MORDM Basics IV: Visualizing ROF-Storage Dynamics (finally)

The previous post described a simple, two-objective test case in which the city of Cary employed risk-of-failure (ROF) triggers as levers to adjust for its preferred tradeoff level between its objectives. The example given showed how ROF triggers allowed Cary to account for future uncertainty in its system inputs, thus enabling it to visualize how their risk appetite would affect their desired outcomes.

In meeting these objectives, different risk thresholds would have affected Cary’s response to extreme events such as floods and droughts, and its ability to fulfill demand. Simply analyzing the tradeoffs between objectives that result from a range of ROF trigger values only presents one side of the story. It is vital to visualize how these performance objectives and tradeoffs manifest in the system’s capacity (pun intended) to store enough water in times of scarcity, and by extension, its ability to fulfill its customers’ demand for water.

Using ROFs allow us to more concretely measure how the dynamics of both storage and demand fulfillment evolve and change over time for a given risk tolerance. In the long term, these dynamics will influence when and where new water infrastructure is built to cope with storage requirements and demand growth, but this is a topic for a future blog post. This week, we will focus on unpacking the dynamic evolution of storage and demand in response to different ROF trigger values.

As a quick refresher, our system is a water supply utility located in Cary, which is a city within the Research Triangle region in North Carolina (Trindade et al, 2017). Cary uses water-use restrictions when a weekly ROF exceeds the threshold of risk that Cary is willing to tolerate (α) during which only 50% of demand is met. More frequent water use restrictions help to maintain the reservoir levels and ensure reliability, which was defined in the previous blog post. However, the decision to implement restrictions (or not) will impact the storage levels of the system. With this in mind, we will first examine storage responds to the triggering of a water use restriction. For context, we consider a scenario in which Cary’s inflow timeseries is only 20% of the original levels. Figure 1 below shows the inflow, demand and storage timeseries for this scenario.

Figure 1: The hydrologic timeseries for Cary given that no water restrictions are implemented in a scenario where inflows are 20% of the original levels.

Cary’s challenge becomes apparent in Figure 1. While inflow decreases over time (fewer peaks), demand is steadily growing and has effectively tripled by the end of the period. This results in periods during which storage levels drop to zero, which occurs once past 2040. Also note that the frequency of low storage peaks have increased in the second half of the period. The following questions can thus be posed:

  1. How does the system’s ROF change with increasing demand and decreasing supply?
  2. How does risk tolerance affect the implementation of water-use restrictions during drought?
  3. How will the system reservoir levels respond to different levels of risk tolerance?
Figure 2: The length of the pink bars denote the nth-week during which the first water use restriction was implemented for a given α-value. This is an indicator of the responsiveness of the system to a drought, or decrease in storage levels. The blue line indicates the percent of storage filled with water.

To answer the first question, it is useful to identify how different values of α affect the first instance of a water-use restriction. Figure 2, generated using ‘‘, demonstrates that lower risk tolerances result in earlier implementations of restrictions. This is reasonable, as an actor who more risk-averse will quickly implement water-use restrictions to maintain reliable levels of storage during a drought. However, an actor who is more willing to tolerate the change of low reservoir levels will delay implementing water use restrictions. The blue line juxtaposed on top of the bars indicates the inflows to the reservoir. After the first period of low flows between weeks 30-40, the plot shows that the amount of inflows do not recover, and is likely insufficient to fill the reservoir to initial levels. With a lower α, an actor is more likely to implement restrictions almost immediately after observing merely a few weeks of low inflows. In contrast, an actor who opts for a higher α will only resort to restrictions after seeing an extended period of low flows during which they can be more certain that restrictions are absolutely necessary.

Answering the second and third questions first require that periods of drought are more definitively quantified. To do this, the standardized streamflow indicator (SSI6) was used. The SSI6 is a method that identifies time periods during which the standardized inflow is less than the 6-month rolling mean (Herman et al, 2016). It detects a drought period when the value of the SSI6 < 0 for three consecutive months and SSI6 < -1 at least once during the three-month period. The juxtaposition of storage-restrictions and the periods of drought will allow us to see where restrictions were imposed and its impacts on reservoir levels for a given demand timeseries.

Figure 3 and Figure 4 are a visualization of how the system’s storage levels responds to drought (the red bars in the lower subplot) by implementing water-use restrictions (the dark red lines in the upper subplot) given α = 1% and α = 15% respectively. Predictably, restrictions coincide with periods of drought as defined by the SSI6. However, with a lower risk tolerance, period of restrictions are longer and more frequent. As Figure 3 shows, an actor with a lower risk tolerance may implement restrictions where only a slight risk of failure exists.

Figure 3: Storage dynamics given α=1%. (Upper subplot) The blue lines indicate the reservoir storage levels in billion gallons per week. The yellow lines are the weekly ROF values, or the likelihood that the percent of water stored will drop below 20% of the reservoir levels. The grey lines indicate where water use restrictions are implemented, and the red dashed line denotes α=2%. (Lower subplot) The zones are where droughts were detected using the SSI6 (Herman et al, 2016) method are highlighted in red.

Compared to α = 1%, an actor who is willing to tolerate higher ROF values (Figure 4 as an example) will implement restrictions less frequently and for shorter periods of time. Although this means that demands are less likely to get disrupted, this also puts water supplies at a higher risk of dropping to critical levels (<20%), as restrictions may not get implemented even during times of drought.

Figure 4: Storage dynamics given α=15%. (Upper subplot) The blue lines indicate the reservoir storage levels in billion gallons per week. The yellow lines are the weekly ROF values, or the likelihood that the percent of water stored will drop below 20% of the reservoir levels. The grey lines indicate where water use restrictions are implemented, and the red dashed line denotes α=15%. (Lower subplot) The zones are where droughts were detected using the SSI6 (Herman et al, 2016) method are highlighted in red.

There is one important thing to note when comparing Figures 3 and 4. When the periods water use restrictions coincide for both α-values (between 2040 and 2050), the actor with a lower tolerance implements water use restrictions at the beginning of both drought periods. This decision makes the biggest difference in terms of the reservoir storage levels. By implementing water use restrictions early and for a longer period of time, Cary’s reservoir levels are consistently kept at levels above 50% of full capacity (given full capacity of 7.45 BG). A different actor with higher risk tolerance resulted in water levels that dropped below the 30% of full capacity during periods of drought.

Although this seems undesirable, recall that the system is said to have failed if the capacity drops below 20% of full capacity. Herein lies the power of using an ROF metric – questions 2 and 3 can be answered by generating storage-restriction response figures as shown in the above figures, which allows an actor to examine the consequences of being varying levels of risk tolerance on their ability to fulfill demand while maintaining sufficient water levels. This ability can improve judgement on how much risk a utility can actually tolerate without adversely impacting the socioeconomic aspects of the systems dependent on a water supply utility. In addition, using ROFs enable a utility to better estimate when new infrastructure really needs to be built, instead of making premature investments as a result of unwarranted risk aversion.

To briefly summarize this blog post, we have shown how different risk tolerance levels affect the decisions made by an actor, and how these decisions in turn impact the system. Not shown here is the ability of an ROF to evolve over time given climate change and the building of new water supply infrastructure. In the next blog post, we will briefly discuss the role of ROFs in mapping out adaptation pathways for a utility, how ROFs form the basis of a dynamic and adaptive pathway and their associated operation policies, and connect this to the concept of the soft path (Gleick, 2002) in water supply management.


Gleick, P., 2002. Water management: Soft water paths. Nature, 418(6896), pp.373-373.

Herman, J., Zeff, H., Lamontagne, J., Reed, P. and Characklis, G., 2016. Synthetic Drought Scenario Generation to Support Bottom-Up Water Supply Vulnerability Assessments. Journal of Water Resources Planning and Management, 142(11), p.04016050.

Trindade, B., Reed, P., Herman, J., Zeff, H. and Characklis, G., 2017. Reducing regional drought vulnerabilities and multi-city robustness conflicts using many-objective optimization under deep uncertainty. Advances in Water Resources, 104, pp.195-209.

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