- Original research
- Open Access
Residue Theorem based soft sliding mode control for wind power generation systems
© The Author(s) 2018
- Received: 20 October 2017
- Accepted: 21 June 2018
- Published: 8 August 2018
This paper proposes a residue theorem based soft sliding mode control strategy for a permanent magnet synchronous generator (PMSG) based wind power generation system (WPGS), to achieve the maximum energy conversion and improved in the system dynamic performance. The main idea is to set a soft dynamic boundary for the controlled variables around a reference point. Thus the controlled variables would lie on a point inside the boundary. The convergence of the operating point is ensured by following the Forward Euler method. The proposed control has been verified via simulation and experiments, compared with conventional sliding mode control (SMC) and proportional integral (PI) control.
- Maximum power point tracking
- Sliding mode control
- Residue theorem
- Wind power generation system
- Permanent magnet synchronous generator
Wind energy is growing fast in the world and is one of the most explored renewable energy sources. The control strategy of WPGS can affect the amount of extracted power from the wind . The wind is a source of energy which is highly affected by the uncertain fluctuations. As a result, wind power generation system can be described as a complex nonlinear system operates in highly uncertain environments which demand a tight control management . The control system of WPGS plays an important role to affect the amount of extracted power from wind. WPGSs need the controller to make MPPT process has fewer overshoots in the speed response and smooth transit between different operation regions .
SMC has been investigated for WPGS due to its robustness and simple structure . Elimination of chattering effect is an area of concern when implementing SMC. Although this can be avoided by a slight change in the sliding mode dynamics, it has been deduced in  that the discontinuity dynamics change the ultimate robustness and accuracy of the sliding mode nature are partially lost.
This paper proposes Residue Theorem based soft sliding mode control for wind power generation systems to improve the energy conversion ratio and the dynamic performance with a better ability to handle the uncertainties. There are two controllers developed with different boundary definition in this paper. The main idea is to set a soft dynamic boundary for controlled variables around a reference point, so that controlled variables lie on a point inside the boundary. The soft dynamic boundary will keep the control signal continuous. The soft changing manner of the boundary allows the speed to settle in reduced damping, and this prevents the WPGS from a mechanical shaft stress and damage. During transient state, as a PMSG accelerating the generated control boundary decreases smoothly, since the generated control boundary depends on the difference between the actual and reference speeds. Then at steady state, the generated control boundary become almost zero. Also, a wind speed estimation algorithm is designed and implemented which provides a solution with the wind speed measurement.
This paper is organised as the following: The first section presents the modelling of PMSG based WPGS. Then WPGS with PI control is illustrated. Besides, SMC is developed for a PMSG based WPGS. The residual control method and the design of the two controllers is shown and explained in this paper. In one controller the proposed method is implemented in the speed control loop of the PMSG, where in the second controller the proposed control method is implemented in speed and dq-axis currents control loops. Also in this section, a wind speed estimation algorithm is proposed and implemented in the WPGS controller. After that, the simulation results and analysis is presented. Also, the comparison between residual and SM controller is undertaken. Following the simulation results, the experimental verification are illustrated to validate the proposed control strategy. The whole paper is concluded in the final section.
By solving Eq. (3) for a TSR from 0 to 2, the WPGS can extract the maximum power from the wind when the TSR is equal to 0.82 and the maximum Cp is 0.221.
2.1 PMSG Model
2.2 Vector control with PI
Proportional and Integral controllers are commonly used in control applications since they are easy to design, implement and tune. However, in WPGS classical PI controllers might not be the desired control strategy as it becomes difficult to handle uncertainties in nonlinear environments. So that the achievement of efficient and robust operations can not guaranteed. Generally, the idea of PI controller is to regulate the error between the measured input and the desired output. This error along with its integral provide a signal for the controller action with respect to time .
Sliding mode control has the advantages of quick response and robustness. It can be defined as a variable structure control strategy based on the feedback and high frequency switching control . Moreover, it is insensitive to system parameter changes, disturbances and load variations . The design of SMC consists of two main stages. The first stage is to achieve the design of a stable sliding surface. The second stage aims to obtain an optimum design of a control law, which forces system operating points to reach a predetermined surface in finite time [13, 14].
3.1 Speed controller design
3.2 Direct current controller design
3.3 Quadrature current controller design
Finally, the values of the d-axis and q-axis voltages are used in order to generate a control signal using pulse width modulation (PWM).
4.1 Residue theorem
The residue theorem is considered as one of the best tools to predict the area under a curve. The Cauchy Theorem stated that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero [16, 17].
4.2 Using residual for speed controller
4.2.1 Controller stability analysis
4.3 Using residual for speed and currents controller
The values of the dq-axis voltages are used to generate a control signal of a pulse width modulation (PWM). The dq-axis voltages are normalized and compared with a triangular signal with an amplitude of 1 and a frequency of 20 kHz.
4.4 Using proposed residual controller with estimated reference speed
where D11=0.0960, D21=−0.0098 and D31=−0.0040. The MPPT controller is implemented in the proposed controller-II which is presented in Section 4.3.
The control diagrams of the two developed MPPT controllers, i.e. Residual for speed controller and Residual for speed and currents controller are shown in Fig. 1a and b respectively. For both the controllers, the dq-model is obtained, and the measurement of wind speed is required. The PMSG voltage and current are measured, and the rotor position is required.
In this paper, the modelled wind turbine is a vertical axial wind turbine(VAWT), and the generator is a PMSG. The WPGS model has been simulated under variable wind speeds. The wind speed starts at 8 m/s then it goes to 10 m/s at t = 1 s. For each wind speed the controllers generate reference speed for optimal operation. The parameters of the PMSG and the VAWT employed in this simulation are illustrated in the Appendix, where the power rating is 500 W.
Figure 5b shows the actual speed tracking the reference speed for the second proposed controller (Residual for speed and currents controller). It is noticeable that the actual speed is perfectly tracking the reference speed. There is a small overshoot appears in the speed response, which can be acceptable. The response speed is very fast and the settling time is at a satisfactory level.
5.1 Comparison with PI and SMC
Comparision between PI, SMC and Residual control
Rise Time (ms)
Settling Time (ms)
Maximum Steady-States error (rad ∖s)
Low Speed Chattering (rad ∖s)
High Speed Chattering (rad ∖s)
Power coefficient (Cp)
According to Table 1, the chattering effect of SMC at low speed is less than the residual controller. However, at high speed, the chattering effect reduces in the residual controller and increases in the SMC. It can be concluded that the residual controller has advantages over both PI and SMC, i.e. more power efficient, better dynamic performance, simple structure and implementation. Although, the SMC is simple structure control. It can be more complicated to overcome the chattering effect.
6.1 Experimental results and analysis
In this paper, a control method based on the residue theorem has been developed for WPGS. By creating a soft boundary around a reference point, the control variables are controlled to lie inside the boundary instead of a particular value. two types of controllers have been designed. For both controllers, the convergence of the manipulating variable to the set point is calculated by the forward Euler method, where the exponential function ensures the convergence for the third controller. The two proposed controllers have been simulated and tested experimentally to validate the control scheme. The soft control strategy developed provides a stable operation which can be partially lost when changing the discontinuity behavior of SMC. The overshoot is completely absent, and the response speed is fast and soft enough. It can be concluded that using the residue control method with the combination of PI controller improves the dynamics performance for the PMSG based WPGS.
The experimental results as well as the simulation results, which is shown in Table 1, show that the proposed controller overcome the limitations of PI and SMC control in terms of overshoots and energy conversion efficiency. Also, a wind speed estimation algorithm has been introduced in this study. It can be concluded that the dynamic performance of the system and the energy conversion ratio have been improved when using the estimated value of the wind speed. Besides, a comparative study between PI, SMC and residual controllers has been presented. Based on the comparison, the residual controller shows improved dynamic performance and higher energy conversion ratio than both PI and SMC.
It can be concluded that this method has been proposed and implemented in VAWT and it can be also implemented in other system as there is no specific design limitations shown. That is because the method is proposed based on mathematical theory and is not a system based. Although, the investigated system is of small-scale, the implementation of the proposed control technique to a medium and large scale system is considered in future.
Wind turbine power
Extracted mechanical power
Permanent magnet flux
Direct and quadrature currents
Mechanical and electrical speeds
Tip speed ratio
Radius of the wind turbine rotor
Direct and quadrature voltages
Direct and quadrature inductance
Mechanical and electrical torques
Viscus friction coefficient
Swept area of the wind turbine
Positive constant matrix
PMSG voltage in α−β coordinate
PMSG back-EMF in α−β coordinate
PMSG current in α−β coordinate
K,ω,d,q,Y,Z,k and c: Constants
Parameters of the WPGS
Maximum Power Coefficient
Stator Winding Resistance
Moment of Inertia
The author would like to thank the Royal commission for Jubail and Yanbu and the University of Liverpool for providing the financial support for this study.
This study has been funded by the Royal Commission for Jubail and Yanbu, Saudi Arabia and the University of Liverpool, UK.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
MA, the main author of this study his contributions toward the idea, mathematical and practical design, analysis, and proof of the idea, test and writing the paper.
LL, the corresponding author her contribution toward the practical test setup and measurements, revising and improving the text.
LJ and WT, the supervisors of the study their contributions toward supervising and guiding the study at all stages, reviewing and improving the text. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
- Johnson, KE, Pao, LY, Balas, MJ, Fingersh, LJ (2006). Control of variable-speed wind turbines: standard and adaptive techniques for maximizing energy capture. IEEE Control Syst, 26(3), 70–81.View ArticleGoogle Scholar
- Howlader, AM, Urasaki, N, Yona, A, Senjyu, T, Saber, AY (2013). Design and implement a digital h robust controller for a mw-class pmsg-based grid-interactive wind energy conversion system. Energies, 6, 42084–2109.View ArticleGoogle Scholar
- Koutroulis, E, & Kalaitzakis, K (2006). Design of a maximum power tracking system for wind-energy-conversion applications. IEEE Trans Ind Electron, 53(2), 486–494.View ArticleGoogle Scholar
- Hu, J, Nian, H, Hu, B, He, Y, Zhu, ZQ (2010). Direct active and reactive power regulation of dfig using sliding-mode control approach. IEEE Trans Energy Convers, 25(4), 1028–1039.View ArticleGoogle Scholar
- Fridman, L, & Levant, A (2002). Higher order sliding modes. Sliding Mode Control Eng, 11, 53–102.Google Scholar
- Burton, T, Jenkins, N, Sharpe, D, Bossanyi, E. (2011). Wind Energy Handbook. Hoboken: Wiley.View ArticleGoogle Scholar
- Morimoto, S, Nakamura, T, Takeda, Y (2003). Power maximization control of variable speed wind generation system using permanent magnet synchronous generator. IEEJ Trans Power Energy, 123, 1573–1579.View ArticleGoogle Scholar
- Jiang, S, Liang, J, Y-d Liu, Yamazaki, K, Fujishima, M (2005). Modeling and cosimulation of fpga-based svpwm control for pmsm. In Industrial Electronics Society, 2005. IECON 2005. 31st Annual Conference of IEEE. https://doi.org/10.1109/IECON.2005.1569133. IEEE, (pp. 6–pp).
- Zentai, A, & Dabóczi, T (2005). Improving motor current control using decoupling technique. In Computer as a Tool, 2005. EUROCON 2005. The International Conference on, vol. 1. https://doi.org/10.1109/EURCON.2005.1629934. IEEE, Belgrade, (pp. 354–357).Google Scholar
- Åström, KJ, & Hägglund, T. (2006). Advanced PID control. Research Triangle Park: ISA-The Instrumentation, Systems, and Automation Society.Google Scholar
- Datta, R, & Ranganathan, V (2003). A method of tracking the peak power points for a variable speed wind energy conversion system. IEEE Trans Energy Convers, 18(1), 163–168.View ArticleGoogle Scholar
- Vrdoljak, K, Perić, N, Petrović, I (2010). Sliding mode based load-frequency control in power systems. Electr Power Syst Res, 80(5), 514–527.View ArticleGoogle Scholar
- Bandyopadhyay, B, Deepak, F, Kim, K-S (2009). Sliding mode control using novel sliding surfaces, vol. 392. https://doi.org/10.1007/978-3-642-03448-0.
- Castaños, F, & Fridman, L (2006). Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans Autom Control, 51(5), 853–858.MathSciNetView ArticleMATHGoogle Scholar
- Europe Texas Instruments (1998). Field orientated control of 3-phase ac-motors. Literature Number: BPRA073.Google Scholar
- Bak, J, & Newman, DJ (2010). Complex analysis. In Complex Analysis. Springer, New York, (pp. 1–20).View ArticleGoogle Scholar
- Cattani, E, & Dickenstein, A (2005). Introduction to residues and resultants. In Solving polynomial equations. ISBN:3540243267. Springer-Verlag Berlin, Heidelberg, (pp. 1–61).Google Scholar
- Sandberg, M (2008). Convergence of the forward euler method for nonconvex differential inclusions. SIAM J Numer Anal, 47(1), 308–320.MathSciNetView ArticleMATHGoogle Scholar
- Higham, DJ, Mao, X, Stuart, AM (2002). Strong convergence of euler-type methods for nonlinear stochastic differential equations. SIAM J Numer Anal, 40(3), 1041–1063.MathSciNetView ArticleMATHGoogle Scholar
- Bjørhus, M (1995). A note on the convergence of discretized dynamic iteration. BIT Numer Math, 35(2), 291–296.MathSciNetView ArticleMATHGoogle Scholar
- Eriksson, K, Estep, D, Johnson, C (2004). Applied Mathematics: Body and Soul, 517–519. https://doi.org/10.1007/978-3-662-05798-8_7.
- Mahdi, AJ. (2011). Development of novel sensorless maximum power point tracking controllers for wind turbine generator systems, Ph.D. dissertation. Liverpool: University of Liverpool.Google Scholar
- Senjyu, T, Tamaki, S, Urasaki, N, Uezato, K, Funabashi, T, Fujita, H (2003). Wind velocity and position sensorless operation for pmsg wind generator. In Power Electronics and Drive Systems, 2003. PEDS 2003. The Fifth International Conference on Power Electronics and Drive Systems, 2003. PEDS 2003, vol. 1. https://doi.org/10.1109/PEDS.2003.1283005. IEEE, (pp. 787–792).
- Harnefors, L, & Nee, H-P (2000). A general algorithm for speed and position estimation of ac motors. IEEE Trans Ind Electron, 47(1), 77–83.View ArticleGoogle Scholar
- Zhao, Y, Qiao, W, Wu, L (2012). Position extraction from a discrete sliding-mode observer for sensorless control of ipmsms. In Industrial Electronics (ISIE), 2012 IEEE International Symposium on Industrial Electronics. IEEE, Hangzhou, (pp. 725–730).View ArticleGoogle Scholar
- Wang, Y, Infield, DG, Stephen, B, Galloway, SJ. (2013). Copula-based model for wind turbine power curve outlier rejection, Wind Energy.Wiley. https://doi.org/10.1002/we.1661.