Uncertainties and risks in reservoir operations under changing hydroclimatic conditions

Uncertainties and risks associated with hydroclimatic variations pose a challenge to the management and planning of water resources systems. This study demonstrates the importance of understanding the changing hydrologic regime of the Feather River Basin (FRB) and its impacts on water resources decision variables (i.e., storage requirement and performance of a water supply reservoir). A simple storage– yield–reliability model (S–Y–R) is used to quantify the risk of the stationary-based designed reservoir under the temporal variation and nonstationarity in N-year blocks of the Feather River Inflow into Lake Oroville (FRI). Furthermore, the potential linkages of the long-term variability in the FRI to climate variations are investigated by applying wavelet spectrum and coherence analysis to the FRI and atmospheric–oceanic indices (e.g., ENSO and PDO). The results show substantial variations in the FRB hydrologic regime over different timescales with episodes of abrupt shifts toward significantly higher storage requirements, and decrease in the reservoir performance during historical periods of high FRI variance and lag-1 serial correlation. Although the mean inflows are high, the storage capacity is increased by (a) 38 and 48% due to the 5 and 20% increase in the FRI variance during the periods 1904– 1953 and 1960–2009, respectively, and (b) 34% due to the increase in the serial correlation coefficient in the period of 1750–1799. Likewise, reservoir performance significantly decreased for the same reasons in the same critical periods. The reliability and resilience dropped to 74 and 29% (1904–1953) and to 76 and 50% (1960–2009 period) due to the increased variance of FRI, while vulnerability reached 70% during the high lag-1 correlations in 1532–1581 and 1564–1613, and 40% in 1904–1953 due to the high FRI variance. Furthermore, the wavelet coherence analysis observes strong associations between the streamflow and climate teleconnection patterns in specific periodic cycles during the same critical periods which link the variability in FRI and decision variables to the hydroclimatic variations. These linkages give a primary indication for the reservoir storage requirement characterization.


INTRODUCTION
Dams and reservoirs are the most pervasive infrastructure elements that help achieve multiple societal objectives: reliable water supplies, flood control, recreation, and hydroelectricity. Large dams, such as Oroville, Hoover, Grand Coulee, and Glen Canyon, have been catalysts for regional socio-economic development. At the same time, the deleterious effects of dam building include harm to ecosystems, dislocation of people and cultures, and inundation of lands. Given this precarious balance, sound approaches to dam operations and management have the potential to ensure water security, as well as support efforts to restore and improve the ecosystem services (Poff & Olden ). which relied on relatively short hydrological records add more challenges to water resources management. This motivated the authors to use the long-term annually resolved reconstructed data sets in the analysis of this study to investigate the surprising changes in the decision variables that occurred in the past and may occur again in the future due to hydroclimatic variations.
In this paper, a simplified reservoir with specified storage and demand is used to quantify the risk of a stationary-based storage estimate and to examine the reservoir performance under the uncertainties and risks of hydroclimatic variation. The analysis presented here uses the Feather River inflow into Lake Oroville (FRI) as a case study to understand the changes in storage requirement and reservoir performance, stemming from the changing hydrological regime of the FRB, for a given level of demand and reliability, and to investigate the potential linkages between the runoff variability and climate variations. By using annual data sets of observed and resolved multi-century long-term reconstructed records of the FRI and climate indices, we answer the following questions: (a) How has the hydrologic regime of the FRB changed historically? (b) What are the relative impacts of embedded temporal variations and nonstationarity to N-year blocks from the historical FRI and climate-driver records on the reservoir storage requirements and performance? (c) Finally, what are the long-term linkages between the historical FRI variability and the climate variations which may affect the system design and performance? We explore these issues using a simplified reservoir storageyield-reliability (S-Y-R) model for reservoir storage estimations, and using the Reliability, Resilience, Vulnerability (RRV) metrics to evaluate the system performance. Further, we apply wavelet and coherence analysis to identify the temporal and spatial periodic patterns as well as to introduce the possible linkages among the hydroclimatic variables.

Reservoir characteristics
The reservoir's function is to regulate the irregular natural flow to provide a regular rate of outflow to serve reservoir objectives. Several quantities are necessary for the reservoir's design and modeling (e.g., storage, yield, demand, etc.). Storage requirement is the volume of storage that is needed to supply a given demand in a specified period under a selected level of reliability. The considered storage in this study is the active storage capacity, the difference between the maximum reservoir capacity at full supply level and the dead storage, the volume of water held below the lowest off-take valve. Active storage S t ranges between zero and a maximum value C imposed by the reservoir size. The target draft or water demand D t is the volume of water withdrawn from a reservoir to meet demand over a selected period. While the reservoir yield, draft or release R t is the abstracted water during the same period of demand, and both have units of volume per specified time. Although the desirable yield is equal to the demand, it may fall below the target draft during the drought period and may exceed it in times of plenty. Yield can be decreased less than the target draft when the level of storage in a reservoir is low.
Conversely, yield may increase over the target draft when the reservoir is full. The system starts to spill if it is filled to its maximum capacity and operating at its maximum level and is fed by an inflow that is higher than its ultimate operating level. The base yield is the only yield component considered in this study. It is the lowest yield recorded when a reservoir is fed by an inflow while attempting to supply water to meet demand under a particular operating policy. However, the maximum abstracted base yield from a reservoir equals the target draft. The reservoir spill W t is the excess volume of water that cannot be stored in the reservoir due to its maximum capacity C which usually occurs during periods of flood. These magnitudes can be determined in the following steady-state equations (Vogel & Stedinger ; McMahon et al. b): (1) where Q t is the net inflow per selected time. In the storageyield relationship, the storage capacity is expressed by a ratio or a percentage of the mean annual flow or as a stan- Throughout this study, the 50-year period  prior to the construction of the Oroville Dam is used to compute the reference hydrology (   The model accounts for the inflow persistence by using the lag-1 serial correlation as follows: where C is the required storage, Z p is the reliability, α is the is the mean annual inflow, σ is the standard deviation, and ρ is the lag-1 serial correlation. Two checks are adopted here to ensure that the storage estimates are consistent with the over-year storage assumption based on standardized net inflow or drift, μ ¼ ((1 À α)=C v ) < 1 and the critical time is greater than 1 year as follows: where n ctrl: is the time taken by the reservoir to empty from a fully filled state.

Criteria of reservoir performance evaluation
The evaluation of reservoir performance in this work is car- Although these criteria were defined based on the assumption of stationarity, the distribution is time-invariant, the present analysis uses them to evaluate the dynamic risk of a reservoir's performance in a changing climate.
Reliability is the number of satisfactory events when the targeted demand is met during the simulation time, and it can be determined as follows: where R s is the time-based reliability, N s is the number of satisfied years or events, and N is the total number of events or the whole period of simulation.
Resilience measures how quickly the reservoir will recover when it has already failed to meet the target draft.
The expression used to find the reservoir resilience in the current study is defined as follows: where r is the resilience, f is the number of individual continuous sequences of failures, and N f is the total duration of all the failures.
The dimensionless vulnerability variable measures the severity of reservoir shortfall during the period of failure.
It is defined here as follows: where υ is the dimensionless vulnerability, s i is the volumetric shortfall during the ith continuous failure sequence,   Table 1). However, the impact of the lag-1 serial correlation coefficient on the storage requirements appears through (1 þ ρ)=(1 À ρ). Changes in the inflow persistence can individually alter the storage estimation, i.e., although the inflow mean was relatively high and the interannual variability was low in the second half of the 15th century (1750-1799), the reservoir storage was 34% higher than the baseline storage due to the effect of high serial correlation values (see Figure 1(c) and 1(d) and Table 1). Further, periods of the 16th and early 17th centuries display an upward trend toward high storage requirements due to the increased values of the lag-1 correlation coefficient during the same periods (Table 1). Thus, the results of the S-Y-R model show that the sensitivity of the reservoir storage requirements is related to the changes in the FRI variance and persistence.

Reservoir performance evaluation
The performance of a hypothetical reservoir with three different capacities is evaluated by using the RRV indices ( Figure 2). Fluctuations in the reliability curves throughout the entire record capture the changes in the hydrological regime of the FRI, as shown in Figure 1. It is clear that the reliability of the water supply improved under a condition of low interannual variability with a relatively high mean state. It is noteworthy that the reliability of the water supply during the periods of 1904-1953 and 1960-2009 reached higher rates of failure to fulfill the demand in a 50-year window throughout the 11 centuries (see Table 2).
These changes in the system reliability are explained by the elevated level of the streamflow variance during these periods (Figure 1(b)). Furthermore, the influence of the streamflow persistence on the reservoir reliability is observed during the periods of high lag-1 serial correlation coefficient values such as in the 16th and 18th centuries when the system became less reliable for water supply (Table 2).
However, the results presented above are limited in the following manner: (a) potential events of shortfalls are counted regardless of the persistence and severity of deficit   Interestingly, in the same two time periods of the 20th century, the system displayed low resilience and high vulnerability in its performance due to the high year-to- year variability in the FRI during these periods, even though they were periods of relatively high mean states, which demonstrates the key role of the runoff variance in reservoir performance (see Figure 1(b) and Table 2). On the other hand, the positive high coefficients of FRI lag-1 serial correlation affected both metrics in the opposite fashion. As such, the system spent more time in failure and became more prone to fail, such as during periods in the 1500s and 1800s (Table 2).
These curves in Figure 2 address the critical contribution of storage requirements in the sustainability of a Contrary to what was expected from the results, the resilience and vulnerability in Figure 2 did not always increase and decrease with an increase in the capacity of the reservoir for a given target draft, e.g., both metrics of the system in the latter periods of the 20th century did not improve relative to the increased storage requirement (Table 2)  To this end, although the large-scale climate drivers (i.e., ENSO and PDO) show linkages with the FRI during the critical periods (Table 1)

SUMMARY AND CONCLUSIONS
The results presented here are obtained by using the annually resolved records and the long-term reconstructed data sets of the FRI and climate indices (e.g., ENSO and The results of the reservoir applications show that the FRI interannual variability and persistence play a key role in the system decision variables. This leads to significant changes in the system decision variables on short to long timescales. As such, the storage requirement during the last three decades is abruptly increased by 50% from the baseline storage. On the other hand, the results show that the storage requirement can be affected by the streamflow persistence, e.g., the 40% higher storage requirements in the late 18th century were caused by the high serial correlation values during that period. In terms of the system performance, reliability and resilience of the reservoir decrease and the system become more vulnerable to shortfalls during the periods of high inflow variability and persistence. Furthermore, the reservoir performance metrics (RRV) responded nonlinearly to the incremental increase in storage requirements during the periods in which variance and lag-1 serial correlation of FRI are high.
By using the wavelet power spectrum and coherence analysis, the authors demonstrate that, in many instances and periods, changes in reservoir storage requirements and performance (reliability, resilience, and vulnerability) could be readily linked to the changing co-variability between the FRI and climate teleconnection patterns. The above-mentioned results allow a qualitative assessment of the relationship between streamflow and climate indices (e.g., ENSO and PDO) which is resolved at low-frequency bands from interannual to multidecadal and centennial.
The coherence estimates shown above provide a clearer interpretation of swings in the decision variables that occur during eras of high coherence between the streamflow and climate drivers at the select timescale. While the results provide an interesting perspective regarding FRI and climatic phenomenon, it is worth noting that the diagnostic studies do not imply causal relations. The moderate co-variability between climate indices and streamflow merits attention, in particular related to the high-frequency atmospheric phenomenon, such as the atmospheric rivers a key moisture delivery mechanism for the US west coast.
The correlations of the FRI-climate indices and the stat- All in all, it is reasonable to draw the conclusion that the reconstructed hydroclimatic records lend useful insights regarding the underlying streamflow variability over different timescales, thus any historical record of shorter length will only contain or represent a fraction of the variability seen here. Therefore, the timescales that are not represented in a record of limited length are likely to be a source of the system's deterioration if they occur in the future. Our ability to anticipate future hydrology and integrate that knowledge into design and planning is thus well-informed by analysis of the type presented here. It is hoped that, alongside other emerging work on the topic of nonstationarity and its applications to water resources planning and management (e.g., Ho et al. ), this work will aid in providing a fresh perspective. Much remains to be done to clarify and adopt systematic approaches to decision-making under uncertain and changing climate conditions.