Cost-Efficient Operation of a Denitrifying Activated Sludge ... - CiteSeerX
Cost-Efficient Operation of a Denitrifying Activated Sludge Process – An Initial Study
April 8, 2005
P¨ar Samuelsson, Bj¨orn Halvarsson and Bengt Carlsson Division of Systems and Control, Department of Information Technology, Uppsala University, PO Box 337, SE-751 05 Uppsala, Sweden E-mail: [email protected], [email protected], [email protected]
Abstract In this paper, possible choices of optimal set-points and cost minimizing control strategies for the denitrification process in an activated sludge process are discussed. In order to compare different criterion functions, simulations utilizing the European COST benchmark are considered. By means of operational maps the results are visualized. It is found that there is a clear set-point area where the process can be said to be efficiently controlled in an economic sense. For most reasonable operating points this optimal area corresponds to a nitrate concentration in the anoxic compartment in the interval 1–3 mg(N)/l. Furthermore, the location of this optimum does not seem to be very sensitive to changes in the ASM1 parameters. With an appropriate nitrate cost function, the legislatory authorities can place this economic optimum in an area where also the effluent regulations are met. It is also discussed how this efficient control can be accomplished.
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Introduction
In recent years, cost minimization has become increasingly important in the control and operation of wastewater treatment plants. In order to run a plant economically, operational costs such as pumping energy, aeration energy and dosage of different chemicals should be minimized. At the same time, the discharges to the recipient should be kept at a low level. Of course, minimizing the operational costs and at the same time treat the wastewater properly may lead to a conflict of interest that must somehow be solved. The main problem is how to keep the effluent discharges below a certain pre-specified limit to the lowest possible cost, see Olsson and Newell (1999). Part of the answer is to design the control algorithms in such a way that the overall operational costs are minimized. This goal can be strived to attain in different ways. As an example, the controller set-points could be separately optimized or the cost could be minimized online by some control strategy, for instance model predictive control (MPC). See Qin and Badgwell (2003) for how MPC can be used in this context. See also Halvarsson (2005) where an overview of the MPC literature is given. In some countries, the authorities charge according to effluent pollution. A possible way to formulate the on-line minimization criterion in such a case is to use a cost function that takes actual costs (energy and chemicals) into account and at the same time economically penalizes the effluent discharges. Over the years, much effort has been put in developing economically efficient control strategies for operation of wastewater treatment plants. An interesting cost function is presented in Carstensen (1994) where the effluent nitrogen is penalized using a piecewise linear discontinuous function. Optimization of nitrate set-points is considered in Yuan et al. (1997) where it is proposed that the nitrate concentration in the anoxic zone of a predenitrifying wastewater treatment plant should be controlled at a low setpoint, with an optimal set-point of about 1 mg(N)/l. In Yuan et al. (2002), the internal recirculation flow rate is used in order to control the denitrification process minimizing a linear effluent discharge criterion. The optimal set-point for the nitrate concentration at the outlet of the anoxic zone is then found to be near 2 mg(N)/l or, at least, in the interval 1–3 mg(N)/l. These ideas are further discussed in Yuan and Keller (2004) where optimal control of the denitrification process is the topic. Also in Ingildsen et al. (2002), an optimization of the dissolved oxygen (DO) and nitrate set-points is made. In Galarza et al. (2001) steady-state operational maps are utilized to examine the feasible operating area for two activated sludge processes with emphasis on sensitivity analysis. Fuzzy control evaluated using multicriteria cost functions is the subject of Cadet et al. (2004). In Vanrolleghem and Gillot (2002), different multi-criteria control strategies are evaluated.
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In this paper, the choice of optimal set-points and cost minimizing control strategies for the denitrification process in an activated sludge process configured for pre-denitrification are evaluated. The manipulated variables (input signals) are the internal recirculation flow rate and the flow rate of an external carbon source and the controlled variables (output signals) are the nitrate concentrations in the last anoxic compartment and the effluent. In order to compare the impact of different criterion functions, stationary simulations utilizing the European COST benchmark (see Copp (2002)) are considered. By means of operational maps the results are visualized. It is found that there is a clear set-point area where the process can be said to be efficiently controlled in an economic sense. It is also discussed how this efficient control can be accomplished. Note that we only study the denitrification process here and hence only consider nitre discharges. A natural extension is to also consider the nitrification process and then consider total nitrogen discharges. The organization of the paper is as follows: In Section 2, the simulation model (the COST benchmark) is briefly described together with the associated operational costs. In Section 3 simulation results are presented using operational maps. The simulation results are discussed and interpreted in Section 4. Finally, in Section 5 the general conclusions are drawn.
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The Model and the Operational Cost Functions
In the simulation study presented in this paper, the COST benchmark model is used, see Jeppson and Pons (2004) for a general survey and Copp (2002) for a more technical description. In the benchmark model, five biological reactors are implemented using the IAWQ activated sludge model No. 1 (ASM1), see Henze et al. (1987). The model plant is pre-denitrifying with two anoxic and three aerated compartments. A secondary settler is also implemented. See Figure 1 for a schematic overview of the benchmark plant.
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Figure 1: Schematic representation of the simulation benchmark bioreactor configuration.
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The aim here is to analyze the stationary operational costs of the denitrification process, and in order to visualize the costs, these are presented in stationary operational maps together with the considered output signals. The output signals are the nitrate concentration in the last anoxic compartan [mg(N)/l], and the nitrate concentration in the effluent, S e ment, SN O NO [mg(N)/l]. The available control handles considered in this paper are the internal recirculation flow rate Qi [m3 /day] and the flow rate of an external carbon source, Qcar [m3 /day]. The carbon source is assumed to be ethanol with a COD of 1.2·106 mg(COD)/l. To express the cost for controlling the denitrification process, a number of partial costs were taken into account: • Pumping costs due to the required pumping energy; • Aeration costs due to the required aeration energy; • External carbon dosage costs; • A possible fee for effluent nitrate discharge. In the benchmark, the total average pumping energy over a certain period of time, T , depends directly on the internal recirculation flow rate Qi and is according to Copp (2002) calculated as 0.04 EP = T
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(1)
expressed in units of kWh/day. In (1), Qr denotes the return sludge flow rate and Qw the excess sludge flow rate, both in units of m3 /day. The average energy in kWh/day required to aerate the last three compartments can in turn be written as EA =
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(2)
where KL ai (t) is the oxygen transfer function in the aerated tank number i in units of h−1 . Further, assuming a prize kcar [EUR/m3 ] on the external carbon source and that an external carbon flow rate, Qcar (t) [m3 /day], is fed into the process during the time interval T , the cost per day of the external carbon flow rate is of course Z 1 t0 +T kcar Qcar (t)dt (3) Ccar = T t0 expressed in EUR/day. Finally, a reasonable way to describe a fee for the effluent nitrate discharge is to let the fee depend on how large mass of nitrate 4
that is discharged per time unit. This depends, of course, on the effluent e (t). Such flow rate Qe (t) and the nitrate concentration in the effluent, SN O a fee can in other words be expressed as (in EUR/day) CN O
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Now, assuming a constant energy prize, kE , the total average cost expressed (in EUR/day) can be calculated during a representative time interval T from (1)-(4) as Ctot = kE (EP + EA ) + Ccar + CN O
(5)
The fee functions for the discharge of nitrate can be chosen1 in several different ways. Here, three different choices are evaluated: 1. No charge is added for the disposal of nitrate and ammonium, i.e. e (t)) ≡ 0. fN O (Qe (t), SN O 2. Effluent nitrate is charged with a constant cost, say ∆α per discharged kg. Such a cost function is given in (6) as e e fN O (Qe (t), SN O (t)) = ∆αSN O (t)Qe (t)
(6)
3. Effluent nitrate is charged with a constant cost per discharged kg, ∆α, up to a legislatory discharge limit for the effluent concentration, αlimit . Above this limit the cost of discharging additional nitrate is ∆β. Exceeding the limit also imposes an additional charge of β0 per volume effluent water. Figure 2 provides an illustration of the cost function and a mathematical description is given in (7). In principle, this cost function is the one proposed by Carstensen (1994). The difference between the cost functions presented in Carstensen (1994) and in this paper is that here only the nitrate discharge is penalized, while in Carstensen (1994) the total inorganic nitrogen is charged.
e e ∆αSN O (t)Qe (t) if SN O (t) ≤ αlimit e e (t)− fN O (Qe (t), SN O (t)) = ∆ααlimit Qe (t) + β0 Qe (t) + ∆β(SN O e −αlimit Qe (t)) if SN O (t) > αlimit 1
Normally, this choice is made by legislatory authorities.
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(7)
Cost per volume
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0
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Figure 2: Cost function for the discharge of nitrogen as proposed by Carstensen (1994). Below the legislatory limit the slope equals ∆α and above it is ∆β.
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Simulation results
The considered cost functions for penalizing nitrogen discharge are evaluated utilizing the COST benchmark. The simulation study begins with a set of simulations where the benchmark WWTP is fed with constant influent flow rate of wastewater, 18 446 m3 /day, giving results for the steady state case. For the last two sets of simulations a decentralised control law where an and Q e Qi controls SN car controls SN O is employed in the control of the O an , and the aerobic effluent nitrate concentration from the anoxic zone, SN O e zone, SN O , in the stationary case, see Figure 10. In the last simulation set the scenario is more realistic since the benchmark WWTP is fed with timevarying influent, see Figures 11–14.
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Simulations with constant influent flow rate
The influent wastewater in the stationary case is assumed to have a concentration of readily biodegradable substrate, SSi , of 60 mg(COD)/l. All of the other parameters adopt the standard values used in the benchmark; these values can be found in for instance Copp (2002) or on the COST homepage (COST June 29, 2004). See Table 1 for other parameter values. 6
Table 1: Selected parameter values used in the simulation studies. All costs are recalculated from swedish currency unit SEK, hence, all the decimals. Here, 1 SEK = 9.11 EUR which implies, e.g., ∆α = 25 SEK/kg NO. Parameter αlimit β0 ∆α ∆β kE kcar
Value 8.0 1.3721 2.7442 8.2327 0.03622 548.85
Unit mg(N)/l EUR/1000 m3 EUR/kg N O EUR/kg N O EUR/kWh EUR/m3
The benchmark has been run for 150 simulation days for a grid of different, constant, input values. Only the last 100 simulation days were considered when evaluating the cost functions to avoid the influence of transients. In Figures 3–9 the benchmark has been run for a grid of different setpoints for the external carbon dosage, Qcar , and the internal recirculation flow rate, Qi . Qcar has been varied between 0 and 3.0 m3 /day in steps of 0.1 m3 /day and Qi between 0 and 100 000 m3 /day in steps of 2500 m3 /day. Consequently, the range of Qi corresponds to about 0–5.4 times the influent flow rate. Figure 3 shows stationary operational maps for different parts of the total cost. The energy cost is the sum of the aeration cost and the pumping cost. The lower right plot shows the total cost when no charge on nitrogen discharge is imposed. Clearly, the carbon dosage cost starts to dominate rather quickly when increasing Qcar . In Figure 4 the stationary operational map is shown for the total cost when no nitrate discharge fee is used. Also, the level curves for the nitrate an and S e , reconcentrations in the anoxic and aerobic compartments, SN O NO spectively, are plotted. The cost-optimal operating point is marked with a star, and, not surprisingly, it corresponds to performing no active nitrate control: Qcar = 0 and Qi = 2500 m3 /day (Qi = 0 results in a slightly higher energy cost since the aeration cost for Qi = 0 is higher than for Qi = 2500 m3 /day, see Figure 3.). On the other hand, if the intention is to not exceed e a certain value of SN O it is easily seen from the map that, for values of e an from around 1–3 mg(N)/l SN O between about 5–15 mg(N)/l, a level of SN O gives the cost-optimal operating points. However, if a lower effluent nitrate e , is desired, S an needs to be controlled at lower values. concentration, SN NO O In fact, as can be seen in Figure 5 where the maximum Qi is four times as large as in the other operational maps (400 000 m3 /day corresponding to 7
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Figure 3: Stationary operational maps depicting different costs for a grid of different values of Qcar and Qi . an at the cost optimal about 21.7 times the influent flow rate), the value of SN O e e point decreases as SN decreases: when S = 14 mg(N)/l the cost optimal O NO an near 3 mg(N)/l, and for S e an point corresponds to SN = 2 mg(N)/l, SN O NO O near 0.3 mg(N)/l gives the cost optimal operating point.
The above discussed cost optimal operating area can be seen as the set of Pareto optimal solutions to the formulated multiobjective optimization problem weighting performance versus process economy. There are several approaches to multiobjective optimization, see the discussion in Section 4. Here, all of the objectives are expressed in one single objective function, the total cost in (5), that is to be minimized in the optimization process. Next, a constant cost per discharged kg effluent nitrate is introduced according to (6). Figure 6 shows this case with a fee of 2.7442 EUR/kg effluent nitrate, i.e. ∆α = 2.7442 EUR/kg. Apparently, the level curves for the total cost change appearance compared to the previous case, particularly for low values of Qcar . This makes it less desirable to select a set-point for an considerable outside the range 1–3 mg(N)/l – a set-point for S an in SN O NO e this interval still gives cost efficient operation for SN O levels between about
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Figure 4: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost without nitrate-charge, dash-dotted e , and dotted lines show the lines show the effluent nitrate concentration, SN O an . The star indicates nitrate concentration from the anoxic compartment, SN O the minimum-cost point. an 5–15 mg(N)/l. Consequently, the importance of the set-point choice for SN O has become larger. The overall cost-optimal operating point, marked with a star, is now located at Qcar = 0 and Qi = 20000 m3 /day. This operating an = 2.3 mg(N)/l and S e point gives SN O N O = 14.2 mg(N)/l.
In Figure 7 the nitrate fee is twice as high, i.e. ∆α = 5.4885 EUR/kg. This moves the cost-optimal operating point to Qcar = 0.7 m3 /day and an = 1.7 mg(N)/l Qi = 52 500 m3 /day with a cost of 1620 EUR/day and SN O e and SN O = 8.8 mg(N)/l and creates a well-defined operational region corresponding to cost efficiency. As before, to keep the total cost low it is an adopt a value in the range between a bit less than 1 advisory to let SN O mg(N)/l and 3 mg(N)/l. This will result in operation with a total cost less than 1650 EUR/day. If, instead, a cost of 1700 EUR/day is accepted the region corresponding to cost efficient operation is much larger, see Figure 7.
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Figure 5: Stationary operational map with the same settings as in Figure 4 except that here the range of Qi is four times as large. By adjusting the nitrate fee in this way, the legislatory authorities are able to move the cost-optimal set-point. However, since there is no direct e , it may be relation between ∆α and the effluent nitrate concentration, SN O hard to choose a proper value of ∆α when the aim is to place the optimal operating point – or operating region – at (or below) a certain effluent nitrate concentration. The third suggested nitrate cost function given by (7) and graphically in Figure 2 has the advantage of changing appearance at a certain predefined concentration of effluent nitrate, αlimit , where a discontinuity is located. If this discontinuity is sufficiently large it is easy for the authorities to place the optimal operating point at a certain predefined effluent nitrate concentration. Figure 8 shows the stationary operational map when using this cost function with parameter values given by Table 1. The optimum point is located at Qcar = 0.9 m3 /day and Qi = 57 500 m3 /day with a total cost an = 1.1 mg(N)/l and S e of 1248 EUR/day, SN O N O is, not surprisingly, near e 8 mg(N)/l: SN O = 7.7 mg(N)/l. (In fact, with a higher resolution it is e = 8 mg(N)/l. easily verified that the optimum point is placed where SN O 10
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Figure 6: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a constant nitrate-charge per kg effluent nitrate, dash-dotted lines show the effluent nitrate concentrae , and dotted lines show the nitrate concentration from the anoxic tion, SN O an . The star indicates the minimum-cost point. Here compartment, SN O ∆α = 2.7442 EUR/kg. Also, a higher resolution reveals that the optimum point rather is located near Qcar = 0.8250 m3 /day and Qi = 63 000 m3 /day with a total cost of an = 1.7 mg(N)/l.) In practice, when treating non1227 EUR/day, and SN O constant influent, it is of course advisory to chose an operating point that is located slightly below the legislatory limit. The region of economic efficient operation, say operation with a total cost below 1300 EUR/day, is relatively an ranging from somewhat over 0.5 mg(N)/l up to near large with values of SN O e 3 mg(N)/l and values of SN O between a bit below 7 mg(N)/l and 9 mg(N)/l. A way to get a sharper limit at αlimit is to enlarge β0 . In Figure 9 β0 equals 5.4885 EUR/1000 m3 effluent water, i.e. four times as large as the e =α previous setting. As seen in the operational map, the limit at SN limit = O 8 mg(N)/l gets a bit sharper and the region where the total cost is below 1300 EUR/day shrinks. Now, most of the operating points in this area gives 11
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Figure 7: Stationary operational map with the same settings as in Figure 6 except that here ∆α = 5.4885 EUR/kg. e SN O below the limit, αlimit = 8 mg(N)/l. The optimal operating point is, of course, the same as with the previous settings.
Furthermore, two PI-controllers were used to control the effluent nitrate an , and the aerobic zone, S e . A concentration from the anoxic zone, SN O NO decentralized (diagonal) regulator structure is used where Qi is employed an and Q e to control SN car is employed to control SN O . Consequently, the O decentralized control can be written as Qi (s) = F1 (s)E1 (s)
(8)
Qcar (s) = F2 (s)E2 (s)
(9)
where E1 (s) is the Laplace transformed control error of the first loop, i.e. an,sp an e1 (t) = SN O (t) − SN O (t), and E2 (s) is the Laplace transformed control e (t) − S e,sp (t) since this process is error of the second loop, i.e. e2 (t) = SN O NO known to have negative gain. These controllers were tuned to give approximately the same rise and settling time and were F1 (s) = 1300 + 12
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Figure 8: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a nitrate-charge according e , and to (7), dash-dotted lines show the effluent nitrate concentration, SN O dotted lines show the nitrate concentration in the last anoxic compartment, an . The additional charge for exceeding the legislatory discharge limit at SN O αlimit = 8.0 mg(N)/l is here β0 = 1.3721 EUR/1000 m3 . The star indicates the minimum-cost point. 0.05 (11) 0.13s Additionally, the anti-windup configuration provided in the benchmark implementation was used. The resulting operational map is shown in Figure 10 where the stars indicate where the cost-optimal point is located for each e . Obviously these optimum points are arranged in a regular level of SN O way. This relation also seems to be valid when varying the ASM1 parameter values. The heterotrophic maximum specific growth rate, µH has been varied between 2.0 day−1 and 6.0 day−1 . It was found that there were only minor differences between the operational maps, and in fact, most of the optimum set-points coincide. The case for µH = 4.0 day−1 is shown in Figure 10. Several other parameter values have also been modified yielding similar results. F2 (s) = 0.05 +
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15
3.2
Simulations with time-varying influent flow rate
The COST benchmark is now evaluated for three different time-varying influent compositions representing dry weather, a storm event, and finally, a rain event. The data files are those supplied for the COST Benchmark plant. For each set of simulations the benchmark plant model has been run according to the description given in Copp (2002) for a grid of different setan and S e , i.e. with constant influent for 150 days followed points for SN O NO by a 14 day simulation set with dryinfluent to generate initial states for the final 14 day simulation with either dryinfluent again or with one of the other two influent files (i.e. storm- or raininfluent). Only the last seven simulation days are used in the evaluation. The results are visualized in operational maps of the total cost in EUR when effluent nitrate is penalized according to the cost function (7) proposed by Carstensen (1994), see Figures 11, 13 and 14. The used parameter values are those listed in Table 1. The heterotrophic maximum specific growth rate (µH ) is selected to 4.0 day−1 . Furthermore, a plot of the quadratic control P e,sp e , i.e. 2 e error for SN O t (SN O (t) − SN O (t)) , is supplied in Figure 12 for the case of dryinfluent. For all of the three influent compositions, there is a clear set-point area where the total cost is lower compared to the surrounding set-point e choices. Clearly, a set-point for SN O in the interval 1–2 mg(N)/l gives an cost efficient operation when SN O set-point is chosen to αlimit , i.e. in this case, 8.0 mg(N)/l. This recommendation is in accordance with the steadystate results discussed in the previous section. Furthermore, the plot of the e quadratic error for SN O in Figure 12 also suggests that roughly the same e interval, 1–2 mg(N)/l, is optimal for the SN O set-point for dryinfluent. For the other two influent compositions, raininfluent and storminfluent, the corresponding quadratic error plots look somewhat different suggesting a bit e ; these plots are omitted here. lower set-point values for SN O
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Figure 11: Operational map depicting total cost for a grid of different sete an points of SN O and SN O when using PI controllers and dryinfluent. The e stars indicate for each level of SN O the minimum-cost point.
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Figure 13: Operational map depicting total cost for a grid of different sete an points of SN O and SN O when using PI controllers and raininfluent. The e stars indicate for each level of SN O the minimum-cost point.
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Figure 14: Operational map depicting total cost for a grid of different sete an points of SN O and SN O when using PI controllers and storminfluent. The e stars indicate for each level of SN O the minimum-cost point.
20
4
Discussion
Considering the case when no charge on effluent nitrate is imposed, from Figure 4 it is clear that the cost for dosing an external carbon source dominates the total cost. It is also clear that the operational area with respect to effluent nitrate is divided in two parts with different gain characteristics, e , there is a cost-optimal and that for each desired value (set-point) of SN O point in the operational map corresponding to a certain value of the nitrate an . This point naturally concentration in the last anoxic compartment, SN O also corresponds to certain stationary values of the input signals, Qcar and Qi . As indicated in Section 3 the cost-optimal point corresponds to values of an in the interval 1–3 mg(N)/l for a range of values of S e SN O N O between about 5 mg(N)/l and 15 mg(N)/l. The location of this point is not very sensitive to variations in the ASM1 parameters. Similar conclusions are also drawn e by Yuan and Keller (2004). However, according to Section 3, if lower SN O an levels is desired the control strategy must impose SN O levels lower than 1 e mg(N)/l. Such low values of SN O may not be desirable for other reasons an may inhibit the denitrification since the corresponding low values of SN O process in a real plant, see Yuan et al. (2002). By including a nitrogen disposal charge in the total cost function the resulting problem becomes a multiobjective optimization problem involving process economy as well as performance. There are several possible approaches to multiobjective optimization. It is common to include all objectives in one single objective function that is minimized, as is outlined here in this paper. However, it is difficult to formulate a single objective function that fairly reflects all of the underlying objectives. Another multiobjective optimization approach is to minimize the objective functions in order of importance, see for instance Kerrigan and Maciejowski (2002), Fleming (1999), Elia and Dahleh (1997), Aggelogiannaki et al. (2004) and Maciejowski (2002). All of the above discussed multiobjective optimization approaches are a priori methods. This means that the judgement and preferences of the decision maker is specified in advance, before the multiobjective problem is solved. On the contrary, with an a posteriori approach a set of Pareto optimal points is generated and then the decision maker selects the most preferred solution from this set. These and other approaches are surveyed by Shibayev (2004). From Figure 4, some questions may rise. The first is which control structures that can be expected to give a good performance in terms of disturbance rejection and set-point tracking. In different operating points different control structure selections may be suitable. As indicated by e.g. Figure 4, in the area of the operational map corresponding to cost-efficient nitrate e an control, SN O is mostly affected by the input signal Qcar , while SN O is af21
fected by both input signals in this area. This situation in particular is further discussed by Samuelsson et al. (2005) and also to some extent by Yuan and Keller (2004). The second question is how to design the actual control law in order to minimize operational costs. Below, some possibilities for this control design are discussed: • Use two different control loops (for instance PI-controllers) to control e an SN O and SN O separately. Given the desired nitrate effluent concentrae , the optimal set-point of S an can easily be found from Figtion, SN O NO e ure 4. For instance, if the set-point for SN O is chosen as 8 mg(N)/l, the an optimal set-point for SN O is close to 1.5 mg(N)/l. If decentralized cone an trol is to be used, Qcar should control SN O and Qi SN O . Other control structures may, however, yield a better performance, see Samuelsson et al. (2005). • Another possibility is to use a constant high internal recirculation flow rate Qi and to use only Qcar in order to control the nitrate effluent concentration. Since Qi has a much smaller impact on the total cost than Qcar , this would render a close to cost-optimal operation. This possibility has also been mentioned by Ingildsen (2002) • To achieve a cost-optimal performance, the total cost could be minimized on-line using quadratic criteria yielding for example LQG or MPC controllers. Such a criterion could be of the form Z T V = eT (t)Q1 e(t) + uT (t)Q2 u(t)dt (12) 0
where e is a column vector containing the control errors and u the input signals. The weighting matrix Q2 can be chosen to reflect the costs for the different input signals and Q1 can be seen as a performance weight. The difficulty with this criterion is how to weight control performance against cost minimization, i.e. how to choose the matrix Q1 . From the prior knowledge obtained from Figure 4, the elements of the matrix Q2 could be chosen in an ad hoc manner. Since the external carbon source is much more expensive than the pumping of the internal recirculation flow rate, a rule of thumb could be to choose Q2 as a diagonal matrix with the element corresponding to Qcar significantly larger than the element corresponding to Qi . Such a choice clearly penalizes a large value of Qcar in the criterion. • A simple grid search could be performed on-line until the optimal point is reached. This method is simple and has the advantage that no operational map and thereby no model is required. One such optimization algorithm is presented by Ayesa et al. (1998). This algorithm is employed to minimize a global penalty function combining effluent requirements and costs. 22
For the case when the nitrate discharge is penalized with a constant charge per kg, see (6), it is seen from Figures 6 and 7 that this creates a minimum in the total cost function (5). For some corresponding stationary e , this minimum is of course located where the operational costs value of SN O are as low as possible. The main drawback using such a cost function for automatic control is that it is hard to relate the location of the minimum to the nitrate fee, ∆α, and thereby hard to say which set-point a certain fee results in. This problem is overcome if instead the fee function is chosen according to Carstensen (1994). The location of the discontinuity of the fee function (7) (see also Figure 2) immediately determines the optimal sete . Using this fee in the total cost is a point for the effluent nitrate, SN O convenient way to achieve cost optimality for a certain set-point of effluent e , see Figures 8–9, 11 and 13–14. Minimizing this total cost nitrate, SN O function on-line using some automatic control strategy would be a good way to impose the importance of good performance via penalizing the effluent discharge into the control design. If the discharge of nitrogen over a certain legislatory limit is directly associated with a higher fee, this could clearly motivate the use of more advanced control strategies. The impact of the fee in the control design is also easy to understand even for people with a limited knowledge in automatic control, compared to the related matter of choosing a performance weight in some quadratic criterion.
5
Conclusions
In this paper a bioreactor model describing a pre-denitrifying wastewater treatment plant is studied from a process economic point of view. The impact of different nitrate cost functions on the location of the cost-optimal operating point were examined. The main conclusion is that a nitrate concentration in the anoxic compartment in the interval 1–3 mg(N)/l gives cost-efficient operation for most reasonable operating points. Furthermore, the location of this optimum is not very sensitive to changes in the ASM1 parameters. This gives a clear range of operating points where the process can be said to be cost-efficiently controlled. With an appropriate nitrate cost function, the legislatory authorities can place this economic optimum in an area where also the effluent regulations are met. Such a cost function would therefore be suitable for model based automatic control.
Acknowledgements This work has been financially supported by the EC 6th Framework programme as a Specific Targeted Research or Innovation Project (HipCon, Contract number NMP2-CT-2003-505467). Please note that the European Community is not liable for any use that may be made of the information contained herein. 23
References Aggelogiannaki, E., H. Sarimveis and G. Bafas (2004). Multiobjective constrained MPC with simultaneous closed loop identification for mimo processes. In: European Symposium On Computer-Aided Process Engineering-14: 37th European Symposium of the Working Party on Computer-Aided Process Engineering (A. Barbosa-Povoa and H. Matos, Eds.). Ayesa, E., B. Goya, A. Larrea, L. Larrea and A. Rivas (1998). Selection of operational strategies in activated sludge processes based on optimization algorithms. Water Science and Technology 37(12), 327–344. Cadet, C., J. F. Beteau and S. Carlos Hernandez (2004). Multicriteria control strategy for cost/quality compromise in wastewater treatment plants. Control Engineering Practice 12(3), 335–347. Carstensen, J. (1994). Identification of Wastewater Processes. PhD thesis. Institute of Mathematical Modelling, Technical University of Denmark. Copp, J. B. (2002). EUR 19993 – COST Action 624 – The COST simulation benchmark – Description and simulator manual. European Communities. Luxembourg. Editorial. COST (June 29, 2004). Homepage for the COST simulation benchmark. http://www.ensic.inpl-nancy.fr/COSTWWTP/. Elia, N. and A. Dahleh (1997). Controller design with multiple objectives. IEEE Transactions on Automatic Control 42(5), 596–613. Fleming, P. (1999). Designing control systems with multiple objectives. In: IEE Colloquium on Advances in Control Technology. pp. 4/1–4/4. Galarza, A., E. Ayesa, M. T. Linaza, A. Rivas and A. Salterain (2001). Application of mathematical tools to improve the design and operation of activated sludge plants. Case study: The new WWTP of GalindoBilbao, part ii: Operational strategies and automatic controllers. Water Science and Technology 43(7), 167–174. Halvarsson, B. (2005). Model Predictive Control - An overview. HIPCON Report number HIP0 5-37-v1-R. Uppsala University. Henze, M., C. P. L. Grady Jr., W. Gujer, G. v. R. Marais and T. Matsuo (1987). Activated sludge model no. 1. Scientific and Technical Report No. 1. IAWPRC, London. Ingildsen, P. (2002). Realising Full-Scale Control in Wastewater Treatment Systems Using In Situ Nutrient Sensors. PhD thesis. Lund University, Sweden. 24
Ingildsen, P., G. Olsson and Z. Yuan (2002). A hedging point strategy – balancing effluent quality, economy and robustness in the control of wastewater treatment plants. Water Science and Technology 45(4–5), 317– 324. Jeppson, U. and M-N Pons (2004). The COST benchmark simulation model– current state and future perspective. Control Engineering Practice 12(3), 299–304. Editorial. Kerrigan, E.C. and J.M. Maciejowski (2002). Designing model predictive controllers with prioritised constraints and objectives. In: IEEE Conference on computer aided Control system design. pp. 33–38. Maciejowski, J.M. (2002). Predictive Control with Constraints. Prentice Hall. Olsson, G. and B. Newell (1999). Wastewater Treatment Systems. IWA Publishing. Qin, S. Joe and T. A. Badgwell (2003). A survey of industrial model predictive control technology. Control Engineering Practice 11(7), 733–764. Samuelsson, P., B. Halvarsson and B. Carlsson (2005). Interaction analysis and control structure selection in a wastewater treatment plant model. IEEE Transactions on Control Systems Technology. Accepted. Shibayev, S. (2004). Multicriteria optimisation: concepts and methods a survey. HIPCON report number LSE-WP7-1 v.1. London School of Economics. Vanrolleghem, P. A. and S. Gillot (2002). Robustness and economic measures as control benchmark performance criteria. Water Science and Technology 45(4–5), 117–126. Yuan, Z., A. Oehmen and P. Ingildsen (2002). Control of nitrate recirculation flow in predenitrification systems. Water Science and Technology 45(4– 5), 317–324. Yuan, Z. and J. Keller (2004). Achieving optimal conditions for nitrogen removal using on-line sensors and control. In: Proceedings of the 2nd IWA Leading-Edge Conference on Water and Wastewater Treatment Technologies. Prague, Czech Republic. Yuan, Z., H. Bogaert, P. Vanrolleghem, C. Thoeye, G. Vansteenkiste and W. Verstraete (1997). Control of external carbon addition to predenitrifying systems. Journal of Environmental Engineering 123(11), 1080– 1086.
25
April 8, 2005
P¨ar Samuelsson, Bj¨orn Halvarsson and Bengt Carlsson Division of Systems and Control, Department of Information Technology, Uppsala University, PO Box 337, SE-751 05 Uppsala, Sweden E-mail: [email protected], [email protected], [email protected]
Abstract In this paper, possible choices of optimal set-points and cost minimizing control strategies for the denitrification process in an activated sludge process are discussed. In order to compare different criterion functions, simulations utilizing the European COST benchmark are considered. By means of operational maps the results are visualized. It is found that there is a clear set-point area where the process can be said to be efficiently controlled in an economic sense. For most reasonable operating points this optimal area corresponds to a nitrate concentration in the anoxic compartment in the interval 1–3 mg(N)/l. Furthermore, the location of this optimum does not seem to be very sensitive to changes in the ASM1 parameters. With an appropriate nitrate cost function, the legislatory authorities can place this economic optimum in an area where also the effluent regulations are met. It is also discussed how this efficient control can be accomplished.
1
1
Introduction
In recent years, cost minimization has become increasingly important in the control and operation of wastewater treatment plants. In order to run a plant economically, operational costs such as pumping energy, aeration energy and dosage of different chemicals should be minimized. At the same time, the discharges to the recipient should be kept at a low level. Of course, minimizing the operational costs and at the same time treat the wastewater properly may lead to a conflict of interest that must somehow be solved. The main problem is how to keep the effluent discharges below a certain pre-specified limit to the lowest possible cost, see Olsson and Newell (1999). Part of the answer is to design the control algorithms in such a way that the overall operational costs are minimized. This goal can be strived to attain in different ways. As an example, the controller set-points could be separately optimized or the cost could be minimized online by some control strategy, for instance model predictive control (MPC). See Qin and Badgwell (2003) for how MPC can be used in this context. See also Halvarsson (2005) where an overview of the MPC literature is given. In some countries, the authorities charge according to effluent pollution. A possible way to formulate the on-line minimization criterion in such a case is to use a cost function that takes actual costs (energy and chemicals) into account and at the same time economically penalizes the effluent discharges. Over the years, much effort has been put in developing economically efficient control strategies for operation of wastewater treatment plants. An interesting cost function is presented in Carstensen (1994) where the effluent nitrogen is penalized using a piecewise linear discontinuous function. Optimization of nitrate set-points is considered in Yuan et al. (1997) where it is proposed that the nitrate concentration in the anoxic zone of a predenitrifying wastewater treatment plant should be controlled at a low setpoint, with an optimal set-point of about 1 mg(N)/l. In Yuan et al. (2002), the internal recirculation flow rate is used in order to control the denitrification process minimizing a linear effluent discharge criterion. The optimal set-point for the nitrate concentration at the outlet of the anoxic zone is then found to be near 2 mg(N)/l or, at least, in the interval 1–3 mg(N)/l. These ideas are further discussed in Yuan and Keller (2004) where optimal control of the denitrification process is the topic. Also in Ingildsen et al. (2002), an optimization of the dissolved oxygen (DO) and nitrate set-points is made. In Galarza et al. (2001) steady-state operational maps are utilized to examine the feasible operating area for two activated sludge processes with emphasis on sensitivity analysis. Fuzzy control evaluated using multicriteria cost functions is the subject of Cadet et al. (2004). In Vanrolleghem and Gillot (2002), different multi-criteria control strategies are evaluated.
2
In this paper, the choice of optimal set-points and cost minimizing control strategies for the denitrification process in an activated sludge process configured for pre-denitrification are evaluated. The manipulated variables (input signals) are the internal recirculation flow rate and the flow rate of an external carbon source and the controlled variables (output signals) are the nitrate concentrations in the last anoxic compartment and the effluent. In order to compare the impact of different criterion functions, stationary simulations utilizing the European COST benchmark (see Copp (2002)) are considered. By means of operational maps the results are visualized. It is found that there is a clear set-point area where the process can be said to be efficiently controlled in an economic sense. It is also discussed how this efficient control can be accomplished. Note that we only study the denitrification process here and hence only consider nitre discharges. A natural extension is to also consider the nitrification process and then consider total nitrogen discharges. The organization of the paper is as follows: In Section 2, the simulation model (the COST benchmark) is briefly described together with the associated operational costs. In Section 3 simulation results are presented using operational maps. The simulation results are discussed and interpreted in Section 4. Finally, in Section 5 the general conclusions are drawn.
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The Model and the Operational Cost Functions
In the simulation study presented in this paper, the COST benchmark model is used, see Jeppson and Pons (2004) for a general survey and Copp (2002) for a more technical description. In the benchmark model, five biological reactors are implemented using the IAWQ activated sludge model No. 1 (ASM1), see Henze et al. (1987). The model plant is pre-denitrifying with two anoxic and three aerated compartments. A secondary settler is also implemented. See Figure 1 for a schematic overview of the benchmark plant.
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The aim here is to analyze the stationary operational costs of the denitrification process, and in order to visualize the costs, these are presented in stationary operational maps together with the considered output signals. The output signals are the nitrate concentration in the last anoxic compartan [mg(N)/l], and the nitrate concentration in the effluent, S e ment, SN O NO [mg(N)/l]. The available control handles considered in this paper are the internal recirculation flow rate Qi [m3 /day] and the flow rate of an external carbon source, Qcar [m3 /day]. The carbon source is assumed to be ethanol with a COD of 1.2·106 mg(COD)/l. To express the cost for controlling the denitrification process, a number of partial costs were taken into account: • Pumping costs due to the required pumping energy; • Aeration costs due to the required aeration energy; • External carbon dosage costs; • A possible fee for effluent nitrate discharge. In the benchmark, the total average pumping energy over a certain period of time, T , depends directly on the internal recirculation flow rate Qi and is according to Copp (2002) calculated as 0.04 EP = T
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expressed in units of kWh/day. In (1), Qr denotes the return sludge flow rate and Qw the excess sludge flow rate, both in units of m3 /day. The average energy in kWh/day required to aerate the last three compartments can in turn be written as EA =
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where KL ai (t) is the oxygen transfer function in the aerated tank number i in units of h−1 . Further, assuming a prize kcar [EUR/m3 ] on the external carbon source and that an external carbon flow rate, Qcar (t) [m3 /day], is fed into the process during the time interval T , the cost per day of the external carbon flow rate is of course Z 1 t0 +T kcar Qcar (t)dt (3) Ccar = T t0 expressed in EUR/day. Finally, a reasonable way to describe a fee for the effluent nitrate discharge is to let the fee depend on how large mass of nitrate 4
that is discharged per time unit. This depends, of course, on the effluent e (t). Such flow rate Qe (t) and the nitrate concentration in the effluent, SN O a fee can in other words be expressed as (in EUR/day) CN O
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Now, assuming a constant energy prize, kE , the total average cost expressed (in EUR/day) can be calculated during a representative time interval T from (1)-(4) as Ctot = kE (EP + EA ) + Ccar + CN O
(5)
The fee functions for the discharge of nitrate can be chosen1 in several different ways. Here, three different choices are evaluated: 1. No charge is added for the disposal of nitrate and ammonium, i.e. e (t)) ≡ 0. fN O (Qe (t), SN O 2. Effluent nitrate is charged with a constant cost, say ∆α per discharged kg. Such a cost function is given in (6) as e e fN O (Qe (t), SN O (t)) = ∆αSN O (t)Qe (t)
(6)
3. Effluent nitrate is charged with a constant cost per discharged kg, ∆α, up to a legislatory discharge limit for the effluent concentration, αlimit . Above this limit the cost of discharging additional nitrate is ∆β. Exceeding the limit also imposes an additional charge of β0 per volume effluent water. Figure 2 provides an illustration of the cost function and a mathematical description is given in (7). In principle, this cost function is the one proposed by Carstensen (1994). The difference between the cost functions presented in Carstensen (1994) and in this paper is that here only the nitrate discharge is penalized, while in Carstensen (1994) the total inorganic nitrogen is charged.
e e ∆αSN O (t)Qe (t) if SN O (t) ≤ αlimit e e (t)− fN O (Qe (t), SN O (t)) = ∆ααlimit Qe (t) + β0 Qe (t) + ∆β(SN O e −αlimit Qe (t)) if SN O (t) > αlimit 1
Normally, this choice is made by legislatory authorities.
5
(7)
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∆β
β
0
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Effluent concentration
Figure 2: Cost function for the discharge of nitrogen as proposed by Carstensen (1994). Below the legislatory limit the slope equals ∆α and above it is ∆β.
3
Simulation results
The considered cost functions for penalizing nitrogen discharge are evaluated utilizing the COST benchmark. The simulation study begins with a set of simulations where the benchmark WWTP is fed with constant influent flow rate of wastewater, 18 446 m3 /day, giving results for the steady state case. For the last two sets of simulations a decentralised control law where an and Q e Qi controls SN car controls SN O is employed in the control of the O an , and the aerobic effluent nitrate concentration from the anoxic zone, SN O e zone, SN O , in the stationary case, see Figure 10. In the last simulation set the scenario is more realistic since the benchmark WWTP is fed with timevarying influent, see Figures 11–14.
3.1
Simulations with constant influent flow rate
The influent wastewater in the stationary case is assumed to have a concentration of readily biodegradable substrate, SSi , of 60 mg(COD)/l. All of the other parameters adopt the standard values used in the benchmark; these values can be found in for instance Copp (2002) or on the COST homepage (COST June 29, 2004). See Table 1 for other parameter values. 6
Table 1: Selected parameter values used in the simulation studies. All costs are recalculated from swedish currency unit SEK, hence, all the decimals. Here, 1 SEK = 9.11 EUR which implies, e.g., ∆α = 25 SEK/kg NO. Parameter αlimit β0 ∆α ∆β kE kcar
Value 8.0 1.3721 2.7442 8.2327 0.03622 548.85
Unit mg(N)/l EUR/1000 m3 EUR/kg N O EUR/kg N O EUR/kWh EUR/m3
The benchmark has been run for 150 simulation days for a grid of different, constant, input values. Only the last 100 simulation days were considered when evaluating the cost functions to avoid the influence of transients. In Figures 3–9 the benchmark has been run for a grid of different setpoints for the external carbon dosage, Qcar , and the internal recirculation flow rate, Qi . Qcar has been varied between 0 and 3.0 m3 /day in steps of 0.1 m3 /day and Qi between 0 and 100 000 m3 /day in steps of 2500 m3 /day. Consequently, the range of Qi corresponds to about 0–5.4 times the influent flow rate. Figure 3 shows stationary operational maps for different parts of the total cost. The energy cost is the sum of the aeration cost and the pumping cost. The lower right plot shows the total cost when no charge on nitrogen discharge is imposed. Clearly, the carbon dosage cost starts to dominate rather quickly when increasing Qcar . In Figure 4 the stationary operational map is shown for the total cost when no nitrate discharge fee is used. Also, the level curves for the nitrate an and S e , reconcentrations in the anoxic and aerobic compartments, SN O NO spectively, are plotted. The cost-optimal operating point is marked with a star, and, not surprisingly, it corresponds to performing no active nitrate control: Qcar = 0 and Qi = 2500 m3 /day (Qi = 0 results in a slightly higher energy cost since the aeration cost for Qi = 0 is higher than for Qi = 2500 m3 /day, see Figure 3.). On the other hand, if the intention is to not exceed e a certain value of SN O it is easily seen from the map that, for values of e an from around 1–3 mg(N)/l SN O between about 5–15 mg(N)/l, a level of SN O gives the cost-optimal operating points. However, if a lower effluent nitrate e , is desired, S an needs to be controlled at lower values. concentration, SN NO O In fact, as can be seen in Figure 5 where the maximum Qi is four times as large as in the other operational maps (400 000 m3 /day corresponding to 7
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Figure 3: Stationary operational maps depicting different costs for a grid of different values of Qcar and Qi . an at the cost optimal about 21.7 times the influent flow rate), the value of SN O e e point decreases as SN decreases: when S = 14 mg(N)/l the cost optimal O NO an near 3 mg(N)/l, and for S e an point corresponds to SN = 2 mg(N)/l, SN O NO O near 0.3 mg(N)/l gives the cost optimal operating point.
The above discussed cost optimal operating area can be seen as the set of Pareto optimal solutions to the formulated multiobjective optimization problem weighting performance versus process economy. There are several approaches to multiobjective optimization, see the discussion in Section 4. Here, all of the objectives are expressed in one single objective function, the total cost in (5), that is to be minimized in the optimization process. Next, a constant cost per discharged kg effluent nitrate is introduced according to (6). Figure 6 shows this case with a fee of 2.7442 EUR/kg effluent nitrate, i.e. ∆α = 2.7442 EUR/kg. Apparently, the level curves for the total cost change appearance compared to the previous case, particularly for low values of Qcar . This makes it less desirable to select a set-point for an considerable outside the range 1–3 mg(N)/l – a set-point for S an in SN O NO e this interval still gives cost efficient operation for SN O levels between about
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Figure 4: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost without nitrate-charge, dash-dotted e , and dotted lines show the lines show the effluent nitrate concentration, SN O an . The star indicates nitrate concentration from the anoxic compartment, SN O the minimum-cost point. an 5–15 mg(N)/l. Consequently, the importance of the set-point choice for SN O has become larger. The overall cost-optimal operating point, marked with a star, is now located at Qcar = 0 and Qi = 20000 m3 /day. This operating an = 2.3 mg(N)/l and S e point gives SN O N O = 14.2 mg(N)/l.
In Figure 7 the nitrate fee is twice as high, i.e. ∆α = 5.4885 EUR/kg. This moves the cost-optimal operating point to Qcar = 0.7 m3 /day and an = 1.7 mg(N)/l Qi = 52 500 m3 /day with a cost of 1620 EUR/day and SN O e and SN O = 8.8 mg(N)/l and creates a well-defined operational region corresponding to cost efficiency. As before, to keep the total cost low it is an adopt a value in the range between a bit less than 1 advisory to let SN O mg(N)/l and 3 mg(N)/l. This will result in operation with a total cost less than 1650 EUR/day. If, instead, a cost of 1700 EUR/day is accepted the region corresponding to cost efficient operation is much larger, see Figure 7.
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Figure 5: Stationary operational map with the same settings as in Figure 4 except that here the range of Qi is four times as large. By adjusting the nitrate fee in this way, the legislatory authorities are able to move the cost-optimal set-point. However, since there is no direct e , it may be relation between ∆α and the effluent nitrate concentration, SN O hard to choose a proper value of ∆α when the aim is to place the optimal operating point – or operating region – at (or below) a certain effluent nitrate concentration. The third suggested nitrate cost function given by (7) and graphically in Figure 2 has the advantage of changing appearance at a certain predefined concentration of effluent nitrate, αlimit , where a discontinuity is located. If this discontinuity is sufficiently large it is easy for the authorities to place the optimal operating point at a certain predefined effluent nitrate concentration. Figure 8 shows the stationary operational map when using this cost function with parameter values given by Table 1. The optimum point is located at Qcar = 0.9 m3 /day and Qi = 57 500 m3 /day with a total cost an = 1.1 mg(N)/l and S e of 1248 EUR/day, SN O N O is, not surprisingly, near e 8 mg(N)/l: SN O = 7.7 mg(N)/l. (In fact, with a higher resolution it is e = 8 mg(N)/l. easily verified that the optimum point is placed where SN O 10
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Figure 6: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a constant nitrate-charge per kg effluent nitrate, dash-dotted lines show the effluent nitrate concentrae , and dotted lines show the nitrate concentration from the anoxic tion, SN O an . The star indicates the minimum-cost point. Here compartment, SN O ∆α = 2.7442 EUR/kg. Also, a higher resolution reveals that the optimum point rather is located near Qcar = 0.8250 m3 /day and Qi = 63 000 m3 /day with a total cost of an = 1.7 mg(N)/l.) In practice, when treating non1227 EUR/day, and SN O constant influent, it is of course advisory to chose an operating point that is located slightly below the legislatory limit. The region of economic efficient operation, say operation with a total cost below 1300 EUR/day, is relatively an ranging from somewhat over 0.5 mg(N)/l up to near large with values of SN O e 3 mg(N)/l and values of SN O between a bit below 7 mg(N)/l and 9 mg(N)/l. A way to get a sharper limit at αlimit is to enlarge β0 . In Figure 9 β0 equals 5.4885 EUR/1000 m3 effluent water, i.e. four times as large as the e =α previous setting. As seen in the operational map, the limit at SN limit = O 8 mg(N)/l gets a bit sharper and the region where the total cost is below 1300 EUR/day shrinks. Now, most of the operating points in this area gives 11
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Figure 7: Stationary operational map with the same settings as in Figure 6 except that here ∆α = 5.4885 EUR/kg. e SN O below the limit, αlimit = 8 mg(N)/l. The optimal operating point is, of course, the same as with the previous settings.
Furthermore, two PI-controllers were used to control the effluent nitrate an , and the aerobic zone, S e . A concentration from the anoxic zone, SN O NO decentralized (diagonal) regulator structure is used where Qi is employed an and Q e to control SN car is employed to control SN O . Consequently, the O decentralized control can be written as Qi (s) = F1 (s)E1 (s)
(8)
Qcar (s) = F2 (s)E2 (s)
(9)
where E1 (s) is the Laplace transformed control error of the first loop, i.e. an,sp an e1 (t) = SN O (t) − SN O (t), and E2 (s) is the Laplace transformed control e (t) − S e,sp (t) since this process is error of the second loop, i.e. e2 (t) = SN O NO known to have negative gain. These controllers were tuned to give approximately the same rise and settling time and were F1 (s) = 1300 + 12
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Figure 8: Stationary operational map for a grid of different values of Qcar and Qi . Solid lines show the total cost including a nitrate-charge according e , and to (7), dash-dotted lines show the effluent nitrate concentration, SN O dotted lines show the nitrate concentration in the last anoxic compartment, an . The additional charge for exceeding the legislatory discharge limit at SN O αlimit = 8.0 mg(N)/l is here β0 = 1.3721 EUR/1000 m3 . The star indicates the minimum-cost point. 0.05 (11) 0.13s Additionally, the anti-windup configuration provided in the benchmark implementation was used. The resulting operational map is shown in Figure 10 where the stars indicate where the cost-optimal point is located for each e . Obviously these optimum points are arranged in a regular level of SN O way. This relation also seems to be valid when varying the ASM1 parameter values. The heterotrophic maximum specific growth rate, µH has been varied between 2.0 day−1 and 6.0 day−1 . It was found that there were only minor differences between the operational maps, and in fact, most of the optimum set-points coincide. The case for µH = 4.0 day−1 is shown in Figure 10. Several other parameter values have also been modified yielding similar results. F2 (s) = 0.05 +
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Figure 9: Stationary operational map with the same settings as in Figure 8 except that here the additional charge for exceeding the legislatory discharge limit is β0 = 5.4885 EUR/1000 m3 .
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Figure 10: Stationary operational map depicting total cost without nitratee an charge for a grid of different set-points of SN O and SN O when using PI e controllers. The stars indicate for each level of SN O the minimum-cost point. Here µH = 4.0 day−1 .
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Simulations with time-varying influent flow rate
The COST benchmark is now evaluated for three different time-varying influent compositions representing dry weather, a storm event, and finally, a rain event. The data files are those supplied for the COST Benchmark plant. For each set of simulations the benchmark plant model has been run according to the description given in Copp (2002) for a grid of different setan and S e , i.e. with constant influent for 150 days followed points for SN O NO by a 14 day simulation set with dryinfluent to generate initial states for the final 14 day simulation with either dryinfluent again or with one of the other two influent files (i.e. storm- or raininfluent). Only the last seven simulation days are used in the evaluation. The results are visualized in operational maps of the total cost in EUR when effluent nitrate is penalized according to the cost function (7) proposed by Carstensen (1994), see Figures 11, 13 and 14. The used parameter values are those listed in Table 1. The heterotrophic maximum specific growth rate (µH ) is selected to 4.0 day−1 . Furthermore, a plot of the quadratic control P e,sp e , i.e. 2 e error for SN O t (SN O (t) − SN O (t)) , is supplied in Figure 12 for the case of dryinfluent. For all of the three influent compositions, there is a clear set-point area where the total cost is lower compared to the surrounding set-point e choices. Clearly, a set-point for SN O in the interval 1–2 mg(N)/l gives an cost efficient operation when SN O set-point is chosen to αlimit , i.e. in this case, 8.0 mg(N)/l. This recommendation is in accordance with the steadystate results discussed in the previous section. Furthermore, the plot of the e quadratic error for SN O in Figure 12 also suggests that roughly the same e interval, 1–2 mg(N)/l, is optimal for the SN O set-point for dryinfluent. For the other two influent compositions, raininfluent and storminfluent, the corresponding quadratic error plots look somewhat different suggesting a bit e ; these plots are omitted here. lower set-point values for SN O
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Figure 11: Operational map depicting total cost for a grid of different sete an points of SN O and SN O when using PI controllers and dryinfluent. The e stars indicate for each level of SN O the minimum-cost point.
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Figure 13: Operational map depicting total cost for a grid of different sete an points of SN O and SN O when using PI controllers and raininfluent. The e stars indicate for each level of SN O the minimum-cost point.
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4
Discussion
Considering the case when no charge on effluent nitrate is imposed, from Figure 4 it is clear that the cost for dosing an external carbon source dominates the total cost. It is also clear that the operational area with respect to effluent nitrate is divided in two parts with different gain characteristics, e , there is a cost-optimal and that for each desired value (set-point) of SN O point in the operational map corresponding to a certain value of the nitrate an . This point naturally concentration in the last anoxic compartment, SN O also corresponds to certain stationary values of the input signals, Qcar and Qi . As indicated in Section 3 the cost-optimal point corresponds to values of an in the interval 1–3 mg(N)/l for a range of values of S e SN O N O between about 5 mg(N)/l and 15 mg(N)/l. The location of this point is not very sensitive to variations in the ASM1 parameters. Similar conclusions are also drawn e by Yuan and Keller (2004). However, according to Section 3, if lower SN O an levels is desired the control strategy must impose SN O levels lower than 1 e mg(N)/l. Such low values of SN O may not be desirable for other reasons an may inhibit the denitrification since the corresponding low values of SN O process in a real plant, see Yuan et al. (2002). By including a nitrogen disposal charge in the total cost function the resulting problem becomes a multiobjective optimization problem involving process economy as well as performance. There are several possible approaches to multiobjective optimization. It is common to include all objectives in one single objective function that is minimized, as is outlined here in this paper. However, it is difficult to formulate a single objective function that fairly reflects all of the underlying objectives. Another multiobjective optimization approach is to minimize the objective functions in order of importance, see for instance Kerrigan and Maciejowski (2002), Fleming (1999), Elia and Dahleh (1997), Aggelogiannaki et al. (2004) and Maciejowski (2002). All of the above discussed multiobjective optimization approaches are a priori methods. This means that the judgement and preferences of the decision maker is specified in advance, before the multiobjective problem is solved. On the contrary, with an a posteriori approach a set of Pareto optimal points is generated and then the decision maker selects the most preferred solution from this set. These and other approaches are surveyed by Shibayev (2004). From Figure 4, some questions may rise. The first is which control structures that can be expected to give a good performance in terms of disturbance rejection and set-point tracking. In different operating points different control structure selections may be suitable. As indicated by e.g. Figure 4, in the area of the operational map corresponding to cost-efficient nitrate e an control, SN O is mostly affected by the input signal Qcar , while SN O is af21
fected by both input signals in this area. This situation in particular is further discussed by Samuelsson et al. (2005) and also to some extent by Yuan and Keller (2004). The second question is how to design the actual control law in order to minimize operational costs. Below, some possibilities for this control design are discussed: • Use two different control loops (for instance PI-controllers) to control e an SN O and SN O separately. Given the desired nitrate effluent concentrae , the optimal set-point of S an can easily be found from Figtion, SN O NO e ure 4. For instance, if the set-point for SN O is chosen as 8 mg(N)/l, the an optimal set-point for SN O is close to 1.5 mg(N)/l. If decentralized cone an trol is to be used, Qcar should control SN O and Qi SN O . Other control structures may, however, yield a better performance, see Samuelsson et al. (2005). • Another possibility is to use a constant high internal recirculation flow rate Qi and to use only Qcar in order to control the nitrate effluent concentration. Since Qi has a much smaller impact on the total cost than Qcar , this would render a close to cost-optimal operation. This possibility has also been mentioned by Ingildsen (2002) • To achieve a cost-optimal performance, the total cost could be minimized on-line using quadratic criteria yielding for example LQG or MPC controllers. Such a criterion could be of the form Z T V = eT (t)Q1 e(t) + uT (t)Q2 u(t)dt (12) 0
where e is a column vector containing the control errors and u the input signals. The weighting matrix Q2 can be chosen to reflect the costs for the different input signals and Q1 can be seen as a performance weight. The difficulty with this criterion is how to weight control performance against cost minimization, i.e. how to choose the matrix Q1 . From the prior knowledge obtained from Figure 4, the elements of the matrix Q2 could be chosen in an ad hoc manner. Since the external carbon source is much more expensive than the pumping of the internal recirculation flow rate, a rule of thumb could be to choose Q2 as a diagonal matrix with the element corresponding to Qcar significantly larger than the element corresponding to Qi . Such a choice clearly penalizes a large value of Qcar in the criterion. • A simple grid search could be performed on-line until the optimal point is reached. This method is simple and has the advantage that no operational map and thereby no model is required. One such optimization algorithm is presented by Ayesa et al. (1998). This algorithm is employed to minimize a global penalty function combining effluent requirements and costs. 22
For the case when the nitrate discharge is penalized with a constant charge per kg, see (6), it is seen from Figures 6 and 7 that this creates a minimum in the total cost function (5). For some corresponding stationary e , this minimum is of course located where the operational costs value of SN O are as low as possible. The main drawback using such a cost function for automatic control is that it is hard to relate the location of the minimum to the nitrate fee, ∆α, and thereby hard to say which set-point a certain fee results in. This problem is overcome if instead the fee function is chosen according to Carstensen (1994). The location of the discontinuity of the fee function (7) (see also Figure 2) immediately determines the optimal sete . Using this fee in the total cost is a point for the effluent nitrate, SN O convenient way to achieve cost optimality for a certain set-point of effluent e , see Figures 8–9, 11 and 13–14. Minimizing this total cost nitrate, SN O function on-line using some automatic control strategy would be a good way to impose the importance of good performance via penalizing the effluent discharge into the control design. If the discharge of nitrogen over a certain legislatory limit is directly associated with a higher fee, this could clearly motivate the use of more advanced control strategies. The impact of the fee in the control design is also easy to understand even for people with a limited knowledge in automatic control, compared to the related matter of choosing a performance weight in some quadratic criterion.
5
Conclusions
In this paper a bioreactor model describing a pre-denitrifying wastewater treatment plant is studied from a process economic point of view. The impact of different nitrate cost functions on the location of the cost-optimal operating point were examined. The main conclusion is that a nitrate concentration in the anoxic compartment in the interval 1–3 mg(N)/l gives cost-efficient operation for most reasonable operating points. Furthermore, the location of this optimum is not very sensitive to changes in the ASM1 parameters. This gives a clear range of operating points where the process can be said to be cost-efficiently controlled. With an appropriate nitrate cost function, the legislatory authorities can place this economic optimum in an area where also the effluent regulations are met. Such a cost function would therefore be suitable for model based automatic control.
Acknowledgements This work has been financially supported by the EC 6th Framework programme as a Specific Targeted Research or Innovation Project (HipCon, Contract number NMP2-CT-2003-505467). Please note that the European Community is not liable for any use that may be made of the information contained herein. 23
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