# A Review of Sources of Uncertainty in Optimization Objectives of Water Distribution Systems

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## Abstract

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## 1. Introduction

## 2. Sources of Uncertainty and Their Relative Importance in Relation to Optimization Objectives

- Model/algorithm uncertainty (e.g., pipe roughness values or chemical reaction rates);
- Data/input related uncertainty (e.g., uncertainty in water demands due to natural variability, population growth and/or climate change); and
- Human related uncertainty (e.g., lack of knowledge about the network or ambiguity in the framing or decision making process).

#### 2.1. Water Distribution System Planning and Design

#### 2.2. Water Distribution System Rehabilitation

#### 2.3. Water Distribution System Operations

## 3. Review of Literature on Sources of Uncertainty on Optimization Objectives

#### 3.1. Water Distribution System Planning and Design

#### 3.2. Water Distribution System Rehabilitation

#### 3.3. Water Distribution System Operations

## 4. Conclusions

## 5. Recommendations for Further Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Steps involved in optimizing the planning, design or rehabilitation of WDS and the various sources of uncertainty.

**Figure 2.**Steps involved in Optimizing the Operations of WDS and the Various Sources of Uncertainty.

**Table 1.**Relative importance of the various sources of uncertainty on each objective (three ticks indicates a high level of importance, two ticks a medium level and one tick a low level of importance; n/a means not applicable).

Sources of Uncertainty | |||||||||
---|---|---|---|---|---|---|---|---|---|

Optimization Category | Optimization Objectives | Model | Optimization Method | Data—Randomness in Demand | Data—Randomness in Other Variables (e.g., Mechanical, Water Quality, …) | Data—Long-term Changes in Demand | Data—Long-term Changes in Other Variables | Human—Lack of Knowledge | Human—Stakeholders Preferences |

WDS planning/design | Min capital cost | ✓ | ✓ | ✓✓ | ✓ | ✓✓ | ✓✓ | ✓✓✓ | ✓ |

Min life-cycle cost | ✓✓ | ✓ | ✓✓✓ | ✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | |

Min life-cycle energy/emissions | ✓✓ | ✓ | ✓✓✓ | ✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | |

Max flexibility | ✓✓✓ | ✓ | ✓✓✓ | ✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓✓✓ | |

Max hydraulic performance (e.g., hydraulic capacity, system reliability) | ✓ | ✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓✓ | |

Max water quality performance | ✓✓✓ | ✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | |

WDS rehabilitation | Min capital cost | ✓ | ✓ | ✓ | ✓✓ | ✓✓ | ✓✓ | ✓✓✓ | ✓ |

Min operation cost | ✓ | ✓ | ✓✓ | ✓✓ | ✓✓✓ | ✓ | ✓✓✓ | ✓ | |

Min life-cycle cost | ✓ | ✓ | ✓✓ | ✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | |

Min life-cycle energy/emissions | ✓ | ✓ | ✓✓ | ✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | |

Min water loss | ✓ | ✓ | ✓ | ✓ | ✓✓ | ✓ | ✓✓✓ | ✓ | |

Max hydraulic performance (e.g., hydraulic capacity, supply reliability) | ✓ | ✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | |

WDS operation | Min life cycle cost | ✓ | ✓ | ✓✓ | ✓✓ | n/a | n/a | ✓✓✓ | ✓ |

Min operating cost | ✓ | ✓ | ✓✓ | ✓✓ | n/a | n/a | ✓✓✓ | ✓ | |

Max hydraulic efficiency (e.g., pump power, pump switches) | ✓ | ✓ | ✓✓ | ✓✓ | n/a | n/a | ✓✓✓ | ✓ | |

Min energy consumption/GHG emissions | ✓✓ | ✓ | ✓✓✓ | ✓✓ | n/a | n/a | ✓✓✓ | ✓ | |

Max water quality | ✓✓✓ | ✓ | ✓✓✓ | ✓✓ | n/a | n/a | ✓✓✓ | ✓ | |

Min average/maximum water age | ✓ | ✓ | ✓✓✓ | ✓✓ | n/a | n/a | ✓✓✓ | ✓ |

**Table 2.**Summary of the Extent to which each Source of Uncertainty in Relation to each Objective is Investigated in Previous Literature (three ticks indicates significant coverage, two ticks a moderate level of coverage and one tick little or no coverage; n/a means not applicable).

Sources of Uncertainty | |||||||||
---|---|---|---|---|---|---|---|---|---|

Optimization Category | Optimization Objectives | Model (e.g., Model Structure and Parameter, e.g., Pipe Roughness or Chemical Reaction Rates for Water Quality Considerations) | Optimization Algorithm (e.g., Algorithm and Algorithm Parameters) | Data—Randomness (Natural Variability) in Demand | Data—Randomness (Natural Variability) in Other Variables (Mechanical, Water Quality, …) | Data—Long-term Changes in Demand (Due to Population Growth, Climate Change) | Data - Long-term Changes in Other Variables, e.g., Carbon Tax, Hazards | Human—Lack of Knowledge | Human—Stakeholders Preferences |

WDS planning/design | Min capital cost | ✓✓✓ | ✓✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | ✓ | ✓ |

Min life-cycle cost | ✓✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Min life-cycle energy/emissions | ✓ | ✓ | ✓✓✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Max flexibility | ✓ | ✓✓✓ | ✓✓✓ | ✓ | ✓✓ | ✓ | ✓ | ✓ | |

Max hydraulic performance (e.g., hydraulic capacity, system reliability) | ✓✓✓ | ✓✓✓ | ✓✓✓ | ✓✓ | ✓✓ | ✓✓ | ✓ | ✓ | |

Max water quality performance | ✓✓✓ | ✓✓✓ | ✓✓✓ | ✓✓ | ✓✓✓ | ✓ | ✓✓ | ✓ | |

WDS rehabilitation | Min capital cost | ✓ | ✓✓ | ✓ | ✓ | ✓✓ | ✓✓ | ✓✓ | ✓✓ |

Min operation cost | ✓ | ✓ | ✓ | ✓ | ✓✓ | ✓✓ | ✓✓ | ✓ | |

Min life-cycle cost | ✓ | ✓ | ✓✓✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Min life-cycle energy/emissions | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Min water loss | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Max hydraulic performance (e.g., hydraulic capacity, system reliability) | ✓ | ✓✓✓ | ✓✓✓ | ✓✓✓ | ✓ | ✓✓ | ✓ | ✓✓✓ | |

WDS operation | Min life cycle cost | ✓ | ✓✓✓ | ✓ | ✓✓ | n/a | n/a | ✓ | ✓ |

Min operating cost | ✓ | ✓✓✓ | ✓✓✓ | ✓✓ | n/a | n/a | ✓ | ✓ | |

Max hydraulic efficiency (e.g., pump power, pump switches) | ✓ | ✓✓ | ✓✓ | ✓✓ | n/a | n/a | ✓ | ✓ | |

Min energy consumption/GHG emissions | ✓ | ✓ | ✓ | ✓ | n/a | n/a | ✓ | ✓ | |

Max water quality | ✓✓ | ✓ | ✓✓ | ✓ | n/a | n/a | ✓ | ✓ | |

Min average/maximum water age | ✓ | ✓ | ✓ | ✓ | n/a | n/a | ✓ | ✓ |

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## Share and Cite

**MDPI and ACS Style**

Dandy, G.; Wu, W.; Simpson, A.; Leonard, M. A Review of Sources of Uncertainty in Optimization Objectives of Water Distribution Systems. *Water* **2023**, *15*, 136.
https://doi.org/10.3390/w15010136

**AMA Style**

Dandy G, Wu W, Simpson A, Leonard M. A Review of Sources of Uncertainty in Optimization Objectives of Water Distribution Systems. *Water*. 2023; 15(1):136.
https://doi.org/10.3390/w15010136

**Chicago/Turabian Style**

Dandy, Graeme, Wenyan Wu, Angus Simpson, and Michael Leonard. 2023. "A Review of Sources of Uncertainty in Optimization Objectives of Water Distribution Systems" *Water* 15, no. 1: 136.
https://doi.org/10.3390/w15010136