Graphical analysis of cumulative reservoir inflows and outflows, suggested by Rippl, is a fundamental tool for reservoir analysis. (1)
Figure 1 shows the maximum yield in each drawdown period from a reservoir of a given capacity. (2) When water supplies must be assured, or when a sudden loss of water supplies cannot be permitted, one might limit water deliveries from a reservoir to the lowest yield found in Figure 1, often called the "safe yield". (3) If this is done, the full capability of the reservoir to deliver water is used only once in the historic period of record.
When a reservoir operates for maximum yield in each drawdown period, as is sketched in Figure 1, its storage is fully used in each drawdown period. The reservoir empties in every drawdown period just as the reservoir inflows begin to exceed water demands. This is impractical without absolute knowledge of future reservoir inflows. These dilemmas, that obtaining maximum yields from a reservoir in every drawdown period leads to unreliable water deliveries, and that limiting water deliveries to "safe yield" fails to exploit the capabilities of reservoirs, are discussed here.
Figure 2 shows cumulative reservoir inflows and outflows for one drawdown season. For the drawdown period that begins now, when now is at point a and the reservoir is full, we wish to know "what reservoir yield can be expected over the coming drawdown period?". The yield or the release rate that can be sustained throughout the drawdown period is:
Yield(over time t) =
reservoir capacity + cumulative inflow (over time t)
Reservoir capacity is known, but at a (now) neither the future cumulative reservoir inflow in the drawdown period nor the time period t are known.
Future reservoir inflows depend on the watershed state (snowpacks, soil moistures, etc.) at a, and on the future weather over the drawdown period from a to b. The current watershed state at a is known. (4) Future weather is unknown, but can be handled by using equally likely "alternate future" weather --- weather observed in the watershed in the same period of the year in the past. Reservoir inflows are simulated from a, using the watershed state at a, and all available historic, or alternate future, weather sequences. If 75 years of historic weather are available, 75 different reservoir yields are found for the coming drawdown period, and a frequency distribution for the reservoir yield, specific to the coming drawdown period, can be plotted. The reservoir yield at the 98%, 95% or other exceedance levels can be found. Reservoir managers can then select a water delivery rate as a drawdown period begins, with knowledge of the risk that the delivery rate cannot be sustained.
The relative importance of the current watershed state and future weather, for reservoir yields in a drawdown period, depends on climate, watershed and reservoir characteristics. For some reservoirs the watershed state as the drawdown season begins is the most important factor, and for other reservoirs future weather may be dominant. (5)
. . . Albert Einstein
Taking Einstein at his word, is the use of the current watershed state, and hydrologic simulation based on "alternate futures", necessary? Why could we not use observed historic streamflows in the drawdown period as "alternate futures"? We need to examine observed data on watersheds to answer these questions.
Figure 3 shows the historic streamflow observed
from June 1st to December 31st for the Cedar River near
The observed 1989 streamflows and the 1989 streamflows simulated with 1941 initial conditions, shown
in Figure 3, could be plotted cumulatively. This analysis could be repeated for
many historic years. This would create a family of cumulative inflow plots of
observed June through December streamflows, and a
family of plots of simulated June through December streamflows
based on June 1, 1941 initial conditions. Figure 4 shows the minimum and
maximum limits of these two families of plots. The historic streamflows
were observed at the Cedar River near
There are 45 samples of cumulative streamflow between the minimum and maximum limits in Figure 4. These 45 samples could be used to plot a frequency distribution of the observed or simulated cumulative streamflows for any date from June 1st to December 31st. Figure 5 shows the frequency distributions of the cumulative observed and simulated streamflows in Figure 4, for December 31st.
The frequency distribution of cumulative streamflows at any future time, at June 1st plus n months in Figure 4, approximates the probability density function (pdf) for cumulative streamflows that would exist at that time. Frequency distributions for both simulated and observed historic cumulative streamflows, at June 30, one month in the future, at September 30, four months in the future, and at May 31st, 12 months in the future, are shown in Figure 6. Figures 3, 4, 5 and 6 show that future forecast streamflows are strongly influenced by the initial watershed state. (8)
. . . James Joyce, Ulysses
Does it matter that the forecast streamflow pdfs are different than the historic streamflow pdfs? Reservoirs are operated at now, and the watershed state at now alters future streamflow pdfs. Optimization of reservoir operations to meet objectives for water supply, hydro generation or environmental goals, is done continuously, and it is always done at now.
Optimization of operational objectives is straightforward when all future streamflows are assumed to be known. Upper limit benchmarks for operational objectives, e.g. maximum energy generation, minimum flood damages, etc., are found when future streamflows are assumed known in optimization. Lower limit benchmarks for operational objectives are found when historic streamflows are used as alternate futures in optimization. Using historic streamflows assumes that nothing is known about the watershed at now. Future streamflows are never known precisely, but more precise knowledge of future streamflows enhances operational objectives, and less precise knowledge of future streamflows inhibits operational objectives. (9) The benefits of hydrologic forecast modeling in reservoir operations, together with the benefits of all of the field work that supports forecast modeling (snow pillows, meteorologic gages, streamgages, etc.) stem from the difference between the forecast and the historic pdfs in Figure 6.
. . . Robert Frost, Pertinax
Illustrations of form nourish intuition about watershed behavior. Quantitative results like those in Figure 6 are interesting since they show behavior that is rarely documented.
A transition takes place in Figure 6. The pdf for cumulative forecast streamflows close to June 1st (now) has a much lower standard deviation and variance than the pdf for the observed flows. Over time, the effect of watershed initial conditions, or of the watershed state at now, damps out, (10) and the mean and the variance of the forecast and historic flows become similar. At some future time the pdfs for forecast and historic flows will merge.
The results in Figures 3, 4, 5 and 6 are logical given hydrologic processes. Historic streamflows from June 1st to December 31st are influenced by a wide range of June 1st watershed states, and this variability in initial watershed states increases the variance of subsequent streamflows. Forecast streamflows that use a single June 1st watershed state have a lower variance than the historic streamflows. The lower variance is most noticeable near June 1st (now). The variability in June 1st watershed states, from one year to the next, will cause forecast streamflows to have a higher or lower median than the historic streamflows, but the forecast streamflows will always have a lower variance.
The transition in Figure 6 illustrates persistence of streamflows due to the watershed state at now. Streamflow persistence is watershed dependent. Persistence is minimal in watersheds with shallow soils and limited sustained baseflow from perched aquifers. Persistence is much larger in watersheds with snow, a high range in soil moistures over time, and perched aquifers with substantial storage. High streamflow persistence and a strong watershed state/streamflow interdependence lead to lengthy transitions from Figure 6(a) to Figure 6(c).
...quoted by Robert Fulgrum
The person quoted by Fulgrum must have an extraordinary tolerance for risk! When does the risk become too great?
All but the most conservative reservoir operations risk water delivery deficits. To make decisions based on risk a reservoir operator must first understand what the risks are. Decisions depend on probabilities of future reservoir inflows, a topic that is foreign to many who are affected by reduced water deliveries. Decisions to restrict water deliveries to reduce deficit risks must be made in advance: Delaying a decision to reduce water deliveries until a deficit is imminent increases the severity of water delivery restrictions. A demand reduction - exceedance probability plot, similar to those introduced by R. L. Moore, can help to clarify risks of reduced water deliveries for water supply or irrigation reservoirs. (11)
As a drawdown period continues the demand reduction consequences of deficits
will change, even if the deficit magnitudes remain the same. Figure 7 compares
deficit exceedance probability plots for
As the drawdown period advances, the demand reductions that need to be undertaken to prevent a sudden loss of water deliveries become more stringent even when the deficit risk remains the same. A 10,000 ac-ft deficit may require a 20% demand reduction over 6 months, a 40% demand reduction over 3 months, or an 80% demand reduction over 1 1/2 months.
Trends in deficits or demand reduction during the drawdown season are
important. Are conditions getting better or worse? An analysis of the 1992
drawdown season in
Deficit exceedance probability plots and demand reduction - exceedance probability plots both tend to become more volatile toward the end of the drawdown season.
Many people have contributed to our work on simulation and reservoir operations. Laura Marino at Hydrocomp is the architect of the Seafm modeling system, and of subsequent forecast/analysis systems. Charles D. D. Howard of Charles Howard and Assoc. (Victoria, B. C.) has participated in the development of these modeling systems, and has contributed many ideas and criticisms. Tom Johanson of the Seattle Water Department has given us valuable ideas and suggestions based on his experiences using modeling in day-to-day reservoir operations.