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_______________________________________________________________________________ 8th IWA Symposium on Systems Analysis and Integrated Assessment
Solution to an optimization problem with applications to control wastewater treatment processes B. Carlsson, Å. Nordenborg and M. Lundgren Department of Information Technology, Uppsala University, P.O. Box 337, 751 05, Uppsala (E-mail:
[email protected]) Abstract In the paper the problem to control a class of processes in a resource optimal way is considered. The optimization is done under the constraint that the daily (for example) mean value of the effluent discharge is below a certain threshold. Key assumptions are that the magnitude of the control signal to the plant is proportional to the quantity that should be minimized (for example consumption of energy), the process efficiency decreases with increasing magnitude of the control signal, and that the disturbance is additive. We show that for static processes fulfilling the assumptions it is optimal to use a constant input signal. Some practical aspects of the results are discussed and a strategy to control effluent ammonia is outlined. Keywords: optimal control; energy efficiency; real-time control; biological nitrogen removal
INTRODUCTION Operation of a wastewater treatment plant (WWTP) is a challenging task where the effluent discharge requirements should be satisfied at minimal operation costs (energy, chemicals, labor, and maintenance), see Olsson (2006). In order to improve the operation of WWTPs more extensive use of mathematical models, instrumentation, control and automation (ICA) have been introduced over the years. The choice of control strategies range from simple on/off control, PID-control to complex model based predictive controllers. A popular tool to evaluate control strategies is the IWA/COST Simulation Benchmark (BSM1) where a simulation model, a plant layout, performance criteria and test procedures are defined (Copp, 2002). This allows for a rigorous evaluation of control strategies, see for example, Stare et al (2007). In order to optimize a wastewater treatment step, it is rather common to formulate a criterion and solve for the optimal plant layout or control strategy. See, for example, Rivas (2008), Alsina (2008) and Chachuat (2001). Often, it is impossible to obtain an analytical solution to the optimization problem and also detailed calibrated mathematical model of the process is needed. The effluent regulations for a WWTP are commonly based on flow proportional mean values. It is hence not required to consistently trying to keep the effluent discharge to a given value. This opens up for an interesting optimization problem, namely what sequence of control signals minimizes the resource consumption (for example, energy consumptions for aeration) and at the same time keeps the (daily) average of the effluent discharge equal to a certain threshold? In this paper we show that for a class of processes it is optimal to keep the input signal to the process constant. This gives a formal proof to the intuitive feeling that it would be possible to save resources by not controlling a process too tight. Practical implications of the results are discussed and a strategy for energy efficient aeration control in a nitrifying activated sludge process is outlined
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METHODS Statement of the problem Consider a unit process in, for example, a WWTP, depicted in Figure 1. The control signal (control handle) is denoted u(k), the output signal (typically an effluent discharge component) is y(k). The argument k is an integer representing discrete time samples. The disturbance to the plant d(k) may typically consist of changes in flow rates and influent concentrations.
Figure 1. A general sub process in a WWTP.
The classical way to control the output y(k) is to use feedback and/or feedforward control Numerous applications in WWTPs use this basic principle, for example to control dissolved oxygen concentration. Often, the set point in such a control strategy is set to the required effluent standard even though the effluent requirement may be based on a flow proportional daily (weekly/monthly/yearly) average. See Ayesa et al (2006) for an example of when averages are considered. The problem we want to solve is as follows: What sequence of control signals minimizes the resource consumption and at the same time keeps the (daily) average of the effluent discharge equal to a certain threshold? We believe that an answer to this question is of great interest in order to run a WWTP plant as efficient as possible. Main result Consider the following plant model y (k ) f (u (k )) d (k )
(1)
where f is a static nonlinear function which is assumed to be (i) smooth, (ii) has a strictly negative derivative, and (iii) has a second derivative which is strictly positive. Assumption (iii) means that the function is strictly convex and that the (relative) efficiency decreases when u is increased. This is a common property in many unit processes in, for example, biological wastewater treatment where the growth rate of microorganisms (and treatment results) is described by Monod functions. Next consider the following optimisation problem for the plant (1): Minimize
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c u (k )
(2)
k 1
subject to the constraint
g
1 N
N
y (k )
(3)
k 1
The criterion (2) is of interest to minimize when the magnitude of the control signal is proportional to consumption of resources. A very common example is that u is proportional to energy consumption. The constraint (3) means that the mean value of the output should be equal to a given threshold α. In practice we may instead use a flow proportional mean value. In practice, the constraint is g ≤ α but it is clear that (2) is minimized if we use an equality constraint. The output signal is typical an effluent concentration (possible flow weighted) and the choice of α is then a mean value which should be fulfilled. The time window N and α may be obtained from legal regulations. However, it can be both easier to tune the control signal (see below) and increase the robustness by choosing N lower than the official regulation. A natural choice is to let N corresponds to 24 hours. The solution to the optimization problem is presented in the following theorem: Theorem 1: Consider the optimisation problem (2)-(3) for the plant (1) under the assumptions (i)-(iii). The optimal control input sequence is then given by u(k)= u*, k=1,2... N
(4)
where u* is a constant. Proof of Theorem 1: See Carlsson (2010) The important conclusion from Theorem 1 is that it is shown that it is in fact optimal in the sense of (2)-(3) to let the input u(k) to the process be constant. Some practical considerations are given in the next section. RESULTS AND DISCUSSION Comments In practice, the optimal value u* of the control signal is seldom known a priori. Often, however, the disturbances to the plant have a periodic pattern. It is then possible to estimate u* by a recursive algorithm. A simple strategy is uo(n)=uo(n-1)+K ( (n-1) - α),
n=1,2…
(5)
where (n-1) is the mean value obtained during the previous evaluation time, say the last _______________________________________________________________________________ Watermatex 2011: Conference proceedings 742
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day, when the control signal was chosen as uo(n-1). The new control signal (to be applied during the next N samples) is denoted uo(n). The gain K is a user choice which determines the tracking speed versus noise sensitivity.
In a practical implementation more constraints than (3) may occur. For example, there may be a limit ymax on the maximum value of y(k) so that y(k)
