Stochastic Reservoir Theory: An
Outline of the State of the Art as
Understood by Applied
Probabilists
Anis, A.A. and Lloyd, E.H.
IIASA Research Report
September 1975
Anis, A.A. and Lloyd, E.H. (1975) Stochastic Reservoir Theory: An Outline of the State of the Art as Understood
by Applied Probabilists. IIASA Research Report. IIASA, Laxenburg, Austria, RR-75-030 Copyright © September
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STOCHASTIC RESERVOIR THEORY:
AN OUTLINE
O F THE STATE O F THE ART
UNDERSTOOD
BY A P P L I E D P R O B A B I L I S T S
A.A.
E.H.
Anis
Lloyd
September 1975
R e s e a r c h R e p o r t s are p u b l i c a t i o n s r e p o r t i n g
o n t h e w o r k of t h e a u t h o r s . A n y v i e w s o r
c o n c l u s i o n s a r e t h o s e of t h e a u t h o r s , and
do n o t n e c e s s a r i l y r e f l e c t those of I I A S A .
Stochastic Reservoir Theory: An Outline
of the State of the Art as Understood
1
by Applied Probabilists
A.A. Anis
1.
2
and E.H. Lloyd
3
Introduction
The relation between water-storage systems in the real
world and the system encompassed by the basic theory (due to
Moran [15,16]may be described as follows in Table 1.
~t first sight this model would appear to rest on simplifications that are so drastic as to render it devoid of
mathematical interest or of practical potentialities. In fact,
however, this is not the case. As far as purely mathematical
developments are concerned, the interest that these simplifications have aroused is amply illustrated by the so-called
"Dam Theory," in which the basic Moran theory was transformed
largely by Moran [I41 himself and by D. G. Kendall [ I 01 into
a sophisticated corpus of pure mathematics dealing with continuous-state, continuous-time stochastic processes. (For
comprehensive contemporary surveys see Gani [61, Moran [151,
and Prabha [ 22I
.
It is true that the "reservoirs" in Dam Theory are of
infinite capacity, that the release occurs at unit rate, and
that the "water" involved posesses no inertia and no correlation structure of any kind, so that the inflow rate at t + Bt
is statistically independent of the inflow rate at time t,
however small the increment Bt. This lack of realism does not
detract from the beauty of the mathematics involved, but it does
limit the possibilities of applying the theory to the real world.
.-
-
his
report is an expanded version of a seminar presented by the authors at IIASA, in i4ay 1975.
in
Shams University, Cairo, Er~y~t.
3~niversitYof Lancaster, UK.
-2-
Table 1.
Real World Situation
Basic Moran theory
Storage
mechanism:
System of interconnected
reservoirs
Single reservoir
Inflow
process
Continuous in state and
time
Discrete in state
and time
Seasonal
Non-seasonal
Autocorrelated
Independent increments
Losses
Evaporation, seepage
Ignored
Capacity
Time-dependent due
to silting
Constant
Release
procedure
Continuous in state and
time
Discrete in state
and time
Related to past, current
and predicted future
contents
Constant rate of
release
A second avenue of development, which is directed specifically towards engineering applicability, is also possible. This
development views the basic Moran theory as an extremely ingenious abstraction from reality, so constructed as to allow
modifications which are capable of removing practically all the
restraints listed in Table 1 above. In particular, seasonality
in inflow and outflow processes may be accommodated; the errors
involved in approximating continuity by discreteness may be reduced to an acceptable level by working in appropriate units;
flexible release rules may be built in; and, most important,
the original requirement of mutual independence in the
sequence of inflows may be abandoned, and realistic autocorrelation structures incorporated by using Markovian approximations of arbitrary complexity (Kaczmarek [8], Lloyd [I21 )
What this development of the theory produces is the
probabilistic structure of the sequence of storage levels and
overspills in the reservoir -- both structures being obtained
in terms of the size of the reservoir, the inflow characteristics and the release policy. (See for example Odoom and
Lloyd [191 , Lloyd [I21 , Anis and El-Naggar [2], Gani [51 ,
Ali Khan [1 1 , Anis and Lloyd [3], Phatarfod and Mardia [21];
and review articles by Gani [7] and Lloyd [131 ) Thus the
effect of varying the release policy may be determined, with a
view to optimizing various performance characteristics.
The theory can therefore be said to have reached a fairly
satisfactory form, and regarded as a probabilistic model.
The same cannot perhaps be said of the statistical estimation
and modelling procedures needed for the practical application
of the theory, and since the usefulness of the theory must
depend on the reliability of the numerical estimates of the
probability distribution and autocorrelation structure of the
inflow process, it is the authors' opinion that particular
attention ought now to be concentrated on these statistical
problems. This is discussed further in Section 9.
.
.
2.
The Basic Moran Model:
Inde~endentInflows
In this section we describe the simplest version of the
model, as outlined by the following diagram of the reservoir
(see Figure 1). The llscheduling"or "programming" required by
the fact that continuous time is being approximated by discrete
time is shown in Figure 2.
Available Inflows {xt}(1ID)
Additional
"intermediate Storage"
W
Effective
Capacity
!
Contents {zt}
0
Desired release Dt = w
Actual release Dt = Wt < w
Figure 1.
time epochs
Figure 2.
Here Xg denotes the acceptable part of the inflow, in
the light of the restraints imposed by the finite capacity of
the reservoir. Similarly Dg (= W ) is the feasible part of
t
the desired outflow w, taking into account the fact that the
reservoir may contain insufficient water to meet the whole of
the demand.
In the Moran sequencing illustrated above, the outflow
W is supposed to occur after the inflow has been completed.
t
The inflow quantities Xt = 0.1 ,..., the outflow quantities
= 0 ~ 1 , . ,w, and the storage quantities Zt = 0 , l ,... ,c
Wt
are all quantized, all being expressed as integer multiples of
a common unit. The "intermediate storage" indicated in the
figure is required for the operation of the program.
The inflow process {X
is supposed to be IID: that is,
t
the random variables
..
are supposed to be mutually independent, and identically distributed.
Taking account of the finite size of the reservoir and of
the sequencing imposed by the "program" we may formulate the
following stochastic difference equation for the storage
process {zt1:
which we shall abbreviate when necessary in the form
Here the constant, w, represents the desired outflow Dt.
The actual outflow Wt is given by
so t h a t
The a c t u a l l y a c c e p t e d i n f l o w i n t o t h e r e s e r v o i r d u r i n g
(t,t
+
I ) , a s d i s t i n c t from t h e a v a i l a b l e i n f l o w X t ,
is
and t h e q u a n t i t y l o s t t h r o u g h o v e r f l o w i s
I t i s worth p o i n t i n g o u t t h a t , w h i l e p r e s e n t a t i o n s o f
t h i s t h e o r y o f t e n c o n c e n t r a t e on t h e d e t e r m i n a t i o n of t h e
s t o r a g e p r o c e s s { z t l , b o t h t h e o u t f l o w p r o c e s s {wtl and t h e
s p i l l a g e p r o c e s s {St} a r e p r o b a b l y more i m p o r t a n t i n a p p l i c a t i o n s , p a r t i c u l a r l y when it i s d e s i r e d t o o p t i m i z e t h e
r e l e a s e p o l i c y , s i n c e t h e f u n c t i o n t o b e o p t i m i z e d w i l l norm a l l y depend d i r e c t l y on t h e s e two p r o c e s s e s .
(An a l t e r n a t i v e "programming,11 which- may b e d e s c r i b e d a s
a n e t i n f l o w scheme, and which d i s p e n s e s w i t h t h e need t o
i n t r o d u c e a n i n t e r m e d i a t e s t o r a g e zone, i s shown i n F i g u r e 3.
Here t h e t o t a l a c c e p t a b l e i n f l o w X z t h a t o c c u r s d u r i n g ( t , t + 1 )
and t h e t o t a l o u t f l o w Wt
a r e assumed t o b e s p r e a d o v e r t h e
e n t i r e i n t e r v a l ( w i t h s u i t a b l e m o d i f i c a t i o n s when a boundary
i s reached) both t a k i n g p l a c e a t c o n s t a n t r a t e during t h a t
i n t e r v a l , s o t h a t t h e y combine t o form a s i n g l e " i n f l o w " of
The
magnitude Xt
W t , t h i s b e i n g n e g a t i v e i f Xt < Wt.
s t o c h a s t i c difference equation f o r {ztl i s then
-
which coincides with the equation (1) obtained for Moran's
own program. Similarly the actual outflow Wt and the spillage
St are the same as under Moran's programming.)
Lt
Lt+l
I
I
f) time
Figure 3.
3.
The Storage Process
{ztI
{xtI
for the Basic Moran Model
to be an IID process, and D+
AS before, we take
Then {zt] is a lag-1 Markov Chain. For. since (by (2))
=
w.
where
the information relating to Zt,Zt-l,..., is suppressable on
account of the assumed structure of {xtI. Since (4) depends
on s but does not depend on sl,s",,.., the result follows. Thus
where
and
q(r,s)
= P(Zt+,
=
r ( z t = s)
,
f o r r , s = 0.1
,...,c .
I n a n o b v i o u s m a t r i x n o t a t i o n t h i s may b e w r i t t e n
-5t + l
-
-
QCt
t = 0,1,...
whence
Thus t h e d i s t r i b u t i o n v e c t o r
L~
of storage a t t i m e t i s
determined i n t e r m s o f t h e i n i t i a l c o n d i t i o n s -0
5
'
In a l l
Q may b e assumed
" r e a l i s t i c " s i t u a t i o n s t h e t r a n s i t i o n matrix -
t o be e r g o d i c , whence, f o r s u f f i c i e n t l y l a r g e v a l u e s of t ,
5 = 5 where
-t
5 b e i n g t h e " e q u i l i b r i u m d i s t r i b u t i o n " v e c t o r which i s t h e
-
unique p o s i t i v e n o r m a l i z e d s o l u t i o n of t h e honogeneous l i n e a r
a l g e b r a i c system
The m a t r i x Q, of o r d e r ( c
+
1)
x
(c
+
1)
,
h a s a s i t s ( r ,s )
element
and f o r any g i v e n f u n c t i o n h ( * ) , t h i s i s e a s i l y o b t a i n e d
i n terms o f t h e i n £ low d i s t r i b u t i o n P (Xt
j = O , l , . . . , L3.
= j)
= f ( j ), say,
A s a s i m p l e example, t a k i n g w = 1 , w e have:
(for r = 1, 2 , . .
.,c)
and
4.
The Y i e l d P r o c e s s i n t h e B a s i c Iloran PIodel
The y i e l d i s t h e a c t u a l q u a n t i t y r e l e a s e d from t h e
r e s e r v o i r , d e f i n e d by ( 2 ) a s
Wt
= min
(Zt
+
Xt,w)
Because of a) the "lumping" of Z-states implied by this
function, and b) the addition process Zt + Xt, the yield
process {W t is not Markovian. For most purposes it is best
studied as a function defined on the llarkov Chain {zt), with
conditional probabilities
This is, for each r and s, a well-defined function of the
distribution of Xt, which allows us to evaluate (for example)
the expected yield at time t as
For other purposes it may be convenient to use the fact
that the pair {Zt,Xt} forms a bivariate lag-1 Markov Chain.
5.
The Basic Moran Model with Flexible Release Policy
If one modifies the release policy Dt so that, instead
of being a fixed constant w, Dt is, for example, a function
of the current values of Z and of X--say a monotone nondecreasing function of Zt + Xt--the effect of this on the
theory outlined in Section 5 is merely to modify the transiQ, without altering the Markovian structure of
tion matrix {zt}. Equation (I), whether under Moran programming or netinflow programming, is replaced by
Zt+l
say.
=
min (zt
+
Xt,c + Dt) - min (Zt + XttDt)
This still holds good, and Zt maintains its simple
(12)
Markovian character, if Dt also depends on some additional
random variable Yt which may be correlated with Xt, provided
that the sequence Yt consists of mutually independent elements.
The actual outflow Wt is given by
the analogue of (2), and the spillage becomes
st
=
max (zt + Xt
-
Dt
-
c,O)
.
(14)
If for example Dt is defined by the following Table 2:
Table 2.
we may construct Table 3 with entries such as the following
(for, say, c = 8) as found in Table 3 below. Row (a) indicates
a situation giving no spillage, (b) one where five units
are spilled, (c) one where several combinations of Z and Xt
t
lead to the same value of Zt+l, in which case we have
whereas q(O,O) = f(O), a single term.
6.
The Basic Moran Model Operating Seasonally
The effect of working with a multi-season year may be
adequately illustrated in terms of a two-season year (See Table 4):
Table 3 .
Dt
Zt+l
Table 4 .
Year t
Year t
Season
0
Epoch:
Storage
distrib.
vectors
Transition
matrix
Season
Season
1
0
i
+
1
Season
1
Year t
+
- - -
2
Here
z (t) is the storage
where Q = Q0Q 1, and at the beginning of season 1 of year t.
year storage sequence at this season is
Chain, with transition matrix Q = QoQ1,
theory applies.
7.
distribution vector
Clearly the year-toa homogeneous llarkov
and the preceding
Reservoir Theory with Correlated Inflows:
Basic Version
F4oran's basic theory is applicable as a first approximation, more or less without modification, to a reservoir providing
year-to-year storage, in which there is a well-defined
"inflow season" with no outflow, followed by a well-defined
and relatively short "outflow interval," a situation approximated by the conditions on the Nile at Aswan. This approximation holds only to the extent that inter-year inflow correlations can be neglected, which may not be totally unreasonable
for a one-year time scale. The approximation becomes increasingly inaccurate if one reduced the working interval from a
year to a quarter, or a month, or less, and it becomes essential to provide a theory that allows the inflow process to have
an autocorrelation structure. It was pointed out simultaneously
in independent publications in 1963 by Kaczmarek [8] and by
Lloyd that this could be done by approximating the actual
inflow process by a Markov Chain.
In the simplest form of this theory we may consider a
discrete-state/discrete-time reservoir, with stationary
inflow process ( ~ ~ where
1 , Xt is a finite homogeneous lag-1
Markov C h a i n w i t h e r g o d i c t r a n s i t i o n m a t r i x B = b
where brs = P(Xt+l =
(Xt
rlxt
say,
= s ) , with r,s = 0,1,...,n
b e i n g assumed < n f o r a l l t ) and t h e d i s t r i b u t i o n v e c t o r
is 5 , where 5 i s t h e unique non-negative normalized
s o l u t i o n o f t h e homogeneous s y s t e m
o f Xt
With a r e l e a s e p o l i c y Dt which may b e a f u n c t i o n o f Z t ,
and Xt-l,
t h i s s t o c h a s t i c d i f f e r e n c e equation f o r {zt)
i s t h e same a s i n ( 1 2 ) , w i t h t h e s u p p l e m e n t a r y i n f o r m a t i o n t h a t
Zt-l,
Xt
{ x t ) i s a Markov C h a i n .
I t i s e a s y t o show, by a n a r g u m e n t
e n t i r e l y a n a l o g o u s t o t h a t employed i n S e c t i o n 3 , t h a t { z t )
i s no l o n g e r a Markov C h a i n , b u t t h e o r d s r e d p a i r { z t ,xtl
f o r m s a b i v a r i a t e l a g - 1 ~ a r k o vC h a i n , t h a t i s , t h a t
i s independent of i ' , j l , i " , j " ,
where
...,.
The v e c t o r :
represents the joint distribution vector of Zt and Xt, and this
is determined by a vector equation of the form
M is the relevant transition matrix, of order
where (c + 1 ) (n + 1 ) x (c + I ) (n + 1 )
The structure of M
- is obtained
from the stochastic equation for {zt) and the inflow transition
matrix. If for example we take the simple release policy
Dt = 1, the equation (17) in partitioned form becomes
.
(n + 1 )
where the M
- (r,s) are submatrices of order (n + 3 )
which may most easily be defined in terms of the following
representation of the inflow transition matrix B. Let
where bS represents the s-column of B, that is
and let
Then, in formal agreement with (8), we find
,
for r
=
1,2,. ..,c
and
Equation (17) has as its solution:
giving the joint distribution vector of Zt and Xt in terms of
the initial vector L, this being the analogue for Markovian
inflows of the equation (5) for independent inflows. The
analogue of (6) is given by the statement that, for large
where ~r is the "joint equilibrium
values of t, T~ 2 3
distribution" vector which is the unique normalized non-negative
solution to the homogeneous system
(M-I)
-Tr
= 0
.
One extracts the distribution Z from the joint distrit
bution by using the result that
where, for given r, the terms P(Zt
=
rrXt = s) are elements of
t h e v e c t o r ~ ( r of
)
the vector
8.
zt
( 3 6 ) , t h i s i n t u r n b e i n g a s u b v e c t o r of
g i v e n by ( 21 )
.
R e s e r v o i r Theory w i t h C o r r e l a t e d I n f l o w s :
Elaborations
The t h e o r y d e s c r i b e d i n S e c t i o n 7 r e f e r s t o a s t a t i o n a r y
lag-1 Markovian i n f l o w p r o c e s s , and a c o n s t a n t - v a l u e r e l e a s e
policy.
Dt
The r e p l a c e m e n t of a c o n s t a n t r e l e a s e by a r e l e a s e
depending on Z t , X t , Z t - l
random e l e m e n t s Yt
t h e m a t r i x M.
and Xt-l
and p o s s i b l y f u r t h e r
i s a c h i e v e d by s u i t a b l y modifying
and Yt-l
T h i s d o e s n o t a l t e r t h e s t r u c t u r e of t h e
( A s h a s been p o i n t e d o u t by Kaczmarek [9]
{ Z ,Xt} process.
t
t h e s t o r a g e p r o c e s s { z t } becomes a lag-2 Markov c h a i n p r o v i d e d
t h a t t h e e q u a t i o n ( 1 2 ) c a n be s o l v e d t o g i v e a unique v a l u e of
Xt
f o r e a c h p a i r ( Z t , Zt + l
)
.)
F u r t h e r , t h e i n t r o d u c t i o n of s e a s o n a l l y v a r y i n g i n f l o w
and o u t f l o w p r o c e s s e s may be a c h i e v e d by u s i n g t h e m a t r i x
p r o d u c t t e c h n i q u e d e s c r i b e d i n S e c t i o n 6.
A s i n t h e c a s e of
i n d e p e n d e n t i n f l o w s , once one h a s o b t a i n e d t h e d i s t r i b u t i o n of
one may o b t a i n t h a t of t h e a c t u a l r e l e a s e Wt and t h e
t
s p i l l a g e S t , and u t i l i z e t h e s e i f d e s i r e d i n o p t i m i z a t i o n
Z
studies.
We have spoken s o f a r of a lag-1 Markovian i n f l o w .
G e n e r a l i z a t i o n s t o a m u l t i - l a g Markovian i n f l o w a r e immediate:
w i t h , f o r example, a l a g - 2 Markov i n f l o w one c o n s i d e r s t h e
q(t) = {Z
.
X X
T h i s w i l l b e a lag-2
t ' t' t-1
( t r i v a r i a t e ) Markov c h a i n .
S i m i l a r l y we may accommodate a
three-vector
m u l t i v a r i a t e inflow process.
Suppose f o r example we have a
b i v a r i a t e i n f l o w p r o c e s s {xt )
(2) 1,
,Xt
i n which t h e components
a r e cross-correlated a s well a s s e r i a l l y correlated.
This
i n f l o w p r o c e s s w i l l be s p e c i f i e d by an a p p r o p r i a t e t r a n s i t i o n
m a t r i x whose e l e m e n t s r e p r e s e n t t h e c o n d i t i o n a l p r o b a b i l i t i e s
u s i n g any s u i t a b l e o r d e r i n g c o n v e n t i o n , f o r example t h e
following Table 5.
Table 5.
Values a t t i m e t
x(l)
0
0 l...n
0
...
1
...
0 l...n
.
I
I
I
n
I
I
0
1
value
at
1
time
t + l
n
n
0 l...n
I
I
I
I
0
-
n
--
I
- - -
I
o
I
1
I
--
I
I
I
n
I n t h i s example t h e t r i p l e t { z ~ , x : ' )
,xi2) 1 would
( t r i v a r i a t e ) lag-1 Markov c h a i n (Lloyd [ I 1 ]
9.
C l o s i n s Remarks:
,
be a
Anis and Lloyd [ 4 1
Pers~ectives
I n t h e a u t h o r s ' o p i n i o n , t h e . p r o b a b i l i s t i c framework
.
provided by the model described above is adequate for practical
purposes, and the next steps ought to be concerned with the
statistical problems of specifying families of few-parameter
inflow transition matrix models and estimating their parameters.
In the case of a nonseasonal univariate lag-1 inflow process,
for example, it is likely that the available information will
be best adapted to estimating the inflow distribution vector
p (which is of course a standard and well-understood statistical procedure) and the first few autocorrelation coefficients,
say P1 and P2* A model involving these directly would be
particularly welcome. In the case of where an exponential
autocorrelation function
were thought to be appropriate, the Pegran [ 2 0 1 transition
model
that is
where
and
1, r =
S
cS(r,s) =
0,
otherwise
satisfies these requirements. Further development along
these lines, making m a discrete-state and discrete-time
framework, and avoiding the difficulties inherent in the use
of transformations of the normal autoregressive model,
(Moran, [ 1 7 1 ) , would be highly desirable, particularly
in the case of multivariate inflows.
Acknowledgements
The authors wish to express their indebtedness to IIASA,
whose guests they were whilst this report was being prepared.
-21
-
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