Torsten Wik

Distribution and activity of ammonia oxidizing bacteria in a large full-scale trickling filter

The biofilm in a full-scale nitrifying trickling filter (NTF) treating municipal wastewater has been investigated with microbiological methods using fluorescence in situ hybridization (FISH) with 16S rRNA oligonucleotide probes in combination with confocal laser scanning microscopy (CLSM) and mathematical modeling using a dynamic multi-species biofilm reactor model. Ammonia oxidizing bacteria (AOB) were found to belong to the genus Nitrosomonas at different depths in the NTF at every sampling occasion, corresponding to different long-term operational conditions for the NTF. Both the measurements and the corresponding simulated predictions showed the same general trend of a decrease with filter depth of the amount of biofilm, the proportion of AOB to all bacteria and the total amount of AOB. The latter decreased by several times from top to bottom of the NTF. Measurements and simulations of potential ammonium oxidizing activity in the biofilm also showed a decreasing activity with depth in the NTF, which generally was operating at close to complete nitrification. However, no difference was observed when the activity was normalized to the amount of biofilm, despite decreasing proportions of AOB to all bacteria with depth in the NTF. This could be explained by diffusion limitations in the biofilm from the upper parts of the NTF according to the biofilm reactor model. The relatively good agreement between the simulations and the measurements shows that the kind of biofilm reactor model used can qualitatively describe an averaged behavior and averaged composition of the biofilm in the reactor.

Steady-state solution of a two-species biofilm problem

Through a thorough investigation of the boundary conditions for a general two‐species biofilm model, a simple and fast method for solving the steady‐state case is developed and presented. The methods used may be extended to biofilm models in which more than two species are considered. Four different sets of boundary conditions are possible for the two‐species biofilm model. Each set is shown to be asymptotically stable. A biofilm model describing the competition between autotrophic and heterotrophic bacteria and a biofilm model considering only Nitrosomonas and Nitrobacter are used for illustration. A parameter Lcrit, critical film thickness for bacterial coexistence, is introduced from which criteria on the bulk concentrations for coexistence are derived. From these criteria it is seen that the thinner the biofilm, the more restrictive the conditions are for steady‐state coexistence. For thin biofilms there may, in many cases, be no point in considering more than one species in the biofilm model. Furthermore, the gradients of the bacterial concentrations are in many cases negligible in thin biofilms, and the biofilm may then be assumed to be homogeneous. The criteria on the bulk concentrations together with the four sets of boundary conditions provide the necessary information for a direct solution of the steady‐state two‐species biofilm model by means of an ordinary differential and algebraic equation solver.

Adsorption and denitrification in nitrifying trickling filters

Abstract Step and pulse response experiments carried out on large pilot-scale nitrifying trickling filters show that adsorption and desorption of ammonium, as well as denitrification, may occur in the biofilm. These phenomena should then be taken into account when modeling the fast reactor dynamics. The observed adsorption and desorption cause the transients of the effluent ammonium concentration to become significantly slower than expected from measured residence time distributions. Sampling for measurements of stationary nitrification rates should then be carried out several mean residence times after changes in operating conditions. A physically based model of simultaneous nitrification, denitrification, adsorption and desorption in the biofilm is presented. Simulated responses to changes in influent concentrations agree with the experimental data. From the simulations it is also evident that the effects of adsorption on the responses are considerable although the actual amounts of adsorbed ammonium are small relative to amounts observed for activated sludge.

Electrochemical Estimation and Control for Lithium-Ion Battery Health-Aware Fast Charging

Fast charging strategies have gained an increasing interest toward the convenience of battery applications but may unduly degrade or damage the batteries. To harness these competing objectives, including safety, lifetime, and charging time, this paper proposes a health-aware fast charging strategy synthesized from electrochemical system modeling and advanced control theory. The battery charging problem is formulated in a linear time-varying model predictive control algorithm. In this algorithm, a control-oriented electrochemical–thermal model is developed to predict the system dynamics. Constraints are explicitly imposed on physically meaningful state variables to protect the battery from hazardous operations. A moving horizon estimation algorithm is employed to monitor battery internal state information. Illustrative results demonstrate that the proposed charging strategy is able to largely reduce the charging time from its benchmarks while ensuring the satisfaction of health-related constraints.

Feedforward feedback controller design for uncertain systems

We describe an optimization method for design of combined feedforward and feedback controllers when the plant model is uncertain. It is demonstrated that the feedback design and the feedforward design have to be made jointly for the performance to be optimal. The uncertainties used in the synthesis are given as intervals for the parameters with corresponding probability density functions. These are used in the evaluation of the objective function, which is the expected value of the effect of load disturbances. The minimization is subject to constraints on the sensitivity function and the controller response to reference signals and measurement noise. By changing the constraints the trade-off between performance, robustness and actuation is elucidated in the same manner as for plants with no explicit uncertainties. Depending on the problem character, i.e. SISO or MIMO, number of uncertain parameters and size of nominal closed loop in the uncertainty formulation, three different methods to guarantee the constraints are suggested: A direct evaluation, Horowitz-Sidi bounds with a Horowitz-Sidi test, or use of the structured singular value for robust performance. We also derive a second order approximation of the objective function to use when the number of uncertain parameters is high. The methods are illustrated on a nonlinear and uncertain control problem, namely the external carbon addition in predenitrification wastewater treatment plants.

Global Controller Optimization Using Horowitz Bounds

Abstract A procedure for global optimization of PID type controller parameters for SISO plants with model uncertainty is presented. Robustness to the uncertainties is guaranteed by the use of Horowitz bounds, which are used as constraints when low frequency performance is optimized. The basic idea of both the optimization and the parameter tuning is to formulate separate criteria for low, mid and high frequency closed loop properties. The trade-off between stability margins, high frequency robustness and low frequency performance is then elucidated and, hence, the final choice of parameters is facilitated. The optimization problems are non-convex and ill-conditioned and we use a combination of new global and standard local optimization algorithms available in the TOMLAB optimization environment to solve the problem. The method does not rely on a good initial guess and converges fast and robustly. It is applied to a controller structure comparison for a plant with an uncertain mechanical resonance. For a given control activity and stability margin as well as identical tuning parameters it is shown that a PID controller achieves slightly improved low frequency performance compared to an ℋ ∞ controller based on loop-shaping. The reason for this somewhat surprising result is the roll-off in the ℋ ∞ controller, which adds additional high frequency robustness compared to the PID controller. Computationally, a factor of 10–20 has been gained compared to an earlier, less general, version of the procedure.

Nitrification in a tertiary trickling filter at high hydraulic loads - pilot plant operation and mathematical modelling

A large pilot scale trickling filter with an established nitrifying biofilm on a cross flow plastic material having a high specific surface area has been used to evaluate the short term effects of changes in hydraulic load and influent ammonium concentration. The experiments were conducted as 15 randomized 22 factorial design experiments where the hydraulic load was set at 5.7 or 11.3 m/h and the influent ammonium concentration was set at 8 or 16 gN/m3 independently of the hydraulic load. An evaluation of the factorial design experiments revealed no difference in nitrification rate between a high hydraulic load and a low ammonium concentration and vice versa for the two setups with the same ammonium load. The short term dynamics were investigated by pulse experiments and step changes in ammonium load. These investigations revealed that the amount of water in the trickling filter was approximately independent of the flow and that stable conditions after a drastic change in the influent conditions were achieved in the time order given by the residence time distribution. A mathematical model based on a general multispecies biofilm model was used for comparison with the experimental data. The model simulations showed good correlation with experimental data indicating a feasibility for multispecies biofilm models also for large scale processes.

On the infinite time solution to state-constrained stochastic optimal control problems

A method is presented for solving the infinite time Hamilton-Jacobi-Bellman (HJB) equation for certain state-constrained stochastic problems. The HJB equation is reformulated as an eigenvalue problem, such that the principal eigenvalue corresponds to the expected cost per unit time, and the corresponding eigenfunction gives the value function (up to an additive constant) for the optimal control policy. The eigenvalue problem is linear and hence there are fast numerical methods available for finding the solution.

Non-parametric convex identification of extended generalized Prandtl-Ishlinskii models

The Generalized Prandtl-Ishlinskii model (GPI) of hysteresis has wide applicability, partly because of its capability to model asymmetric hysteresis. It is characterized by three unknown functions. Today, GPI models are typically identified through trial and error by ad hoc methods, presuming parameterized expressions for these functions and then using nonlinear least squares to determine the parameters, with concurrent problems of convergence, a dependence on the initial parameter guess, and local minima. Except for the aggregated hysteresis input-output fit the result gives no information as to whether the functions chosen are appropriate or not. Here we present a method to circumvent these problems for a more general class of hysteresis models. First, we introduce an extended GPI model (XGPI), where an additional memoryless function is placed in parallel to the GPI model. This further widens the applicability, allowing, for example, arbitrary orientation of the hysteresis loop. For such models it is shown how its four separate mappings can be identified by convex optimization. Appropriate single-valued functions can then be fitted individually to the resulting mappings and, if necessary, the function parameters found can be fine-tuned using nonlinear least squares on input-output data. The method is applied to simulated data and experimental data from a magnetoelastic torque sensor, and the results are favorably compared to the results of another commonly used hysteresis model.

AN EIGENVALUE APPROACH TO INFINITE-HORIZON OPTIMAL CONTROL

Abstract A method for finding optimal control policies for first order state-constrained, stochastic dynamic systems in continuous time is presented. The method relies on solution of the Hamilton-Jacobi-Bellman equation, which includes a diffusion term related to the stochastic disturbance in the model. A variable transformation is applied that turns the infinite-horizon optimal control problem into a linear eigenvalue problem in state-space. The method is demonstrated on a buffer control problem for a fuel cell-supercapacitor system. The obtained closed-form solution explains the shape of previous heuristically found control laws for this type of problem.