This is the continuation of the Analysis of the Risk of Consent Failure of UK Wastewater Treatment Plants series from Alan Bland. I hope you enjoy:

In the early 1990s, when this project was undertaken, the established sewage treatment works discharge consents were those originally issued under the 1963 Water Resources Act, referred to as WRA consents. For very small discharges the consent would be “descriptive”, for example requiring that the visual impact of the discharge on the environment be negligible, but for larger discharges the consent would be numeric. Numeric consents covered BOD 5 , SS and sometimes NH 4 . In catchments containing industrial premises, the consent determinands might also include heavy metals (Pb, Zn, Cu, Hg etc.). In all cases, these WRA numeric consents were framed to require 95% compliance with the consent value, which in many cases was the so-called “Royal Commission Standard” of 30mg/l SS and 20mg/l BOD.

Compliance sampling was carried out initially by the NRA and later by the EA. In principle, the dates chosen for sampling were random, to avoid any systematic bias, but in practice weekends and public holidays were avoided. The number of samples taken per year varied according to the size of the discharge. This created a statistical problem. How many samples could be allowed to fail the consent value before the discharge was found to be non-compliant? At first inspection the answer might be thought to be obvious; 95% compliance means only 5 out of every 100 samples should be allowed to fail, so if 20 samples are taken then only 1 sample failure should be allowed. But this frequency of sampling at thousands of discharges across the UK would be very expensive and, furthermore, a single sample failure could be unrepresentative. The consent is a legal requirement; if non-compliance was contested in court, could it be proven that a single failure showed the discharge was non-compliant? Generalising this, there are four cases to consider:

The discharge is actually compliant, but the sampling indicates non-compliance The discharge is compliant and the sampling also indicates compliance The discharge is non-compliant, but the sampling indicates compliance The discharge is non-compliant and the sampling indicates non-compliance

What was needed was a method of analysing samples that reliably identifies case 4 but ignores the other three cases. This gives the benefit of any doubt to the discharger, whose discharge may actually be non-compliant, as in case 3. To meet the need, a device called the Look Up Table (or LUT) was introduced.

The LUT utilises the Binomial probability distribution, which deals with the likelihood of successes and failures. The probability distribution formula is b(x: n,P) = n C x *Px*(1-P)n-x , where P = probability of success (which in this context is a failed sample), x is the number of successes (failed samples), n is the number of trials (in this context the number of samples per year), n C x is the number of combinations of n samples taken x at a time and b is the probability of achieving exactly x successes (failed samples). For our purposes we need the cumulative probability distribution, which adds together the probabilities of achieving up to and including x successes.

Using this distribution, the number of samples required to demonstrate non-compliance is indicated in the LUT below:

Number of Samples Taken Per Year Number of Failed Samples Allowed

4 to 7 1

8 to 16 2

17 to 28 3

29 to 40 4

41 to 53 5 (and so on)

These numbers are selected from the cumulative Binomial probability distribution such that it is 95% probable that, for example, more than 4 sample failures in 35 samples taken in a year would indicate the true level of compliance with the consent is less than 95%. Note that the LUT even caters for very small numbers of samples taken in the year (between 4 and 7 samples), but still provides assurance that more than one failed sample could confidently be declared as indicating non-compliance of the discharge.

The next posting will examine the problems the LUT presents for a water company trying to achieve compliance within the limitations of its fixed assets.