3 is verified by Figure 7. Furthermore, the proposed scheme was further verified for the suppression of current harmonics. Here, we take the Aphase stator current of generator as an example. The Aphase stator current waveform is shown in Figure 8. Obviously, the proposed method also can reduce the current distortion.
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To further verify the effectiveness of the proposed method, the system responses under variable wind speed must be observed and analyzed. As Figure 9(a) shows, the wind speed varied from 10 m/s to 12.5m/s. At this time, the system response curves under the variable wind speed are shown in Figures 9(b)–9(h). It is clear that the results demonstrated the correctness and effectiveness of the proposed approach again. Figures 9(c) and 9(f) show that the wind energy utilization coefficient C_{P} is always maintained at its maximum value (0.476) and the DClink voltage U_{dc} always fluctuates around U_{dc_ref} (1800V), whether the wind speed changes. From Figure 9(d), the gridconnected active power is always less than the electromagnetic power due to the copper loss.
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5. Conclusion
A novel PItype SMC was proposed to ensure efficient and stability of PMSGbased WECS, in this paper. The presented strategy with a strong robustness for the uncertainties and disturbances of the system can make the closedloop system globally stable and more performant, compared with the conventional control method. Finally, a 2MW WECS simulation test platform which is based on the MATLAB/MATLAB/SIMULINK/SimPowerSysterms environment was built to verify the maturity and effectiveness of this presented control algorithm. The simulation results indicated that the new method is able to reduce the current distortion and torque fluctuation, which has important actual significance for the control of practical wind turbine.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research was supported by Key R&D projects in Shaanxi (Grant no. 2017GY061).
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Copyright © 2019 Jun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.