Journal of Science & Technology 101 (2014) 025-029
Grey-Box Identification of Steam Boiler Using Linear State-Space Model and Closed-Loop Data
Trinh Thi Khanh Ly'-^, Hoang Minh Son '•*
' " Hanoi University of Science and Technology, No I, Dai Co Viet Sir, Hai Ba Trung, Ha Noi, Viet Nam '^'Electric Power University
Received. March 03, 2014; accepted April 22, 2014 Abstract
This paper presents a grey-box identification approach for developing a linear state-space model with the physical state variables of steam boilers The mam concept of this research is to combine theoretical modeling principles with the closed-loop identification technique The structure of the model was determined by theoretical analysis of the boiler dynamics. The parameter estimation algorithm was derived based on the prediction error method and Gauss-Newton iterative method. This method was applied to a steam boiler al Phu tAy Plant revealing its effectiveness in the closed-loop identification problem.
Keywords Grey-box identification, prediction error method, Gauss-Newton method, boiler modeling I. Introduction
Steam boiler is an important component found in many industrial processes. Several dynamic models of the boiler system have been developed employing different methods such as theoreucal modeling {or physical modeling, modeling from the first principles) [1],[2], black-box idendfication [3],[4] and grey-box identification [5],[6] The models obtained from the first principles provide good insights into the system behavior but they are quite complex. On the other hand, black-box models are built from experimental data and require no a prion knowledge. However, the drawback of black- box approach is the lost of physical interpretation, as in the case of state-space models where both system order and stale vanables may not associate with physical characteristics of the system. Grey-box identificafion aims at combining theoretical modeling with experimental-based parameter estimation, thus provides a balance between fidelity and simplicity.
This technique is suitable to build dynamic models required for advanced control methods such as model-based predictive control.
Hitherto, several grey-box methods have been proposed for steam boilers using open-loop data [6]- [8]. Yet in the practice, we are often interested in modeling of systems operating under closed-loop control, for which many conventional open-loop methods would fail. Some closed-loop idenfificafion techniques have been applied for steam boiler as reported in [3],[4],[9]-[ll], However, for such a complicated system, they used black-box approaches.
For advanced process control, a state-space model is usually preferable to deal with high-order, multiple-mput multiple-output (MIMO) systems Thus, in this paper we aim al combining a closed- loop idendfication technique based on prediction enor method [14] and Gauss-Newton iterative algorithm, and theorefical modeling to derive a linear MIMO state-space model with physical state variables of the steam boiler The physical model equations are utilized to determine the state-space model strucnire. Then, the model parameters are estimated fi-om closed-loop data by employing the prediction enor method and Gauss-Newton iterative algorithm.
The obtained theoretical results are applied to develop a dynamical model of the steam boiler at Phu My plant [12] In this real boiler system, we focus on the master processes as shown in Fig 1, where the mam output vanables are temperamre and pressure of the superheated steam, and the inputs are tiie fuel flow rate, spray water flow rate and steam flow rate.
2. The Physical Model Equations and The Identiflcation Model
In the following, we employ a linearized state- space model as published in [1], in which the material and the energy balance equations were derived with appropriate simplifications [[2],[5],[ 13]], AD,, = AD -I-AD.
-DCJ.^
+^Q,,,+[h,-h^^)AD^
(1) (2)
• Conesponding author: Tel.: (+84) 3869.2005
E-mail address, [email protected] vn -0,Cp,A7;;-(/(^3-A^]AO^,-
Journal of Science & Technology 101 (2014) 025-029
1. Simplified scheme of a drum-type steam
The noise e(k) represents uncertainties such as unknown disturbances, noises and mode! inaccuracy.
We model e(i)' as a white noise process with zero mean and stadstically independent of past input and output data, even in the case of closed-loop operation.
The covariance mafi-ix E[e{k)e^(k)] = R should be also estimated for use by different control design methods.
From physical considerations, certain elements of the parameter matrices are zeros. For a fix operating point, these matrices have constant
APj -=Kj,AP^., +k^^ -^A
— t^, = — ( A D - A D , (4) (5)
dt
^,
where Dj and D^i are steam mass flow rates &om the steam drum and the secondary superheater (kg/h), Ddi IS attemperator spray flow rate (kg/h), Df is fiiel flow rate (kg/h), Vst^Vsi are steam volumes m the primary and secondary superheater (m^); Aj, /ij;, hs2 and /ij^are specific enthalpies of the steam in the drum, the steam in the primary and secondary superheater, and the spray water (kj/kg), respecdvely; Qsh is heat supplied to the superheater from the ftimace, P j is drum pressure (bar), Ps2 is superheater steam pressme (bar), Tsi and T,,2 are steam temperatures at the outlet of the primary and the secondary superheater (°C); Kr IS pressure coefficient, the over-bar symbol ( * ) denotes the values at operating point.
The set of lineanzed equations (1-6) can be rewritten in matrix form:
m = A^x{f) + B^u{t)
y(f) = Cx{t) ' where mput, output and state variables are:
((-[AD^ADrf.AD,,]'', >'=[A7;,JAP,J]' X = [A7;, A7;,J A/^jf
Smce the identification process is based on sampled data, we should use a discrete-time model of the form
x(k + ]) = Ax{l<) + Bu{k) + e(k)
y{k)^Cx{k) ^^' The parameter matrices A, B and C are system
matrix, input matrix and output matrix, resprectively.
h ^ ''9
c-\
ro 1 o| [0 0 i jDefme the parameter vector 8 as:
6 = [a„ .,a,,i|,,.,,;><,f
Considermg that all chosen state variables of the steam boiler are commonly measureable, the parameter estimation problem is now expressed in the Imear regression model
x(k + \)^(p'{k)e + e{k) (9) For a data set, k =
written in the matrix form:
1 N, Eq. (9) could be (10)
[eiN)\
3. The Parameter Estimation Algonthm In the following, we derive the iterative algorithm to estimate the parameter matrix by using the available input and state variables from closed- loop experiment. The noise e(k) is unconelated with past data, but due to the feedback control it is directly conelated with present and future data, making the traditional identification method failed. Ljung stated that the prediction enor method were the best approach under closed-loop conditions [14],
Let 0 denote an estimate of 6. Since e is a"
white noise process with zero mean, then:
x{k + l) = <p^(k)d (11) IS the best predictor with the prediction error:
Journal of Science & Technology 101 (2014) 025-029
E{k + \) = xik + \)-(p^(k)e
The sequence of prediction enors is stacked into the vector:
The cost fianction to be minimized i
Employing the Gauss-Newton method, the Hessian of V(d) is approximated by J^J where J is the Jacobian matnx:
The computation procedure for parameter estimation is derived as follows.
Algorithm 1
a) Initial step (i=0): Calculate the initial value 9^
usmg the least squares algorithm' /» = (*'"*)-'5>''
k^p\
(18)
(16) The matrix <D may become singular or poorly conditioned and hence there exist problems with computing its inverse. Consequently, regularization should be applied, in which case (16) becomes:
/> = ((t»'"0 + ;?/)-'(I)'"
where the regularization parameter ^ is chosen to be small(;? = lO-'-10~'),
b) Repeat following calculations for each step i:
- Compute E, using Eq. (13).
- Compute ff,+i by
0„,^d_+7j,PE, (17) where >j is the adjustable step length, which could be
detemuned by algonthm in [14].
c) Terminate the procedure if | 4 + , - ^ , | < 5 for a small value i5 > 0, or if the number of iterations i exceeds a limit.
Note that for the ideal case 0 = 0, the prediction error e(k) will be the same as e(k). Thus, for a sufficiently large N, the covariance of e{k) is estimated by.
4. Identification of Steam Boiler at Phu My Plant For the identification purpose, the first 2000 samples are used to estimate the model, while the next 2000 samples are used for model validafion.
This experiment was carried out in closed-loop operation using a sampling time Ts=3s. which was the smallest scan time technically realizable by the data collection system
Applymg Algorithm 1, the parameter matrices are obtained as:
"0 6889 -0.1384-0.6708"
- 0 0027 0.9678 -0.2945
0 0 0
0-1195 0.1152 0.1155 0.0379 0,0028 0.0332 0.0034 0 0124 -0 0023
The covariance matrix of the noise is estimated as.
"0.9919 0,0064 0 0015"
K= 0.0064 0 0029 0.0001 0.0015 0,0001 0-0040
One method to validate the obtained model is using a Variance Accounted For (VAF) percentage [14]. The VAF is used to measure the fitness between the measured output y{k) and the output ^(ft) simulated from the model.
' var(;.) j - (19) In our case, we use the measured state vanables mstead of output variable. As shown in Table 1, the VAF for the three state vanables indicate very good fitness
Using the validation data, the prediction enors or residuals of the three state variables are shown in Fig. 2, The small residuals (relatively to measurement values) which are randomly distributed around zero like a white noise process indicate a good modeling quality.
Table J Model validation results
96.46 98 49
Journal of Science & Technoli^y 101 (2014) 025-029
• hiNl#A«ril||lyV*^^
^w^y^V^*M#wHtlWl^^
Fig. 2. Prediction errors
Fig. 3. Auto-conelations of residuals
.»
I " " I ! " " * K ; ! " " ; ' ! " " ' " ! ; ! ! I I " ' '
'j-l.'l -'
. . . , , , „ . M " . M > " ' t " " l ' " > l > ' . ' t ' > " l
1 -i 1 S IS IS ^ ^ ^ , 3)
Fig, 4. Cross-con-elafions between residuals and input data
Another common technique for model validation is the conelation analysis of the residuals [14] A good model should produce small residuals diat are weakly conelated with previous residuals and with the input data. Fig 3 shows the auto-conelation
functions of the residuals. The distribution confirms that the prediction enors behave very much like a white noise process. Fig 4 shows the cross- conelations between different inputs and residuals.
The very small values reveal that the residuals are almost unconelated to the mput data.
Beside evaluation based on the residuals, the elements in the covariance matnx R obtained above should also give us some interpretations. All the small values except Rw indicate that uncertainties including measurement noise and model inaccuracy associated with the second and the third slate variables are negligible. The distinguished value of Riu however, reveals that the uncertainty associated with the first state variable is significant. This fact also reflects well the characteristics of the physical variables, i.e. the outlet temperature of the first superheater has most complicated dynamics and uncertainty, especially under changing operating conditions. This is the reason why the temperature of the superheated steam should be set under tight control. From modeling viewpoint, the remaining uncertainty suggests us to consider incorporating an appropriate nonlinear model structure This should be the interests of our future research.
5. Conclusion
This paper presents a grey-box identification approach to obtain a linear state-space model of steam boilers, including a noise model. It is important to point out that the model structure is derived based on the physical analysis. The parameter matrices and the covariance matrix of the noise are estimated from closed-loop data by using prediction enor method and the Gauss-Newton iterative algorithm. The iterative algorithm greatly improves the data fihiess and can deal with the problems of closed-loop identificafion.
The proposed method was applied to develop a state-space mode! of a steam boiler at Phu My plant with satisfactory results. This model is suitable for the purposes of advanced control design or dynamic real-time optimization.
The limitation of our current work is that the linear model is valid for a pre-defined operating condition (i.e. for a fix load). Of course, we can apply multi-model approach to cover different operating condifions. Another direction for our future work is to extend the proposed technique to achieve a single nonlmear state-space model that is valid for a wide operatmg range.
Journal of Science & Technology 101 (2014) 025-029 Acknowledgment
This work was supported by the AUN/SEED- Net Collaborative Research Program with Industry (CRI).
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