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Reliability Concepts in Reservoir Design

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A reservoir's function is to regulate natural inflows, which vary irregularly, to provide outflows at a more regular rate that is determined by water demand for one or more uses (water supply, irrigation, hydropower), temporarily storing the surplus, when inflows exceed outflows. Reservoir reliability is defined as the probability that the reservoir will perform the required function, i.e. provide the outflow required to satisfy the water demand, at a specified period of time under stated conditions. The traditional reservoir design procedures are more commonly based on empirical approaches. It is shown, however, that the reliability concept is a more rational basis, and provides easy and accurate computational procedures, for reservoir design and operation. Under some simplified assumptions, a simple explicit expression relating reservoir size, yield and reliability is extracted. This expression can be used for preliminary stages of a reservoir design. For more detailed and accurate studies, a generalized solution procedure based on stochastic simulation of inputs is outlined.
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The Encyclopedia of Water
Reliability Concepts in Reservoir Design (SW-776)
Demetris Koutsoyiannis
Department of Water Resources, Faculty of Civil Engineering, National Technical University, Athens,
Heroon Polytechneiou 5, GR-157 80 Zographou, Greece
dk@hydro.ntua.gr
Introduction
The reliability of a system is defined to be the probability that a system will perform the
required function for a specified period of time under stated conditions (Chow et al., 1988, p.
434). Reliability is the complement of probability of failure (or risk), i.e. the probability that
the “loading” will exceed the “capacity”. Denoting α the reliability, β the probability of failure
and P[ω] the probability of an event ω, the mathematical expression of this definition is
a := P[L(t) < C(t); t Π] =: 1 – β (1)
where L(t) and C(t) represent the loading and capacity, respectively, at time t, within a certain
time period Π (e.g. a year). Failure of a system can be classified as structural failure and
performance failure (Tung, 1996, p. 7.3). Structural failure involves damage of the structure
or facility, hindering its ability to function as desired in the future, whereas performance
failure does not necessarily involve structural damage but rather inability of the system to
perform as desired at some time within the period of interest, which results in temporary
unfavorable consequences.
Reservoir dynamics
A reservoir’s function is to regulate natural inflows, which vary irregularly, to provide
outflows at a more regular rate that is determined by water demand for one or more uses
(water supply, irrigation, hydropower), temporarily storing the surplus, when inflows exceed
outflows. The reservoir dynamics are more conveniently expressed in discrete rather than
continuous time. The quantities that are necessary to describe dynamics are the following:
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 2
Storage St. More precisely known as active storage, it is the volume of water stored, at
time t, above the minimum level, which is determined either technically (i.e. as the level of
the lowest valve of off-take) or legally by a decree imposing rules for a reservoir’s operation.
Active storage St ranges between zero and a maximum value c imposed by the reservoir size,
which corresponds to the level of the spillway crest (or some specified level above it in case
that sluice gates are constructed over the spillway). During floods, excess water is routed
through the spillway, which causes temporary storage above the normal limit c. This is known
as flood control storage. Water storage below the minimum level is known as dead or inactive
storage and it serves two main purposes: It provides volume for sediment accumulation and
environmental protection, as it protects the habitat of the reservoir during dry periods by
hindering complete emptying. Associated to the last function is also the conservation of the
quality of landscape. This article is focused on the design of the active storage of a reservoir;
some notes on the additional storage zones are contained in the last section of the article.
Net inflow Xt. It is the algebraic sum of cumulative inflows to the reservoir from time t – 1
to time t, minus the losses during the same time period. Inflows include runoff from the
catchment upstream of the reservoir (typically, the main component of inflows), rainfall to the
surface area of the reservoir and, possibly, water artificially conveyed from other sources (e.g.
inter-basin transfers through tunnels or pipe-lines). Losses include evaporation from the
surface area of the reservoir and possibly seepage to groundwater and leakage under or
through the dam.
Water demand, δt. It is the sum of all water requirements for the different water uses
served by the reservoir for the time period (t – 1, t). The demand may vary with time (e.g. due
to seasonal agricultural demand or due to some rule, usually based on the quantity of water in
the reservoir).
Release, Rt. Also known as draft, withdrawal or outflow, it is the actual amount of water
taken from the reservoir in an attempt to satisfy water demand during the time period (t – 1, t).
When there is a sufficient amount of water in the reservoir, Rt equals demand δt; otherwise
Rt < δt.
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 3
Spill, Wt. It is the excess water that, during times of floods and simultaneously high
reservoir storage, cannot be stored in the reservoir due to the upper reservoir storage limit c.
The reservoir dynamics are easily expressed by means of the mass conservation, or
equivalently, water balance equation. Considering that St is limited between 0 and c the water
balance equation is easily formulated as
St = max[0, min(St – 1 + Xt δt, c)] (2)
In addition, the release is determined as
Rt = min(St – 1 + Xt, δt) (3)
and the spill as
Wt = St – 1St + Xt Rt = max[0, St – 1 + Xt δtc)] (4)
Equations (2)-(4) apply when the inflow and withdrawal occur at constant rates throughout
the period (t, t 1) – this could be called the “steady” model. A simple modification to the
equations allows for the case where the inflow (or withdrawal) is highly seasonal so can (in
the limit) be modeled as a sudden occurrence; this could be called the “sudden” model. These
two models (called “simultaneous” and “staggered” by Pegram (1980)) bound all the behavior
observed in real reservoirs.
Definition of reliability applied to reservoir
Now, the above stated general definition of reliability (also known as dependability; e.g.
Raudkivi, 1979, p. 312) can be applied to a reservoir. It is observed that the failure of a
reservoir’s function is a performance failure, i.e., a failure to meet the water demand. At time t
the loading is the water demand δt and the capacity is the sum of St – 1 (storage at time t – 1)
and Xt (inflow from time t – 1 to t). Thus, application of (1) yields
a = 1 – β = P[δt < St – 1 + Xt] (5)
Considering (2) and (3), the following equivalent and more convenient expressions are found
a = P[St > 0], β = P[St = 0] (6)
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 4
and
a = P[Rt = δt], β = P[Rt < δt] (7)
In this context, the demand δt is regarded as a known quantity at any time instant t. All
other involved quantities, namely St, Xt, Rt, and Wt, are regarded as random variables. Given
the storage capacity c, the demand δt and the probability distribution and autocorrelation
functions of the input Xt, in theory, the probability distributions of output variables St, Rt, and
Wt can be determined in terms of that of Xt; this, however, is not an easy task due to the
nonlinearity of the dynamics expressed in equations (2)-(4). Theoretically, once the
distribution function of St or Rt has been determined, the reliability α is directly obtained from
(6) or (7), respectively. However, although the theoretically based calculations unavoidably
involve the complete knowledge of the distribution function of St and Rt, the reliability α can
be determined in an alternative, much simpler, manner. That is, under the assumptions of
stationarity and ergodicity, a can be estimated from a historical (for an existing reservoir) or
synthesized (via simulation) time series of storage st or release rt with adequate length n.
(Here lower case symbols were used for values of the random variables St and Rt.)
Specifically, the estimate of a based on (6) is
α = 1
n
t = 1
n
[1 – U(–st)] (8)
where U(x) is the Heaviside’s unit step function, with U(x) = 1 for x 0 and U(x) = 0 for
x < 0. Correspondingly, the estimate of a based on (7) is
α = 1
n
t = 1
n
U(rtδt)] (9)
The purpose of the sum in (8) (or (9)) is to count the periods where storage is not zero (or
release rt equals the demand δt). Thus, reliability is expressed as the proportion of time steps
in which the system performs as desired. For this reason, α has also been termed time-based
reliability. Here it must be observed that although (5)-(7) are all mathematically equivalent to
each other, as are (8) and (9), when applied to historical time series they may result in
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 5
different estimates. For example, a city’s water supply may not be allowed to empty, as
restrictions on releases are applied before this situation is reached. In such a case, (8) may
result in the erroneous estimation α = 1, whereas (9) will estimate the reliability correctly if
the desired demand, before restrictions, is entered into the calculations. Thus, (9) is preferable
when dealing with historical time series but in mathematical simulations both are equivalent;
moreover, application of (8) is faster as it does not require simulation of releases at all (only
(2) needs to be applied).
The stationarity assumption that was inherent in the above analysis is satisfactory when the
time step is a year (either calendar or hydrological). However, the annual time step is usually
too large and hides the variation of both inflows and demand within a year, which may result
in a failure some time within the year that is recovered in the end of the year. Therefore, a
smaller time step (e.g. monthly) is usually chosen, so that one year corresponds to k > 1 (e.g.
12) time steps. At this finer time scale all processes depend on the time step in a periodic
manner, that is, they are cyclostationary. Yet the reliability and failure probability are usually
expressed at the annual scale, in which stationarity is redeemed. To shift from the finer scale
to the annual scale, the rule adopted is that a failure occurring in one or more finer-scale time
steps is regarded as a failure for the year. With this rule, (6) becomes
a΄ = P
I
i = 1
k
(St – i > 0)
, β΄ = P
U
i = 1
k
St – i = 0
(10)
where to distinguish from the time-based reliability the symbols a΄ and β΄ were used instead
of α and β, whereas the symbol ‘I’ indicates that all of the following events should occur
simultaneously and ‘U’ indicates that any of the following events should occur. In a similar
manner, (7) becomes
a΄ = P
I
i = 1
k
(Rt – i = δt – i)
, β΄ = P
U
i = 1
k
Rt – i < δt – i
(11)
or alternatively
a΄ = P
i = 1
k
Rt – i =
i = 1
k
δt – i
, β΄ = P
i = 1
k
Rt – i <
i = 1
k
δt – i
(12)
Likewise, (8) and (9) become
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 6
a΄ = k
n
p = 1
n / k
min{[1 – U(–st)]; t = k (p – 1) + 1, …, k p} (13)
a΄ = k
n
p = 1
n / k
min{U(rtδt) ; t = k (p – 1) + 1, …, k p} (14)
respectively. The sums in (13) and (14) count the number of years in which no failure has
occurred. Apparently, a΄ and β΄ provide information on the occurrence of a failure within a
year and not on the time period the failure lasted. Therefore, they have been known as
occurrence-based reliability and failure probability, respectively. An overall indication of the
time period that failures last within an average year can be obtained by applying equation (8)
or (9) and estimating β, the time-based probability of failure.
Apart from the occurrence-based and time-based reliability, an additional reliability
measure has been often used, which is not expressed in terms of probability (and thus,
literally does not comply with the general definition of reliability). This is the so-called
volumetric or quantity-based reliability, expressed as the ratio of the average release to
demand, i.e.,
aV = 1 – βV = E
i = 1
k
Rt
/
i = 1
k
δt (15)
Given that a failure occurring in a year does not extend over the whole year, and, in
addition, the release during the failure is not necessarily zero but some positive quantity
smaller than demand, it is easily concluded that
a΄ α aV (16)
Among the three measures of reliability, the most important and most frequently used is
the severest, i.e., the occurrence-based reliability a΄. Another means for expressing virtually
the same concept is the return period or recurrence interval of emptiness, T. This is the mean
time between two consecutive empty states of the reservoir and it is none other than the
reciprocal of the probability of failure β΄ (Pegram, 1980), i.e.,
T := 1 / β΄ = 1 / (1 – a΄) (17)
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 7
which is expressed in years (given that a΄ and β΄ are expressed on annual time scale). The
concept of return period of emptiness of a reservoir is similar to that typically used for design
floods. The difference is that in design floods, failure is the exceedance of the magnitude of
design flood, whereas in a reservoir failure is the emptying of the reservoir. Typical design
values of reliability and return period for reservoir design are a΄ = 99% (T = 100 years) for
municipal water supply reservoirs, a΄ = 70-85% (T = 3.3-6.7 years) for irrigation reservoirs in
subhumid climates and a΄ = 80-95% (T = 5-20 years) for irrigation reservoirs in arid climates
(Raudkivi, 1979. p. 313).
Traditional reservoir design procedures
Most hydraulic structures, e.g. flood protection works, drainage networks, etc., whose load is
randomly varying, have been designed on a probabilistic basis, adopting a certain reliability
level or, equivalently, a certain return period for the design flood. Traditionally, however, this
has not been the case in reservoir design, which has rarely been based on a sound probabilistic
basis. This is witnessed even from the terminology traditionally used. For example, the use of
the term firm yield implies a non-probabilistic, or failure-free concept. Specifically, firm
yield of a reservoir has been defined to be the draft or withdrawal that lowers the water
content in a reservoir from a full condition to its minimum allowable level just once during
the critical historical drought. (McMahon, 1993, p. 27.8; Chow et al., 1988, p. 534). It has
been characterized as essentially the no-failure yield (McMahon, 1993, p. 27.8). In a
probabilistic context, however, any draft has a non zero probability of failure (unless the
demand is less than a hypothetical lower bound of the inflow distribution, which can be
plausible only for perennial streams; this is unusual).
Several procedures have been widely used in reservoir design, which are rather
deterministic and not consistent with the reliability concept. The most common has been mass
curve analysis and its variations. A mass curve is a plot of cumulative inflow volumes
(typically based on historical discharge records) as a function of time. Using this plot the firm
yield, as well as the required reservoir storage to attain this firm yield, can be determined
graphically. In addition, the method can determine the required storage for a smaller target
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 8
release. This graphical method was developed one hundred and twenty years ago (Rippl,
1883) and has been widely used until now, although criticized (e.g. Schultz, 1976) for
providing no information on the probability of failure and for the fact that the reservoir
capacity determined by this method increases with the arbitrary length of available observed
inflow data. As shown by Feller (1951) this increase is asymptotically proportional to the
square root of the length of record.
A first variation of the method is its application with synthetic, rather than observed, data
(e.g. Schultz, 1976). This eliminates the drawback of the arbitrary length of record and also
provides some measure of uncertainty by applying the same procedure with different
generated synthetic series. However, this kind of description of uncertainty is not consistent
with a rational definition of reliability (e.g. that of the previous section).
A more theoretical flavor for the method has been given by the so-called range analysis,
commenced with the work of Hurst (1951; see also Kottegoda, 1980, p. 184). Range is
essentially the algebraic difference of the maximum and minimum departures of the mass
curve from the straight line that joins its starting and ending points. The range concept has
greatly contributed to the understanding and description of the so-called Hurst phenomenon
(see the entry SW-434 – Hydrologic Persistence and the Hurst Phenomenon) in hydrology,
climatology and other geophysical sciences. Applied to a reservoir, the range represents the
required storage of a reservoir operating without any spill or other loss and providing a
constant outflow equal to the mean flow. Obviously, this is an oversimplification of a real
reservoir. On the other hand, the range concept involves complexity in estimation, and
simpler and more efficient methods have been proposed which can be used instead of range
analysis (Koutsoyiannis, 2002).
An additional design method is the so-called sequent-peak analysis (Mays and Tung, 1992,
p. 274; Mays, 2001, p. 400). Essentially, it is a tabulated version of mass-curve analysis and
can also incorporate in the calculations, apart from runoff, the effect of precipitation,
evaporation and leakage. The method does not involve the reliability concept, nor does it
consider spills from the reservoir.
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 9
Simplified reliability-based procedures for reservoir design
As already mentioned, analytical determination of reliability in the general case of a reservoir
fed by inflows with seasonality and arbitrary probability distribution and autocorrelation
functions, is a very difficult, if not impossible, task (Pegram et al., 1980). Therefore, existing
analyses have been based on several simplifications. However, the results of such analyses are
very useful, at least for the initial stage of reservoir design. The typical simplifying
assumptions are to:
neglect secondary inflows (precipitation) and losses (evaporation, leakage);
neglect seasonality by the adoption of an annual time step;
neglect autocorrelation and assume that inflows are independent in time;
use a specific distribution function for inflows, typically two-parameter such as
normal, lognormal or gamma.
The objective of such probability-based theoretical analyses is to determine the relation of
the following three quantities:
reservoir size c; it is usually standardized as κ := c / σ, where σ is the standard
deviation of annual net inflow Xt;
demand δ, which is assumed constant for all years; it is usually standardized as ε :=
(µδ) / σ, where µ is the mean of annual net inflow Xt; ε has been termed the
standardized inflow (Hurst, 1951; McMahon, 1993, p. 27.7) or the drift (Pegram,
1980);
probability, expressed either as reliability α, probability of failure β, or return period T;
because of the annual time step used, the time-based reliability (equations (6)-(7)) is
identical to the occurrence-based reliability (equations (10)-(12)).
The first among the probabilistic approaches used in such analyses is the discretization of
reservoir storage into several zones, each representing a certain state, and the use of a Markov
chain model to represent transitions from state to state (Moran; 1959; Zsuffa and Gálai, 1987;
see also Kottegoda, 1980, p. 264).
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 10
A second method is stochastic (Monte Carlo) simulation, in which a long synthetic series
of inflows is generated from the appropriate distribution function and then transformed into a
series of storage values using equation (2); reliability is then easily determined from the
storage time series using (8). Gould (1960) using this method was able to propose a reservoir
size-yield-reliability formula, fitted to 240 sets of Monte Carlo simulations for various
combinations of demand, reservoir size and skewness of gamma distributed inflows. This is
(ε + 0.15) [κ + d1(α, γ)] = d2(α, γ) (18)
where d1 and d2 are coefficients depending on reliability α and skewness γ, and are given by
nomographs (see also Raudkivi, 1979. p. 323). McMahon and Mein (1986) adapted this
formula to more explicitly indicate the reliability; this can be estimated from the standardized
normal variate zα corresponding to α, using equation
zα = 2 ε (κ + d(α) σ/µ) (19)
where d is a coefficient depending on reliability α, and is given by a table (see also McMahon,
1993, p. 27.14).
A more accurate and rigorous theoretical methodology to estimate the reservoir size-yield-
reliability relationship has been developed by Pegram (1980). This has been based on finite-
difference and integral equations, which employ the reservoir dynamics equation (2) in a
probabilistic context to determine the return period of emptiness. Pegram applied his
methodology, for normal, lognormal and discrete inputs both independent and serially
correlated. His results, when compared to those of the Gould method (equations (18)-(19))
indicate that the latter underestimates the reservoir size required to attain a certain reliability
level. Using Pegram’s results for normally distributed inflows, which were verified and
expanded here with extended simulations, the following approximate relationship has been
established:
ln(T – 1) = 2 (ε + 0.25) (κ + 0.5)0.8 (20)
This is valid for T > 2 (α > 0.5) and can be alternatively written as
ln(T – 1) = –ln(1/α – 1) = (2/σ1.8) (µ + 0.25σδ) (c + 0.5σ)0.8 (21)
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 11
For known mean µ and standard deviation σ of inflows, (21) can directly yield either the
reliability α for known reservoir size and demand, or the reservoir size (c) for given demand
and reliability, or the demand (δ) that can be met with a given reliability for known reservoir
size. Equation (20) is graphically depicted in Figure 1 in comparison with Pegram’s exact
results. This equation can be suggested for preliminary estimations, but it should be applied
with caution for the reasons explained in the next section.
1.0
0.8
0.6
0.4
0.2
0.1
0
1
2
3
4
5
6
7
8
Design return period (T, years)
Standardized reservoir size (κ)
510 20 50 100
80 90 95 98 99
Design reliability level (α, %)
Figure 1 A simple representation of reservoir size-yield-reliability: Standardized reservoir size (κ = Κ / σ)
required to achieve a certain drift (ε = (µδ)/σ) with a certain reliability level, for independent inputs normally
distributed. Lines are constructed from equation (20) whereas plotted points are theoretical results by Pegram
(1980) for ε = 0.2 (triangles), 0.4 (diamonds), 0.6 (squares), 0.8 (circles) and 1.0 (stars).
Effects of inflow characteristics to reservoir size
As explained earlier, simplified design procedures such as the one using equation (20), are
based on a number of abridging assumptions about the inflows. Significant differences may
appear if these assumptions are not valid. More specifically, what may cause significant
departures from (20) are the hydrologic persistence, especially the long-term one, and the
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 12
seasonal distribution of inflow and demand. Less significant differences are caused by the
skewness of inflows and secondary inflows and losses.
The effect of hydrologic persistence is demonstrated in Figure 2, which depicts the
standardized reservoir size (κ) required to achieve two combinations of drift and reliability (ε
= 0.8, α = 98% and ε = 0.2, α = 90%) versus the lag-one autocorrelation coefficient, ρ. Two
cases of hydrologic persistence have been examined, short-term and long-term. In the case of
short-term persistence the inflows were assumed to follow the autoregressive process of order
1 (AR(1) or Markov), whereas in the case of long-term persistence they were assumed to
follow the fractional Gaussian noise (FGN) process with Hurst exponent H = ln(2 + 2ρ) / ln 4.
Obviously, the effect of persistence is very significant, especially in the case of long-term
persistence and high demand (low drift ε). Thus, the required reservoir size for ε = 0.2 and α =
90% is c = 2.4 σ when ρ = 0 (H = 0.5) and becomes 4 times larger (c = 9.6 σ) when ρ = 0.4 (H
= 0.74).
0
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5
Lag one autocorrelation (ρ)
Standardized reservoir size (κ)
Short-term percistence
(AR(1))
Long-term percistence
(FGN)
Figure 2 Effect of short-term and long-term hydrologic persistence of inflows to the standardized reservoir size
(κ) required to achieve a drift ε = 0.8 with a reliability level α = 98% (continuous lines) and a drift ε = 0.2 with a
reliability level α = 90% (dashed lines). Results are obtained by simulation.
To quantify the effect of seasonal variation of inflow and demand, it is observed that in the
worst case, the total annual inflow comes before the beginning of withdrawal (the “sudden”
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 13
model). The reservoir dynamics in (2) assume that inflow and withdrawal are distributed
evenly during a year (the “steady” model). If inflow precedes withdrawal, (2) should be
modified to read St = max[0, min(St – 1 + Xt, c) – δt]. It is then easily determined that an extra
storage equal to δ is required in addition to that estimated from (21). This worst case however
is not very realistic, except in arid regions. In real world cases, the extra storage capacity
required (in addition to that estimated from (21)) is a percentage of δ. This can be as high as
50% for water supply reservoirs and 80% for irrigation reservoirs in semiarid regions.
The effect of the skewness of inflows is demonstrated in Figure 3, which depicts the
standardized reservoir size (κ) required to achieve three combinations of drift and reliability (ε
= 0.8, α = 98%; ε = 0.2, α = 90%; and ε = 0.2, α = 98%) versus the coefficient of skewness,
Cs. It can be observed that the effect of skewness is not significant; for low draft (high drift ε)
this effect can be beneficial (lowering of required storage) but for high draft the required
storage is practically insensitive to skewness.
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Coefficient of skewness of inflows (Cs)
Standardized reservoir size (κ)
Figure 3 Effect of the skewness of inflows to the standardized reservoir size (κ) required to achieve a drift ε =
0.8 with a reliability level α = 98% (continuous line), a drift ε = 0.2 with a reliability level α = 90% (dashed line),
and a drift ε = 0.2 with a reliability level α = 98% (dotted line). Results are obtained by simulation using
independent two-parameter gamma distributed inflows.
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 14
Generalized simulation procedure for reliability-based reservoir design
The simplest general procedure to estimate the reservoir size-yield-reliability relationship in
an accurate and detailed manner, and for any arbitrary inflow characteristics is stochastic
(Monte Carlo) simulation. A simplified simulation procedure is outlined as follows:
1. Generate a series of inflows Xt at an appropriate timescale (e.g. monthly) using an
appropriate stochastic model (e.g. Koutsoyiannis, 2000, 2001; see also the entry SW-
913 – Stochastic Simulation in Water Resources).
2. Assume a reservoir size c.
3. Calculate a series of reservoir storages using (2).
4. Estimate the reliability using (13).
5. Repeat steps 3-4 for different reservoir sizes.
This procedure is very simple to execute even in a tabulated form on a spreadsheet.
The drawback of stochastic simulation is that it requires a vast simulation length to find
accurate results. It can be shown that the required number of simulated time steps (e.g.
months) to estimate the occurrence-based failure probability β΄ with an acceptable error ±ε β΄
and confidence γ is
n = k (z(1 + γ) / 2 / ε)2 (1 / β΄ – 1) (22)
where k is the number of time steps per year and zp is the p-quantile of the standard normal
distribution. For instance, for k = 12, γ = 95% ( z(1 + γ) / 2 = 1.96), ε = 10% and β΄ = 0.01 this
yields n = 456 000 months (38 000 years). Today, this is not a major problem as the required
computer time for such a simulation length can be less than one second in a common PC (but
not in a spreadsheet environment, which requires much more time).
The above-described simplified procedure can be extended to a detailed simulation
procedure, which includes, in addition to runoff, the precipitation, evaporation and leakage of
the reservoir. In this case, the level-area-volume and the level-leakage relationships are
required to establish the functions a(S) and l(S) which yield the reservoir area a and the
leakage l for any storage S (usually using interpolation from arrays of tabulated values). In
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 15
this case, the runoff Q, has to be expressed in equivalent depth units, as are precipitation P
and evaporation E. If f is the catchment area, the net inflow becomes
Xt = Qt [fa(St – 1)] + (PtEt) a(St – 1) – l(St – 1) (23)
where it was implicitly assumed that variations of the reservoir area and leakage within a time
step are not large, so that a(St – 1) and l(St – 1) can be assumed as representative for the entire
period within a time step. When the detailed simulation procedure is employed, three time
series, instead of one, have to be synthesized. Obviously, Qt and Pt are cross-correlated and
therefore they cannot be generated independently of each other; a bivariate stochastic model is
needed in this case. On the other hand, Et can be generated independently of the other two
series.
Obviously, the simulation procedure, either the simplified or the detailed one, can be
directly applied with historical, rather than synthesized inputs. However, due to small record
length, the accuracy of results in this case will not be satisfactory.
Extension of the simulation method, combined with optimization, for a multiple reservoir
system has been discussed by Nalbantis and Koutsoyiannis (1997).
Design of additional storage zones of a reservoir
The design of the flood control storage is typically reliability-based but on a very different
context from that described above for the active storage. A failure of the flood routing
function of a reservoir is not a performance failure but a structural one: an overtopping of the
dam due to a severe flood can result in collapse of the dam. Therefore much lower levels of
probability of failure are adopted, of the order of 10–3-10–6. The typical steps here are (a)
estimation of a design storm, based on statistical analysis of rainfall, for an appropriate return
period (103-106 years); (b) estimation of the inflow hydrograph using an appropriate rainfall-
runoff model; and (c) routing of this hydrograph through the spillway and estimation of the
outflow hydrograph and the maximum water level. Implicit assumptions in the entire
procedure, like the assumption that the reservoir is full at the beginning of the flood, decrease
the risk further. It must be noted that several procedures have been proposed that are
D. Koutsoyiannis, Reliability Concepts in Reservoir Design (SW-776) 16
supposedly risk-free. These are based on the so-called probable maximum precipitation
concept. However, it has been argued that a risk-free procedure is an illusion and the value of
probable maximum precipitation can be exceeded with a certain probability (e.g. of the order
of 10–5; Koutsoyiannis, 1999).
The sizing of the dead storage has been typically based on the expected sediment
accumulation in the reservoir for a certain design period (e.g. of the order of 102 years).
Because of the large design period and the accumulation character of this process, the
approach to this problem is very different. Expected values, rather than probabilities are
involved in the calculations. The additional objectives that the dead storage serves, i.e.
environmental protection (protection of the habitat of the reservoir during dry periods) and
conservation of the quality of landscape, have not been given special attention, until now, and
have not been considered in the design procedure. One would expect that, some years after
construction, these additional objectives would not be served adequately because the water in
the dead volume would be reduced due to sediment accumulation. Fortunately, however, this
has not been the case: the implicit assumption that sediments will reach the bottom of
reservoir near the dam is not verified. Thus, in large reservoirs, sediment accumulation occurs
mainly in the active zone of reservoir (near the entrance of the river to the reservoir) and
much less in the dead storage. The unfavorable consequence is the reduction of the active
storage. These problems need to be further investigated in future reservoir designs.
Acknowledgment
The review by the editor and the comments by G.G.S. Pegram are gratefully appreciated.
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