# Calculation Proposal for the Economic Level of Apparent Losses (ELAL) in a Water Supply System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Components of Apparent Losses

- Intervention independent apparent losses. This category is related to the unavoidable level of losses in a system, no matter the number and frequency of the interventions regularly carried out. These losses could only be reduced if there is a substantial change in an essential element of the system (water metering technology, installation conditions of the meters, a variation of water meter suppliers quality, etc.), but they would not be affected if, for example, customers’ meters are replaced more frequently. When expressed in annual terms, this category is called the Intervention Independent Annual Apparent Losses (IIAAL, in m
^{3}/year). - Intervention dependent apparent losses. This category is related to the amount of losses that depends on the intervention policies carried out by the water company and grows when interventions are delayed over time. On the contrary, more frequent interventions requiring greater investments by the utility lead to smaller volumes of apparent losses. When expressed in annual terms, this category is called the Intervention Dependent Annual Apparent Losses (IDAAL, in m
^{3}/year).

^{3}/year).

#### 2.1. IIAAL—Intervention Independent Annual Apparent Losses

- Unavoidable measuring errors ($IIAA{L}_{MES}$), which are associated with the minimum (initial) measuring the error of brand new water meters. The magnitude of this term depends not only on the metering technology but also on the consumption characteristics of the users, as defined by their water consumption flow rate probability distribution function [24].

- Unavoidable illegal uses of water ($IIAA{L}_{ILL}$). Even if great effort and resources are put in place to avoid the illegal uses of water, there will always be a minimum volume of water taken from the system without the company’s knowledge or authorization. This minimum volume is mainly related to local socio-cultural and economic conditions.
- Systematic data handling errors ($IIAA{L}_{DH}$). This component is typically caused by data manipulations performed by the water utility when actual meter readings are not available or are noticeably wrong. Frequently, it is associated with incorrect water consumption calculation procedures and incorrect estimations of meter readings. Consequently, unless these procedures are changed, the magnitude of this term will remain approximately constant over time and be independent of the frequency of the intervention activities. However, only when consumption calculation procedures are extremely imprecise does the magnitude of this component compared to the previous ones prove significant.

#### 2.1.1. IIAAL_{MES}—Intervention Independent Annual Apparent Losses Due to Unavoidable Measuring Errors

^{3}/year) is the Annual Consumption Volume of all the customers using this meter type. As a convention, the weighted error of a meter type i, ${\epsilon}_{i}\left(t\right)$, t years after installation, has a negative value when the meter is under-registering water consumption from a user, and a positive value if it is over-registering [31]. Consequently, the negative symbol in front of the equation compensates the negative sign of the error to obtain positive figures of apparent losses when meters are under-registering water consumption.

^{3}/year) may be considered instead. The relationship between $AC{V}_{i}$ and $AR{V}_{i}$ is defined in Equation (2).

^{3}/year), can be obtained based on the average consumption of customers belonging to group i, the failure rate probabilities of the installed meters, and the average time between failure and repair/replacement of the meter, as defined by Equation (3):

#### 2.1.2. IIAAL_{ILL}—Intervention Independent Annual Apparent Losses Due to Illegal Uses of Water

^{3}/year), whereas the second one will be considered below as an intervention dependent component.

^{3}/year) is the Annual Consumption Volume of all the customers belonging to group j.

#### 2.1.3. IIAAL_{DH}—Intervention Independent Annual Apparent Losses Due to Systematic Data Handling Errors

^{3}/year). Quite frequently, the magnitude of $IIAA{L}_{DH}$ is similar every year and independent of the type or age of the meters installed [10]. $IIAA{L}_{DH}$ will only change if the current calculations and working procedures used by the utility are changed. As a general rule, the more advanced and automated the water meter reading practices, the lower the importance of this component. Unfortunately, the actual magnitude of this term can only be obtained if a comprehensive audit of the commercial system is conducted. The $IIAA{L}_{DH}$ is calculated as shown by Equation (6):

^{3}/(meter × year) and it’s assumed to be the same for the whole utility) and $N{M}_{i}$ is the total number of meters belonging to type i (thus, ${\displaystyle \sum}_{i}}N{M}_{i$ is the actual number of meters in the utility).

#### 2.2. IDAAL—Intervention Dependent Annual Apparent Losses

^{3}/year). Similar to the concepts applied to find the economic level of real losses, more frequent interventions by the utility reduce the magnitude of this component while delayed interventions will lead to higher losses. In the case of apparent losses, the reason for this can be easily found in the growing nature of the losses; water meters as mechanical devices subject to wear tend to underperform with time. Consequently, according to this principle, the under-registration of water consumption by the water meter increases with time or with totalized volume. A similar concept can also be applied to solid state (non-mechanical) water meters with electronic components. However, in this case, the reason for an increased under-registration of water consumption is related to the increasing frequency of failure of electronic components and battery power with age.

- Intervention Dependent Apparent Losses associated with measuring errors ($IDAA{L}_{MES}$). This component accounts for the amount of water not registered by a functional water meter. The rate of increase of the volumes not registered by the meters every year depends on the manufacturing and design quality of the meters and the working conditions in the field.
- Intervention Dependent Apparent Losses associated with illegal consumption ($IDAA{L}_{ILL}$). Unauthorized uses of water in the system caused by meter tampering and illegal connections are included within this category. The magnitude of the volume stolen every year from the water company is directly related to the inspection frequency of the connections [38].

#### 2.2.1. IDAAL_{MES}—Intervention Dependent Annual Apparent Losses Associated with Measuring Errors

^{3}/year

^{2}). This parameter can be obtained by multiplying the degradation rate of the weighted error of each meter type, $AD{R}_{i}$ (%/year), by the total annual consumption volume of the meters belonging to this group, $AC{V}_{i}$ (m

^{3}/year).

^{3}/year), will be directly related to the average age of the meters of this type ($t$) and the annual rate of growth of the unmeasured volumes, as defined in Equation (9).

- -
- Employ a single value of the $NRRA{L}_{MES}^{\ast}$ for all meters in the system regardless of their type.
- -
- Use an average meter age for all meters in the system, ${t}^{\ast}$.$$IDAA{L}_{MES}^{\ast}\left({t}^{\ast}\right)=NRRA{L}_{MES}^{\ast}\times {t}^{\ast}$$

#### 2.2.2. IDAAL_{ILL}—Intervention Dependent Annual Apparent Losses Associated with Illegal Connections

^{3}/year

^{2}):

^{3}/year), will be obtained as

## 3. Policies to Reduce Apparent Losses and Related Costs

## 4. Calculation of Economic Level of Apparent Losses

^{3}/year) is defined as the magnitude of apparent losses for which the total costs, calculated as the sum of the control and reduction policies and the utility’s cost of the water losses, reach a minimum.

^{3}) is the selling Price of Water. At this point, it should be noted that the exact calculation of this price may become very complex when block tariffs are applied. For a precise calculation, the $AC{W}_{ME{S}_{i}}$ and $AC{W}_{IL{L}_{j}}$ should always consider the selling price of the last cubic meter sold to each customer.

_{MES}or AIB

_{ILL}) and the annual cost of water ($AC{W}_{ME{S}_{i}}$ or $AC{W}_{IL{L}_{j}}$):

^{3}/year) are, respectively, the optimum levels of each intervention dependent component of apparent losses.

- Intervention costs (from Equation (15) for $AI{B}_{ME{S}_{i}}^{OPT}$ and Equation (16) for $AI{B}_{IL{L}_{j}}^{OPT}$):$$AI{B}_{ME{S}_{i}}^{OPT}=\frac{1}{2}\frac{TI{C}_{ME{S}_{i}}\times NRRA{L}_{ME{S}_{i}}}{\overline{IDAA{L}_{ME{S}_{i}}^{OPT}}}$$$$AI{B}_{IL{L}_{j}}^{OPT}=\frac{1}{2}\frac{TI{C}_{IL{L}_{j}}\times NRRA{L}_{IL{L}_{j}}}{\overline{IDAA{L}_{IL{L}_{j}}^{OPT}}}$$$$AI{B}^{OPT}={\displaystyle {\displaystyle \sum}_{i}}AI{B}_{ME{S}_{i}}^{OPT}+{\displaystyle {\displaystyle \sum}_{j}}AI{B}_{IL{L}_{j}}^{OPT}$$
- Optimum intervention periods (from Equations (15), (21), and (24) for ${T}_{ME{S}_{i}}^{OPT}$, and Equations (16), (22), and (25) for ${T}_{IL{L}_{j}}^{OPT}$):$${T}_{ME{S}_{i}}^{OPT}=\frac{TI{C}_{ME{S}_{i}}}{AI{B}_{ME{S}_{i}}^{OPT}}=\sqrt{\frac{2\times TI{C}_{ME{S}_{i}}}{NRRA{L}_{ME{S}_{i}}\times PW}}$$$${T}_{IL{L}_{j}}^{OPT}=\sqrt{\frac{2\times TI{C}_{IL{L}_{j}}}{NRRA{L}_{IL{L}_{j}}\times PW}}$$
- Percentage of water meters or customers’ connections annually replaced or inspected, respectively, by each policy:$$PAI{M}_{ME{S}_{i}}^{OPT}=\frac{1}{{T}_{ME{S}_{i}}^{OPT}}\times 100$$$$PAI{M}_{IL{L}_{j}}^{OPT}=\frac{1}{{T}_{IL{L}_{j}}^{OPT}}\times 100$$

## 5. Apparent Loss Indicators

- Apparent Losses Index ($ALI$). This indicator is a measure of how the current apparent losses compare to the minimum achievable value. This indicator should always be greater than one.$$ALI=\frac{CAAL}{IIAAL}$$
- Apparent Losses Economic Index ($ALEI$). This indicator measures how close the utility losses are with respect to the economic level of losses. The target for a properly managed water utility is to attain a value for the $ALEI$ as close to one as possible.$$ALEI=\frac{CAAL}{ELAL}$$
- Apparent Losses Economic Potential Index ($ALEPI$). The $ALEPI$ is not directly related to the water loss management policies of the utility. It measures how far the economic level is from the unavoidable level of losses.$$ALEPI=\frac{ELAL}{IIAAL}$$

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

ACV | Annual Consumption Volume (m^{3}/year); |

ACW_{ILL} | Annual Cost of Water lost due to illegal connections (€/year); |

ACW_{MES} | Annual Cost of Water lost due to meter inaccuracies (€/year); |

ADMFAL | Annual Detected Meter Failures Apparent Losses (m^{3}/year); |

ADR | Annual Degradation Rate of the weighted error (%/year); |

AIB | Annual Intervention Budget (€/year); |

AIB_{ILL} | Annual Inspection Budget of customers’ connections (€/year); |

AIB_{MES} | Annual Intervention Budget due to meter replacement (€/year); |

AICR | Annual Illegal Consumption Increasing Rate (%/year); |

AIF | Average Illegal use Frequency (%/year); |

ALEI | Apparent Losses Economic Index (-); |

ALEPI | Apparent Losses Economic Potential Index (-); |

ALI | Apparent Losses Index (-); |

AMFF | Annual Meter Failure Frequency (%/year); |

ART | Average Repair Time between the occurrence and resolution of a meter failure (years); |

ARV | Annual Registered Volume (m^{3}/year); |

CAAL | Current Annual Apparent Losses (m^{3}/year); |

DHE | Data Handling Error parameter (m^{3}/(meter × year); |

ELAL | Economic Level of Apparent Losses (m^{3}/year); |

$\epsilon \left(t\right)$ | Average Weighted Error depending on time (%); |

$\epsilon \left(0\right)$ | Average Initial Weighted Error (%); |

i | Meter type; |

ICF | Infrastructure Condition Factor (-); |

IDAAL | Intervention Dependent Annual Apparent Losses (m^{3}/year); |

IDAAL_{ILL} | Intervention Dependent Annual Apparent Losses due to illegal connections (m^{3}/year); |

IDAAL_{MES} | Intervention Dependent Annual Apparent Losses due to meter inaccuracies (m^{3}/year); |

IIAAL | Intervention Independent Annual Apparent Losses (m^{3}/year); |

IIAAL_{DH} | Intervention Independent Annual Apparent Losses due to data handling errors (m^{3}/year); |

IIAAL_{ILL} | Intervention Independent Annual Apparent Losses due to illegal connections (m^{3}/year); |

IIAAL_{MES} | Intervention Independent Annual Apparent Losses due to meter inaccuracies (m^{3}/year); |

j | Customer group; |

NM | Number of Meters installed (meters); |

NRRAL_{ILL} | Natural Rate of Rise of Apparent Losses due to illegal connections (m^{3}/year^{2}); |

NRRAL_{MES} | Natural Rate of Rise of Apparent Losses due to meter inaccuracies (m^{3}/ year^{2}); |

PAIM_{ILL} | Percentage of Annually Inspected Meters (%/year); |

PAIM_{MES} | Percentage of Annually Replaced Meters (%/year); |

PW | Selling Price of water (€/m^{3}); |

t | Time Period (years); |

${T}_{ILL}$ | Time required to inspect all water connections (years); |

${T}_{MES}$ | Time required to replace all water meters (years); |

TAC | Total Annual Cost (€/year); |

TAC_{ILL} | Total Annual Cost of the policy associated with the inspection of customers’ connections (€/year); |

TAC_{MES} | Total Annual Cost of the policy associated with meter replacements (€/year) |

TIC_{ILL} | Total Intervention Cost of inspections (€); |

TIC_{MES} | Total Intervention Cost of replacements (€); |

UAUV | Unavoidable Annual Unmeasured Volume (m^{3}/year); |

## Appendix A. Case Study 1—Example for a Water Utility with Basic Data

#### Appendix A.1. General Data

USER TYPE | RESIDENTIAL |
---|---|

METER TYPE | VELOCITY |

NM (meters) | 30,000 |

ACV (m^{3}/year) | 4,500,000 |

$\epsilon \left(0\right)$(%) | −5 |

AMFF (%/year) | 0.5 |

ART (years) | 0.7 |

AIF (%/year) | 0.3 |

ADR (%/year) | 0.5 |

AICR (%/year) | 0.15 |

T_{MES} (years) | 14 |

T_{ILL} (years) | 3 |

UIC_{MES} (€/meter) | 32 |

UIC_{ILL} (€/meter) | 5 |

ICF | 1.21 |

DHE (m^{3}/(meter × year)) | 0.180 |

PW (€/m^{3}) | 0.9 |

#### Appendix A.2. IIAAL—(Current) Intervention Independent Annual Apparent Losses

#### Appendix A.3. IDAAL—(Current) Intervention Dependent Annual Apparent Losses

#### Appendix A.4. CAAL—Current Annual Apparent Losses

**Figure A1.**Case study 1—Time evolution of the various components of apparent losses (IIAAL, IDAAL

_{MES}, and IDAAL

_{ILL}).

#### Appendix A.5. TAC—Total Annual Costs of Current Control Policies

#### Appendix A.6. ELAL—Control Policies Frequency and the Economic Level of Apparent Losses

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}$ | ||

288,000 m^{3}/year | ||

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}$ | |

306,900 m^{3}/year | 13,500 m^{3}/year | |

$\mathit{C}\mathit{A}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{D}\mathit{H}}$ | |

474,525 m^{3}/year | 5400 m^{3}/year | |

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}}$ | ||

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}\mathit{L}}$ | 157,500 m^{3}/year | |

167,625 m^{3}/year | $\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}}$ | |

10,125 m^{3}/year | ||

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}$ | ||

288,000 m^{3}/year | ||

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}$ | |

306,900 m^{3}/year | 13,500 m^{3}/year | |

$\mathit{E}\mathit{L}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{D}\mathit{H}}$ | |

440,162 m^{3}/year | 5400 m^{3}/year | |

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}^{\mathit{O}\mathit{P}\mathit{T}}}$ | ||

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}^{\mathit{O}\mathit{P}\mathit{T}}}$ | 109,545 m^{3}/year | |

133,262 m^{3}/year | $\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}^{\mathit{O}\mathit{P}\mathit{T}}}$ | |

23,717 m^{3}/year |

AIB_{MES}(€/year) | AIB_{ILL}(€/year) | ACW_{MES}(€/year) | ACW_{ILL}(€/year) | TAC_{MES}(€/year) | TAC_{ILL}(€/year) | TAC (€/year) | |
---|---|---|---|---|---|---|---|

Current | 68,571 | 50,000 | 141,750 | 9113 | 210,321 | 59,113 | 269,434 |

Minimized | 98,590 | 21,345 | 98,590 | 21,345 | 197,180 | 42,691 | 239,871 |

Equation | Variable | Calculation | Result |
---|---|---|---|

(31) | $ALI$ | $\frac{CAAL}{IIAAL}$ | 1.55 |

(32) | $ALEI$ | $\frac{CAAL}{ELAL}$ | 1.08 |

(33) | $ALEPI$ | $\frac{ELAL}{IIAAL}$ | 1.43 |

#### Appendix A.7. Case Study 1—Conclusions

## Appendix B. Case Study 2—Example for a Water Utility with Complete Data

#### Appendix B.1. General Data

USER TYPE | RESIDENTIAL | SMALL ICI | MEDIUM-LARGE ICI | |
---|---|---|---|---|

METER TYPE | VELOCITY | VOLUMETRIC | VELOCITY | VELOCITY |

NM_{i} (meters) | 30,000 | 17,250 | 2500 | 250 |

ACV_{i} (m^{3}/year) | 4,500,000 | 2,587,500 | 1,250,000 | 750,000 |

${\mathit{\epsilon}}_{\mathit{i}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$(%) | −5 | −1 | −3 | 0 |

AMFF_{i} (%/year) | 0.5 | 0.5 | 0.5 | 0.2 |

ART_{i} (year) | 0.7 | 0.7 | 0.7 | 0.2 |

AIF_{j} (%/year) | 0.3 | 0.3 | 0.4 | 0 |

ADR_{i} (%/year) | 0.5 | 0.5 | 0.3 | 0.1 |

AICR_{j} (%/year) | 0.15 | 0.15 | 0.05 | 0 |

T_{MESi} (years) | 14 | 14 | 14 | 14 |

T_{ILLj} (years) | 3 | 3 | 3 | 3 |

UIC_{MESi} (€/meter) | 32 | 37 | 62 | 400 |

UIC_{ILLj} (€/meter) | 5 | 5 | 5 | 50 |

ICF | 1.21 | |||

DHE (m^{3}/(meter × year)) | 0.18 | |||

PW (€/m^{3}) | 0.9 |

#### Appendix B.2. IIAAL—(Current) Intervention Independent Annual Apparent Losses

#### Appendix B.3. IDAAL—(Current) Intervention Dependent Annual Apparent Losses

Equation | Variable | Calculation | Type of Customer/Type of Meter | |||
---|---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | ||||

Velocity | Volumetric | Velocity | Velocity | |||

(8) | $NRRA{L}_{ME{S}_{i}}$ | $AD{R}_{i}/100\times AC{V}_{i}$ | 22,500 | 12,938 | 3750 | 750 |

(13) | $\overline{IDAA{L}_{ME{S}_{i}}}$ | $\frac{1}{2}NRAA{L}_{ME{S}_{i}}\times {T}_{ME{S}_{i}}$ | 157,500 | 90,563 | 26,250 | 5250 |

Equation | Variable | Calculation | Type of Customer | ||
---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | |||

(11) | $NRRA{L}_{IL{L}_{j}}$ | $AIC{R}_{j}/100\times AC{V}_{j}$ | 10,631 | 625 | - |

(14) | $\overline{IDA{L}_{IL{L}_{j}}}$ | $\frac{1}{2}NRRA{L}_{IL{L}_{j}}\times {T}_{IL{L}_{j}}$ | 15,947 | 938 | - |

#### Appendix B.4. CAAL—Current Annual Apparent Losses

#### Appendix B.5. TAC—Total Annual Costs of Current Control Policies

Equation | Variable | Calculation | Type of Customer/Type of Meter | |||
---|---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | ||||

Velocity | Volumetric | Velocity | Velocity | |||

- | $TI{C}_{ME{S}_{i}}$ | $UI{C}_{MES\_i}\times N{M}_{i}$ | 960,000 | 638,250 | 155,000 | 100,000 |

(15) | $AI{B}_{ME{S}_{i}}$ | $\frac{TI{C}_{ME{S}_{i}}}{{T}_{ME{S}_{i}}}$ | 68,571 | 45,589 | 11,071 | 7143 |

(17) | $AC{W}_{ME{S}_{i}}$ | $PW\times \overline{IDAA{L}_{ME{S}_{i}}}$ | 141,750 | 81,506 | 23,625 | 4725 |

(19) | $TA{C}_{ME{S}_{i}}$ | $AI{B}_{ME{S}_{i}}+AC{W}_{ME{S}_{i}}$ | 210,321 | 127,096 | 34,696 | 11,868 |

Equation | Variable | Calculation | Type of Customer | ||
---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | |||

- | $TI{C}_{IL{L}_{j}}$ | $UI{C}_{ILLj}\times N{M}_{j}$ | 236,250 | 12,500 | 12,500 |

(16) | $AI{B}_{IL{L}_{j}}$ | $\frac{TI{C}_{IL{L}_{j}}}{{T}_{IL{L}_{j}}}$ | 78,750 | 4167 | 4167 |

(18) | $AC{W}_{IL{L}_{j}}$ | $PW\times \overline{IDAA{L}_{IL{L}_{j}}}$ | 14,352 | 844 | 0 |

(20) | $TA{C}_{IL{L}_{j}}$ | $AI{B}_{IL{L}_{j}}+AC{W}_{IL{L}_{j}}$ | 93,102 | 5010 | 4167 |

#### Appendix B.6. ELAL—Control Policies Frequency and the Economic Level of Apparent

Equation | Variable | Calculation | Type of Customer/Type of Meter | |||
---|---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | ||||

Velocity | Volumetric | Velocity | Velocity | |||

(21) | $\overline{IDAA{L}_{ME{S}_{i}}^{OPT}}$ | $\sqrt{\frac{TI{C}_{ME{S}_{i}}\times NRRA{L}_{ME{S}_{i}}}{2\times PW}}$ | 109,545 | 67,731 | 17,970 | 6455 |

(27) | ${T}_{ME{S}_{i}}^{OPT}$ | $\sqrt{\frac{2\times TI{C}_{ME{S}_{i}}}{NRRA{L}_{ME{S}_{i}}\times PW}}$ | 9.74 | 10.47 | 9.58 | 17.21 |

(29) | $PAI{M}_{ME{S}_{i}}^{OPT}$ | $\frac{1}{{T}_{ME{S}_{i}}^{OPT}}\times 100$ | 10.3% | 9.6% | 10.4% | 5.8% |

Equation | Variable | Calculation | Type of Customer/Type of Meter | |||
---|---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | ||||

Velocity | Volumetric | Velocity | Velocity | |||

(15) | $AI{B}_{ME{S}_{i}}^{OPT}$ | $\frac{TI{C}_{ME{S}_{i}}}{{T}_{ME{S}_{i}}^{OPT}}$ | 98,590 | 60,957 | 16,173 | 5809 |

(17) | $AC{W}_{ME{S}_{i}}^{OPT}$ | $PW\times \overline{IDAA{L}_{ME{S}_{i}}^{OPT}}$ | 98,590 | 60,957 | 16,173 | 5809 |

(19) | $TA{C}_{ME{S}_{i}}^{OPT}$ | $AI{B}_{ME{S}_{i}}^{OPT}+AC{W}_{ME{S}_{i}}^{OPT}$ | 197,180 | 121,915 | 32,346 | 11,619 |

Equation | Variable | Calculation | Type of Customer | ||
---|---|---|---|---|---|

Residential | Small ICI | Med-Large ICI | |||

(22) | $\overline{IDAA{L}_{IL{L}_{j}}^{OPT}}$ | $\sqrt{\frac{TI{C}_{IL{L}_{j}}\times NRRA{L}_{IL{L}_{j}}}{2\times PW}}$ | 37,354 | 2083 | - |

(28) | ${T}_{IL{L}_{j}}^{OPT}$ | $\sqrt{\frac{2\times TI{C}_{IL{L}_{j}}}{NRRA{L}_{IL{L}_{j}}\times PW}}$ | 7.03 | 6.67 | - |

(30) | $PAI{M}_{IL{L}_{j}}^{OPT}$ | $\frac{1}{{T}_{IL{L}_{j}}^{OPT}}\times 100$ | 14.2% | 15.0% | - |

Equation | Variable | Calculation | Type of Customer | ||
---|---|---|---|---|---|

Residential | Small ICI | Med-Larg ICI | |||

(16) | $AI{B}_{IL{L}_{j}}^{OPT}$ | $\frac{TI{C}_{IL{L}_{j}}}{{T}_{IL{L}_{j}}^{OPT}}$ | 33,619 | 1875 | - |

(18) | $AC{W}_{IL{L}_{j}}^{OPT}$ | $PW\times \overline{IDAA{L}_{IL{L}_{j}}^{OPT}}$ | 33,619 | 1875 | - |

(20) | $TA{C}_{IL{L}_{j}}^{OPT}$ | $AI{B}_{IL{L}_{j}}^{OPT}+AC{W}_{IL{L}_{j}}^{OPT}$ | 67,238 | 3750 | - |

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}$ | ||

378,415 m^{3}/year | ||

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}$ | |

413,678 m^{3}/year | 26,263 m^{3}/year | |

$\mathit{C}\mathit{A}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{D}\mathit{H}}$ | |

710,124 m^{3}/year | 9000 m^{3}/year | |

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}}$ | ||

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}\mathit{L}}$ | 279,563 m^{3}/year | |

296,447 m^{3}/year | $\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}}$ | |

16,884 m^{3}/year | ||

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}$ | ||

378,415 m^{3}/year | ||

$\mathit{I}\mathit{I}\mathit{A}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}$ | |

413,678 m^{3}/year | 26,263 m^{3}/year | |

$\mathit{E}\mathit{L}\mathit{A}\mathit{L}$ | $\mathit{I}\mathit{I}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{D}\mathit{H}}$ | |

654,815 m^{3}/year | 9000 m^{3}/year | |

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{M}\mathit{E}\mathit{S}}^{\mathit{O}\mathit{P}\mathit{T}}}$ | ||

$\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}^{\mathit{O}\mathit{P}\mathit{T}}}$ | 201,700 m^{3}/year | |

241,138 m^{3}/year | $\overline{\mathit{I}\mathit{D}\mathit{A}\mathit{A}{\mathit{L}}_{\mathit{I}\mathit{L}\mathit{L}}^{\mathit{O}\mathit{P}\mathit{T}}}$ | |

39,438 m^{3}/year |

AIB_{MES}(€/year) | AIB_{ILL}(€/year) | ACW_{MES}(€/year) | ACW_{ILL}(€/year) | TAC_{MES}(€/year) | TAC_{ILL}(€/year) | TAC (€/year) | |
---|---|---|---|---|---|---|---|

Current | 132,375 | 87,083 | 251,606 | 15,196 | 383,981 | 102,279 | 486,261 |

Minimized | 181,560 | 35,494 | 181,530 | 35,494 | 363,060 | 70,988 | 434,048 |

Equation | Variable | Calculation | Result |
---|---|---|---|

(31) | $ALI$ | $\frac{CAAL}{IIAAL}$ | 1.72 |

(32) | $ALEI$ | $\frac{CAAL}{ELAL}$ | 1.09 |

(33) | $ALEPI$ | $\frac{ELAL}{IIAAL}$ | 1.58 |

#### Appendix B.7. ELAL—Sensitivity Analysis

#### Appendix B.8. Case Study 2—Conclusions

## Appendix C. The comparison Between the Variables Used for Calculating the Economic Level of Real Losses vs. Apparent Losses

Variable | Calculation of the Economic Level of REAL Losses (ELL) | Calculation of the Economic Level of APPARENT Losses (ELAL) | |

Apparent Losses Due to Meter Ageing | Apparent Losses Due to Illegal Consumption | ||

Growing trend of losses with time | Natural Rate of Rise of Leakage ($NRRL$) | Natural Rise of Apparent Losses due to meter ageing ($NRRA{L}_{ME{S}_{i}}$) | Natural Rise of Apparent Losses due to illegal consumption ($NRRA{L}_{IL{L}_{j}}$) |

Intervention to reduce current losses to their initial value | Pipe inspection | Meter replacement | Meter/Connection inspection |

Unit intervention cost | Pipe inspection cost ($CI$) | Meter replacement cost ($UI{C}_{ME{S}_{i}}$) | Meter/Connection inspection cost ($UI{C}_{IL{L}_{j}}$) |

Annual intervention cost | Annual budget for intervention ($ABI$) | Annual intervention (replacement) budget ($AI{B}_{ME{S}_{i}}$) | Annual intervention (inspection) budget ($AI{B}_{IL{L}_{j}}$) |

Unit water value | Marginal cost of supply ($MCS$) | Price of water ($PW$) | Price of water ($PW$) |

Total annual costs | Total annual costs ($CT$) | Total annual intervention (replacement) costs ($TA{C}_{ME{S}_{i}}$) | Total annual intervention (inspection) costs ($TA{C}_{IL{L}_{j}}$) |

Total annual intervention cost ($TAC$) | |||

Optimized intervention time period | Economic inspection frequency ($EIF$) | Optimum intervention (replacement) period (${T}_{ME{S}_{i}}^{OPT}$) | Optimum intervention (inspection) period (${T}_{IL{L}_{j}}^{OPT}$) |

Economic level of losses | Economic level of real losses ($ELL$) | Economic level of apparent losses ($ELAL$) | |

Main indicator on losses level | Infrastructure leakage level ($ILI$) | Apparent losses indicator ($ALI$) |

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**Figure 1.**The real and apparent loss components [3].

**Figure 2.**The levels of water losses [6].

**Figure 5.**The time evolution of IDAAL

_{MESi}and IDAAL

_{ILLj}and average values associated with T

_{MESi}and T

_{ILLj}.

**Table 1.**The typical ranges for the initial weighted error of different water meter types [32].

Worst Case (%) | Best Case (%) | |
---|---|---|

Single jet | −5 | −2 |

Oscillating piston | −1 | +0 |

AWWA Single-jet | −7 | −3 |

AWWA Multi-jet | −7 | −3 |

Fluidic | −7 | −5 |

Multi-jet | −6 | −2 |

AWWA Nutating disc | −3 | −1 |

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**MDPI and ACS Style**

Arregui, F.J.; Cobacho, R.; Soriano, J.; Jimenez-Redal, R.
Calculation Proposal for the Economic Level of Apparent Losses (ELAL) in a Water Supply System. *Water* **2018**, *10*, 1809.
https://doi.org/10.3390/w10121809

**AMA Style**

Arregui FJ, Cobacho R, Soriano J, Jimenez-Redal R.
Calculation Proposal for the Economic Level of Apparent Losses (ELAL) in a Water Supply System. *Water*. 2018; 10(12):1809.
https://doi.org/10.3390/w10121809

**Chicago/Turabian Style**

Arregui, Francisco J., Ricardo Cobacho, Javier Soriano, and Ruben Jimenez-Redal.
2018. "Calculation Proposal for the Economic Level of Apparent Losses (ELAL) in a Water Supply System" *Water* 10, no. 12: 1809.
https://doi.org/10.3390/w10121809