Supplemental Data
Reliable, robust and resilient system design framework with application to
wastewater treatment plant control
Chris Sweetapple, Guangtao Fu, David Butler
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DECISION VARIABLES
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Operational decision variables considered in the optimisation are detailed in Table S1.
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Table S1: Operational decision variables (adapted from Sweetapple et al. (2014))
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Decision variable
Minimum
Maximum
Internal recycle flow rate (m3/d)
51,620
72,268
Wastage flow rate (m3/d)
93.5
506.5
Reactor 1 aeration intensity (/d)
0
24
Reactor 2 aeration intensity (/d)
0
24
Carbon source addition to activated sludge reactor 1
1.5
2.5
Carbon source addition to activated sludge reactor 2
0
0.5
Carbon source addition to activated sludge reactor 5
0
0.5
Dissolved oxygen setpoint (g O2/m3)
0
10
Controller offset
0
240
Controller amplification
0
500
Controller integral time constant
0.0005
0.035
Gain factor for reactor 3 aeration
0
1
Gain factor for reactor 5 aeration
0
1
5
2
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ROBUSTNESS ASSESSMENT
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Robustness indicators used in the case study quantify characteristics of the system response
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curve (performance measure vs disturbance magnitude) and are based on an adaptation of the
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methodology detailed by Mens et al. (2011). Calculation of robustness indicators by Mens et
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al. (2011) includes sections of the response curve deemed to correspond with resilience,
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which contradicts the notion the robustness is a component of resilience (Bruneau and
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Reinhorn 2006) – in this work, therefore, a distinction is made between performance
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providing an acceptable level of service (which contributes to assessment of robustness) and
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performance which is unacceptable but not beyond the point of no recovery (corresponding
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with non-robust behaviour and used in resilience assessment). The failure point (illustrated in
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Fig. S.1) is now defined as the point at which the system fails to provide satisfactory
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performance (as in Hashimoto et al. (1982) and Kjeldsen and Rosbjerg (2004)), and the point
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of no recovery is the point at which the system is no longer able to return to an acceptable
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level of service.
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Robustness indicators corresponding to instances in which the maximum acceptable system
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output is and is not exceeded under the specified maximum disturbance magnitude, and how
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these relate to the system response curve are shown in Fig. S.1a and Fig. S.1b respectively. In
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Fig. S.1a, the failure point is the normalised disturbance magnitude corresponding to the point
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at which the maximum acceptable output is exceeded, whereas in Fig. S.1b there is no failure
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point as acceptable performance is maintained under the maximum disturbance. In both cases
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the response severity is equal to the total area under the response curve.
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Able to return to
acceptable
performance
Acceptable
performance
System performance measure
(normalised with respect to compliance limit)
a) Maximum acceptable output exceeded
1
Maximum
acceptable output
Maximum
acceptable output
Response severity =
shaded area
Response severity =
shaded area
Failure
point
0
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b) Maximum acceptable output not exceeded
1
Disturbance magnitude
(normalised within analysis range)
1
0
Disturbance magnitude
(normalised within analysis range)
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Fig. S.1: Robustness indicator components (shown in bold type) for instances in which the
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maximum acceptable system output is: a) exceeded, and b) not exceeded, under disturbances
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of a specified maximum magnitude
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In this study, plant performance is evaluated under eleven different percentage magnitudes of
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change for each uncertain parameter (influent flow rate, total nitrogen, COD and
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temperature), distributed evenly across the total range considered, to provide an estimation of
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each response curve. To enable comparison of robustness indicators for different performance
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indicators and different classes of disturbance, disturbance magnitudes are normalised with
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respect to the maximum disturbance in each case and model outputs are normalised with
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respect to the acceptable range (i.e. from zero to the failure limit). Therefore, all disturbance
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magnitudes are reported as values from 0-1 and model outputs which are compliant or better
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than the base case fall in the range 0-1; outputs which exceed the acceptable limit (i.e. fail)
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take a value greater than one.
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RESILIENCE ASSESSMENT
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Resilience measures
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Resilience assessment measures used in this study are based on application of the resilience
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metrics used by Wang and Blackmore (2009) to the concept of using a response curve
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(system performance as a function of disturbance magnitude) for comparison of solutions
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(Mens et al. 2011).
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Metrics which can be used when the characteristics (probability and magnitude) of a threat
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are known and can be modelled include ‘resilience against crossing a performance threshold’
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(RV) and ‘resilience for system response and recovery’ (RT1 and RT2) (Wang and Blackmore
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2009). RV is a measure of the mean performance deficit and takes into account both the
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magnitude and duration of performance failures, although is unable to differentiate between
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failures of low magnitude/long duration and those of high magnitude/short duration. RT1 and
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RT2 relate to the rate at which the system recovers following disturbances and are defined by
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Wang and Blackmore (2009) as the inverse of the mean time in failure state and inverse of
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maximum time in failure state respectively. It has been suggested that the definition based on
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maximum time in failure state (RT2) is best, since the mean failure duration may be affected
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by the presence of short, insignificant events (Kundzewicz and Kindler 1995). An example
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plot of system response as a function of time when subject to disturbances, given in Fig. S.2,
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illustrates the components contributing to each indicator.
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System response
Unable to
return to
acceptable
performance
Failures
Level of no recovery
Able to return
to acceptable
performance
Maximum
acceptable output
Ei
Ti
Acceptable
performance
Fk
Base performance
i
Time
Resilience for system response and recovery based
on behaviour in these regions
Resilience against crossing a performance
threshold based on behaviour in this region
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Fig. S.2: Visual representation of resilience components, based on system response with
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respect to time
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The performance metrics RV and RT2 are adapted for use in this work, giving performance
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indicators Pdefecit and Pduration,max respectively. Firstly, RV is modified to account for instances
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in which the system output exceeds a specified limit (which cannot occur in the original
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example):
𝑃𝑑𝑒𝑓𝑖𝑐𝑖𝑡
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∑𝑁
𝑖=1(𝑇𝑖 − 𝐸𝑖 )
=
∑𝑁
𝑖=1 𝑇𝑖
Eq.S.1
where:
Pdefecit
=
Performance measure based on mean performance deficit
Ti
=
Threshold (maximum acceptable output) at time step i
Ei
=
Threshold exceedance at time step i
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= max(0, Ti – outputi)
N
=
Number of time steps
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Note that Pdeficit may take a negative value where the threshold is a maximum rather than
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minimum acceptable output and the observed output exceeds this by more than 100%.
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In the form given by Wang and Blackmore (2009), the RT2 metric takes a value of infinity
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when no failures are observed under a specific threat (due to the use of an inverse function).
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This makes any further manipulation or use of the value challenging. An alternative would be
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to simply use the mean performance deficit, mean time in failure state and maximum time in
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failure state (without the inverse); however, in this case a lower value would imply higher
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resilience, which is counterintuitive. The following measure is therefore proposed, which
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takes a value of less than one when failures occur and tends towards one as failure duration
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approaches zero. Constant failure yields a value of zero.
𝑃𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛,𝑚𝑎𝑥 =
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𝑇𝑡𝑜𝑡𝑎𝑙 − max (𝐹𝑘 )
𝑘=1…𝑀
𝑇𝑡𝑜𝑡𝑎𝑙
Eq.S.2
where:
Pduration,max
=
Performance measure based on maximum failure duration
Ne
=
Number of times failure state is entered
k
=
Failure number
Fk
=
Duration of failure event k
Ttotal
=
Total duration of evaluation period
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These performance metrics can only be calculated when a threat can be modelled, which
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requires a disturbance magnitude to be specified. However, this is not always possible and,
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for extreme events, there is likely to be a high degree of uncertainty. It would be more useful
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if resilience to a specific threat could be assessed without the need to know the magnitude
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and probability of this threat.
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It is, therefore, proposed that a plot of system performance against disturbance magnitude,
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similar to that proposed by Mens et al. (2011) for robustness, is used for comparison of the
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resilience of different solutions. In this application, however, the performance metrics used
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are based on indicators detailed above: Pdeficit and Pduration,max. A maximum disturbance
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magnitude must be still be specified, but performance metrics are calculated and plotted for
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all disturbance magnitudes up to this value, yielding a response curve of the form given in
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Fig. S.3.
System performance measure
(Pdeficit, Pduration,mean or Pduration,max
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1
Resilience indicator =
shaded area
0
1
Disturbance magnitude
(normalised within analysis range)
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Fig. S.3: Response curve used for calculation of resilience indicators
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The area under the response curve provides a measure of the system's resilience to a given
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threat, with a higher value representing a more resilient system and a value of one implying
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full resilience to a specific type of threat with a specified maximum magnitude. Three
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resilience indicators can be calculated, depending on the chosen system performance
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measure: Rdeficit and Rduration,max correspond with the performance measures Pdeficit and
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Pduration,max respectively. In the case of a particularly severe response, it is theoretically
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possible that Rdeficit takes a negative value, since Pdeficit can be negative.
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Assessment methodology
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This study provides measures of specified resilience for the WWTP, i.e. resilience of a
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specified output to a specified threat. Outputs considered are:
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1. BOD5 (threshold 25 g/m3)
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2. COD (threshold 125 g/m3)
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3. TSS (threshold 35 g/m3)
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4. Total nitrogen (threshold 15 g/m3)
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Threats are as follows, where given percentage changes are relative to standard loading
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conditions (e.g. 15°C wastewater may be reduced by up to 14.985°C, i.e. to a temperature of
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approximately 0°C):
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1. Increased influent flow rate (up to 100%)
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2. Increased influent total nitrogen concentration (up to 100%)
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3. Decreased influent COD concentration (up to 99.9%)
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4. Decreased temperature (up to 99.9%)
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For each solution, 32 response curves of the form shown in Fig. S.3 are required to give
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Rdeficit and Rduration,max values for every combination of threat and output. Each response curve
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is approximated using the results of 21 dynamic simulations in which the disturbance
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magnitude is increased from 0% to 100% of the maximum disturbance at 5% intervals.
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Resilience indicators are equal to the area under the corresponding curve, calculated as
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follows:
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𝑅𝑑𝑒𝑓𝑖𝑐𝑖𝑡 = 0.05 ∑(𝑃𝑑𝑒𝑓𝑖𝑐𝑖𝑡,𝑘+1 + 𝑃𝑑𝑒𝑓𝑖𝑐𝑖𝑡,𝑘 )
Eq.S.3
𝑘=1
10
𝑅𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛,𝑚𝑎𝑥 = 0.05 ∑(𝑃𝑢𝑟𝑎𝑡𝑖𝑜𝑛,𝑚𝑒𝑎𝑛,𝑘+1 + 𝑃𝑢𝑟𝑎𝑡𝑖𝑜𝑛,𝑚𝑒𝑎𝑛,𝑘 )
Eq.S.4
𝑘=1
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where k represents the disturbance magnitude step number.
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This yields 16 Rdeficit values and 16 Rduration,max values for every solution (one for each
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combination of threat and output), of which the minima are used to provide an overall
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representation of resilience.
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ROBUSTNESS OF OPTIMISED SOLUTIONS
Fig. S.4: Effluent BOD5 response severities of non-dominated solutions bettering base case
GHG emissions under default conditions
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Fig. S.5: Effluent COD response severities of non-dominated solutions bettering base case
GHG emissions under default conditions
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Fig. S.6: Effluent TSS response severities of non-dominated solutions bettering base case GHG
emissions under default conditions
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Fig. S.7: Effluent total nitrogen response severities of non-dominated solutions bettering base
case GHG emissions under default conditions, with solutions which reach failure point circled
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d)
Fig. S.8: Worst case effluent quality response severities for non-dominated solutions which
better the base case GHG emissions under default conditions, with solutions which reach
failure point circled
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RESILIENCE OF OPTIMISED SOLUTIONS
Fig. S.9: Resilience (Rdeficit) of effluent quality compliance to decreased temperature for nondominated solutions bettering base case GHG emissions under standard loading and providing
acceptable robustness
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Fig. S.10: Resilience (Rdeficit) of effluent quality compliance to increased influent flow rate
for non-dominated solutions bettering base case GHG emissions under standard loading and
providing acceptable robustness
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Fig. S.11: Resilience (Rdeficit) of effluent quality compliance to increased influent total
nitrogen for non-dominated solutions bettering base case GHG emissions under standard
loading and providing acceptable robustness
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Fig. S.12: Resilience (Rdeficit) of effluent quality compliance to decreased influent COD for
non-dominated solutions bettering base case GHG emissions under standard loading and
providing acceptable robustness
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