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Model-based adaptive sliding mode control of the subcritical boiler-turbine system with uncertainties

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As higher requirements are proposed for the load regulation and efficiency enhancement, the control performance of boiler-turbine systems has become much more important. In this paper, a novel robust control approach is proposed to improve the coordinated control performance for subcritical boiler-turbine units.  To capture the key features of the boiler-turbine system, a nonlinear control-oriented model is established and validated with the history operation data of a 300 MW unit. To achieve system linearization and decoupling, an adaptive feedback linearization strategy is proposed, which could asymptotically eliminate the linearization error caused by the model uncertainties. Based on the linearized boiler-turbine system, a second-order sliding mode controller is designed with the super-twisting algorithm. Moreover, the closed-loop system is proved robustly stable with respect to uncertainties and disturbances. Simulation results are presented to illustrate the effectiveness of the proposed control scheme, which achieves excellent tracking performance, strong robustness and chattering reduction.

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Model-based adaptive sliding mode control of the subcritical boiler-turbine system with uncertainties

  1. 1. Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Practice article Model-based adaptive sliding mode control of the subcritical boiler-turbine system with uncertainties Zhen Tiana , Jingqi Yuana,∗ , Liang Xub , Xiang Zhanga , Jingcheng Wanga a Department of Automation, Shanghai Jiao Tong University, Dongchuan Road 800, Minhang District, Shanghai 200240, China b China Ship Development and Design Center, Shanghai, Huaning Road 2931, Minhang District, Shanghai 201108, China A R T I C L E I N F O Keywords: Coal-fired power plant Control-oriented modeling Sliding mode control Adaptive feedback linearization Chattering reduction A B S T R A C T As higher requirements are proposed for the load regulation and efficiency enhancement, the control perfor- mance of boiler-turbine systems has become much more important. In this paper, a novel robust control ap- proach is proposed to improve the coordinated control performance for subcritical boiler-turbine units. To capture the key features of the boiler-turbine system, a nonlinear control-oriented model is established and validated with the history operation data of a 300 MW unit. To achieve system linearization and decoupling, an adaptive feedback linearization strategy is proposed, which could asymptotically eliminate the linearization error caused by the model uncertainties. Based on the linearized boiler-turbine system, a second-order sliding mode controller is designed with the super-twisting algorithm. Moreover, the closed-loop system is proved ro- bustly stable with respect to uncertainties and disturbances. Simulation results are presented to illustrate the effectiveness of the proposed control scheme, which achieves excellent tracking performance, strong robustness and chattering reduction. 1. Introduction In recent decades, renewable power generation technologies have been developed rapidly. However, coal-fired power plants are still playing the major electricity suppliers in most developing countries [1]. The control system performance is one of the most important factors related to the efficiency and safety of boiler-turbine units [2]. The ob- jective of the coordinated control system (CCS) is to regulate the power output of the unit to meet the power demand. Specifically, the power output regulation is required as fast as 1.5%–2% of the full load per minute, and the main steam pressure needs to be stabilized within a limited range of −0.4 ∼ +0.4 MPa around its set value [3]. For a practical boiler-turbine unit, various uncertainties and disturbances occurred in the operation need to be considered [4], such as the heating value variation of the feed coal, the contamination of the heat ex- changer surfaces, the soot blowing operation, etc. Therefore, the ro- bustness of the CCS is of great importance to achieve satisfactory con- trol performance. In order to improve the CCS performance, two ways are commonly investigated. On the one hand, an appropriate control- oriented model with a balance between fidelity and simplicity needs to be built, which must cover the key features of the boiler-turbine system. On the other hand, the robust coordinated controller is studied to address the nonlinearity, intercoupling and uncertainties related to the boiler-turbine system. Over past decades, several models have been established to describe the complex dynamics of the boiler-turbine system. In Refs. [5,6], full mechanism models are developed based on the basic mass and energy balance, which can only be applied for the detailed process simulation due to its complicated iterative calculation. Another typical model is built with the experimental operation data of a 160 MW oil-fired power plant [7]. However, the oil-fired power plants differ from large-scale coal-fired power plants in many aspects, e.g., the time delay, the var- iance of the fuel heating value, the combustion system, etc. Similarly, a dynamic model is built for once-through boiler-turbine units [8]. Some improved models have also been found in the literature, such as the Takagi–Sugeno (TS) fuzzy model [9], piecewise affine model [10] and data-driven model [11]. However, the model parameters of these data- driven models are lack of explicit physical significance, which may lead to difficulty in dynamic analysis and model transplantation. In recent years, various control approaches have been studied to improve the control performance of boiler-turbine units, which are summarized as follows: (1) Linear control methods. So far, the conventional proportional https://doi.org/10.1016/j.isatra.2018.05.012 Received 11 October 2017; Received in revised form 27 April 2018; Accepted 16 May 2018 ∗ Corresponding author. E-mail addresses: tianzhen9032-@sjtu.edu.cn (Z. Tian), jqyuan@sjtu.edu.cn (J. Yuan), xuliang34@163.com (L. Xu), zhangxiangzi0071@163.com (X. Zhang), jcwang@sjtu.edu.cn (J. Wang). ISA Transactions 79 (2018) 161–171 Available online 25 May 2018 0019-0578/ © 2018 ISA. Published by Elsevier Ltd. All rights reserved. T
  2. 2. integral derivative (PID) control approaches are commonly adopted for boiler-turbine units, such as the cascade PID control [12], fuzzy PID control [13] and internal-mode-based PID control [14]. Based on a locally linearized model, an L1 adaptive control strategy is proposed to address the nonlinearity and model uncertainties [15]. To achieve disturbance attenuation, a passivity-based H∞ controller is designed with respect to a linear boiler-turbine model [16]. However, the performance of these linear control approaches may deteriorate for large-scale operating units due to the neglect of nonlinearity and intercoupling properties. (2) Nonlinear control methods. Various nonlinear control methods have been applied to the nonlinear boiler-turbine model proposed in Ref. [7]. Feedback linearization based control methods are stu- died for the nonlinear boiler-turbine system [17,18]. With the successive on-line model linearization, a nonlinear predictive con- troller is designed to address the nonlinearity [19]. Some robust control approaches are proposed to achieve disturbance rejection, such as the active disturbance rejection control (ADRC) [3], sliding mode control [4] and backstepping control [20]. These nonlinear control methods may improve the CCS performance for the oil-fired units from different aspects. However, their effectiveness for prac- tical boiler-turbine system needs to be further verified while con- sidering the dynamics difference between the oil-fired units and coal-fired units. (3) Model-free control methods. With the development of data-driven modeling, some model-free control approaches have been in- vestigated. A predictive controller is designed based on the data- driven model by using fuzzy clustering and subspace methods [21]. To adapt to the load variation, a predictive controller is proposed based on the iterative learning principle [22]. Intelligent controllers are developed based on the neural network methods [23,24]. Even though these control approaches do not require mathematical models, they may encounter feasibility and reliability problem in practice for their complicated implementation. Sliding mode control (SMC) approach is extensively studied for its merits of excellent performance and inherent robustness [25]. For nonlinear systems, several nonlinear SMC approaches have been stu- died [26,27], but most of them are limited to special structural con- straints, such as the holomorphic and single-input conditions. Moreover, the reaching condition and stability analysis become much more complicated for nonlinear sliding mode controllers. With the help of feedback linearization technique, linear control methods could also be extended to nonlinear systems [17]. Thus, the linearization-based sliding mode control approach may facilitate the controller design for many nonlinear industrial processes [28]. In this paper, an adaptive sliding mode control approach is proposed for the boiler-turbine system to address the model uncertainties and external disturbances. Three attractive features may be found in this contribution. Firstly, an appropriate control-oriented model is built and validated, which covers the key dynamics of the subcritical boiler-tur- bine units under wide-range operation. Besides, an adaptive feedback linearization strategy is proposed to eliminate the linearization error caused by the model uncertainties. Last but not the least, a super- twisting sliding mode controller is designed to address the uncertainties and disturbances, which is free from the heavy computation and chat- tering trouble. The rest of this paper is organized as follows: Section 2 develops a control-oriented model for the subcritical boiler-turbine system. In Section 3, an adaptive sliding mode controller is designed and the closed-loop stability is proved. Simulation results with three different control approaches are presented in Section 4. Section 5 gives the concluding remarks of this paper. 2. Modeling of the drum-type boiler-turbine system A schematic diagram of the subcritical boiler-turbine system is shown in Fig. 1. As seen, the preheated feed water enters the boiler drum firstly, and then enters the waterwall through the downcomer. The working fluid absorbs heat from the waterwall and becomes satu- rated mixture of water and steam. Next, the mixture led to the drum is separated into saturated steam and water by the separator. The satu- rated steam leaves the drum and passes through multi-stage super- heaters, becoming superheated steam or main steam for the high- pressure cylinder (HPC). The steam from the HPC exit is reheated in multi-stage reheaters and then fed into the intermediate-pressure cy- linder (IPC). The steam from the IPC exit is fed into the low-pressure cylinder (LPC). Eventually, the steam working in the turbine drives the synchronous generator to produce electric power. This paper mainly concerns the control-oriented modeling of the Fig. 1. Schematic of a 300 MW subcritical power plant (The data are corresponding to 100% BMCR working conditions approximately). Z. Tian et al. ISA Transactions 79 (2018) 161–171 162
  3. 3. integrated boiler-turbine system, which is simplified as a two-input two-output nonlinear system [3]. Two input variables are the governor opening value of the steam turbine u1 and the mass flow rate of the feed coal u2; Two output variables include the power output y1 and the main steam pressure y2. According to the energy conversion sequence, the system dynamics is typically divided into three parts, i.e., the coal-heat conversion, the heat-steam conversion and the steam-power conversion processes. 2.1. Coal-heat conversion process Before combustion in the furnace, the coal is pulverized in the coal mill. The mass balance for the coal pulverizer is [29]: = − + −T dD dt D μ t τ( )f f f B (1) where, μB and Df are the mass flow rate of the feed coal and the coal entering the boiler furnace, respectively; Tf is a time constant and τ denotes the time delay of the feed coal transportation. The heat released by the coal combustion is mainly composed of two parts, i.e., the heat absorbed by the working fluid in the waterwall and the heat carried away by the flue gas. Thus, the static energy balance in the furnace is represented as [30]: − + + = + + +D Q q D h D h Q D h D h D h(1 )f net ar a a f f s fg fg sl sl fa fa, 4 (2) where, Qnet, ar the lower heating value (LHV) of the feed coal, which can be acquired by offline assay or online identification [30]; Qs represents the heat absorption rate of the working fluid in the waterwall; q4 the heat loss rate caused by the mechanical incomplete combustion; D, ρ and h indicate the mass flow rate, the density and the specific enthalpy, respectively; The subscripts a, fg, sl and fa refer to the inlet air, the flue gas, the slag and the fly ash, respectively; hf and ha denote the sensible specific enthalpy of the feed coal and inlet air, respectively. The flue gas energy obtained from the combustion process is re- presented as follows: = − − = −Q D h D h D h D h h( )fg fg fg f f a a fg fg in (3) where hin denotes the specific enthalpy of the flue gas under environ- mental temperature. According to the operation data analysis, Qfg is found highly associated with the power output N [30] and approxi- mated with a linear polynomial function: = +Q λ λ Nfg 10 11 (4) Substituting Eq. (3) into Eq. (1) and neglecting the slight sensible heat of the slag and fly ash, then Qs is acquired: = − −Q D Q q Q(1 )s f net ar fg, 4 (5) 2.2. Heat-steam conversion process The heat transformation equations for the waterwall and drum [5]: − =D h h Q( )gs s fw s (6) − = =D D d V ρ dt V dρ dp dp dt ( ) gs s D D D D D D (7) where, Dgs and Ds represent the mass flow rate of the saturated steam in the waterwall and the saturated steam leaving the drum, respectively; VD, ρD and pD denote the total volume of the saturated steam in the drum and waterwall, the density of the saturated steam and the drum pressure, respectively; hfw and hs are the specific enthalpy of the feed- water and the saturated steam in the drum, respectively. The thermodynamic parameters (ρD and hs) of the saturated steam in the drum are uniquely determined by the drum pressure pD, which are approximated as following polynomial functions: = + +ρ λ λ p λ pD D D20 21 22 2 (8) = + +h λ λ p λ ps D D30 31 32 2 (9) Since the feedwater is subcooled and its temperature is almost constant, the specific enthalpy hfw of the feedwater is only related to the drum pressure pD, and represented as: = + +h λ λ p λ pfw D D40 41 42 2 (10) The mass balance equation for the steam in superheaters [3]: − = =D D d V ρ dt V dρ dt ( ) s ms T T T T (11) where, Dms, ρT and pT denote the mass flow rate, the density and the pressure of the main steam, respectively; VT is the total volume of the steam in superheaters. Under actual operation conditions, the main steam temperature Tms is controlled within a quite narrow range around 540 °C. Therefore, ρT may be regarded as the function of pT, and the discrete values of ρT are calculated over the routine operating range (pT: 8–20 MPa) according to the IAPWS-IF97 formulation [31]. The correlation between ρT and pT is approximated by a linear polynomial function: = +ρ λ λ pT T50 51 (12) Thus, = ∂ ∂ dρ dt ρ p dp dt T T T T T ms (13) According to the Darcy-Weisbach equation [5], Ds is calculated as = −D C ρ p ps D D T0 (14) where C0 is the damping coefficient of the superheater pipes. Under a given stable load condition, C0 could be calculated as = − C D ρ p p sref Dref Dref T ref 0 (15) where the subscript ref means the respective parameters under a given stable load condition. Combining Eqs. (11)–(15), the dynamic equation of the main steam pressure is obtained: ∂ ∂ = − −V ρ p dp dt C ρ p p DT T T T D D T ms0 (16) 2.3. Steam-power conversion process The governor opening value of the steam turbine is defined as [3]: =μ p pT T 1 (17) where μT∈[0,1] and p1 is the steam pressure of the first governing stage. The throttle regulation of the turbine may be regarded as a first-order inertia process [3], thus Eq. (17) becomes: + =T dp dt p μ pe T T 1 1 (18) where Te is a time constant of the turbine governing. Based on the energy balance for the whole boiler-turbine system, the power output of the generator is represented as [29]: = = + − − ∑ − = − ∑ + − = = = = N η η E η η D h D h h D h D h η η D h α h D h h α D D ( ( ) ) [ ( ) ( )] / , 1,2...9 m g in m g ms ms r r HE i i i LE LE m g ms ms i i i r r HE i i ms 1 8 1 9 (19) Z. Tian et al. ISA Transactions 79 (2018) 161–171 163
  4. 4. where N represents the power output; Ein the net energy entering into the turbine; αi the relative mass flow rate of the i-th extracted steam. Dr, Di and DLE denote the mass flow rates of the reheated steam, the i-th extracted steam and the exhaust steam of the LPC, respectively. hr, hHE, hLE and hi represent the specific enthalpies of the reheated steam, the steam at the HPC outlet, the exhaust steam from LPC outlet and the i-th extracted steam, respectively. Obviously, DLE = D9, hLE = h9. The shaft efficiency ηm and the generator efficiency ηg are found quasi-constant under different load conditions according to the operation experiences. For a 300 MW unit, ηm and ηg are 0.99 and 0.985, respectively [32]. Operating data reveals that the total heat absorption rate of the reheated steam in reheaters is approximately proportional to the power output [30]: − =D h h k N( )r r HE 1 (20) Then Eq. (19) becomes = − ∑ − == N h α h D k k D ( )ms i i i ms η η ms 1 9 1 1 2 m g (21) where k2 is an equivalent coefficient to be identified. The mass flow rate of the main steam satisfies the Flügel equation [29]: = − − D D p p p p T T ms ref ref ref ref1 2 2 2 1, 2 2, 2 (22) where, p2 denotes the exhaust pressure of HPC under actual operating condition; p1, ref, p2, ref, Tref and Dref represent the first governing stage pressure, the exhaust pressure of HPC, the main steam temperature and the mass flow rate of the main steam under the referential operation condition, respectively. Based on the history operation data, p2/p1 is found not greater than 1/5. Without considering the variation of the main steam temperature, Eq. (22) is simplified as: = − ≈ = − − D p p p p k p 1 ( / ) · · ms D p p T T D p p T T 2 1 2 1 1 3 1 ref ref ref ref ref ref ref ref 1 2 2 2 1 2 2 2 (23) where k3 is a steady gain. Combining Eqs. (19), (21) and (23), the dynamic equation of the power output is acquired: = − +T dN dt N k k μ pe T T2 3 (24) 2.4. Control-oriented model and model parameters determination Combining Eqs. (1), (7), (16) and (24), the control-oriented model of the boiler-turbine system is represented as follows: ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ = − + = − − = − − = − + − ∂ ∂ − − − T N k k μ p V C ρ p p k μ p V C ρ p p T D μ t τ( ) e dN dt T T T ρ p dp dt D D T T T D dρ dp dp dt D Q q Q h h D D T f dD dt f B 2 3 0 3 (1 ) 0 T T T D D D f net ar fg s fw f , 4 (25) Utilizing the history operation data, the coefficients of these poly- nomial functions shown in Eqs. (4) and (8)–(10) are obtained by re- gression. The calculation results are shown as follows: λ10 = 3397.9, λ11 = 1374.8, λ20 = 55.41, λ21 = −5.16, λ22 = 1374.8, λ30 = 2734.1, λ31 = 13.17, λ32 = −1.43, λ40 = 1187.1, λ41 = −0.41, λ42 = 0.008, λ50 = −4.67 and λ51 = 3.28. Define the state variables x = [N, pT, pD, Df ] T , the manipulating variables u = [μT, μB(t-τ)] T and the output variables y = [N, pT] T , Eq. (25) may be written in the affine form as follows: ⎧ ⎨⎩ = + = x x g x u y h x f˙ ( ) ( ) ( ) (26) where, = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ − − − − − − ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ − ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = f x g x h x x x x x x x x x x x x ( ) , ( ) 0 0 0 0 0 ( ) [ ] T c c k c c c c Q c c T k T k c T T 1 1 3 2 4 3 2 1 4 2 2 1 1 2 e a n b m a b fg b m f e n f 4 5 3 (27) and the model parameters are represented as follows: = = = − = ∂ ∂ = − = c C ρ c V dρ dp c h h c V ρ p k Q q k k k , , , , (1 ), a D b D D D m s fw n T T T net ar 0 4 , 4 5 2 3 The time constants τ, Te and Tf are estimated according to the ex- perience of the onsite operators. Owing to the complicated relationship in Eq. (21) and the unknown structure parameters of the boiler and superheaters, the accurate values of the model parameters k2, VD and VT are not easy to be acquired by an analytic or empiric method. In order to improve the model accuracy, the optimization method is applied for the parameters identification of k2, VD and VT. To minimize the differ- ence between the real value and the model-based simulated value of the power output, an optimization problem is defined as follows: = ∑ − − ⎧ ⎨ ⎩ ≤ ≤ ≤ ≤ ≤ ≤ = y y y yJ s t k V V min ( ˆ ) ( ˆ ) . . 0.5 2 20 50 100 180 k V V i n i i T i i D T { , , } 1 2 D T2 where yi and yˆi denote the i-th sampling data of the real value and the model-based simulated value of the power output, respectively. The constraints are determined by the practical operation conditions of the unit. Considering the superiority of the genetic algorithm (GA), such as the global optimization and high efficiency, GA is adopted to solve the above optimization problem. The parameters of GA are set as follows: the population number is 100, the individual length is 20, the crossover probability is 0.7 and the mutation probability is 0.01. All model parameters related to Eq. (27) are shown in Table 1. 3. Coordinated control strategy design To facilitate the nonlinear controller design, the feedback linear- ization technique is adopted first. Then, a linear sliding mode controller is designed with respect to the linearized and decoupled boiler-turbine system. 3.1. Feedback linearization 3.1.1. Conventional feedback linearization method In order to achieve complete linearization, the total relative degree must equal to the system dimension [33]. To satisfy this condition, the dynamic extension method is adopted. The motivation of the dynamic extension is to change the system relative degree via choosing appro- priate system states and input variables. The new system state x and control input w are defined as follows: = = x w N p p μ μ D [ ] [ ˙ ] T D T T T f T (28) With the new system states and control input, the dynamic Z. Tian et al. ISA Transactions 79 (2018) 161–171 164
  5. 5. equations of the boiler-turbine system denoted as Eq. (26) are con- verted into the form as follows: ⎧ ⎨⎩ = + = x x g x w y h x f˙ ( ) ( ) ( ) ͠ ͠ (29) where, = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ − + − − − − − ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = = ∼ f x g x x x h x x x x x x x x x x x x g g h h x x ( ) 0 , ( ) [ ( ) ( )] 0 0 0 0 0 1 0 ( ) [ ( ) ( )] [ ] ͠ ͠͠ T k T c c k c c c Q c c k c c T T 1 1 2 4 3 2 2 4 3 2 1 2 1 2 1 2 e e a n n a b fg b m b m 5 3 4 (30) The relationship between the manipulating variables u and the re- defined control input w is = ⎡ ⎣ ⎢ + ⎤ ⎦ ⎥u w T s 0 0 1 s f 1 (31) Now the total relative degree of the expanded system (29) is four, thus the complete linearization condition is satisfied. Taking the second-order derivative of y along system (29), = +y F x M x w¨ ( ) ( ) (32) where, = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ + + − + − ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎢− ⎤ ⎦ ⎥ ⎥ = × ⎡ ⎣ + − + ⎤ ⎦ − − − − − − − + + − ( ) F x x x M x x x x x L h L h x x L L h L L h L L h L L h x x c x x x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 19.76 (0.0276 0.15) f f x k x x T k c x x x k k x x c T k x x c c k x x x c c k x x c x x c c Q c c c c g f g f g f g f k T k c c k c c c g x x x x 2 1 2 2 2 2 2 2 3 2 1 1 2 2 2 2 3 3 2 (0.0138 0.15 4.7) 2 ͠ ͠ ͠ ͠ ͠ ͠ ͠ ͠ ͠ ͠ e a n e a a n a n g n fg b m a b e n g n b m 1 5 2 4 2 5 4 3 4 5 3 4 2 2 3 2 4 2 2 2 3 4 3 4 2 3 2 4 2 3 2 1 2 1 2 5 3 4 3 2 3 3 2 (33) According to the real operation condition, x2 does not equal to zero and thus the decoupled matrix M(x) is nonsingular. Define an auxiliary variable v = [v1, v2] T , a conventional feedback is represented as: = −−w M x v F x( )[ ( )]1 (34) By substituting Eq. (34) into Eq. (31), the nonlinear system (31) is converted into a second-order pseudo-linear system as follows: =y v¨ (35) 3.1.2. Design of the adaptive feedback linearization The conventional feedback linearization denoted as Eq. (34) re- quires the rigid mathematical model of the nonlinear control plant. However, the modeling error is generally inevitable for most complex industrial processes, which may bring to prominent linearization error and thus deteriorate the control performance. In this section, an adaptive feedback linearization approach is proposed to address the linearization error caused by the modeling errors. If the information of the linearization error is available, then a feedback loop could be in- troduced so that the linearization error is asymptotically driven to zero. In order to obtain the linearization error, the nominal pseudo-linear system (35) is chosen as the reference model, =y v¨m (36) According to Eq. (32), a practical boiler-turbine system with un- certainties could be represented as = + + + +y F x F x M x M x w D¨ ( ) Δ ( ) ( ( ) Δ ( )) (37) where ΔF(x) and ΔM(x) represent the model uncertainties, and D de- notes the disturbances. Define the linearization error em = y - ym, then the sliding mode function σ is designed as = +σ e αe˙m m (38) where α is a positive constant. Taking the time derivative of σ based on Eqs. (36) and (37), = + + + +σ φ x e F x D M x M x w˙ ( , ) Δ ( ) ( ( ) Δ ( ))m where = = − +φ x e x e x e F x v αeφ φ( , ) [ ( , ) ( , )] ( ) ˙m m m T m1 2 (39) And assuming that each element of φ(x, em) is bounded by a known nonlinear function xζ ( ) such that <x e xφ ζ( , ) ( )i m . An adaptive feedback control law is designed as: = + = − = − + + − − w w w w M x x e w M x ρσ η σ ψ x e γ σ φ( ) ( , ) ( )[ sgn( ) ( , ) sgn( )] eq d eq m d m 1 1 (40) where, ρ = diag {ρ1, ρ2}, η = diag {η1, η2}, γ = diag {γ1, γ2} and ψ(x, em) = diag {|φ1(x, em)|, |φ2(x, em)|}. When em is driven to zero by the control law denoted as Eq. (40), the linearization error is completely eliminated so that the boiler-turbine system is converted into a pseudo-linear system as Eq. (35). Thus, the coordinated controller could be designed with respect to the second- order linear system (35). With the adaptive feedback linearization, the boiler-turbine system model is decoupled as two independent single- input single-output (SISO) subsystems, i.e., the power output subsystem and the main steam pressure subsystem. Table 1 Model parameters. Parameter type Parameter value Unit Physical significance Empirical parameters τ = 10 s Delay time of the coal feeder Tf = 60 s Time constant of the pulverizer Te = 10 s Time constant of the turbine governing Constant parameters k5 = 26.09 – Correlation coefficient VT = 146.1 m3 Steam volume in superheaters VD = 38.3 m3 Steam volume in the drum k2 = 1.27 kJ/kg Equivalent coefficient k3 = 21.62 – Steady gain Time-variant parameters k4 = 0.964Qnet,ar kJ/kg Effective heat value of the feedcoal ca = 19.76 × (0.0138pD 2 +0.15pD+4.7) – Damping coefficient cb = VD(1.06pT −5.16) – Mass storage coefficient cm = −1.46 pD 2 +12.25 pD +2629.19 kJ/kg Enthalpy difference between the saturated steam and feedwater cn = 3.28VT – Mass storage coefficient Qfg = 1374.8 N + 3397.9 kW Energy of the flue gas at the furnace exit Z. Tian et al. ISA Transactions 79 (2018) 161–171 165
  6. 6. 3.2. Sliding mode controller design The tracking error between the reference model outputs and their set values are defined as = −e y ym r where, e = [e1, e2]T , yr = [yr1, yr2]T and ym = [ym1, ym2]T . ym and yr denote the reference model output and the set value of system output, respectively. The subscript 1 and 2 represent the corresponding com- ponents of the power output subsystem and the main steam pressure subsystem, respectively. Since the same controller configuration is ap- plicable for the two subsystems, only the controller design of the power output subsystem is presented in this section. The response of conventional SMC generally includes two phase, i.e., the reaching phase and sliding phase. The time response before the sliding variables converging to zero (sliding mode surface) is the so- called reaching phase, and the time response while the sliding variables maintaining on the sliding mode surface is called the sliding phase. The conventional SMC is only robust when the system states moving on the sliding phase. In order to eliminate the reaching phase and achieve global robustness, a time-varying sliding mode function s1 is designed as = +s s I t( )1 0 1 (41) where s1 is composed of two terms and represented as = +s c e e˙0 1 1 1 (42) = ⎧ ⎨⎩ + ≤ ≤ > I t A t B t T( ) , 0 t T 1 1 0, 1 (43) where c1 is a positive coefficient, A1 and B1 are two appropriate con- stants. The optimal selection of the parameters A1, B1 and T are avail- able in Ref. [34]: = = − = A U e B e U e T e U ·sgn[ (0)], sgn[ (0)] 2 (0) and 2 (0) / . 1 max 1 1 1 max 1 1 max where Umax is the upper bound of the control input v1. When Umax and e1(0) are determined, the optimal parameters of s1 are acquired. To reduce the chattering phenomenon, the super-twisting algorithm in Ref. [35] is adopted, which renders a sliding mode control law as follows: = + = − − ⎧ ⎨⎩ ≤ > = − + = − v v v v y c e A t T t T v m s sgn s z z m s ¨ ˙ , 0, ( ) ˙ sgn( ) eq sw eq r sw 1 1 1 1 1 1 1 1 1 1 1 1/2 1 2 1 (44) where z is an intermediate variable, m1 and m2 are two controller parameters. A block diagram of the proposed robust coordinated control strategy is summarized in Fig. 2. As seen, the proposed adaptive second-order sliding mode controller (ASSMC) mainly consists of two components, i.e., the adaptive feedback linearization and the second-order sliding mode controller (SSMC). To eliminate the linearization error, an adaptive feedback linearization method is developed based on the model reference adaptive control principle, which is composed of Eqs. (38)–(40). For the linearized boiler-turbine system, a second-order sliding mode controller is designed with the super-twisting algorithm, which is composed of Eqs. (31), (40) and (44). The control law is only related to the information of the system states, the set values and the reference model outputs, which are all measurable or calculable. Therefore, the designed control law is implementable in practice. 3.3. Stability analysis 3.3.1. Closed-loop stability of the adaptive feedback linearization In general, the boiler-turbine system states are bounded in a sub- space of R4 and the unmodelled dynamics depends on the system states, thus it is reasonable to assume that the model uncertainties and dis- turbances are also bounded. The following two assumptions are in- troduced for the stability proof. Assumption 1. [36]. The uncertain control input matrix in Eq. (37) is multiplicative as follows: = =−M x M x M x M M Mˆ ( ) Δ ( ) ( ) [ ˆ ˆ ... ˆ ]1 2 m 1 (45) where Mˆ i is the i-th column of M xˆ ( ). Assumption 2. [36]. Since the system states and disturbances existing in the practical operation are bounded, it is reasonably assumed the model uncertainties and disturbances are bounded such that ⎧ ⎨ ⎪ ⎩ ⎪ + < < < > < < > ∀ = x x x F D a M b and mb b M c and c i m Δ ( ) max ( ˆ ( ) ) 1, 0 ˆ ( ) 1 0 , 1,..., i i i j ij i i i ii i i (46) where ΔFi(x) and Di denote the i-th row of the uncertain matrix ΔF(x) and the disturbances D in Eq. (37), respectively; xMˆ ( )ij denotes the element located in the i-th row and the j-th column of the matrix M xˆ ( ); ai, bi and ci are the corresponding upper bounds of ΔFi(x), the i-th row of M xˆ ( ) and the element xMˆ ( )ij , respectively. Theorem 1. For the boiler-turbine system (29) with the sliding mode function (38) and the control law (40), if the controller parameters η, ρ and γ satisfy the following conditions: ⎧ ⎨ ⎪ ⎩ ⎪ > > > = − − η ρ γ i0 , 1,2 i a c i i b c 1 2 1 i i i i (47) And the above assumptions are satisfied, then the adaptive feedback linearization loop is asymptotically convergent. Proof: Choosing V1 = σT σ/2 as the Lyapunov function candidate, the time derivative of V1 along the sliding function σ is Fig. 2. Structure of the coordinated control system. Z. Tian et al. ISA Transactions 79 (2018) 161–171 166
  7. 7. ∑= = = V σ σ σ σ˙ ˙ ˙T i i i1 1 2 (48) According to Assumption 1, there is = + − − − − ∑ − − − = ⎡ ⎣ ⎢ + − ∑ ⎤ ⎦ ⎥ − + − + + + = = σ σ σ F x D ρ σ η σ σ γ ψ x e σ σ σ M ψ x e ρ M x σ M x η σ σ γ M x ψ x e σ σ F x D M ψ x e σ ρ M x σ η γ ψ x e M x η γ M x ψ x e σ ˙ (Δ ( ) ) sgn( ) ( , ) sgn( ) ˆ ( , ) ˆ ( ) ˆ ( ) sgn( ) ˆ ( ) ( , ) sgn( ) Δ ( ) ˆ ( , ) (1 ˆ ( )) ( ( , ) ˆ ( ) ˆ ( ) ( , ) ) i i i i i i i i i i i i m i i i j ij j m i ii i ii i i i i ii i m i i i i j ij j m i i ii i i i i m ii i i ii i m i 2 1 2 2 1 2 2 (49) Based on Assumption 2 and Eq. (47), one yields < + − − − + − − = − − − − − − − − < σ σ a b ζ x σ ρ c σ η γ ζ x c η c γ ζ x σ η c a σ γ c b ζ x σ ρ c σ ˙ ( 2 ( ) ) (1 ) ( ( ) ( ) ) [ (1 ) ] [ (1 ) 2 ] ( ) (1 ) 0 i i i i i i i i i i i i i i i i i i i i i i i i i i 2 2 (50) Thus V˙1 < 0, which illustrates that the sliding mode function σ will converge to zero in finite time. According to the definition of σ, the linearization error em will asymptotically converge to zero. 3.3.2. Stability of the closed-loop system Although the adaptive feedback linearization approach is applied, the linearization error needs to be considered before it converging to zero. To simplify the presentation, the linearization error is regarded as a lumped disturbance term injected into the linear system (38), and the uncertain linear system is represented as = +y v d¨m1 1 1 (51) where d1 denotes the lumped disturbance. Assumption 3. According to Theorem 1, the linearization error is bounded owing to its convergence. Therefore, the disturbance in Eq. (51) is assumed globally bounded such that ≤d δ s1 1 1/2 , where δ is a positive constant. Theorem 2. For the uncertain linear system (51 with the sliding mode function (41) and the control law (44), if Assumption 3 is satisfied and the controller parameters m1 and m2 in Eq. (44) meet the following conditions: > > + − m δ m m δm δ m δ 2 , 5 4 2( 2 ) 1 2 1 1 2 1 (52) then the global sliding mode exits. Namely, for any t∈[0,∞), there is s1(t) = s˙1(t) = 0. Furthermore, the closed-loop system is asymptotically stable, i.e., = = →∞ →∞ e elim lim ˙ 0 t t 1 1 . Proof. The proof is divided into the following two parts. (1) The convergence of the sliding function s1(t) Based on the control law denoted as Eq. (48), taking the time de- rivative of s1(t), = + = + + = + s t s I t e e I t d v ˙ ( ) ˙ ˙ ( ) c ˙ ¨ ˙ ( ) sw 1 0 1 1 1 1 1 (53) where, = − + = − v m s sgn s z z m s ( ) ˙ sgn( ) sw1 1 1 1/2 1 2 1 (54) A Lyapunov function candidate V2 is selected as = + + − = V m s z m s s z ξ Pξ 2 ( sgn( ) ) T 2 2 1 1 2 2 1 2 1 1 1/2 1 2 (55) where, ξ = [ǀs1ǀ1/2 sgn(s1), z] T and = ⎡ ⎣ ⎢ + − − ⎤ ⎦ ⎥ P m m m m 4 2 1 2 2 1 2 1 1 . Taking the time derivative of V2, there is = − +V s ξ Qξ d s ε ξ˙ 1 T T 2 1 1/2 1 1 1/2 where, = ⎡ ⎣ ⎢ + − − ⎤ ⎦ ⎥ = ⎡ ⎣ + − ⎤ ⎦ Q m m m m ε m m m 2 1 , 2 m T 2 2 1 2 1 1 2 1 2 1 2 1 2 1 1 According to Assumption 3 and Eq. (52), the following equation holds ≤ − ≤ ∼ V s ξ Qξ˙ 1 2 0T 2 1 1/2 where, = ⎡ ⎣ ⎢ + − + − + − + ⎤ ⎦ ⎥ ∼ Q m m m m m δ m δ m δ 2 (4 / ) 2 2 1 2 1 2 2 1 1 1 1 Therefore, the sliding mode function s1 will converge to zero in finite time. (2) The convergence of the tracking error e1. When the system states start moving on the sliding mode surface, there is s1 = s˙1 = 0. Namely, = + =s s I t˙ ˙ ˙ ( ) 01 0 (56) Based on Eqs. (41) and (42), the explicit solution of Eq. (56) is obtained as follows: For t∈[0,T], = ⎡ ⎣ + − ⎤ ⎦ − − − + = −⎡ ⎣ + − ⎤ ⎦ − − e t e c t t e t c e B c t ( ) (0) exp( ) ˙ ( ) (0) exp( ) B c A c A c B c A c A c A c 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 (57) For t∈(T,∞], = − − = − − − e t e T c t T e t c e T c t T ( ) ( )exp( ( )) ˙ ( ) ( )exp( ( )) 1 1 1 1 1 1 1 (58) According to Eqs. (57) and (58), it is easy to prove that = = →∞ →∞ e t e tlim ( ) lim ˙ ( ) 0. t t 1 1 Hence, the closed-loop system is asymptotically stable. 4. Simulation results and discussion 4.1. Validation of the established boiler-turbine model A group of operation data, derived from a 300 MW subcritical coal- fired power plant, is applied to test the accuracy of the developed boiler-turbine system model. The boiler-turbine system model is im- plemented on a PC with the MATLAB/Simulink toolkit, where the model parameters in Table 1 are adopted. The sampling data of the practical manipulated variables serves as the model input, and the si- mulated system output is obtained by solving the model equations. Three sets of data derived from three different months are applied for the simulation test and the data sampling interval is 5 s. To illustrate the Z. Tian et al. ISA Transactions 79 (2018) 161–171 167
  8. 8. proposed model does not rely on the initial conditions, the initial values of the system states are deliberately chosen as different for the three simulation cases. As shown in Fig. 3(a)–(c), it is found that the model- based simulation results are well in coincidence with the measured data. As seen, the established nonlinear model covers the key features of the boiler-turbine system under the wide-range load variation. The average relative model errors of the three cases are acquired: = = = e e e[1.68%, 1.73%, 1.62%] , [1.45%, 1.36%, 1.40%] , [1.62%, 1.54%, 1.55%] r, a r, b r, c T T T where the subscripts a, b and c refer to the three simulation results shown in Fig. 3 (a) - (c), respectively. The model errors are found less than 1.8%, which demonstrates the proposed control-oriented model is precise enough for the purpose of controller design. 4.2. Performance comparison of three coordinated controllers According to the real operation data, the initial system states are set as N(0) = 190 MW, pD(0) = 13.67 MPa, pT (0) = 12.97 MPa and Df (0) = 30 kg/s. The set values of the power output and main steam pressure are Nr = 200 MW, pT, r = 13.5 MPa, respectively. The practical operation condition suggests the following input constraints: ⎧ ⎨ ⎩ ≤ ≤ ≤ ≤ ≤ μ μ μ 0.1 1 ˙ 0.1 14 62 t t B The parameters of the adaptive second-order sliding mode con- troller (ASSMC) are selected as m1 = [12,15], m2 = [800, 750], c = [1,10], A = [600, 0.53], B = [−34, −1], α = [1,10], η = diag {3, 1}, γ = diag {1, 0.5} and ρ = diag {0.5, 0.2}. To illustrate the effec- tiveness of the proposed ASSMC, it is compared with the conventional PID and SMC control approaches. To obtain more objective comparison between different controllers, the parameters of each controller is tuned well to achieve its optimal control performance. The parameters of PID controller are chosen as kP = [3, 1.2], kI = [0.001, 0], and kD = [10,10]. For the conventional SMC, the sliding mode function are designed as = ς e l e e y −y = +˙c 1 c c r Based on Eqs. (31) and (34), the CSMC control law u is represented as = ⎡ ⎣ ⎢ + ⎤ ⎦ ⎥ = −− u w w M x v F x v y l e l ς l sat T s = + + + ς 0 0 1 ( )[ ( )] ¨ ˙ ( )r 1 c 2 3 s f 1 1 where the parameters of CSMC are selected as l1 = [10,15], l2 = [1, 0.5], l3 = [1.5, 0.2] and ω = 0.1. The saturation function sat (·) is re- presented as = ⎧ ⎨⎩ ≤ sat ς ς ω ς ω sign ς otherwise ( ) / , ( ), . Case 1. Nominal performance Without considering any uncertainties, the step responses of the power output and main steam pressure are shown in Fig. 4. It is seen that the time response of ASSMC is faster than both CSMC and PID, and achieves zero steady tracking error without overshoots. The trajectories of the sliding mode variables are depicted in Fig. 5. It is observed that the sliding mode function trajectories of the CSMC starts from the given initial point and then asymptotically converge to zero. However, the sliding mode function trajectories of the ASSMC maintain on the sliding mode surface (s = 0) from the very beginning, which indicates the reaching phase is eliminated. The variations of the manipulating vari- ables are shown in Fig. 6. It is observed that the chattering of the ma- nipulating variables of ASSMC is lighter than that of CSMC. Fig. 3. Validation of the developed control-oriented model. (a): Mar. 12, 2012; (b): Jul. 6, 2012; (c): Nov. 14, 2012. Z. Tian et al. ISA Transactions 79 (2018) 161–171 168
  9. 9. Case 2. Time response in the presence of external disturbance The boiler-turbine system is easily subjected to various disturbances in practical operation. Typically, it is assumed that the external dis- turbances are equivalently injected into the control input channel, which is represented as: = = d t d t 5 sin cos 1 2 where d1 and d2 denote the external disturbances adding on v1 and v2, respectively. The simulation is carried out without changing the controller con- figurations. The time responses of the three controllers are displayed in Fig. 7. It is found that the proposed ASSMC achieves good disturbance rejection, which maintains excellent dynamic performance and zero steady tracking errors. However, the CSMC presents small overshoots and oscillations. The periodic oscillations and prominent steady tracking errors illustrate that the tracking performance of the PID controller is deteriorated dramatically in the presence of external dis- turbances. The trajectories of the sliding variables are shown as Fig. 8, which implies that the CSMC is sensitive to the external disturbance during the reaching phase. Case 3. Time response in the presence of model mismatch The model mismatch, generally including the parameters pertur- bation and modeling errors, is considered to test the robustness of the coordinated controller. In this case, the model parameters variations and modeling uncertainties are taken as a typical model mismatch, Fig. 4. Time response of the system outputs (without uncertainties). Fig. 5. Trajectories of the sliding variables (without uncertainties). Fig. 6. Manipulating variables (without uncertainties). Fig. 7. Time response of the system outputs in the presence of disturbances. Fig. 8. Trajectories of the sliding variables in the presence of external dis- turbance. Z. Tian et al. ISA Transactions 79 (2018) 161–171 169
  10. 10. which is assumed as follows: = + = + = + = + = + = + = + =f x k k t k k t k k t V V t V V t Q Q t k k t x x x x x ˆ (1 0.1 cos ), ˆ (1 0.1 cos ), ˆ (1 0.1 cos ) ˆ (1 0.2 sin ), ˆ (1 0.2 sin ), ˆ (1 0.2 sin ) ˆ (1 0.3 cos ) Δ ( ) [10 sin 0.2 sin 0.2 cos 0] T T D D fg fg T 1 1 2 2 3 3 4 4 1 2 3 3 2 where kˆ denotes the time-varying parameter k; Δf(x) represents the unmodelled term. Fig. 9 shows the time response of system outputs in the presence of model mismatch. As seen, the ASSMC achieves fast response and zero steady tracking error. However, the PID control approach almost loses its tracking ability owing to the periodic oscillation and large tracking errors. The remarkable overshoots and steady tracking errors illustrate the performance degeneration of the CSMC in the presence of model mismatch. The simulation results demonstrate the proposed ASSMC achieve strong robustness with respect to the model uncertainties. The trajectories of the sliding variables are displayed in Fig. 10, which il- lustrates that the proposed ASSMC is global robustness because the reaching phase is eliminated. 4.3. Tracking performance under wide-range load condition The real set values with wide-range variation is considered to test the load tracking ability of the proposed control scheme. The cascade PID controller is adopted for a practical 300 MW coal-fired unit. The unit employs sliding pressure operation and the load ranges from 150 MW to 300 MW. The real operation data of the unit over the time period of 24 h are shown in Fig. 11 (a). It is seen that the tracking performance of the onsite PID control, especially for the main steam pressure, is not satisfactory owing to the large tracking errors. The Fig. 9. Time response of the system outputs in the presence of model mismatch. Fig. 10. Trajectories of the sliding variables in the presence of model mismatch. Fig. 11. Tracking performance comparison under wide-range load variation. Z. Tian et al. ISA Transactions 79 (2018) 161–171 170
  11. 11. absolute value of the maximal tracking error of the main steam pressure even comes to 1.5 MPa, which is further beyond its allowable range. The same set values are used for simulation of the CSMC and ASSMC approaches. The tracking processes of the CSMC and the ASSMC are depicted in Fig. 11 (b) and (c), respectively. Compared with the tracking response of the onsite PID, the tracking errors of CSMC and ASSMC are both reduced remarkably, especially for the main steam pressure. The absolute relative tracking errors of the three control strategies are displayed in Table 2, which demonstrates that the ASSMC achieve better tracking performance under the wide-range load varia- tion. 5. Conclusion In this paper, a nonlinear control-oriented model of the subcritical boiler-turbine system is established and validated over a wide operating range. This model considers not only the nonlinearity of the boiler- turbine system but also the model parameters variations due to the varying thermodynamic properties of the working fluid. Moreover, the model parameters have explicit physical significance and the mean absolute relative model errors are found less than 2%. To facilitate the nonlinear controller design, an adaptive feedback linearization method is proposed by introducing a novel feedback strategy, which could eliminate the linearization error caused by model uncertainties. To address the uncertainties and external disturbances, a chattering-free sliding mode controller is designed with the super-twisting algorithm. Simulation results are presented to illustrate the good properties of the proposed control scheme, such as the fast response, small tracking error and strong robustness. 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Control strategies Onsite PID CSMC ASSMC Maximum absolute relative tracking error for N (%) 2.38 0.52 0.30 Average absolute relative tracking error for N (%) 0.33 0.16 0.01 Maximum absolute relative tracking error for pT (%) 16.86 4.38 1.20 Average absolute relative tracking error for pT (%) 2.75 0.29 0.08 Z. Tian et al. ISA Transactions 79 (2018) 161–171 171

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