Model Identification for Firing Burner Plant Based On ...

Proceedings of the International Conference on Electrical Engineering and Informatics Institut Teknologi Bandung, Indonesia June 17-19, 2007

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Model Identification for Firing Burner Plant Based On Linear Parameter Varying Systems 1

Heri Subagiyo 1*, Bambang Riyanto T. 2

Program Studi Teknik Elektronika, Politeknik Caltex Riau Jl. Umbansari, Rumbai, Pekanbaru 28265, Indonesia Telp. +62-761-53939 [email protected] 2 Laboratorium Sistem Kendali dan Komputer, Sekolah Teknik Elektro dan Informatika, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia Tel. +62-22-250 0960 [email protected] Abstract System or process identification is a mathematical modeling of systems (processes) from test or experimental data. Process models obtained from identification process can be used for operator training simulator, analysis and design of safety systems, and design of process and control systems. This paper presents the Linear Parameter Varying (LPV) model identification for firing burner plant. The LPV model is adopted considering the fact that firing burner plant has several operating conditions (start up, normal, and shutdown). By assuming that on every operating condition there are parameters changes, the LPV model is suitable for covering all operating conditions. The firing burner plant is modeled as LPV systems with 2nd order ARX model structure. Identification algorithm used in the identification process is Recursive Least Square (RLS). Data needed for identification is taken from DCS historian data of the process with sampling time of 1 minute. The identification result is simulated and validated with the measured data. The process models obtained by this technique show reasonably fit with the mean value of 95.53 %. This is satisfactory result for further work on the LPV control system application or process simulator.

1. Introduction System or process identification is a mathematical modeling of systems (processes) from test or experimental data [1]. Process model obtained from identification process can be used for process simulation, analysis and design of control systems, design of safety systems. Most of the existing works on process simulation is based on LTI, which is satisfactory for a number of systems. However, system identification based on LTI model appears to be of limited value when the plant operating conditions significantly varies. One of the effective methods to handle varying operating conditions in the plant is to employ Linear Parameter Varying (LPV) model [3, 4]. LPV constitutes linear dynamic systems whose state space matrices depend on parameters which may vary in time. Such a system was recently studied within the context of gain-scheduling control analysis and design which provides stability and performance guarantee along the trajectory of the varying parameters. System identification of primary reforming process in an ammonia plant was recently studied for simulation purpose by the second author in [2], based on LTI model. Simulation result shows that the model obtained was satisfactory for reforming process operation on normal operating condition. Simulation for start up operating condition has not been successfully conducted as high overshoots of step responses appear in several process models which exceeds maximum operation limit. A careful observation of firing burner in reforming process reveals that some nonlinearity occurs in the system. The term Linear Parameter Varying (LPV) system was originally coined by Shamma dan Athans [3] within the

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context of gain-scheduling design. LPV model appears in many physical modeling and control problems, such as aerospace, vehicle system applications [4], stall and surge control of jet engine [5] and flutter suppression [6]. This paper presents the LPV model identification for firing burner plant of primary reformer section in an ammonia plant based on historical data recorded in a distributed control system (DCS). The LPV model is chosen from the observation that ammonia plant is operated along several operating conditions (start up, normal, shutdown, and emergency operations) and that in each operating conditions there are significant parameter changes. LPV identification process adopted in this paper is based on Recursive Least Square (RLS), a technique originally developed by Bamieh [7].

2. Firing Burner Plant This section gives an overview of firing burner plant of primary reformer section in an ammonia plant. The primary reformer section consists of 144 catalyst tube in the two radiant section and side firing burner systems. The reaction in the catalyst tube needs energy in the form of heat. Heat needed in primary reformer section is supplied from fuel gas burning radiation at burners. These burners attached at four side reformer walls, where each side has 6 rows and 16 columns burners. Sensible flue gas heat from the primary reformer radiant section is also used to heat several media in the waste heat recovery section [8]. Fuel for the firing consists of two fuel gases: natural gas as main fuel and mixed gas (MDEA gas from CO2 removal section mixed with let down gas, inert gas, and purge gas

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from synthesis loop unit). Natural gas and mixed gas flows to dual nozzle type burner through fuel header. Each burner is completed with special type butterfly valve to control firing profile. Firing burner control system is shown in Figure 1.

Parameter function { ai }, { bi } are assumed to be a linear combinations of a set of known fixed basis functions {f1, ..., fN}.

ai ( p) = ai1 f1 ( p) + …… + aiN f N ( p)

(5)

bi ( p) = b f ( p) + …… + b f N ( p)

(6)

1 i 1

N i

k

l

where the constants ai , bi are real numbers. Thus any particular model is completely characterized by the real k

l

numbers { ai } and { bi }. The goal of a parametric identification scheme in this paper is then to find these constants from process measurement data. For this general framework, many choices are possible for the function f1 ( p ) . In particular if we consider a polynomial dependence, then the function simply powers of p

Figure1: Firing Burner Control System

f1 ( p ) = p l −1

3. LPV Model Identification

f1 ( p) are

l = 1,..., N

(7)

In this case the parameter functions are rewritten as Linear Parameter Varying (LPV) system is a special class of nonlinear system [9], where the system coefficients are rational function of a parameter. In LPV model, system matrices depends on one or more time-varying parameters and hence represents a family of LTV systems (one for each parameter trajectory) [10]. A discrete time LPV systems represented in the state space form as [7]: x(t + 1) = A( p (t )) x(t ) + B ( p (t ))u (t ) y (t ) = C ( p (t )) x(t ) + D ( p (t ))u (t )

ai ( p) = ai1 + ai2 p + …… + aiN p N −1 bi ( p) = b + b p + …… + b p 1 i

J = J (Θ ) =

where u is input, y is output, i.e. (δy ) k := y (k − 1) , and

δ

⎡ a11 ⎢ 1 ⎢ a2 ⎢ ⎢ Θ := ⎢a1na ⎢ b1 ⎢ 0 ⎢ ⎢ b1 ⎣ nb

(2) is delay operator,

B (δ , p ) = b0 ( p ) + b1 ( p )δ + …… + bnb ( p )δ nb

(3)

A(δ , p) = 1 + a1 ( p )δ + …… + a na ( p )δ na

(4)

n = na + nb + 1 is the number of parametric functions to be identified. Assume that the varying parameter p is a function of discrete time ( p = p (k ) ).

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N −1

(8) (9)

{

1 T ∑ E ε (k , Θ) 2 T k =0

}

(10)

where Θ is n x N matrix that contains coefficients to be identified:

LPV systems model may also be characterized in the form of [7]:

A(δ , p ) y ( k ) = B (δ , p )u (k )

N i

In recursive least mean square (RLS) identification algorithm it is required to minimize errors between measurement variables and estimation variables which is known as loss function

(1)

As shown by Eq. (1), LPV constitutes linear dynamic systems whose state space matrices depend on parameters which may vary in time. These parameters represent nonlinearity of the system.

2 i

and

… … … … …

a1N ⎤ ⎥ a2N ⎥ ⎥ ⎥ anaN ⎥ b0N ⎥ ⎥ ⎥ N ⎥ bnb ⎦

(11)

ε ( k , Θ ) , called prediction error, is defined as

ε ( k , Θ ) = y k − Θ, Ψ k

(12)

where yk is measurement data (output of the system) and Θ, Ψ k is estimation output. An extended regressor Ψ, is build by past I/O data and parameter trajectories,

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⎡ − y k −1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢− y k −na ⎥ Ψ k := φ k π k := ⎢ ⎥ [ f1 ( p k ) ⎢ uk ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ u k −nb ⎥⎦

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The functional dependence in p of the parameters is given by polynomials of order N = 3: f 2 ( pk ) …

f N ( p k )]

(13)

ˆ is parameter coefficients matrix which is In this paper, Θ k estimated at time k. Following RLS algorithm and under assumption of [7], identification algorithm is given by: ˆ , Ψ = y − trace(Θ ˆ * Ψ ) (14) ε k = yk − Θ k −1 k k k −1 k ˆ =Θ ˆ Θ k k −1 + K k ε k

K k = Pk {Ψ k }

Pk = Pk −1 − Pk −1

(15) (16)

Ψk ⊗ Ψk Pk −1 1 + Ψ k , Pk −1 Ψ k

ˆ =Θ lim Θ 0 k

(20)

bmn ( p) = bmn1 + bmn 2 p + bmn 3 p 2

(21)

In this paper, we use parameter trajectory sequence in the form of p k = sin(0.0001 * k ) . This trajectory sequence satisfies the condition of persistency excitation theorem [7]. Plot of measured and simulated model outputs are shown in Figures 2 and 3. From these figures, it is shown that estimated model approximate the real process behaviour reasonably well. This is particularly true for the case in which there is a quick change in process response, indicating that the models have picked up essential features of the true process.

(17)

Simulated Model (orde 2, p = sin(k)) & Measured Output for TI2026.PNT 1200

where P is covariance matrix expressed in recursive form in Equation (17). Convergence condition of the identification algorithm is attained if Θ 0 is the true value, thus k →∞

aij ( p ) = aij1 + aij 2 p + aij 3 p 2

(18)

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4. LPV Identification Results Measured Output Simulated Model Output

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This section gives an example of application of LPV model identification for firing burner plant of primary reformer section in an ammonia plant Kaltim-4, PT Pupuk Kaltim. By examining the reformer diagrams and operation described in [8], it is known that firing burner plant is a multivariable process. The process has eight inputs and nineteen outputs. In this paper, we only focus on two outputs. These outputs are TI2026.PNT (a point temperature measurement on primary reformer radiant section) and TI2023.PNT (a point temperature measurement on waste heat recovery section). Data needed for identification process is taken from DCS historian data from February 15, 2006 to March 3, 2006 with sampling time of 1 minute. All operating conditions of the process (start up, normal, and shut down) are covered in the duration considered. The model used in the identification process is discretetime LPV system of 2nd-order characterized in the following representation form:

y ( k ) + A 1 ( p ) y (k − 1) + A 2 ( p ) y ( k − 2) = B 1 ( p )u(k − 1) + B 2 ( p )u( k − 2)

(19)

⎡ y1 (k − 1) ⎤ ⎡u1 (k − 1) ⎤ ⎢ ⎥ ⎢ ⎥ where y (k − 1) = , ⎢ ⎥ u(k − 1) = ⎢ ⎥ ⎢⎣ y19 (k − 1)⎥⎦ ⎢⎣u 8 (k − 1)⎥⎦

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0

0

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0.4

0.6

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1

1.2

1.4

1.6

1.8

2 4

x 10

Figure 2: Time responses of simulated and measured output TI2026 with p=sin(0.0001*k) Simulated Model (orde 2, p = sin(k)) & Measured Output for TI2023.PNT 250

200

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Measured Output Simulated Model Output

0

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Figure 3: Time responses of simulated and measured output TI2023 with p=sin(0.0001*k)

The models obtained using LPV identification process give model fit of 95% and of 71.69% for TI2026 and TI2023

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output, respectively. Here, model fit is calculated using the equation of model fit in [11]. However, from plots depicted in Figures 2 and 3, it can be seen that simulated model outputs tend to oscillate, particularly for the case in which there is slow response changes. It indicates that frequency of parameter function is too high for the process considered. Decreasing the frequency of parameter function with p k = sin(0.000001* k ) gives better results. Plot of measurement and simulated model outputs for the new parameter function are shown in Figures 4 and 5. It can be seen from these figures that better LPV identification was obtained. The LPV model output responses approximate satisfactorily measured responses for all cases of fast and slow time-response changes. The LPV model obtained provides model fit of 98.52% and of 92.94% for TI2026 and TI2023 output, respectively. The other outputs have similar results. The mean value of fit from all outputs is 95.52 %, which is acceptable.

5. Conclusion This paper addressed system identification of firing burner plant, based on LPV representation. In general, the models obtained by using the LPV identification and RLS algorithm with parameter trajectory sequence of sufficiently low frequency provides reasonable fit with the mean value of 95.52 %. This is satisfactory result for further work on the LPV control system application or process simulator.

Acknowledgements The authors would like to thank PT. Pupuk Kaltim, Tbk for providing the data needed.

References [1]

Zhu, Yucai, Multivariable System Identification for Process Control, Oxford: Pergamon, 2001.

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[2]

R. T. Bambang and E. Hendarwin, “Modeling and Dynamic Simulation of Reforming Process Control in Ammonia Plant”, Proc. 2006 Asian Control Conf., Bali, 2006

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[3]

J. Shamma and M. Athans, “Guaranteed properties of gain scheduled control for linear parameter varying plants”, Automatica, Vol. 27 No. 3, 1991, pp. 559-564.

[4]

J. Bokor and G. Balas, "Linear parameter varying systems: A geometric theory and applications", Preprints 16th IFAC World Congress, Prague, Czech Republic, July 2005, pp. 69-79.

[5]

B. Bamieh, L. Giarr´e, T. Raimondi, D. Bauso, M. Lodato and D. Rosa, “LPV Model Identification For The Stall And Surge Control Of Jet Engine”, 15th IFAC Symposium on Automatic Control in Aerospace, Bologna, Italy, September 2001.

[6]

H.Y Sutarto* and M.Fadly, “LPV Model Identification for Flutter Suppression Application”, Proc. of the 6th Asian Control Conference (ASCC), Bali, Indonesia, 2006.

[7]

B. Bamieh, L. Giarre, “Identification of Linear Parameter Varying Models”, International. Journal of Robust and Non-Linear Control, Vol. 12, 2002, pp. 841-853.

[8]

Nugraha B.E. Irianto, Kaltim-4 Ammonia Plant Operation Guide, PT. Pupuk Kalimantan Timur, Tbk, Bontang, 2002.

[9]

M. Nemani, R. Ravikanth, B.A. Bamieh, ”Identification of Linear Parametrically Varying Systems”, Proc. of the 34th IEEE Control and Decision Conference Vol. 3, New Orleans, Lousiana, December 1995, pp. 2990-2995.

Simulated Model (orde 2, p = sin(k)) & Measured Output for TI2026.PNT

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Figure 4: Time responses of simulated and measured output TI2026 with p=sin(0.000001*k) Simulated Model (orde 2, p = sin(k)) & Measured Output for TI2023.PNT 250

[10] L.H. Lee, K. Polla, “Identification of Linear Parameter-Varying Systems via LFTs”, Proc. of the 35th Conference on Decision and Control, Kobe, Japan, December 1996.

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[11] L. Ljung, System Identification Toolbox for Use with MATLAB, MathWorks Inc., Massachusetts, 2002.

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Figure 5: Time responses of simulated and measured output TI2023 with p=sin(0.000001*k)

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