9. CO S IDER lNG UNCERTAINTy
9.2 Sensitivity Analysis
ba i � r d scribing aet i at d sludge process in the mode! ' see Chapters 3 and 6).
J lenee, in this formulation, stoichiometric and composition parameters are imbedded in the model. Thus, kinet ic parameters, only, remain as th parameters associated with
M3 mode l and the are a major source of uncertainty.
The remaining parameters in the model are divided into two groups. The fIrst parameters associated with constraints on aerat ion tank volume. These involve uneertaint , like the parameter describing effIciency of diffuser, which depends on diffuser type, and depth at which air pumped this uncertainty has a minor effect on model behavior since it does not contribute in the calculations directly. The second group is parameters associated with object ive funct ion calculation. Capital recovery factor, cost index, base cost index, operating and maintenance wages, and electricity cost, all cannot be considered as uncertain parameters since designer chose these parameters based on current situation and if changed their rate of change is very low which allows another analysis to be conducted with the new parameters values. I n contrast, the other two parameters associated with the calculation o f power consumption cost of pumping, i.e., pumping head and pumping efficiency, are uncertain parameters. They are subjected to variabil ity during operation and hence affect the cost of pumping signifIcantly.
Summarizing the above d iscussion, uncertainty in the introduced optimization model is attributed to several sources. The main source is the infl uent wastewater characteristics including flowrate, pollutant concentrations, and kinetic parameters.
There are other parameters which involve a level of uncertainty but they are of less importance. These are SVI, pumping head and pumping efficiency o f primary sludge pumps and secondary sludge pumps.
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c nquered by either con idering the peak flo rate a the design flo rate or applying a prop r afety factor to the in fluent flowrate. This will account for the variability and will produce a reliable de ign in terms of influent flowrate.
In contra t to influ nt flowrate, influent po llutant concentration shows an -ob iou effi ct on model performance. Di fferent designs and effluent conditions were obtained with different combinat ions of influent characterist ics. Design and effluent were obviously altered even when onJy one in fluent characteristic was changed. The best way in dealing with variabi l ity of influent characteristics is finding a probability di tribut ion that best describes their viabilities. Such an approach was followed by Rou seau et a1. (200 1 ), as mentioned previously. I n that study for every component of influent a triangular distribution was imposed between minimum and maximum
alues calculated according to extensive measurements conducted on several wastewater treatment plants. Then Monte Carlo simulat ion was used to conduct the uncertainty analysis ( Rousseau et aI., 200 1 ).
The remaining random parameters are the k inetic parameters. It has been mentioned that these parameters are random due to estimation and due to their temperature dependency. I n Chapter 8, the model performance for different sets of k i netic parameters at various temperatures was assessed. In generaL temperature variation affects k inetic parameters which in turn affect the model performance significantly. I n t hat analysis, kinetic parameters were all changed according to a method discussed in Chapter 6 to show the values at certain temperatures. However, this is a rare situation which does not happen in real life. Conventionally, a certain parameter might assume a value above or lower than the expected value. I n such a situation the model response is questioned. The answer would provide an insight about the value ( importance) of kinetic parameters uncertainty. I n this sectio n, sensitivity o f a model to kinetic parameters variations at low and high temperatures is explored.
At low temperature, 20°C, the k inetic parameters are assigned values suggested by Henze et a1. (2000) and shown on Table 6.5. H igh temperature is considered at 40°C and the corresponding k inetic parameters are given in Table 6.6.
For every parameter, at 20 and 40°C, three runs were conducted, one at the suggested parameter value, another at 50% of this suggested value, and the third at 1 50% of it.
At each run other values were kept at their original values. The base model i l lustrated
111 hapter 7 was considered the base [or aU conducted runs. Table 9. 1 shows the percentage change in object ive function (total cost) due to ±50% change in parameter alue. A minu sign indicate a reduction in cost while positive one indicates an mcrea , r ferred to values obtained (Table 8. 1 ) at zero variability of each relevant temperature.
Table 9. 1 : Percentage c h a nge in opt i m a l total co t due to va riations i n k i netic Earameters
Kinetic Al 20°C At 40°C
Parameters -50% +50% -50% +50%
kH -0. 1 5 1 1 0.0469 -0.06 1 5 0.0 1 64
K \ 0.0630 -0.056 1 0.0222 -0.0222
ksro 0.0296 -0.0065 -0.0 1 68 0.0048
K s 0.0083 -0.0083 0.0068 -0.0068 Ksro -0.0006 -0.0002 -0.0 1 39 0.0 1 28 J.iH -0.0047 -0.0004 0.03 1 8 -0.0099
KNHI 0.0000 0.0000 -0.0006 0.0007
bH.01 - 1 .5 1 46 1 .8623 -2.0542 0.6479
bSro.01 -0.060 1 0.057 1 -0.0285 0.0263
J.iA 1 ] .338 0.0959 -0. 1 320 0.0323
KA.NHI 0. 1 0 1 9 1 .2096 0.0428 -0.0420
bA.OJ 0.0042 0.3355 0.0030 -0.0368
I t is obvious from the table that variability of kinetic parameters has different effect on the model optimum solution. Even for the same parameter, effect at low temperatures differs from the effect at high temperatures. There is no general trend that can be drawn. Moreover, all the changes are negligible except the ones imposed by the variability of bH JiA, and KA. The effect of changes imposed by bH and KA are still small ( less than 2%) and can be neglected. The most apparent effect is due to a reduction in JiA assumed value by 50% at 20°C. The assumed value at this temperature is 1 .0, which means if JiA becomes for a reason or another 0.5 then a system with 1 1 .3% higher cost is required to achieve the same treatment requirements. This indicates that the system is very sensitive to this parameter and any weak estimation of it would lead to system failure.
Sensitivity of the model to JiA makes sense. JiA is the autotrophic maximum growth rate which is responsible for nitrification in the act ivated sludge process. The developed model has been assumed to perform complete nitrification that lowers the concentration o f ammonium/ammonia nitrogen i n the effluent to less than or equal to 1 .0. It is well known that the growth rate of autotrophic biomass is naturally very
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low. hen any alterat ion in thi growth rate ( ariabi lity of )1 1) would affect ign ificant Iy the nitrification process which in many cases limits the solution..
especiall at low temperatures as has been indicated before. I n the case shown, the decrease in the growth rate required the system to increase the SR T to allow more time for nitrificat ion. Hence the system cost increased significantly.
It sh u ld be noted that the discussed sensitivity analysis above was conducted
b varying one parameter at a time. However in reality all parameters might show d i fferent alue at the same t ime. And hence the combined effect on system performance will be totall different. The aim of the above sensit ivity analysis was only to compare the impact of variability of various individual parameters.
I n summary, the system is most sensitive to variabil ity o f influent characterist ics and maximum growth rate of autotrophic biomass (;.LA). Variability of these parameters should be considered in the design of activated sludge plants.
I gnoring such variabi lity would imply a serious risk and possibility of failure is expected. Hence such variabil ity is considered in the coming example of uncertainty based design. I n contrast, variability of other parameters also exists, however their influence compared to influent characteristics and )1A influence is minor. I n a comprehensive analysis, all random parameters should be considered because a combined effect can be expected.