# Optimal Control of an Ultraviolet Water Disinfection System
## 1 Introduction
As a result, theoretical models have been developed that concern the variation of both the fluid velocity field and UV radiation intensity according to the distance from the radiation source (see [7]). These models can be used in the design of UV systems and to assist CFD simulations. They can be used to calculate, for example, the received radiation dose with good accuracy at any point in the irradiation room at a certain time. The models consider many parameters of the UVDS and the water quality (e.g., transmittance). In conjunction with a kinetic model and validation and calibration measurements, these models can determine the bacterial reduction (abatement) that should be inside certain limits. For large UV systems, this approach seems to be unavoidable. When the UVDS has a small irradiation room, and some constraints are met (including water quality), practical identification can result in a dynamic model whose output variable is bacterial reduction (see [5]). The control inputs may be, for example, the water flow rate and the UV light intensity of the source because they achieve a control action. Data obtained by conducting a series of experimental measurements are used with an identification procedure. Usually, specific software assists the identification method to construct the dynamic model. The result is a pair of transfer functions corresponding to the two influence channels. The current study adopted a typical disinfection water system, described in [5]. In addition to the equipment's technical parameters, we also used data presented in the paper mentioned above, which proposed a linear dynamic model. For this UVDS, Section 2 presents a nonlinear dynamic model, which was used to generate a method that calculates the bacterial abatement (reduction). Optimal control requires a well-stated optimization problem, which includes components that must be defined: the dynamic model, the optimum criterion, and types of constraints (see [12; 13]). Section 3 proposes the statement of an optimal control problem devoted to the UVDS under consideration. It is a bi-criteria optimization (see [14]) problem whose solution has to meet two constraints:  
* the final value of bacterial abatement should be equal to an imposed value (disinfection criterion);
* the control inputs minimize the energy consumed for irradiation (energy criterion). To transform our problem into a standard optimization problem, we adopted a new optimum criterion that is a weighted combination of the two above criteria. In this new description, the optimal control problem is denoted by UV-OCP1. This statement may be considered a contribution of this work, particularly because it can be easily adapted to a different UVDS. Section 4 proposes an optimization algorithm devoted to solving UV-OCP1. It is a meta-heuristic algorithm (see [15; 16; 17]) based on hybrid topology particle swarm optimization (HTPSO) [18; 19], an improved version of PSO (see [20; 21]) with better communication abilities among particles. In addition to the \"social neighborhood\", other known techniques were used to improve the algorithm: adaptation of particle speed, a method to detect the convergence, and a simple deterministic method to initialize the particle position. Section 4.2 and the files attached to this paper provide a detailed description of this algorithm and its implementation, which could be easily adapted to a different UVDS. The proposed algorithm called HTPSO_UV1 receives, as input data, the imposed bacterial abatement value and determines the setpoint optimal values: the flow rate and UV light intensity. The implemented program also yields the values of the two optimal criteria. The examination of optimal solutions supposes that HTPSO_UV1 has been carried out for a range of abatement values. Execution of the series was conducted to answer the following questions: Is there an optimal solution for any given bacterial abatement value? To what extent is the disinfection criterion met? To what extent is the energy criterion met?