Data-Driven Disturbance Rejection Predictive Control for SCR

Article pubs.acs.org/IECR

Data-Driven Disturbance Rejection Predictive Control for SCR Denitrification System Xiao Wu,*,† Jiong Shen,† Shuanzhu Sun,‡ Yiguo Li,† and Kwang Y. Lee§ †

Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, Southeast University, Nanjing, 210096, China Jiangsu Frontier Electric Technology Co. Ltd., Nanjing, 211102, China § Department of Electrical and Computer Engineering, Baylor University, One Bear Place #97356, Waco, Texas 76798-7356, United States ‡

ABSTRACT: This paper develops a data-driven disturbance rejection predictive controller (DRPC) for the selective catalytic reduction (SCR) denitrification system in a coal-fired power plant by using the technique of subspace identification (SID). First, to alleviate the modeling difficulties of the model predictive control (MPC), a subspace predictive controller is constructed from the input−output data of the system by directly adopting the subspace matrices as the predictor. Then, following the same subspace method, a disturbance observer (DOB) can also be designed, which estimates both the external disturbances and plant behavior variations. Unlike the conventional DOB-based control system, the disturbance estimation signal is fed back to the subspace predictive controller to improve the accuracy of the prediction, and the integrated DRPC is finally constructed. The resulting controller can remove the effect of unknown disturbances and modeling mismatches quickly while satisfying the input constraints in the optimization. Case studies demonstrate the advantage and effectiveness of the proposed approach.

1. INTRODUCTION Selective catalytic reduction (SCR) denitrification is an essential technique in coal-fired power plants to reduce the nitrogen oxides (NOx) in the flue gas. Its basic working principle is simple; liquid ammonia is sprayed into the flue gas, reacts with the NOx, and converts it to N2 and H2O in the presence of catalysts. The SCR denitrification technique has some distinctive advantages such as simple device structure, the small physical size, and little influence on boiler operation. Therefore, in the context of serious environmental issues, employing the SCR denitrification technique is now the rule rather than an option in China’s power industry.1 However, control of the SCR denitrification system is challenging due to its properties, such as large inertia, time delay, unknown disturbances, and plant behavior variations. Consequently, the PI/PID based controllers are no longer sufficient in meeting performance specifications, even if they are tuned well under the nominal plant operating conditions. Thus, various control strategies have been explored in recent years.2 Dolanc and Strmcnik et al.3 developed a first-principle model of the boiler to estimate the NOx flow rate at the outlet of the boiler; the model is then utilized to design a feed-forward controller for the SCR denitrification system. Zhou et al.4 built an optimal controller for the SCR system on the basis of an identified neural network (NN) model. To overcome the large inertia of the SCR denitrification system and achieve an optimal control performance under the physical limitation of the ammonia © XXXX American Chemical Society

spray valve, many MPCs, such as the generalized predictive controller (GPC),5 dynamic matrix controller (DMC),6 and NN based nonlinear predictive controller,7 have been developed recently, with an emphasis on optimality, constraint, stability, and so on. Their simulation results indicate that better dynamic control performance can be achieved compared to the conventional PID controller. Under the traditional design framework, developing a model is the first and foremost important step in controller design; however, the modeling process is complex and introduces unavoidable modeling mismatches which will greatly degrade the control performance. Among the aforementioned SCR denitrification controller designs, it is interesting to note that models identified from the input−output data are extensively used.4−7 If the models are developed from the data, the information contained in the model is no more than what is contained in the original data, which implies that the control system can be directly built from the data, bypassing the modeling step. For these reasons, this paper proposes to utilize the subspace identification (SID) method to develop a predictive controller for the SCR denitrification system directly from the input−output data. The SID is a noniterative robust identification method Received: September 16, 2015 Revised: May 3, 2016 Accepted: May 3, 2016

A

DOI: 10.1021/acs.iecr.5b03468 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research which can avoid local minima and convergence problems,8−10 and moreover, the subspace matrices can directly be used to predict the future output without further developing the model parameters.11−13 Therefore, the intermediate modeling effort for the conventional model predictive control (MPC) and resulting modeling mismatch can be avoided. Another important issue in designing controls for the SCR denitrification system is handling of the time-varying behavior and various uncertainties of the plant: (i) The variations of coal quality and combustion condition in the boiler will change the NOx concentration at the outlet of the boiler. (ii) The concentration of injected ammonia may vary, due to the control and measurement errors in the ammonia evaporation system. (iii) The variation of flue gas flow rate and temperature and the decaying catalyst activity will influence the reaction rate of denitrification. (iv) The system equipment such as the ammonia spray valve are wearing out easily. The aforementioned disturbances are all unknown to the system or hard to measure accurately, which will all interrupt the operation of the SCR denitrification system severely. In spite of the effectiveness of predictive controls, they alone cannot react fast enough to reject the disturbances in the system. When strong disturbances are involved, the predicted output will deviate far from the real output; thus the performance of the predictive controller will be degraded greatly and may even cause system instability. The disturbance observer (DOB) based control is one kind of active disturbance rejection method, which makes use of the inverse model of the plant to estimate the effects of unknown disturbances and modeling mismatches. The estimation can then be used as a feedforward signal in the control system to achieve a fast rejection of disturbances.14 Because of its simplicity and capability to compensate for unknown disturbances, the DOB-based control has been developed steadily and used widely in process control in the past few years.15−20 However, as a model based approach, the performance of the DOB-based control is heavily dependent on the model, and moreover, for the ordinary DOB-MPC system, because the disturbance estimation is added later as a supplementary feedforward signal to the control system which was already designed optimally by the MPC, it will violate the optimality of the predictive control system. Given these reasons, we present this paper to address the control problems of the SCR denitrification system using only the input−output data, making the following contributions and advantages to the existing literature: (1) A data-driven method is proposed to develop a disturbance observer, extending the applicability of subspace identification. (2) A novel data-driven disturbance rejection predictive controller is developed for the SCR denitrification system, in which the disturbance estimation is included in the predictive controller design to make an optimal prediction. Owing to the advantages of both DOB and predictive control, their combination can remove the effect of unknown disturbances and plant behavior variations quickly while guaranteeing the optimal performance. The proposed DRPC is applied to the SCR denitrification system in a simplified 600 MW coal-fired power plant simulator. The remainder of this paper is organized as follows: Section 2

introduces the SCR denitrification system and its dynamics. The integrated data-driven DRPC is presented in section 3. Simulation of case studies is given in section 4, and conclusions are drawn in section 5.

2. SYSTEM DESCRIPTION Figure 1 shows the schematic diagram of the SCR denitrification system of a 600 MW coal-fired power plant. The SCR

Figure 1. Schematic diagram of the SCR denitrification process.

devices are installed between the economizer and air preheater of the boiler. The vaporized and diluted ammonia (NH3) is evenly injected into the flue gas through the injection grid, which then reacts with the NOx within the catalytic bed under 280−420 °C temperature conditions, yielding harmless nitrogen and water vapor. The model of this SCR denitrification system is developed with both physical and empirical methods based on field data obtained from a series of identification experiments. It is validated in the MATLAB environment as used in a power plant simulator. Because of the 100 mg/m3 NOx emission standard (GB 13223-2011) and the aforementioned reaction temperature, the NH3 slip is very small and not considered for most of the coal-fired power plant in China. The central task of the SCR denitrification system is to control the NOx emission (controlled variable) by regulating the NH3 injection flow rate (manipulated variable). However, compared to the quick response of the turbinegenerator unit (time scale of seconds), many processes in the boiler unit are relatively slow, called large inertia processes (time scale of several to tens of minutes). In the SCR denitrification system, the influence of NH3 injection flow has a large inertia and time-delay property as shown in Figure 2 for a step response test. Moreover, the flue gas variation, equipment wear, environmental changes, etc. cause unknown disturbances and behavior variations to the system. Therefore, advanced control techniques are called for to improve the SCR denitrification system.

3. DATA-DRIVEN DISTURBANCE REJECTION PREDICTIVE CONTROL FOR SCR DENITRIFICATION SYSTEM The MPC refers to a class of control approaches which utilizes an explicit process model to predict the future response of a B


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a data-driven disturbance rejection predictive control (DRPC) is proposed in this paper by integrating the controller and DOB, where the input−output plant data are directly used in the DOB and controller design using the SID approach. 3.1. Design of Subspace Predictive Controller. Consider the objective function: J = (yf̂ − r f )T Q f (yf̂ − r f ) + ΔuTf R f Δuf

(1)

where Qf = QTf > 0 and Rf = RTf > 0 are weighting matrices of output and input, respectively, and rf = [rkT+ 1 rkT+ 2 ... rkT+ Ny]T is the desired output trajectory. Given the past output−input data wp = [yk T− N + 1 ... yTk ; ukT− N + 1 ... uTk ]T, the f uture predictive output ŷf = [ŷkT+ 1 ŷkT+ 2 ... ŷkT+ Ny]T can be estimated by yf̂ = lwwp + luuf

(2)

following the basic idea of SID,11 where uf = [ukT+ 1 ukT+ 2 ... uk T+ Nu]T is the f uture control input, lw = Lw(1:lNy,:) and lu = Lu(1:lNy,1:mNu) are prediction matrices in MATLAB expression, Ny and Nu, Ny ≥ Nu, are, respectively, the prediction horizon and the control horizon, l and m are the dimensions of the output and input, respectively, and N is the row block number of the data Hankel matrices in the SID (see Appendix). Note that, through the SID, the subspace matrices Lw and Lu can be constructed directly from the input−output data; thus the prediction of the output can be simply achieved by eq 2 without further developing the model, therefore the remaining modeling procedure required for the conventional MPC and resulting modeling mismatch can be avoided. The construction method of Lw and Lu using the SID is shown in Appendix. To achieve an offset free tracking performance of the datadriven predictive controller, an integral action can be included by using a difference operator Δ = 1 − z−1 in the output prediction 2:

Figure 2. Step response of SCR denitrification system. Shows about 15 s input time delay and inertia property with about 3 min response time.

plant and calculate the control inputs through the minimization of a dynamic objective function within the predictive horizon. Owing to its distinguished advantages such as handling constraints and dealing with systems with large inertia behaviors, MPC has made a significant impact on the power plant control.2 To further improve the disturbance rejection property, the DOB technique has been proposed recently and employed in the MPC control system. The basic structure and working principal of the conventional DOB-based MPC control system is shown in Figure 3, where a predesigned inverse model of the plant is used

Δyf̂ = lwΔwp + luΔuf

(3)

Thus, we can have yf̂ = yk + ζlwΔwp + ζluΔuf

(4)

where yk = [ ykT ykT ... ykT ]T

Figure 3. Conventional DOB-based MPC control system.

and

to estimate the control input corresponding to the measured system output. Because the effect of unknown disturbances is also reflected in the control input estimation, by comparing it with the actual control input, the equivalent disturbance as an additive input can be estimated. The disturbance estimation is then added to the control signal calculated by the MPC to compensate for the effect of disturbances. The detailed approaches of DOB can be found in Li et al.14 and are not repeated here. The conventional DOB-based MPC method has two main problems: first, both the design of MPC and DOB rely on the mathematical model of the plant, and second, the modification of the MPC with the disturbance estimation signal violates the optimality of the MPC and degrades the performance, especially when the input constraints are involved. Therefore,

⎡I ⎢ I ζ=⎢ ⎢⋮ ⎢ ⎣I

0 ... 0 ⎤ ⎥ I ... 0 ⎥ ⋮ ⋱ ⋮⎥ ⎥ I ... I ⎦

The input magnitude constraint (umin,umax) as well as the input rate constraint (Δumin,Δumax) can be imposed as ⎡I ⎤ ⎡I ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ I ⎥(u − u ) ≤ ζ Δu ≤ ⎢ I ⎥(u − u ) k f k ⎢⋮⎥ min ⎢⋮⎥ max ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ I I C

(5)


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Industrial & Engineering Chemistry Research ⎡I ⎤ ⎡I ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ I ⎥Δu ≤ Δu ≤ ⎢ I ⎥Δu f ⎢⋮⎥ min ⎢⋮⎥ max ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ I I

(6)

Substituting eq 4 into the objective function eq 1, and at every sampling time, by minimizing eq 1 subject to eqs 5 and 6 through quadratic programming, Δuf can be calculated, and the value of uk+1 can be obtained and applied to the plant. 3.2. Design of Subspace DOB. The essential idea of the DOB is to estimate all the effects of unknown disturbances and lump them at the input side, which is usually achieved by using an inverse plant model. However, eq 2 shows that by identifying the subspace matrices Lw and Lu, the relationship between input and output can directly be built and the control input can be estimated by an inverse form of eq 2: uk̂ = (ludob)−1(yk − lwdobwpdob)

(7)

T T T T T where wdob p = [yk − N... yk − 1; uk − N ... uk − 1] is the past output dob dob and input, lw = Lw(1:l,:), and lu = Lu(1:l,1:m). Remark 3.1: To ensure the identifiability of the subspace DOB, the following conditions need to be satisfied: (i) The system is of minimum phase. (ii) The input uk is uncorrelated with the system noise. (iii) uk is persistently exciting on the order of 2N. (iv) The number of measurements is sufficiently large. Given the behavior of the SCR denitrification system, the input time delay τ must be considered and separated from the minimum phase part of the system in the subspace DOB design (τ can be estimated directly from a step response test of the system as shown in Figure 2). Therefore, a data set {U = [u0, ..., u2N+j−2], Y = [yτ, ..., y2N+j−2+τ]} is chosen in the DOB subspace matrices identification, and wdob = [yTk − N ... yTk − 1; uTk − N − τ ... p T T uk − 1 − τ] has to be constructed to estimate the control input. Similar to the conventional DOB, a low-pass filter with a steady-state gain of 1 is employed to filter out the high frequency noise, and how to devise it is introduced well in Chen et al.16 The detailed schematic diagram of the data-driven DOB is shown in Figure 4, where disturbances are expressed in terms of plant variations and input and output disturbances. To avoid the difficulty of handling various unknown disturbances acted on different parts of the control system, for example, the input disturbance Dm(z), the output disturbance Dc(z), and the effect of plant behavior variation Dp(z) as shown in Figure 4a, an equivalent block (Figure 4b) can be derived by moving the effect of Dm(z) to the output side as an equivalent disturbance D̅ m(z) = Dm(z) G(z)z−τ. A lumped disturbance D(z) = D̅ m(z) + Dp(z) + Dc(z) can then be used to include all different kinds of disturbances. Therefore, the system output becomes

Y (z) = U (z) G(z)z −τ + D(z)

Figure 4. Schematic diagram of the subspace DOB: (a) initial form, (b) equivalent form.

where Q(z) is a first-order filter. Substituting eq 8 into eq 9 yields D̂ (z) =Q (z)(U (z)z−τ + G−1(z) D(z) − U (z)z −τ ) =Q (z) G−1(z) D(z) (10)

Define D̃ (z) as the error between the real disturbance D(z) and the equivalent disturbance estimation D̂ (z) moved to the output side by multiplying the transfer function G(z)z−τ: D̃ (z) =D(z) − G(z)z −1 D̂ (z) =(1 − Q (z)z −1) D(z)

By using the final-value theorem, it can be determined that d (̃ ∞) = lim(1 − Q (z) z −1) lim(z − 1)D(z) z→1

z→1

−1

= lim(1 − Q (z) z ) d(∞) z→1

(12)

Therefore, if the steady-state gain of the filter Q(z) is selected as 1, we have

d(̃ ∞) = 0

(13)

which illustrates that the subspace DOB makes it possible to observe the lumped unknown disturbances. 3.3. Integrated Data-Driven Disturbance Rejection Predictive Control. As shown in Figure 4, under the ordinary DOB-based control system design framework, the disturbance estimation D̂ (z) is utilized as a supplementary feedforward signal and is directly added into the control input C(z) already calculated by the controller. Under this design framework, if C(z) is calculated by an optimal controller such as the subspace predictive controller (SPC) developed in section 3.1, the optimality of the original controller will be broken by the

(8)

Recall that the subspace identification eq 7 can be viewed as an inverse of the plant model without time delay, G−1(z); thus the equivalent disturbance estimation reflected at the input side can be expressed by D̂ (z) = Q (z)(Y (z) G−1(z) − U (z)z −τ )

(11)

(9) D


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The proposed controller is tested and compared with other types of predictive controllers. For the proposed predictive controller, the sampling time is set as 30 s, and a prediction horizon Ny = 300 s and control horizon Nu = 300 s are adopted. The weighting diagonal matrices Q f and Rf are given as Q f = INy and Rf = 100 × INu. The input constraints are umax = 1 and umin = 0 and Δumax = 0.01/s and Δumin = −0.01/s due to the physical limitations of the NH3 injection valve. The filter is selected as Q(s) = 1/(5s + 1)4. Three case studies are simulated using various disturbances, which describe the control challenges of the SCR denitrification system: Case 1: The first case is to show the performance of the controllers for disturbances in the inlet NOx concentration. At t = 10 min, the set point of the SCR denitrification system changes from 60 mg/m3 to 100 mg/m3, then at t = 25 min and t = 60 min, an unknown step-type disturbance dinlet_NOx = 20 mg/m3 and unknown sinusoid-type disturbance dinlet_NOx = 20 sin(0.2(t − 60)) act on the inlet NOx concentration, respectively, reflecting the frequent variation of coal quality and combustion conditions in the boiler. Three predictive controllers are used for comparison: (1) Proposed data-driven disturbance rejection controller (without the integral action) (2) Conventional MPC + DOB controller (same controller parameters, without the integral action) (3) Conventional MPC (same controller parameters, without the integral action) The SCR denitrification system responses and corresponding control actions are shown in Figure 6. The results show that

supplementary input. To overcome this issue, we propose to consider the disturbance estimation signal D̂ (z) early in the predictive controller design stage to improve the accuracy of the output prediction, resulting in an integrated data-driven disturbance rejection predictive controller. Suppose the following state space disturbance model can be used in the prediction: ⎧ x = Ax + Bu + Ed ̂ ⎪ k+1 k k k ⎨ ⎪ y = Cxk + Duk + Fdk̂ ⎩ k

(14)

where d̂k is the disturbance estimation signal in the time domain, we can rewrite eq 14 as ⎧ ̃ ⎪ xk + 1 = Axk + Buk̃ ⎨ ⎪ ̃ k̃ ⎩ yk = Cxk + Du

(15)

where B̃ = [B E] and D̃ = [D F] are the augmented system matrices with DOB and ũk = [uTk d̂Tk ] is the augmented input. The model eq 15 shows that, for combined input data and the estimated disturbances, the SID algorithm introduced in the Appendix can be extended to find the augmented subspace matrices L̃ w and L̃ u, in which the estimated disturbances can be included in the prediction: yf̂ = lw̃ wp̃ + lũ uf̃

(16)

where w̃ p = [yTk−N+1... yTk ; ũTk−N+1 ... ũTk ]T is the past output and the augmented input data, ũf = [ũTk+1 ũTk+2 ... ũTk+Nu]T is the future augmented control input, and lw̃ = L̃ w(1:lNy,:) and lw̃ = L̃ w(1:lNy,1:mNu) are the augmented prediction matrices. Therefore, a data-driven disturbance rejection predictive controller (DRPC) with estimated disturbances augmented in the prediction can be constructed. The structure of the overall data-driven DRPC system is shown in Figure 5.

Figure 5. Data-driven disturbance rejection predictive control system: DRPC optimizes the plant augmented with DOB. Figure 6. Case 1: Performance of the SCR denitrification system for the unknown disturbances in the inlet NOx concentration (solid red, proposed data-driven DRPC; dashed blue, MPC+DOB; dotted black, MPC; dotted-dashed green, reference).

Remark 3.2: Note that at current time instant k, the future disturbances have not occurred and acted on the system yet; thus it is impossible to estimate their values. Therefore, we assume them fixed with current disturbances estimation d̂k over the control horizon Nu. This simplicity is commonly used in practice.

without the disturbances, the three predictive controllers have similar control performance, which can tightly and quickly control the outlet NOx concentration tracking the set point. However, because no integral action is included, in the case of unknown disturbances, significant control offset occurs for the MPC (the NOx concentration is 115.7 mg/m3 for the step

4. SIMULATION RESULTS This section demonstrates the data-driven DRPC design for the SCR denitrification system using subspace identification. E


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Industrial & Engineering Chemistry Research disturbance, and fluctuating strongly between 87.9 and 112.1 mg/m3 for the sinusoid disturbance) which makes the power plant severely violate the environmental regulation. On the other hand, both the proposed DRPC and the MPC +DOB have better performance compared with the MPC alone. Among them the proposed data-driven DRPC has the best performance, which can remove the effect of step disturbances quickly and restrain the outlet NOx emission to a small range (98.2−101.8 mg/m3) under the sinusoid disturbance. With the help of subspace DOB, the MPC +DOB also has a good performance; however, since the control sequence calculated by MPC is compensated by the disturbance estimation directly, the optimality of the MPC is violated and the performance is degraded. Moreover, the tuning of Qf and Rf is no longer possible in order to achieve a balance between fast output tracking and small input variation for the MPC+DOB. Therefore, a slower step disturbance rejection performance compared with the DRPC and larger outlet NOx emission fluctuation range (97.8−102.2 mg/m3) under the sinusoid disturbance can be found in the figure. It should also be noted that, for the MPC+DOB, additional computation for system model matrix estimation and state observer design are needed, which increase the controller design complexity and computational burden. Case 2: The second case is to show the performance of the controllers under the disturbances on the NH3 injection side. At t = 10 min, the set point of the SCR denitrification system changes from 120 mg/m3 to 70 mg/m3, then at t = 35 min, an unknown step-type disturbance du = 0.2 is acted on the NH3 injection due to the wear and tear of the valve, and for t = 65− 75 min, an unknown slope-type disturbance du = 0.2−0.04 × (t − 65) is acted upon representing a decrease of NH3 concentration, due to the fault in NH3 evaporation and NH3/air mixing devices. Three predictive controllers are used for comparison: (1) Proposed data-driven DRPC (with integral action) (2) Conventional subspace predictive controller (SPC)+DOB (same controller parameters, with integral action) (3) Conventional SPC (same controller parameters, with integral action) The simulation results are shown in Figure 7. The results show that all three predictive controllers can achieve satisfactory performance in the nominal case; however, the two DOB-based predictive controllers can reject the disturbances quickly than the SPC, in which only the integral action is used. Similar to case 1, the results also show that, since the optimality of the predictive controller can be kept in the proposed controller, the data-driven DRPC performs better than the SPC+DOB. Case 3: The last case is to further test the disturbance rejection property of the proposed DRPC under significant unknown compound disturbances and measurement noise. It is assumed that the SCR system is working at a 100 mg/m3 operating point initially, and at t = 10 min, the system gain is increased 20%, representing the SCR reaction rate variation due to the changes in flue gas flow rate, temperature, or the catalyst activity, then at t = 30 min and t = 55 min, unknown step-type disturbances du = 0.2 and dy = 20 are acted on the NH3 injection valve and inlet NOx concentration, respectively. Moreover, random measurement noise with amplitude 1 mg/m3 is added to the system output. The controllers tested in case 2 are used for comparison; the simulation result is shown in Figure 8.

Figure 7. Case 2: Performance of the SCR denitrification system under the unknown disturbances at the NH3 injection side (solid red, proposed data-driven DRPC; dashed blue, SPC+DOB; dotted black, SPC; dotted-dashed green, reference).

Figure 8. Case 3: Performance of the SCR denitrification system under the unknown compound disturbances (solid red, proposed data-driven DRPC; dashed blue, SPC+DOB; dotted black, SPC; dotted-dashed green, reference).

Table 1. Quantitative Comparison for the Tested Controllers

F


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where the superscripts p and f signify past and f uture, respectively. The input and innovation data Hankel matrices, U and E, can be constructed in a similar format. Then by stacking up the model eq A1 with input−output data for a number of steps, these Hankel matrices can be used to develop the following subspace matrix equations:10

The result shows that under the random measurement noise, the proposed controller can still handle the compound unknown disturbances more effectively than the SPC+DOB and SPC and can regulate the outlet NOx concentration back to the given set point quickly; thus, the power plant can operate in accordance with the state environmental regulations. A quantitative comparison for the tested controllers in the three cases is shown in Table 1, where the integral square index is employed to show the output and input performances, i.e., N N εy = ∑k = 1 (yk − rk)2 and εu = ∑k = 1 Δuk 2 .

5. CONCLUSION In order to solve the control problem of the SCR denitrification system in the coal-fired power plant, a data-driven disturbance rejection predictive control (DRPC) strategy is proposed using the subspace identification (SID) method. To alleviate the modeling difficulties of MPC and DOB, a subspace predictive controller and disturbance observer are constructed directly from the input−output data of the system by making use of the subspace matrices in the estimation. The integrated data-driven DRPC system is then formed by augmenting the disturbance estimation into the subspace predictor to make optimal prediction of the system output. Owing to the advantages of both DOB and predictive control, the DRPC can remove the effect of unknown disturbances and modeling mismatches quickly while guaranteeing the optimal performance. The advantages and effectiveness of the proposed data-driven DRPC design are demonstrated through the simulations on an SCR denitrification system of a 600 MW power plant. Due to the data-driven nature, the proposed control method is flexible and can easily be adapted to other types of systems without knowing mathematical models of the plant.

y1 ...

y2 ... ... ... yN ...

yN − 1 yN

yN + 1 ...

yN + 1 yN + 2 ... ... ... ... y2N − 1 y2N ...

X f = ΨY Y p + ΨU U p + (A̅ )N X p

(A4)

⎡ D 0 0 ⎢ 0 D ⎢ CB = ⎢ CAB CB D ⎢ ... ... ... ⎢ N−2 − N 3 N ⎣CA B CA B CA − 4 B

0⎤ ⎥ 0⎥ 0⎥ ... ⎥ ⎥ ... D ⎦

... ... ... ...

0⎤ ⎥ 0⎥ 0⎥ ... ⎥ ⎥ ... I ⎦ ... ... ... ...

ΨY = [ A̅ N − 1K A̅ N − 2 K ... AK K ], ̅ ΨU = [ A̅ N − 1B̅ A̅ N − 2 B̅ ... AB ̅ ̅ B̅ ]

in which A̅ = (A − KC) and B̅ = (B − KD). The state matrix X is defined as ⎡ X p ⎤ ⎡ x0 x1 ... xj − 1 ⎤ X=⎢ f⎥=⎢ ⎥ ⎣ X ⎦ ⎣ xN xN + 1 ... xN + j − 1 ⎦

Owing to the stability of the Kalman filter, A̅ N = (A − KC)N → 0 as N → ∞; thus for a large N, eq A4 converges to X f = LN W p

(A1)

(A5) p

where subspace matrices LN and past data matrices W are defined as LN = [ΨY ΨU] and Wp = [(Yp)T (Up)T]T. Substituting eq A5 into eq A2, we have

where uk ∈ ℛm, yk ∈ ℛl, and xk ∈ ℛn denote the system input, output and state, respectively, and A, B, C, and D are the system matrices. The unknown innovation term ek ∈ ℛl is assumed to be zero-mean white noise, and K is the Kalman filter gain. Suppose the input and output data of the system can be obtained consecutively from k = 0 to k = 2N + j − 2; then the following output data Hankel matrix with N-block rows and j-block columns can be built: y1 ...

(A3)

⎡ 0 0 I ⎢ 0 I ⎢ CK HNs = ⎢ CAK CK I ⎢ ... ... ... ⎢ N−2 ⎣CA K CAN − 3K CAN − 4 K

⎪

y0

Y p = ΓN X p + HNdU p + HNsEp

HNd

APPENDIX: SUBSPACE IDENTIFICATION METHOD Assume that an innovation form of the state-space model can be used to describe the system:

⎡ ⎢ ⎢ ⎢ ⎢ p ⎡Y ⎤ ⎢ Y=⎢ f⎥=⎢ ⎣Y ⎦ ⎢ ⎢ ⎢ ⎢ ⎢⎣

(A2)

where ΓN = [CT (CA)T ... (CAN−1)T]T,

■

⎧ xk + 1 = Axk + Buk + Kek ⎨ ⎪ ⎩ yk = Cxk + Duk + ek

Y f = ΓN X f + HNdU f + HNsE f

Y f = LwW p + LuU f + LeE f

(A6)

with the subspace matrices defined by Lw = ΓNLN, Lu = HNd, and Le = HNs. With the conditions that (i) uk is uncorrelated with ek, (ii) uk is persistently exciting on the order of 2N, and (iii) the number of measurements is sufficiently large, i.e., j → ∞, the data Hankel matrices can be decomposed by the QR decomposition as follows:8,11

⎤ ⎥ yj ⎥ ⎥ ... ⎥ yN + j − 2 ⎥ yN + j − 1 ⎥ ⎥ yN + j ⎥ ⎥ ... ⎥ y2N + j − 2 ⎥⎦ yj − 1

⎡W p ⎤ ⎡ R11 0 0 ⎤⎡ Q 1 ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ U f ⎥ = ⎢ R 21 R 22 0 ⎥⎢Q 2 ⎥ ⎢ f⎥ ⎢ ⎥⎢ ⎥ ⎣ Y ⎦ ⎣ R31 R32 R33 ⎦⎢⎣Q 3 ⎥⎦

(A7)

By expanding this equation and comparing it with eq A6, the subspace matrices L = [Lw Lu] can be calculated as G


Article

Industrial & Engineering Chemistry Research ⎡ R11 0 ⎤† ⎥ L = [ R31 R32 ]⎢ ⎣ R 21 R 22 ⎦

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(A8)

where † represents the Moore−Penrose pseudoinverse. The block matrices in eq A6 are identified as Lw = L(:,1:N(m + l)) and Lu = L(:,N(m + l) + 1:end) in the MATLAB expression, representing the first N(m + l) columns and the remaining columns in L, respectively. Note that eq A6 has the potential to be used as a predictor in designing a predictive controller, because the future output is expressed as a function of future input. The predictive expression of the output Ŷf can be written as f

Y ̂ = LwW p + LuU f

(A9)

which implies that by identifying the subspace matrices Lw and L u from the input−output data, a predictor can be constructed.11 The predictor can be used in the predictive control without identifying plant models, thus avoiding the system identification procedure that is required for the MPC and the resulting modeling mismatch.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the National Natural Science Foundation of China (NSFC) under Grant 51506029, Grant 51576040, and Grant 51576041; the Natural Science Foundation of Jiangsu Province, China under Grant BK20150631 and Grant BK20141119; and China Postdoctoral Science Foundation for funding this work.

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REFERENCES

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Data-Driven Disturbance Rejection Predictive Control for SCR

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