C.A. Aceves-Lara*, E. Aguilar-Garnica*, V. Alcaraz-Gonza´lez*, O. Gonza´lez-Reynoso*, J.P. Steyer**, J.L. Dominguez-Beltran* and V. Gonza´lez-A´lvarez* *Department of Chemical Engineering, University of Guadalajara, M. Garcı´a Barraga´n 1451, 44430, Guadalajara, Jalisco, Me´xico (E-mail:
[email protected]) **LBE-INRA Avenue des E´tangs 11100 Narbonne, France Abstract In this work, an optimization method is implemented in an anaerobic digestion model to estimate its kinetic parameters and yield coefficients. This method combines the use of advanced state estimation schemes and powerful nonlinear programming techniques to yield fast and accurate estimates of the aforementioned parameters. In this method, we first implement an asymptotic observer to provide estimates of the non-measured variables (such as biomass concentration) and good guesses for the initial conditions of the parameter estimation algorithm. These results are then used by the successive quadratic programming (SQP) technique to calculate the kinetic parameters and yield coefficients of the anaerobic digestion process. The model, provided with the estimated parameters, is tested with experimental data from a pilot-scale fixed bed reactor treating raw industrial wine distillery wastewater. It is shown that SQP reaches a fast and accurate estimation of the kinetic parameters despite highly noise corrupted experimental data and time varying inputs variables. A statistical analysis is also performed to validate the combined estimation method. Finally, a comparison between the proposed method and the traditional Marquardt technique shows that both yield similar results; however, the calculation time of the traditional technique is considerable higher than that of the proposed method. Keywords Anaerobic digestion; asymptotic observers; kinetic parameters estimation; successive quadratic programming
Introduction
Anaerobic digestion processes are very complex systems that need a detailed and rigorous structure in order to fully understand all the network of independent reactions occurring inside the processes. Many mathematical models at the cell level have been developed and used to predict substrate consumption, cell growth, cell composition, product formation, etc. The progress in understanding of cellular metabolic processes and the regulation system structure for specific pathways, have made it possible to establish mechanistic and structured models including many of the fundamental processes involved in the cellular metabolism of complex biochemical processes (Dochain and Vanrolleghem, 2001; Batstone et al., 2002). Prediction capabilities of such models may be reduced if the parameters are unknown or inaccurate, even if an accurate model structure has been established. The prediction problem is further compounded because of the high variability of the anaerobic digestion process, inherent its biological nature. In this paper we address the parameter estimation problem in anaerobic digestion processes and propose an optimization technique, based on the successive quadratic programming (SQP) and an asymptotic observer to provide fast and accurate estimates of the kinetic parameters and biomass yield coefficients despite noisy measurements, time varying input concentrations and varying operating conditions. An important feature of SQP is that the dynamic model of the process can be introduced as constraints in the optimization
Water Science & Technology Vol 52 No 1-2 pp 419–426 Q 2005 IWA Publishing and the authors
Kinetic parameters estimation in an anaerobic digestion process using successive quadratic programming
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C.A. Aceves-Lara et al.
problem. Therefore, the mass balance is fulfilled at each iteration of the algorithm. However, the implementation of SQP needs to be provided with biomass experimental data. Since biomass concentrations are unmeasured, we propose a modified asymptotic non-linear observer (Bastin and Dochain, 1990) to estimate some parameters that are a function of both, biomass concentrations and biomass yield coefficients. The information provided by the observer is then incorporated to the SQP algorithm to evaluate the parameters. The asymptotic observer is also used to generate preliminary parameter values that result in a considerable economy of calculation time. Finally, we test the proposed technique by using experimental data from a pilot-scale reactor treating raw industrial wine distillery wastewater. The results are compared with those obtained by the Marquardt non-linear regression algorithm.
Methods The AMP1 anaerobic digestion model
The mathematical model that describes the anaerobic digestion process is derived from the mass balances of the following biochemical reactions (Dochain et al., 1991): COD ! VFA2 þ VFA3 þ CO2;ðdÞ þ H 2;ðgÞ VFA2 ! CO2;ðdÞ þ CH 4;ðgÞ
ð1Þ
VFA3 ! CO2;ðdÞ þ VFA2 þ H 2;ðgÞ where, COD is the organic load expressed as the chemical oxygen demand, while VFA2 and VFA3 CO2, CH4 and H2 denote the concentrations of acetic acid and propionic acid, carbon dioxide, methane and hydrogen, respectively. Thus, the mathematical model is described by the following set of ordinary differential equations (Alcaraz et al., 2004): x_ 1 ¼ ðm1 2 aDÞx1 ;
s_1 ¼ Dðs1;in 2 s1 Þ 2 Y11 m1 x1
x_ 2 ¼ ðm2 2 aDÞx2
s_2 ¼ Dðs2;in 2 s2 Þ þ Y12 m1 x1 2 Y13 m2 x2 þ Y14 m3 x3
x_ 3 ¼ ðm3 2 aDÞx3
s_3 ¼ Dðs3;in 2 s3 Þ þ
1 Y5
m1 x1 2
1 Y6
ð2Þ
m3 x3
where D represents the dilution rate (1/d); the parameter a (0 # a # 1) is the biomass fraction which is retained by the reactor bed, i.e., a ¼ 0 for the ideal fixed-bed reactor and a ¼ 1 for the ideal continuous stirred tank reactor. x1 (g/l), x2 (g/l) and x3 (g/l) represent the acidogenic, methanogenic and acetogenic biomass concentrations, respectively, while s1, s2 and s3 are the concentrations of COD (g/l), VFA2 (mmol/l) and VFA3 (mmol/ l) respectively, and Y1, Y6 are the biomass yields coefficients. In this model, it is assumed that the COD degradation (acidogenesis) is described by a Monod kinetics expression and that both, the propionic acid degradation (acetogenesis), under relatively low partial hydrogen pressure, and the acetic acid degradation (methanogenesis), can be described by Haldane expressions; i.e. the specific growth rates m1, m2 and m3 are given by:
m1 ¼
mmax1 s1 ; K s1 þ s 1
m2 ¼
mmax2 s2 K s2 þ s2 þ
s22 K 2i
2
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;
m3 ¼
mmax3 s3 s2
K s3 þ s3 þ K 32
;
ð3Þ
i3
where, mmax,1 (1/d), mmax,2 (1/d) and mmax,3 (1/d) are the maximum specific growth rates for acidogenic, metanogenic and acetogenic biomass, respectively. K s1 (g/l), K s2 (mmol/l) and K S3 (mmol/l) represent their affinity constants and K i2 and K i3 (mmol/l)1/2 are the substrate inhibition constants for the methanogenic and acetogenic process, respectively.
The SQP method
min F ðyðtÞ; q; tÞ
yðtÞ[Rn
subject to gj ðyðtÞ; q; tÞ ¼ 0;
j ¼ 1; … ; m
ð4Þ
yl # yðtÞ # yu
C.A. Aceves-Lara et al.
The SQP algorithm has been extensively discussed in the literature and implemented to solve a great number of optimization problems (including parameter estimation) (Han, 1976; Schittkowski, 1981; Biegler, 1984; Chen and Hwang, 1990; Gonza´lez-Reynoso, 1999). Here, we briefly review the SQP technique and how it is implemented to solve the parameter estimation problem in anaerobic digestion. This technique is based on the following nonlinear programming optimization problem:
where F ðyðtÞ; q; tÞ is the objective function to be minimized which is defined as the error between the experimental data and the prediction of the model. y(t) denotes the state vector and q represents the vector of parameters to be estimated while gj are the constraints which, in this study are represented by the model equations (2) that describe the statespace of the anaerobic digestion process and m is the number of equality constraints. By choosing a suitable discretization algorithm for the differential equations, the optimization problem is transformed into a series of nonlinear programming subproblems where the objective function is defined at each discretized point j as: "
Fj ¼
sj 2 sj
#
tj 2 tj
;
;j ¼ 1…3
ð5Þ
where s represents the substrates vector and t represents a vector of functions that correlates two unknown groups in the model, the unmeasured biomass concentrations and the yield coefficients, that we would like to estimate (the bar above these variables stands for the experimental data while its absence is restricted to model predictions). t is given by: 2
1 Y1
6 1 6 t ¼ 6 2 Y2 4 2 Y15
0 1 Y3
0
32
3 x1 7 6 7 2 Y14 7 x2 7 76 54 5 1 x3 Y 0
ð6Þ
6
By using the above definitions and the discretization of the differential equations the optimization problem results in 18 nonlinear algebraic equations that must be solved simultaneously.
The non-linear asymptotic observer
The following hypotheses are introduced: Hypothesis H1: The dilution rate, D, the three substrates and their corresponding input concentrations are measured continuously. Hypothesis H2: The system is at steady state at t ¼ 0.
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Then, in order to estimate t, and under hypothesis H1, we propose the following asymptotic observer. 3 32 3 2 2 32 3 2 s1;in D z1 0 0 s1 ða 2 1ÞD 2aD 0 0 7 76 7 6 6 76 7 6 76 s2 7 þ 6 0 2aD 0 76 z2 7 þ 6 s2;in D 7 ð7Þ 0 0 ða 2 1ÞD z_ ¼ 6 5 54 5 4 4 54 5 4 s3;in D 0 0 ða 2 1ÞD s3 0 0 2aD z3 C.A. Aceves-Lara et al.
zð0Þ ¼ tð0Þ þ sð0Þ
t ¼z2s It is important to remark that the model (2) fulfills all the structural conditions for the observer design (equation 7), in agreement with Bastin and Dochain (1990) and Alcaraz-Gonza´lez et al. (2003). In the same way, equation (7) also fulfill the sufficient condition for convergence and stability (Alcaraz-Gonza´lez et al., 2003). An important feature of asymptotic observers is that they converge immediately to the true values of the estimated variables provided that good guesses of the initial conditions are given. In this work, reasonably good initial guesses are generated by the same asymptotic observer (7) by using hypothesis H2.
Results and discussion
422
The SQP algorithm and the asymptotic observer were applied to estimate the kinetic parameters of model (2) by using experimental data from a 1 m3 up-flow fixed bed anaerobic digestion reactor using Cloisonylew as the medium for the treatment of raw industrial wine distillery wastewater obtained from the Narbonne area, France (Steyer et al., 2002). We found good initial estimation guesses by applying the asymptotic observer in a number of initial conditions; the resulting “true” initial conditions were: t1 ð0Þ ¼ 17 g=l; t2 ð0Þ ¼ 118 mmol=l and t3 ð0Þ ¼ 33:5 mmol=l: In most applications of the SQP technique, the orthogonal collocation method is often selected to discretise the differential equations and transform them into a set of algebraic equations (Gonza´lez-Reynoso, 1999; Mun˜oz-Ma´rquez, 2002). In this work, we found that the orthogonal collocation method was not suitable for such a purpose due mainly to the nature of the available measurements, which were highly corrupted with noise and the drastic changes introduced by the dilution rate (see Figure 1). Instead, we introduced polynomial functions over certain time intervals to avoid the discontinuities imposed by the actual readings of the anaerobic digester process. Figures 2–4 clearly show how the readily available measurements are corrupted by noise and affected by the time varying dilution rate. Notice that it is not always possible to estimate all the parameters individually. In our case, only the kinetics parameters can be estimated separately. With respect to the yield coefficients, only dimensionless groups of them were estimated. The complete list of estimated parameters is reported in Table 1. In order to test the hypothesis that the experimental points for the k components (k ¼ {COD, VFA2,VFA3}) are properly represented by the model (2) with the parameters described in Table 1, rigorous statistical analysis were performed. Firstly, we calculated the correlation coefficients between the experimental data and the model predictions. The resulting coefficients were 0.976, 0.981 and 0.844 for the COD, VFA2 and VFA3 respectively. Secondly, the Fisher’s test (Froment and Bischoff, 1990), was used to test the validity of the estimated parameters: !, ! n n X X w2ik ð1ik Þ2 p ð8Þ Fk ¼ . F k ðp; nk 2 p; 1 2 bÞ p n 2 p k i¼1 i¼1
C.A. Aceves-Lara et al.
Figure 1 On-line dilution rate data
Figure 2 Comparison between COD experimental data and model predictions
Figure 3 Comparison between VFA2 experimental data and model predictions
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C.A. Aceves-Lara et al. Figure 4 Comparison between VFA3 experimental data and model predictions
where: 1ik ¼ wik 2 w^ ik is the error for the i th data and, the k th component between the i th model prediction (wik), and the i th experimental data points (w^ ik ). nk is the number of data points for the k th component, p is the number of parameters being estimated. F pk is the Fisher distribution with p degrees of freedom in the numerator, nk 2 p degrees of freedom in the denominator and where b is the significance level. The results of the Fisher’s test are shown in Table 2. The values for F pk with b ¼ 0.05, were obtained from tables. As can be seen, F k . p F k ; for all k, which, clearly indicates the excellent fitting capability of the considered model provided with the estimated kinetic parameters. Figures 2– 4 show the comparison between the model predictions with the estimated parameters and the experimental data of COD, VFA2 and VFA3 concentrations. As can be observed, the model provided excellent predictions even in the presence of highly corrupted measurements and operational failures. Indeed, the model correctly described the behavior of the actual anaerobic digestion even when a major disturbance occurred at day 31, resulting in a rapid fall in both, the dilution rate (which actually fell to zero) and the substrate concentration readings. Similar results were achieved by using the Marquardt method proposed by Constantinides and Mostoufi (2000). These results are not shown in this paper. However, it is important to remark that the estimation computational effort by using SQP was considerably less than that invested in the method of Marquardt. In fact, the whole SQP calculation process took 1–2 seconds of CPU time while the Marquardt technique took 1–2 hours of CPU time for the COD parameter estimation and up to 12 hours of CPU time for the acid parameter calculations. This is, perhaps, the main advantage of using SQP over the classical approaches.
Table 1 Model parameters obtained by using the SQP method Parameter
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a mmax1 K s1 mmax2 K s2 K i2 mmax3
Value
0.5059 1.0723 5.0992 0.9172 6.7568 7.6498 0.5548
(1/d) (g/l) (1/d) (mmol/l) (mmol/l)1/2) (1/d)
Parameter
Value
K s3 K i3 Y1/Y2 Y1Y6/Y4Y5 Y6/Y4 Y1/Y5
1.1503 (mmol/l) 7.1346 ((mmol/l)1/2) 2.3987 0.6970 0.1070 1.5902
Table 2 Fisher’s test for the SQP method
FCOD ¼ 76.30 FVFA2 ¼ 97.48 FVFA3 ¼ 7.93
FpCOD ð13; 17; 0:95Þ ¼ 2:35 FpVFA2 ð13; 29; 0:95Þ ¼ 2:08 FpVFA3 ð13; 23; 0:95Þ ¼ 2:18
In this work a methodology that combines the application of an asymptotic observer with a nonlinear programming technique (SQP), was used to determine the kinetic parameters and yield coefficients in an anaerobic digestion process. The optimization problem is solved in such way that the differential equations that describe the anaerobic digestion process are also satisfied. The model provided with these parameters is statically validated and was found to provide excellent estimation results in describing acceptably the concentrations of the COD and VFAs in an actual anaerobic digestion plant. Through rigorous simulation and statistical analysis results it was shown that the SQP technique yields better and faster estimates than those obtained by classical approaches.
C.A. Aceves-Lara et al.
Conclusions
Acknowledgements
The authors gratefully acknowledge the PROMEP program, CONACyT and the TELEMAC project (IST–2000-28156) for the support that made this study possible.
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