**3.5 Tabu search for inverse modeling of biofilm reactor**

The implementation procedure of tabu search is given in Section 1.3. In this work, tabu search is used to determine the parameters of kinetic and film thickness models in association with the validation of the mathematical models using the data of measurements obtained from an experimental fixed bed anaerobic biofilm reactor. The parameter estimation via inverse modeling is carried out by defining an objective function, *J* given by.

$$J = f(a) = \sum\_{i=1}^{l} \left(\mathbb{y}\_i - \hat{\mathbb{y}}\_i\right)^2 \tag{38}$$

where *α* is the vector of parameters, *l* is the number of observations, *yi* is the measured value of the *i* th variable, and ^*yi* is the corresponding predicted value. Tabu search is applied to estimate the parameters of kinetic and film thickness expressions of different modeling configurations with the support of one dimensional (1D) and two dimensional (2D) mathematical models of the biofilm reactor. Random variation in all the parameters is considered to generate neighbors so as to provide the maximum improvement to the current solution. Recency based tabu list with a length of 100 and frequency based tabu list with a length of 50 are employed. A sigmoid function based aspiration criterion, Eq. (2) is employed with *kcenter* as 0.3 and *σ* as 7/*M*, where *M* refers specified number of iterations. An intensification strategy with the coefficient *α* taking the format of a sine function, Eq. (1) is employed, in which the value of *θ* is assigned to be 4.0001. The maximum number of iterations are considered as 100. The termination-on-convergence criteria, Eq. (3) is used as stopping criterion.
