20320130406014 2

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 6, November – December (2013), © IAEME

TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 4, Issue 6, November – December, pp. 134-144
© IAEME: www.iaeme.com/ijciet.asp
Journal Impact Factor (2013): 5.3277 (Calculated by GISI)
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IJCIET
©IAEME

LONG TERM RESERVOIR OPERATION USING EXPLICIT STOCHASTIC
DYNAMIC PROGRAMMING
Safayat Ali Shaikh
Visiting Faculty, Dept. of Civil Engineering, Bengal Engineering and Science University, Shibpur,
P.O.: Botanic Garden, Howrah- 711103, West Bengal, India

ABSTRACT
Uncertain nature of hydrologic parameters in reservoir management has been handled with
explicit stochastic dynamic programming models. Different versions based on the correlation structure
of the inflow sequence are presented and the process of determining the steady state solution is
discussed. These models are then applied to the DV system of reservoirs. Water supply for irrigation,
municipal and industrial use is selected as objective of operation whereas other purposes are treated as
binding constraints. Minimization of sum of square of deviation of release from target during the years
of operation is used as objective function. Performance of the system has been evaluated with the
modern reliability parameters.
Keywords: Explicit stochastic dynamic program, Reservoir operation, Performance evaluation.
1. INTRODUCTION
A reservoir system is built to serve certain specific water related purposes, e.g., water supply for
irrigation, municipal and industrial use, hydroelectric power generation, flood control, navigation, water
quality improvement, recreation etc. The operation problem of such a multipurpose reservoir system is
to release water from the reservoir for different purposes in an optimal manner so as to satisfy a prespecified objective. For a reservoir system, determination of optimal operating policies is a difficult
problem because reservoir operation is a multistage dynamic stochastic control process. Releases are to
be determined in successive stages or time periods (e.g., monthly, weekly, daily) according to the
predetermined objective. This is to be done with the knowledge that present releases from the reservoir
have impacts on the future releases, and in case of integrated operation of a multireservoir system, any
release decision for a particular reservoir will affect the release decisions of the other reservoirs. The
operation problem is stochastic because uncertainties are associated with the system variables (e.g.,
inflows). Future values of these inflows, which form the input to the system, are not precisely known.
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These complexities of the reservoir operation problem require release decisions to be made with an
appropriate simulation or an optimization model. The key to the success of any such model largely rests
on the ability to take the advantage of the system features that lead to simpler mathematical models.
Out of several mathematical techniques ranging from simulation to optimization for optimal operation
of reservoirs, Dynamic Programming (DP) has been recognized as the most efficient solution procedure.
The efficiency of DP is due to the fact that the non-linear and stochastic features that characterize the
operation problem can very well be accommodated in its formulation. Moreover, it has the advantage of
effectively decomposing highly complex problems with a large number of variables into a series of subproblems with few variables. But for multi-reservoir system it suffers from the so-called curse of
dimensionality. With the number of reservoirs in the system increases, computer storage and memory
requirements increase exponentially. This problem becomes more acute when the input in the system
is considered as stochastic. Explicit Stochastic Dynamic Programming (SDP) is the stochastic version
of DP, based on Bellman's Principle of Optimality (Bellman, 1957). It optimizes the expected value of
the objective function, considering the probability of transition of states from one stage to the next.
Major issues in applying SDP are the assumptions regarding the correlation structure of the inflows,
choice of state variables, discretization scheme, steady state solution and most importantly
dimensionality problem.
2. LITERATURE SURVEY
In single reservoir operation problems SDP has been successfully applied by many researchers.
However, application of SDP to multireservoir systems is restricted to a few cases due to excessive
computation time and storage requirement, so some kind of approximation is necessary to make the
problem computationally feasible and in that respect the single reservoir models are useful.EsmaeilBeik and Yu [1984] used a stochastic dynamic programming to develop optimal policies for
operating the multi-purpose pool of ELK City Lake in Kansas, USA, with serially correlated inflows.
Kelman et al.[1990] developed a sampling stochastic dynamic programming (SSDP), a technique
that captures the complex temporal and spatial structure of stream flow process by using a large
number of sample stream flow sequences. Hung et al.[1991], in their paper compared four types of
stochastic dynamic programming for real time reservoir operation, relying on observed or forecasted
inflows. Piccardi and Sonscini-Sessa [1991] presented effects of discretization and inflow corelation assumption on the computation of reliability parameters in reservoir operation problem.
Vedula and Mujumdar [1992] derived optimal operating policy of MalaprabhaReserevoir in
Karnataka, India for irrigation of multiple crops using SDP. In their model reservoir storage, inflow
and soil moisture content in the irrigated area have been treated as state variable.Lee et al. [1992]
evaluated performance of Lake Shelbyville by modified stochastic dynamic programming model
which accounts for the unrepeatable agricultural and property damages and improves the accuracy of
these damages estimates.Tejda-Guibert et al. [1993] determined optimal operation policies for
multireservoir systems by comparing two different approaches using SDP. First they used traditional
approach of determining releases by interpolating in the policy tables produced by SDP and secondly
re-optimized policy which uses the cost to go function generated by SDP to estimate the optimal
release in each period. The problem has been considered as a periodic, stationary, infinite-horizon
problem where future returns are not discounted.
Vasiliadis and Karamouz[1994]developed a demand driven SDPwhere the uncertainties of the
streamflow process, and the forecasts are captured using Bayesian decision theory —probabilities are
continuously updated for each month. Furthermore, monthly demand along with inflow, storage, and
flow forecast are included as hydrologic state variables in the algorithm. Perera and Codner [1996]
developed a generic methodology, using SDP to determine the operating rules. The SDP explicitly
accounts for the stochastic nature of streamflow to the water supply system by considering a theoretical
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probability distribution for streamflow. Archibald et al.[1996] presented an operating policy for a
multireservoir system in which the operating policy for a reservoir is determined by solving a stochastic
dynamic programming model consisting of that reservoir and a two-dimensional representation of the
rest of the system. Sunantara and Ramírez [1997] carried out optimal seasonal multi-crop irrigation
water allocation and optimal stochastic intra-seasonal (daily) irrigation scheduling, using a two-stage
decomposition approach based on a stochastic dynamic programming methodology. Ravikumar and
Venugopal[1998] developed a SDP model to obtain an optimal release policy of a South Indian
Irrigation system. This SDP model considers both demand and inflow as stochastic, and both are
assumed to follow first-order Markov chain model. Third is a simulation model which uses the optimal
release policy from the SDP model.Celeste et al. [2008] solved a monthly operation model by
stochastic-deterministic procedure and applied to Ishitegawa Dam in Japan.
3. DESCRIPTION OF THE SYSTEM UNDER STUDY
DV reservoir project was undertaken in 1948 for unified development of the Damodar River
basin in the states of West Bengal and Jharkhand (erstwhile Bihar) in India. The system consists of
reservoirs Konar and Panchet on river Damodar, reservoirs Tilaiya and Maithon on river Barakar and a
barrage at Durgapur (Fig. 1). The DV reservoir system resembles a general reservoir system with both
series and parallel connections. Different reaches are defined as 1) Reach1: area between reservoirs
Konar and Panchet, 2) Reach2: area between reservoirs Tilaiya and Mithon, 3) Reach3: area
downstream of reservoirs Panchet and Mithonupto Durgapur barrage, and 4) Reach4: area downstream
of Durgapur barrage. Different storage capacities and reservoir identification number are provided in
Table 1.
In Damodar Valley mean annual rainfall is 1295 mm. About 82 percent of the total rainfall occurs
during monsoon period (June to September) and rest occurs during pre-monsoon and post-monsoon
period.For sustenance of aquatic life and ecology in the streams, a minimum quantity (2.1 m3/s) is to be
released from different reservoirs for different reaches and at barrage site all floods are moderated to
7080 m3/s.The mean annual evaporation rates for the reservoirs Konar, Tilaiya, Panchet and Mithon
are 1.496m, 1.773m, 1.547m, and 1.389m respectively, about 50% of the said evaporation takes
place in the four hot months of March, April, May and June. Based on these data, monthly
evaporation rates for each of the four reservoirs are computed. Enroute losses are considered at the
rate of 5% of the flows during monsoon period (June-September) and at 10% during non-monsoon
months (October to May).The reach wise present utilization of M&I demands are 5.865 m3/sec, 0.133
m3/s and 9.181 m3/s for Reach 1, Reach 2 and Reach 3 respectively. Whereas monthly irrigation
demands in m3/s (from October to September) are: 321.5, 6.2, 6.0, 39.0, 50.8, 62.6, 56.7, 0.0, 13.3,
257.2, 141.5 and 265.8 respectively. Except Reservoir 1, the other three reservoirs are associated with
hydro-electric power plants. The operation of the hydelplants are supposed to be a minimum 6 hours per
day. The flood control purpose is considered implicitly by restricting the reservoir storage within the
maximum monsoon storage level so that high inflows can be accommodated in the flood storage space.
Performance of the system has been derived in terms of modern reliability concept as proposed by
Hashimoto et al. [1982]. The values of reliability, resiliency and vulnerability reveal some of the
characteristics of reservoir system performance that can be obtained with reservoir policies that
minimize the specified loss function. Hence these criteria can be used to assist in the evaluation and
selection of alternative policies obtained from different model.

136


Table 1: Storage Capacities of DV Reservoirs
Reservoir
Storage (in Mm3)
Notation
Name
Dead
Conservation
1
Konar
61.09
220.5
2
Tilaiya
74.85
141.6
3
Panchet
170.20
222.05
4
Mithon
164.80
570.95

Flood
54.5
177.5
666.05
384.5

4. FORMULATION OF PROBLEM
4.1 Application of SDP inReservoir Operation.
4.1.1 System Dynamics
In SDP framework, at any stage (‫ݐ‬ሻ, input to the system i.e., inflowሼ‫ݕ‬௧ ሽ to the reservoir is
actually random in nature. Thus the final state (‫ݔ‬௧ାଵ ሻ is not a deterministic function of the initial state
(‫ݔ‬௧ ሻ and the release (‫ݑ‬௧ ሻmade during that stage (‫ݐ‬ሻ. Hence final state cannot be computed uniquely from
the system dynamics given in equation (1). It is only possible to compute a set of ‫ݔ‬௧ାଵ values for a
known possible set of inflowሼ‫ݕ‬௧ ሽvalues
‫ݔ‬௧ାଵ ൌ ‫ݔ‬௧ ൅ ‫ݕ‬௧ െ ‫ݑ‬௧

(1)

At any stage (‫ݐ‬ሻ, input to the system i.e., inflowሼ‫ݕ‬௧ ሽ to the reservoir is actually random in nature.
Thus the final state (‫ݔ‬௧ାଵ ሻ is not a deterministic function of the initial state (‫ݔ‬௧ ሻ and the release
(‫ݑ‬௧ ሻmade during that stage (‫ݐ‬ሻ. Hence final state cannot be computed uniquely from the system
dynamics.
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4.1.2 Objective Function
Evaluation of the systems performance (‫ܮ‬௧ ሻ becomes uncertain as its no unique value can be
obtained. So in SDP framework, when DP (Bellman, 1957) will be applied in reservoir operation
problem, the associated expected real valued objective function can be defined as;
ே

‫ܬ‬ሺܷሻ ൌ ‫ ܧ‬൥෍ ‫ܮ‬௧ ൩

(2)

௧ୀଵ

whereܰ denotes the numbers periods of operation. A solution to this stochastic optimization
problem is a sequence of feasible decisions ࢁ that minimizes ‫ܬ‬ሺࢁሻ.
4.1.3 Transition Probability
The expected value of the objective function is computed from the inflow transition
probabilities. The random inflows to the reservoir may either be serially correlated or serially
independent. When these inflows are serially correlated, i.e. the current period's inflow is dependent on
previous period's inflows; such dependence can be modeled by a Markov chain. Assuming a first order
Markov chain, the inflow transition probabilities can be defined as;
௜
‫݌‬௜௝ ൌ ܲ ሾ‫ݕ‬௧ାଵ ൌ ‫ݕ‬௧ାଵ ห‫ݕ‬௧ ൌ ‫ݕ‬௧ ൧
௥
௝

(3)

4.1.4 Recursive Equations
In SDP formulation, solution to the optimization problem (equation 2) begins with solving the
recursive equations for successive stages. The nature of the recursive equation depends on the
correlation structure of the inflows i.e., whether they are serially correlated or independent. If ‫ܮ‬௧
௜
represents the value of the system performance in stage ‫ ,ݐ‬corresponding to an initial storage volume ‫ݔ‬௧ ,
௝
௞
inflow ‫ݕ‬௧ and a final storage volume ‫ݔ‬௧ାଵ , then considering the inflows to be serially correlated, the
discretized version of the recursive equation can be written as [Loucks et al., 1981; Huang et al., 1991]
௜ ௞
ܸ௜ ൫‫ݔ‬௧ , ‫ݕ‬௧ ൯

ൌ min ቬ‫ܮ‬௧ ൅ ෍
ೕ
௫೟శభ

೗
௬೟శభ

௝
௟
ܸ௧ାଵ ൫‫ݔ‬௧ାଵ , ‫ݕ‬௧ାଵ ൯‫݌‬௞௟ ቭ

(4)

௜ ௞
‫ ݐ‬ൌ ܰ, ܰ െ 1, ܰ െ 2, … ,1; ‫ݔ׊‬௧ , ‫ݕ‬௧ , ‫ݔ‬௧ାଵ ݂݁ܽ‫.݈ܾ݁݅ݏ‬
௝

The superscripts ݅, ݆, ݇, ݈ denote the discretization levels for the corresponding variables. The
௟
term ‫݌‬௞௟ is the transitional probability specifying the conditional probability of inflow ‫ݕ‬௧ାଵ in stage
௞
‫ ݐ‬൅ 1 when the inflow in stage ‫ ݐ‬is ‫ݕ‬௟ .
If there is no correlation between two consecutive inflows, i.e., the current inflow is independent
of the previous one, then the inflow transition probability becomes unconditional probability. Then
equation 4 can be expressed as;
௟
௜ ௞
ܸ௜ ൫‫ݔ‬௧ , ‫ݕ‬௧ ൯ ൌ min ቬ‫ܮ‬௧ ൅ ෍ ܸ௧ାଵ ൫‫ݔ‬௧ାଵ , ‫ݕ‬௧ାଵ ൯‫݌‬௞ ቭ
ೕ
௫೟శభ

೗
௬೟శభ

(5)

௝

௜ ௞
‫ ݐ‬ൌ ܰ, ܰ െ 1, ܰ െ 2, … ,1; ‫ݔ׊‬௧ , ‫ݕ‬௧ , ‫ݔ‬௧ାଵ ݂݁ܽ‫.݈ܾ݁݅ݏ‬

௞
The term ‫݌‬௞ is the unconditional probability ܲ௥ ሾ‫ݕ‬௧ ൌ ‫ݕ‬௧ ሿ.

138

௝


Since at the beginning of stage ‫ ݐ‬the actual value of ‫ݕ‬௧ is unknown, an inflow forecast is to be
used. If reliable forecast is unavailable, the net inflow during the preceding stage, ‫ݕ‬௧ିଵ (known at the
beginning of stage ‫ ) ݐ‬could be used as a state variable instead of‫ݕ‬௧ . Accordingly, the backward
recursive equation may be expressed as [Esmail-Beikand Yu, 1984];
௜ ௞
ܸ௜ ൫‫ݔ‬௧ , ‫ݕ‬௧ିଵ ൯

ൌ min ቬ෍ሼ‫ܮ‬௧ ൅
ೕ
௫೟శభ

೗
௬೟

௝
௟
ܸ௧ାଵ ൫‫ݔ‬௧ାଵ , ‫ݕ‬௧ ൯ሽ‫݌‬௞௟ ቭ

(6)

௜ ௞
‫ ݐ‬ൌ ܰ, ܰ െ 1, ܰ െ 2, … ,1; ‫ݔ׊‬௧ , ‫ݕ‬௧ିଵ , ‫ݔ‬௧ାଵ ݂݁ܽ‫.݈ܾ݁݅ݏ‬
௝

௟
where ‫݌‬௞௟ is the conditional probability of inflow ‫ݕ‬௧ during stage ‫ ,ݐ‬given that the inflow in stage
௞
‫ ݐ‬െ 1 1 is ‫ݕ‬௧ିଵ .

For an independent inflow sequence the above equation 6 will be;
௜
ܸ௜ ൫‫ݔ‬௧ ൯

ൌ min ቬ෍ሼ‫ܮ‬௧ ൅
ೕ
௫೟శభ

where‫݌‬௞ is the unconditional probability.

೗
௬೟

௝
ܸ௧ାଵ ൫‫ݔ‬௧ାଵ ൯ሽ‫݌‬௞ ቭ

(7)

௜
‫ ݐ‬ൌ ܰ, ܰ െ 1, ܰ െ 2, … ,1; ‫ݔ׊‬௧ , ‫ݔ‬௧ାଵ ݂݁ܽ‫.݈ܾ݁݅ݏ‬
௝

In the first two cases (equation 4 and 5), optimal final storage is dependent on each initial
storage and current period's inflow. In the third case (equation 6) optimal final storage is related to each
initial storage and previous period's inflow, and in the fourth case (equation 7), it is related to only initial
storage. To implement these operating policies one has to rely on inflow forecast in the first two cases
whereas in the third and fourth cases inflow forecast can be avoided. However, in each of these different
methods of obtaining the optimal final storage, the precise real time release will depend on the actual
inflow that will occur during the particular stage and hence a constant release throughout the period
cannot be obtained always. The operating policy will determine optimal release values, but as there are
limits on reservoir storage, actual release will vary with low or high inflows.
4.1.5 Steady State Solution
The steady state operating policy can be determined following the successive approximation
procedure suggested by Su andDeininger [1972]. This method computes recursively the minimum
expected yearly loss for successive years and converges uniformly to the steady state solution. The
policy of the cycle immediately before the convergence is taken as the steady state policy.
4.1.6 Dimensionality Problem
In SDP framework, the direct application of the recursive equation for the solution of the
multireservoir operation problem becomes limited due to excessive computational burden, due to
following reasons:
(1) Increased dimension of the state space: The state space is usually two dimensional, containing
joint storage state and joint inflow state. If in a system there are four reservoirs and storage state
and inflow both are discretized into 100 levels, then total number of feasible points in the state
space will be ሺ100 ൈ 100 ൈ 100 ൈ 100ሻ ൈ ሺ100 ൈൈ 100 ൈ 100 ൈ 100ሻ ൌ 10ଵ଺ .
(2) Construction of joint transition probability: For multireservoir system joint transition probability
has to be considered. These joint transition probabilities are based on joint inflow states during
stage ‫ ݐ‬and stage ‫ ݐ‬൅ 1.For a four reservoir problem with each of the four inflow sequences
139


discretized into 100 levels, the size of a joint probability matrix for one stage will be100ସ ൈ
100ସ . A total of 10ଵ଺ elements of the matrix are to be computed from the observed inflow
series. This will require a very long observed series of inflows which may not be available in
practice. For a short series most of the elements of the joint transition probability matrix will be
zero which may lead to non-optimal solution. To avoid this situation, a long synthetic sequence
is needed which should resemble the historical series.
(3) Determination of steady state policy: The solution to the annual operation problem (for 12
monthly periods) has to be repeated for successive years to determine the steady state policy.
This requires large computation time so as to achieve the convergence with a desired level of
accuracy.

4.2 Application of SDP to DV System
There are four reservoirs in DV system. In order to reduce the dimensionality problem
(discussed 4.1.6) the original higher dimensional problem is decomposed into lower dimensional subproblems (Fig. 2) maintaining the spatial continuity and solved (using decomposition algorithm
discussed in section 4.2.1) for each subsystem successively.SDP models of DV system using recursive
equation 4, 5, 6 and 7 are termed as SDP1, SDP2, SDP3 and SDP4 respectively. For evaluation of
performance of a model a loss function is used for this purpose in the form of penalty. The penalty is
associated with the deviation of release (‫ݑ‬௧ ሻ from the target (ܴܶ௧ ሻ.For reservoir 1 and reservoir 2 target
release is the Municipal and Industrial demand in ‘Reach 1’ and ‘Reach 2’ respectively whereas for
reservoir 3 and reservoir 4 there is a combined target demand for municipal and industrial in ‘Reach 3’
and irrigation demand at ‘Reach 4’. The objective is to minimize the sum of square deviation of release
from target. so during stage ‫ ,ݐ‬the generic form of single stage loss function to be used in recursive
equation can be expressed as;
‫ܮ‬௧ ሺ‫ݑ‬௧ ሻ ൌ ۤሺܴܶ௧ െ ‫ݑ‬௧ ሻ/ܴܶ௧ ‫ۥ‬ଶ

4.2.1 Decomposition Algorithm

Fig. 2 Decomposition of DV System
140

(8)

The decomposition algorithm is based on solving one reservoir at a time, from upstream to
downstream while maintain the connectivity. With reference to Fig. 2, the processed can be summarized
as follows:
(1) The procedure starts with optimizing for sub-problem 1. The steady state policy of reservoir 1 is
௝
௜
௞
determined using SDP1 model with the system dynamics; ‫ݔ‬௧ାଵ ሺ1ሻ ൌ ‫ݔ‬௧ ሺ1ሻ ൅ ‫ݕ‬௧ ሺ1ሻ െ
‫ݑ‬௧ ሺ1ሻ െ ݁‫ݒ‬௧ ሺ1ሻ where bracketed number indicates the corresponding sub-problem 1 (for
reservoir 1) and ݁‫ݒ‬௧ ൌ evaporation at stage ‫.ݐ‬
(2) Now sub-problem 2 is selected and steady state policy of reservoir 2 is determined in similar
manner.
(3) After solving for the two upstream reservoirs (sub-problem 1 and sub-problem 2 respectively),
one of the two downstream reservoirs, say sub-problem 3 is selected. First, the total demand
ሺܶ‫ܦ‬ሻ is to be met with the combined release ‫ݑ‬௧ ሺ3ሻ ൅ ‫ݑ‬௧ ሺ4ሻ. Actual sharing of the total
demandamong the two reservoirs cannot be specified as it depends on the prevailing state of the
system. But, as the algorithm optimizes one reservoir at a time, individual demands‫ܦܫ‬௧ ሺ3ሻ and
‫ܦܫ‬௧ ሺ4ሻare to be defined in some way. In the algorithm, the combined demand ܶ‫ ܦ‬is divided in
the ratio of ߚ: ሺ1 െ ߚሻ for reservoirs 3 and 4 respectively. i.e., ‫ܦܫ‬௧ ሺ3ሻ ൌ ߚ ൈ ܶ‫ܦ‬௧ and ‫ܦܫ‬௧ ሺ4ሻ ൌ
ሺ1 െ ߚሻ ൈ ܶ‫ܦ‬௧ .
Secondly, the net inflow received by reservoir 3 gets affected by the release from upstream
′
′
reservoir 1. The net inflow ‫ݕ‬௧ ሺ3ሻ, can be expressed as ‫ݕ‬௧ ሺ3ሻ ൌ ‫ݑ‬௧ ሺ3ሻ ൅ ‫ݕ‬௧ ሺ3ሻ െ ݁݊௧ ሺ3ሻ. As
‫ݕ‬௧ ሺ1ሻ is a random variable, during stage ‫ ,ݐ‬the precise information about the amount of release
′
‫ݑ‬௧ ሺ1ሻis not available. So actual value of ‫ݕ‬௧ ሺ3ሻcannot be computed. To overcome this difficulty,
before solving reservoir 3 a simulation run is made with the reservoir 1 policy and a long
generated inflow sequence ሼ‫ݕ‬௧ ሺ1ሻሽ to determine the sequence of release ሼ‫ݑ‬௧ ሺ1ሻሽ from reservoir
′
1. The net inflow sequence ‫ݕ‬௧ ሺ3ሻ is then determined from the above equation for a similar, long
generated inflow sequence ሼ‫ݕ‬௧ ሺ3ሻሽ. Transition probabilities ‫݌‬௞௟ are then computed from this
௝
′
sequence ‫ݕ‬௧ ሺ3ሻ and the recursive equation is solved with the system dynamics; ‫ݔ‬௧ାଵ ൌ
௜
′௞
‫ݔ‬௧ ሺ3ሻ ൅ ‫ݕ‬௧ ሺ3ሻ െ ‫ݑ‬௧ ሺ3ሻ െ ݁݊௧ ሺ3ሻ, to determine steady state policy of reservoir 3.
(4) Next sub-problem 4 is considered and steady state policy of reservoir 4 is determined in similar
way.
(5) All the above-mentioned steps (1-4) are followed to get steady state policies for model SDP2,
SDP3 and SDP4.
5. COMPARISON, SELECTION AND PERFORMANCE EVALUATION OF SDP MODELS
To verify the performances of the SDP models, following comparisons are performed: (1)
Comparison of objective function values, (2) Comparison of optimal trajectories and (3) Computation of
reliability parameters of the best model.
5.1 Comparison of Objective Function Values
Objective function values are computed in terms of 'Expected Cost' and 'Average Cost' and
presented in Table 2. The term 'Expected Cost' represents minimum expected yearly loss obtained from
the steady state solution. The term 'Average Cost' indicates the average value of the objective function
computed from 44 years of release sequence as obtained from 44 years historical inflow sequence in
simulation with steady state policies (for state discretization level 100 and inflow discretization level
10).The 'Average Cost' values are usually different than the 'Expected Cost'. Because, 'Expected Cost' is
derived using transition probabilities that include all possible inflow magnitudes, but, 'Average Cost' is
associated with a particular sequence of inflows used in deriving the release sequence. This sequence is
usually an observed inflow sequence of few years, which does not contain all possible magnitudes of
141

inflows.It is observed in Table 2 that model SDP1 yielded lowest values of 'Expected Cost' and
'Avgerage Cost' whereas SDP2 is the second lowest for all the reservoirs. Hence, it can be concluded
that SDP1 is the best model and SDP3 is better model.

Model
SDP1
SDP2
SDP3
SDP4

Table 3: Objective Function Value of Different SDP Models
Reservoir 1
Reservoir 2
Reservoir 3
Reservoir 4
Expected Average Expected Average Expected Average Expected Average
Cost
Cost
Cost
Cost
Cost
Cost
Cost
Cost
33.08
33.56
62198.5 72296.7
28.75
66.50
10.55
33.04
66.70
59.04
76799.2 78592.2
41.05
72.29
59.37
78.80
52.65
39.03
75573.3 77329.6
38.19
70.06
42.51
52.92
70.05
62.48
79022.5 85025.2
46.70
77.25
66.90
83.02

5.2 Comparison of Optimal Trajectories
Comparative plots of the trajectories obtained from SDP1 (best model) and SDP3(better model)
are prepared for all reservoirs and presented in Fig. 3(a-d). For convenience, these plots are shown for
′
the first seven years, which include the successive low-flow years (5th and 6th year). In these plots, ‫ݔ‬௧
indicate the non-dimensionalized storage [ൌ ‫ݔ‬௧ /ሺ‫ܥ‬௠௔௫ െ ‫ܥ‬௠௜௡ ሻሿ .

Fig. 3 Plot of Optimal Trajectories
5.3 Computation of Reliability Parameters of the SDP Models
Reliability parametersof SDP1 and SDP2 have been determined using modern concept of
reliability parameters: reliability, resiliency, vulnerability and total deficit. Reliability is defined as
number of periods of deficit (‘ND’), Resiliency is the number of consecutive deficits (‘Res’),
Vulnerability is the maximum number of deficit (‘Vul’) and sum of total amount of deficit during the
142

period of operation is total deficit (‘TotD’). These reliability parameters are computed from a 44 years
release sequence.
Regarding water supply for different uses, it is usually accepted that the municipal demand is to
be met with 100% efficiency, industrial demand with 90% efficiency and irrigation demand with 75%
efficiency. Since permissible values for resiliency (‘Res’), vulnerability (‘Vul’) and total deficit(‘TotD’)
are not well established, a maximum value of 12 is adopted for industrial supply, and a maximum value
of 3 is assumed for irrigation water supply for a particular crop. Three crop seasons (Rabi, hot weather
and kharif crop season) havebeen considered in the analysis. The maximum permissible vulnerability
values are assumed as 90% of the maximum monthly demand and allowable average total deficit per
year values are assumed as 10% for M&I and 25% for irrigation of the total annual demand. In the said
framework reliability parameters are presented in Table 3. Reliability analysis has not been presented
for Reservoir 2, as no deficits were observed.

Mode
l
SDP1
SDP2

N
D
23
40

SDP1
SDP2

3
3

Reservoir 1
Res Vul TotD
6
16.4 234.6
8
16.4 353.4
Rabi Crop
2
16.0 36.9
2
16.0 39.0

Table 3: Reliability Parameters
Reservoir 3 & 4, (M & I)
ND Res
Vul
TotD
31
39

7
30.8
369.1
7
30.8
616.5
Hot Weather Crop
77
4
151.6 4336.0
79
4
151.6 4586.0

Reservoir 3 & 4 (Irrigation)
ND Res
Vul
TotD
95
99
15
17

4
741.3
4
750.2
Kharif Crop
2
750.2
2
761.0

7351.0
7921.2
2948.1
3296.2

6. RESULTS AND DISCUSSION
In Table 3 it is observed that vulnerability (‘Vul’) values exceeded 90 percent for both SDP1 and
SDP2. This happened during the low flow year, when in one month the observed inflow was zero.
Hence these values are not considered for comparison. It is also observed that numeric values of all
reliability parameters other than ‘Vul’ for SDP1 model is less than that of SDP2. Reliability parameters
for SDP1 are also within permissible limit for 44 years historic inflow records. From these observations,
it may be concluded that SDP1 model produced comparative better reliability parameters than SDP2.
7. CONCLUSION
In stochastic framework, the multireservoir operation problem is solved decomposing the
original four-reservoir operation problem to four single reservoir operation problems. For this purpose,
single reservoir SDP models are developed. Inflows to the reservoir are assumed as serially correlated
and transition probability matrices are computed accordingly. Inflow is considered as the second state
variable and accordingly two different SDP models are developed: SDP1 that used current period's
inflow y_t, as the second state variable, and SDP2 that used previous period's inflow y_(t-1) as the
second state variable. From the discussions on comparative behaviors of the trajectories and reliability
parameters, it can be concluded that model SDP1 yielded better results than model SDP2. So, use of y_t
instead of y_(t-1) as the second state variable is a better choice. As the decomposition models yielded
acceptable results in terms of systems performances, these can be used for solving multireservoir
operation models in stochastic framework. Conclusions above are system dependent and applicable to
the DV system in particular.

143

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