Mixed integer optimal control of an intermittently aerated sequencing batch reactor for wastewater treatment
Introduction
Finding optimal aeration profiles, specifically an optimal intermittent aeration profile (OIAP) is a challenging task in waste water treatment (WWT) plants (Spagni and Marsili-Libelli, 2009). A repeated change of aerobic to anoxic conditions and back allows the consumption of biodegradable matter and ammonia while redrate accumulation. This method with a high number of aerobic–anoxic phases, known as bypass nitrification (Turk and Mavinic, 1986), has shown to offer a number of advantages over the standard nitrification–denitrification process. Still, finding the optimal number of both intervals, aerobic/anoxic, as well as their respective lengths represents a complex optimization problem involving integer variables, highly nonlinear dynamics and a tightly constrained operating range. Yet, the effort is justified. On the one hand, OIAP reduces the operation time and the chemical oxygen demand (COD) input by achieving maximal efficiency of bacterial growth and substrate consumption. On the other hand, OIAP economizes energy input but also energy consumption through reduction of the aeration feed volume (Sinha and Annachhatre, 2007).
The semi-continuous sequencing batch reactor (SBR) technology, developed in the seventies (Irvine and Davis, 1971), is widely used given its small footprint, its capacity of treating reduced load charges, variable influent quality and its high flexibility (Mahvi, 2008). The basic SBR system consists of a batch basin where five phases take place: idle, fill, react, settle, draw (Mahvi, 2008). Aerobic phases are defined by simply turning aeration on and off during the fill and the reaction phases. Additionally, operation conditions may be changed from cycle to cycle, which then also requires an adaptation of the OIAP (Cruz Bournazou et al., 2013).
Moreover, as the volume of SBRs is significantly smaller compared to continuous plants, the sensitivity of the process to changes in the quality of the water is significantly higher.
Bypass nitrification (BN) enables denitrification directly from nitrite skipping nitrate accumulation (Sinha and Annachhatre, 2007, Yoo et al., 1999, Chung et al., 2006, Peng and Zhu, 2006). BN also offers a number of advantages including a reduction of the oxygen demand for nitrification by up to 25% and an increase of the denitrification rate by 63% and CO2 emissions by 20% (Turk and Mavinic, 1986, Peng and Zhu, 2006, Turk and Mavinic, 1987, Abeling and Seyfried, 1992).
The use of an intermittent aeration profile (IAP) has been shown to be an effective method to accomplish bypass nitrification (Spagni and Marsili-Libelli, 2009, Yoo et al., 1999, Peng and Zhu, 2006, Katsogiannis et al., 2003, Akın and Ugurlu, 2005, Yang et al., 2007, Yongzhen et al., 2007).
Fig. 1 illustrates the basic concept of the SBR technology with BN. The intermittent aeration leads to an alternation of aerobic and anoxic phases with high frequency. The OIAP defines the combination of parameters (number of aeration intervals, length of each interval) that minimizes time, energy input and COD while maximizing removal nitrogen compounds.
The SBR is a dynamic process involving binary switches in the aeration control, which define aerobic and anoxic phases. The length of each phase is defined by a variable time grid. There is a number of publications dealing with the optimization of the IAP (Cruz Bournazou et al., 2013, Coelho et al., 2000, Chachuat et al., 2005, Fikar et al., 2005, Holenda et al., 2007, Kim et al., 2008, Souza et al., 2008, Balku et al., 2009, Chai and Lie, 2008) and also some based on experimental evidence (Sin et al., 2004). However, none of them considered integer and continuous variables simultaneously. Instead, the adopted solution approach is based on a simplification computing the specific aeration interval length (continuous problem) for a given number of intervals (integer problem). Unfortunately, this independent computation of the interval number and interval length leads to suboptimal solutions (Cruz Bournazou et al., 2013).
In mixed integer nonlinear programming (MINLP) both integer and continuous variables are combined in a nonlinear system such as mechanistic physico-chemical or biological multi-variable models. For the here studied optimal control problems, the models are described by a set of differential equations. The incorporation of a control system which can only adopt fixed values, such as this case, turns the problem into a mixed integer optimal control (MINTOC) problem (Logist et al., 2010).
An algorithm for the solution of MINTOC problems was presented by S. Sager and applied a number of times (Sager, 2005, Sager, 2009). Hereby MINTOC found some popularity among Volterra type problems (Sager, 2005), various vehicle gear control systems (Sager, 2005, Logist et al., 2010), as well as in circadian cycle in-silico simulation (Shaik and Goerecki, 2008) and in engineering or simple medical substance tracking applications (Sager, 2005, Slaby et al., 2007).
Many models for the description of the activated sludge process (ASP) have been published in the last three decades (Hauduc et al., 2013). Although the ASM1 is still the most referred model, it lacks the description of the nitrification–denitrification process in two steps and is therefore unsuitable for simulations of the BN. Instead, models which describe the two step reaction of the nitrification–denitrification process are needed. Therefore, the extended version of the ASM3 has been selected in this work (Kaelin et al., 2009). The extended ASM3, includes seven new differential equations, with a stoichiometric matrix of dimension 15 × 20 (15 differential equations and 20 reaction rates). This version has shown some complication when applied for control purposes. Firstly, the identification of the model parameters is not possible with the information obtained from online measurements. Secondly, it has proved to be very stiff in BN conditions (Cruz Bournazou et al., 2014).
However, the model describes many states that we can in fact neglect for the specific case of SBR processes (Balku et al., 2009, Velmurugan et al., 2010). Thus, a reduced nonlinear version of the extended ASM3, namely the 5-State model, has been developed (Cruz Bournazou et al., 2012). Its capacity to mimic the behavior of the extended ASM3 in complex optimizations, when the process is restricted to the SBR conditions, has been demonstrated (Cruz Bournazou et al., 2013).
Section snippets
Methods
We consider the optimal operation of one batch sequence (one cycle) for a given loading, e.g. wastewater quality. To achieve this, an OIAP must be computed such that: (1) the total batch time is minimized, (2) the effluent (characterized at the end of the batch) complies with environmental regulations, (3) the required aeration (energy input) is minimized, while (4) the switch between anoxic-aerobic phases achieves BN. This requires the solution of a MINTOC problem. There are two factors which
Computation of the optimal intermittent aeration profile
Computation of the OIAP presents a mayor challenge with respect to feasibility of the state variables at the end of the batch time. The constraints set by the process specifications are nonlinear and very sensitive to the quality of the load. To eliminate organic matter as well as ammonia while maintaining low nitrogen oxide concentrations in the reactor, a delicate equilibrium between total aerobic time and aerobic and anoxic intervals is required.
Moreover, the optimal solution is defined by a
Results
In this section, the solution of the MINTOC problem using IPH, see Fig. 4, is presented along with information on the performance of the applied algorithm. Moreover, it is also shown that from the analysis of the computed solution a simplification of the original problem is derived (reduction of the number of optimization variables), which yields very similar results but can be solved at a significantly reduced computing time.
Discussion
For the original and reduced problem, better results were obtained for both, the final and the aeration time compared to previous publications, see Table 4.
Compared to the solution of the original problem, the solution of the reduced problem shows an slightly increased objective function value. However, the computation time on an Intel Pentium CPU [email protected] GHz computer using Matlab 8/Windows 32 bit for the solution of the original and reduced MINTOC problem using IPH was 1 h 32 min and 23 min,
Conclusions
Optimal intermittent aeration profiles (OIAPs) have been computed for SBRs under bypass nitrification using a MINTOC algorithm. The optimal aeration strategies are defined by the number of switches and length of each aerobic and anoxic interval. It has been shown that the efficiency of the SBR process can be improved compared to previously published works in both, the aeration costs (oxygen used) and the total batch time while complying with effluent regulations.
The optimal operation is defined
Variable index
Variable Gn Grid number tk min Time point u Control variable X State variables β Penalty term γ Subdivision coefficient Δt, h min Interval length θ Parameter matrix λ weighting factor τ min Interval time points Φ min Objective Function ω Control function min Integer objective function min Objective function PH ɛSTO,ɛROU Tolerance of rounding and STO Indexes n Grid number k Interval number i Generic component j Reaction number Glossary COD Chemical oxygen demand N Atomic nitrogen composition NL Non linear P Atomic phosphorous composition PH
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