Fuzzy generalised predictive control for a fractional‐order nonlinear hydro‐turbine regulating system
2018; Institution of Engineering and Technology; Volume: 12; Issue: 14 Linguagem: Inglês
10.1049/iet-rpg.2018.5270
ISSN1752-1424
AutoresKe Shi, Bin Wang, Haoyong Chen,
Tópico(s)Adaptive Control of Nonlinear Systems
ResumoIET Renewable Power GenerationVolume 12, Issue 14 p. 1708-1713 Research ArticleFree Access Fuzzy generalised predictive control for a fractional-order nonlinear hydro-turbine regulating system Ke Shi, Ke Shi School of Electric Power, South China University of Technology, Guangzhou, 510641 Guangdong, People's Republic of ChinaSearch for more papers by this authorBin Wang, Corresponding Author Bin Wang binwang@nwsuaf.edu.cn Department of Electrical Engineering, Institute of Water Resources and Hydropower Research, Northwest A&F University, Yangling, 712100 Shaanxi, People's Republic of ChinaSearch for more papers by this authorHaoyong Chen, Haoyong Chen School of Electric Power, South China University of Technology, Guangzhou, 510641 Guangdong, People's Republic of ChinaSearch for more papers by this author Ke Shi, Ke Shi School of Electric Power, South China University of Technology, Guangzhou, 510641 Guangdong, People's Republic of ChinaSearch for more papers by this authorBin Wang, Corresponding Author Bin Wang binwang@nwsuaf.edu.cn Department of Electrical Engineering, Institute of Water Resources and Hydropower Research, Northwest A&F University, Yangling, 712100 Shaanxi, People's Republic of ChinaSearch for more papers by this authorHaoyong Chen, Haoyong Chen School of Electric Power, South China University of Technology, Guangzhou, 510641 Guangdong, People's Republic of ChinaSearch for more papers by this author First published: 21 September 2018 https://doi.org/10.1049/iet-rpg.2018.5270Citations: 16AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study focuses on a fuzzy generalised predictive control method for a fractional-order hydro-turbine regulating system (HTRS). Based on the Grünwald–Letnikov (G–L) definition of fractional calculus and discretisation, the fractional-order hydraulic servo system is transformed into the standard controlled autoregressive integrating moving average (CARMA) model. With the help of fuzzy linearisation theory, the fuzzy predictive model of the integer-order part of the HTRS is presented. Furthermore, by using the fourth-order Runge–Kutta algorithm, the obtained fuzzy predictive model can be easily transformed into the CARMA model. Then, based on the overall CARMA model and the generalised predictive control theory, a novel nonlinear fuzzy generalised predictive controller is designed for the fractional-order HTRS. Finally, numerical simulations are implemented to verify the validity and superiority of the proposed method. It also provides a reference for relevant hydropower systems. 1 Introduction Since hydropower is a clean energy and meets the requirements for sustainable development, its production has become a priority for many countries, and China is no exception. A hydro-turbine regulating system (HTRS) is an important requirement for hydroelectric power stations and includes a penstock, a hydro-turbine, a generator and a governor [1-3]. A normally functioning HTRS is critical for the safe and stable operation of a hydropower station. Therefore, modelling, dynamic analysis, and control of the HTRS have drawn increasing attention [4-7]. For a long period of time, HTRS modelling has been based on integer-order calculus [8-12]. However, the HTRS is a complex coupling, nonlinear and non-minimum phase system. Consequently, it is impractical to describe the complex HTRS using integer-order calculus. In recent years, fractional-order calculus has attracted interest in many fields due to its great potential for describing the strong dependence, memory and viscoelastic attributes of numerous processes and materials [13-15]. Many projects could be better represented by fractional-order calculus, such as wind turbine generators [16, 17], financial systems [18], and brushless DC motors [19]. For this reason, considering the memory characteristics and historical dependence of the hydraulic-servo system, it is necessary to establish a more realistic fractional-order mathematical model for the HTRS. There have been many reports on the dynamic analysis of the HTRS [20-23] as well as the nonlinear control of fractional-order systems, such as sliding mode control [24], pinning control [25], finite-time control [26], backstepping control [27], and fractional-order proportional–integral–derivative (PID) control [28]. Predictive control has the advantages of prediction and online optimisation, which is, in essence, different from traditional control methods. Currently, many results on generalised predictive control (GPC) for linear systems have been presented [29-32]. Most of these studies are focused on integer-order systems. However, the HTRS is a complex nonlinear system. Could linear GPC be applied to the fractional-order nonlinear HTRS? The well-known Takagi–Sugeno (T–S) fuzzy model could approximate nonlinear systems universally [33-35]. The nonlinear model is described through fuzzy rules; then, a certain region of the system state is locally represented by the linearisation description [36, 37]. There have been many results regarding the fuzzy control of nonlinear systems [38-41]. Thus, could we combine the fuzzy technique with a linear GPC scheme for the control of fractional-order nonlinear systems? If applicable, what are the predictive models, the synthesised controller form, and the detailed mathematical derivation? This has not yet been presented. Given the above analysis, some advantages are concluded from this study. First, the G–L definition of fractional calculus is adopted to discrete the fractional-order hydraulic servo system into the standard controlled autoregressive integrating moving average (CARMA) model. Second, the T–S fuzzy model was applied to describe the integer-order part of the HTRS. Then, according to the fourth-order Runge–Kutta algorithm, the fuzzy predictive model could be easily transformed into the CARMA model. Furthermore, based on the overall CARMA model, a new nonlinear fuzzy generalised predictive controller is designed for the fractional-order HTRS. Also, simulation results are consistent with the theoretical analysis. The remaining contents of our paper are organised as follows. In Section 2, a system description of the HTRS is provided. The model transformation and controller design are presented in Section 3. In Section 4, numerical simulations are performed. Section 5 concludes this paper. 2 System description The integer-order HTRS is given as follows [42]: (1) where , and , , , y are the generator rotor angle deviation, the rotational speed relative deviation of the generator, the hydro-turbine output incremental torque deviation and the incremental deviation of the guide vane opening, respectively. Here, the control input is not considered, i.e. . Considering the significant historical reliance of the hydraulic servo system, the following fractional-order hydraulic servo system is adopted [43]: (2) where is the major relay connector response time. In the actual project, due to the electromagnetic interference, equipment deviation and other reasons, random and unsteady disturbance signals often appear in the practical operation. In addition, the residual error between the mathematical model and the actual model often exhibit random and uncertain characteristics. According to (1) and (2), the fractional-order HTRS with external random disturbances could be represented as (3) Here, the parameters of system (3) are as follows: rated angular speed , inertia time constant of the rotating part , generator damping coefficient , quadrature-axis transients potential , inertia time constant of water flow in pressure diversion system , transfer time of turbine servomotor stroke , direct-axis transients reactance of total system , quadrature-axis reactance of total system , busbar voltage , transfer coefficient , transfer coefficient of water head , transfer coefficient of turbine servomotor stroke , fractional order and rand (1) means a random number between 0 and 1. For convenience, let . The time domain of HTRS (3) under the starting operating condition is shown in Fig. 1. Clearly, the system is in irregular and unstable vibrations with an initial value , which needs to be controlled. Fig. 1Open in figure viewerPowerPoint State trajectories of HTRS (3) (a) , (b) , (c) , (d) 3 Controller design 3.1 Fractional-order model transformation The fractional-order model can be described by the following differential equation: (4) where is an arbitrary real coefficient, is the -order differential of output , and is the input. According to the G–L definition of fractional calculus, there is (5) where is the sampling interval. The term can be derived from the following recursive formula: (6) Generally, the fractional-order model only contains zero-order items and fractional-order items, thus the control object can be identified as the following fractional-order model: (7) where the order is , the lag constant is , and K are arbitrary real coefficients. According to the G–L definition of fractional calculus, assume that the present time and is denoted as the time point k. The initial time is taken as the time before the current sampling time, i.e. time point , and the corresponding time is . Then, (5) can be rewritten as (8) From (7) and (8), one can easily obtain (9) From (6), one has , and by substituting it to (9), the CARMA model is presented as (10) 3.2 Integer-order model transformation The Takagi–Sugeno (T–S) fuzzy model is described by the IF–THEN fuzzy rules. Local dynamics in different state space regions is represented over a linear realisation. Then, the combination of the linear model is used to represent the nonlinear system. The T–S fuzzy model is given in the following form: (11) where is the fuzzy set, , m is the number of fuzzy rules, is the state vector, and is the premise variables. Then the total model is given as follows: (12) where (13) (14) By using the fourth-order Runge–Kutta algorithm with sampling interval , the T–S fuzzy model could be easily transformed into the CARMA model. Together with the CARMA model of the fractional-order hydraulic servo system (10), the overall prediction model of the HTRS (3) is obtained 3.3 Nonlinear fuzzy GPC Fig. 2 shows the GPC block diagram. The CARMA model of the HTRS (3) could be represented as (15) where is the state vector, is the control input, and is the white noise. Fig. 2Open in figure viewerPowerPoint GPC block diagram Considering the CARMA model (15) of the HTRS (3), the output prediction model could be given as (16) where (17) The sub-matrix in (17) can be obtained by (18) is the control diversification vector, the control diversification of time point can be obtained by (19) is the predictive output without control change, the predictive output of time point can be obtained by (20) where (21) (22) (23) Considering the CARMA model (15) of the HTRS (3), the reference trajectories are set as (24) where where denotes the softness factor and is the expected output. The quadratic cost function is selected as (25) It is easy to obtain According to , one has (26) Thus, the state trajectories of HTRS (3) will asymptotically converge to the set reference under the control law (26). 4 Simulation results To match the conversion of the integer-order model, the sampling interval of the fractional-order model transformation in Section 3.1, , the CARMA model of the fractional-order hydraulic servo system could be represented with sampling interval (27) To make it convenient to obtain the coefficient matrix for the integer-order part of the HTRS (3), the Maclaurin series is introduced as follows: (28) Considering the boundedness, select and the first three items of the Maclaurin series, one obtains (29) Here, d = 4. Then, the following fuzzy model could be obtained: : IF is (near 0), THEN . : IF is (near ), THEN . where , and . The membership function is presented as Then, the following coefficient matrix could be obtained: Thus, the T–S fuzzy model for the integer-order part of HTRS (3) can be written as (30) By using the fourth-order Runge–Kutta algorithm (31) where (32) and we could easily transform the T–S fuzzy model (30) into the CARMA model (33) It is equal to (34) Together with the CARMA model of the fractional-order hydraulic servo system (10), the overall prediction model of HTRS (3) could be obtained (35) where . To compare the performance of the proposed control scheme, Fig. 3 shows the simulation results of the proposed scheme, controller (9) in [44] as well as the traditional PID control. From Fig. 3, it is obvious that when theproposed fractional-order fuzzy GPC (FFGPC) is applied to the HTRS (3), the system state quickly convergedto the equilibrium point, regarding the system with external random disturbances,which demonstrate the effectiveness and robustness of the designed method. Inaddition, by comparing the proposed scheme with the existing fuzzy control methodand PID control, it is clear that the oscillation frequency is lower and the stabletime is shorter, which implies the rapidness and the advantages of the proposedFFGPC method. Fig. 3Open in figure viewerPowerPoint State trajectories of HTRS (3) with different control methods (a), (b), (c), (d) The comparison of performance indexes is shown in Table 1. Table 1. Performance of the HTRS (3) with different control methods Method State variable Overshot Oscillation Steady speed FFGPC X1 0.011 least fast X2 0.005 X3 0.005 X4 0.003 X1 0.026 fuzzy control X2 0.009 most medium X3 0.007 X4 0.008 X1 0.063 PID control X2 0.005 medium slow X3 0.168 X4 0.123 5 Conclusions and discussion A new nonlinear fuzzy GPC was proposed to stabilise the fractional-order HTRS in this study. Compared with fuzzy control and PID control, the FFGPC shows the advantages of smaller overshoot, less concussion, and faster stability. This means that the turbine can be controlled more quickly and effectively, and it can reduce the bad running state of the turbine and ensure the long-term and effective operation of the turbine. However, the method requires that the fractional-order part in the model can be calculated independently. Besides, the control of this method is obtained through online real-time calculations, whereas the PID control can be designed in advance. Therefore it needs sufficient computing power and fast controller response. These should be studied further. 6 Acknowledgments The first two authors contributed equally to this paper. This work was supported by the Scientific Research Foundation of the National Natural Science Foundation (nos.51509210 and 51479173), the Shaanxi Province Science and Technology Plan (no.2016KTZDNY-01-01), the Science and Technology Project of Shaanxi Provincial Water Resources Department (nos. 2017slkj-2 and 2015slkj-11), the Shaanxi Province KeyResearch and Development Plan (no. 2017NY-112) and the Fundamental Research Fundsfor the Central Universities (no. 2452017244). 7 References 1Kishor, N.: 'Nonlinear predictive control to track deviated power of an identified NNARX model of a hydro plant', Expert Syst. 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