1) The document discusses reservoir sizing and operating policies. It describes methods for determining the required active storage capacity of a reservoir given inflows and demands over time, including the mass curve and sequent peak methods. 2) It then covers the standard operating policy for reservoir release decisions based on current period availability and demand. The policy aims to meet demand to the extent possible given current water availability. 3) Finally, it introduces the concept of deriving an optimal operating policy to maximize meeting demands over the long run, using linear programming to formulate the reservoir operation problem.

1. Chapter Seven: Applications of System Analysis
contents..
Reservoir sizing
Reservoir operating policy

2. 7.1 Reservoir sizing
The annual demand for water at a particular site may be less than the
total inflow there, but the time distribution of the demand may not
match the time distribution of inflow, resulting in surplus in some
periods and deficit in some other periods of the year. A reservoir serves
the purpose of temporarily storing water in periods of excess inflow and
releasing it in periods of low flow so that the demands may be met in all
periods. The problem of reservoir sizing involves determination of the
required storage capacity of the reservoir when inflows and demands in
a sequence of periods are given.

3. Cont.…
The total storage can be divided into three components: dead storage
(for accumulation of sediments), active storage (for conservation
purpose such as water supply and hydropower production), and flood
storage (for reducing flood peak). While each of these components may
be determined by separate modeling studies, we confine ourselves in
this section only to the determination of the active storage capacity of
the reservoir. The inflow to the reservoir is in fact a random variable.
The problem gets complicated if the randomness of the inflow has to be
taken into account.

4. Let us consider the simpler case of deterministic inflow for the purpose
of the present discussion, and also assume that the given inflow
sequence repeats itself. If the length of the inflow sequence is a year, it
means that the inflow in a given (within-the-year) period is the same in
all the years.
7.1.1.Mass curve method
One common method, extensively used in practice, is to determine the
active storage capacity using the Ripple diagram or the mass diagram by
plotting cumulative inflow with time.
Cont.…

5. The method involves finding the maximum positive cumulative
difference between a sequence of reservoir releases (equal to demands)
and historical inflows over a sequence of time periods in which the
demand is constant.
Cont.…

6. If the demand is constant in each time period, the method is quite simple
to apply. When the demand varies across time periods, the procedure
requires a plot of the cumulative deficits in time from the period in
which a deficit sets in, for the duration of the deficit, and finding the
maximum deficit among all such durations. The total deficit duration
containing this maximum deficit is known as the critical period.
Cont.…

7. 7.1.2 The sequent peak
The sequent peak analysis can be applied for constant or varying
demands in time. In this method, we find the maximum cumulative
deficit over adjacent sequences of deficit periods, and determine the
maximum of these cumulative deficits. The inflow sequence is assumed
to repeat and the analysis is carried out over two cycles, or two
consecutive inflow sequences. If the critical period lies towards the end
of an inflow sequence, carrying out the analysis over two cycles ensures
the capture of the maximum value of the cumulative deficit, which
really is the required active storage capacity.

8. Cont.…
The sequent peak algorithm is as follows: Let t denote the time period,
Qt the inflow, and Rt , the required release or demand in period t. Let
Kt be defined as follows:
Kt = Kt–1 + Rt – Qt =0 if positive, otherwise,
with K0 set equal to zero (K0 = 0). Or, Kt may be expressed
conveniently as the maximum of zero and Kt–1 + Rt – Qt , as
Kt = max {0, Kt–1 + Rt – Qt }

9. The values of Kt are computed for each period t for two cycles or
successive inflow sequences.
Let K * = max {Kt } over all t.
Then K * is the required active storage capacity of the reservoir.
The sequent peak method is just an analytical procedure to determine
the maximum cumulative deficit that occurs over time, given the
inflows and demands across time periods.
Cont.…

10. The method is very similar to the mass diagram approach, when the
evaporation losses in the reservoir are neglected. It is obvious that if the
total annual demand exceeds the total annual inflow, no amount of
reservoir storage would satisfy the full demand in all the periods of a
year. This is because of inflow limitation, or lack of available water.
Cont.…

11. Example 7.1.1 (Reservoir capacity with evaporation loss neglected)
Determine the required capacity of a reservoir whose inflows and demands
over a 6-period sequence are as given below (release, Rt = demand, Dt).
Period, t 1 2 3 4 5 6
Inflow, Qt 4 8 7 3 2 0
Demand,Dt 5 0 5 6 2 6
It is assumed that the sequence repeats itself. In this case, total inflow = total
demand in the six periods of the sequence and is equal to 24.
Cont.…

12. Since evaporation loss is neglected, it is possible to determine the
required storage to meet the demands in full. In this case, release equals
demand in each period and there is no spill.
Table7.1 illustrates the computations using the sequent peak Sequent
Peak Method (Evaporation Neglected)
Cont.…

13. The computations in the second cycle repeat after period 5 (hence, are
not shown). The required capacity of the reservoir is max {Kt} = 10.
Note: A reservoir of capacity of K * = 10 will be full at the end of the
third period, and will be empty at the end of the first period of the
following sequence (analyses why?). The intervening period is the
‘critical period.’ Constant demand: If the demand in each period is
constant (say, for instance, the water supply demand for a town), then
Rt = R for all t in the above formulation.
Cont.…

14. It is to be noted that in the absence of losses, the maximum constant
demand, R, which can be met from a given sequence of inflows is equal
to the average of the inflows, Qt, over all t.
A constant demand larger than this magnitude cannot be met, whatever
be the reservoir capacity, due to inadequate quantum of inflow. In the
example illustrated above, the maxi-mum constant demand that can be
met in a time period is 24/6 = 4 units of water. Any higher value is
infeasible due to limitation of the total quantum of inflow, no matter
what the reservoir capacity is.
Cont.…

15. 7.2 Reservoir Operating Policy
A reservoir operating policy is a sequence of release decisions in
operational periods (such as months), specified as a function of the state
of the system. The state of the system in a period is generally defined by
the reservoir storage at the beginning of a period and the inflow to the
reservoir during the period. Once the operating policy is known, the
reservoir operation can be simulated in time with a given inflow
sequence. A number of optimization algorithms have been developed for
deriving reservoir operating policies.

16. However, the most common policy implemented in practice is the so-
called standard operating policy, which is discussed first in this section.
This policy by itself is not based on or derived from any optimization
algorithm.
7.2.1 The standard operating policy
The standard operating policy (SOP) aims to best meet the demand in
each period based on the water availability in that period
Cont.…

17. It thus uses no foresight on what is likely to be the scenario during the
future periods in a year.
Let D and R represent, respectively, the demand and the release in a
period. Let the capacity of the reservoir be K. Then the standard
operating policy for the period is represented as illustrated in Fig.below.
The available water in any period is the sum of the storage, S, at the
beginning of the period, and the inflow Q during the period. The release
is made as per the line OABC in the figure below .
Cont.…

18. Cont.…

19. Figure above implies the following: Along OA: Release = water
available; reservoir will be empty after release. Along AB: Release =
demand; excess water is stored in the reservoir (filling phase). At A:
Reservoir is empty after release. At B: Reservoir is full after release.
Along BC: Release = demand + excess of availability over the capacity
(spill) In other words, the release in any time period is equal to the
availability, S + Q, or demand, D, whichever is less, as long as the
availability does not exceed the sum of the demand and the capacity.
Cont.…

20. It is to be noted that the releases made as per the standard operating
policy are not necessarily optimum as no optimization criterion is used
in the release decisions.
For highly stressed systems (systems in which water availability is less
than the demand in most periods), the standard operating policy
performs poorly in terms of distributing the deficits across the periods in
a year. With evaporation loss E included, the standard operating policy
may be expressed as:
Cont.…

21. • Rt = Dt if St + Qt – Et > Dt = St + Qt – Et , otherwise
• Ot = (St + Qt – Et – Dt ) – K if positive = 0 otherwise
• St + 1 = St + Qt – Et – Rt – Ot , with Rt and Ot determined as above
Cont.…

22. 7.2.2. Optimal operating policy
One of the classical problems in water resources systems modelling is
the derivation of an optimal operating policy for a reservoir to meet a
long-term objective. Modelling techniques to be used depend on
whether the reservoir inflows are treated deterministic or stochastic.
In this chapter only the case of deterministic inflows is dealt with.
Given a reservoir of known capacity K, and a sequence of inflows, the
reservoir operation problem involves determining the sequence of
releases Rt , that optimizes an objective function.

23. In general, the objective function may be a function of the storage
volume and/or the release.
LP Formulation
Consider the simplest objective of meeting the demand to the best extent
possible (the same objective as considered in the standard operating
policy), such that the sum of the demands met over a year is maximum.
This may be formulated as a LP problem as follows:
Cont.…

24. Cont.…
Fig. Single Reservoir Operation

25. Cont.…