Control of an anaerobic digester through normal form of fold bifurcation

A Dynamic Analysis for an Anaerobic Digester: Stability and Bifurcation Branches

“…Each of these four concentrations is associated with a differential equation. The dilution rate, the influent COD concentration, and the influent VFA concentration are considered as stability parameters (see [2,4,[22][23][24]-model IV-, [25]). The nonlinear nature of the specific biomass growth rate leads to multiple equilibria, but only some branch sections correspond to realistic process operation, either normal process operation or biomass washout condition.…”

“…Moreover, the dynamic analysis can also indicate the constraints on the dilution rate that allow us to avoid the biomass washout and keep the digester under normal operation [4]. As it can be noticed from [22][23][24], the interval of the dilution rate that leads to positive and stable features of the equilibrium biomass concentration depends on the influent substrate concentrations.…”

“…The dynamical system (1) has six equilibrium points, but only one of them, the so-called normal operation point (NOP), describes the normal operation of the digester (see [23,26]). The other ones play an important role in the system dynamics and their interactions explain the different behaviors in the bioreactor and its transitions from stable operation to washout phenomena.…”

“…Similarly, 11 = 0 by (3) and 22 = 0 by (4). The eigenvalues of the linearized system are (see [4,23])…”

“…This method is based on the linearization of the vector field and involves the definition of the following factors: (i) equilibrium points, (ii) parameter constraints that lead to expected intervals of the equilibrium state variables, (iii) Jacobian matrix, eigenvalues, and stability of each equilibrium, (iv) coordinates of local bifurcation points, and (v) parameter regions that lead to either desired or undesired system operation. The analytical development of the dynamics has many advantages with respect to the graphical analysis, as can be noticed from [4,8,20,22,23]: (i) it allows a fast computation of equilibrium points, bifurcation points and other properties, (ii) it allows an in-depth, complete, and efficient understanding of the influence of model parameters on the equilibria, bifurcation points and on other characteristics, specially in the case of three or more bifurcation parameters. Indeed, when there are three or more bifurcation parameters, the graphical anlaysis is cumbersome and the analytical results can be more practical.…”