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Water Resources Research
RESEARCH ARTICLE
10.1002/2015WR017250
Key Points:
A new set of release rules are derived
for parallel reservoirs system
Optimality conditions for operation
of parallel system are first obtained
The commonly used policies are
demonstrated as a special case of
derived rules
Correspondence to:
T. Hu,
tshu@whu.edu.cn
Citation:
Zeng, X., T. Hu, L. Xiong, Z. Cao, and
C. Xu (2015), Derivation of operation
rules for reservoirs in parallel with joint
water demand, Water Resour. Res., 51,
9539–9563, doi:10.1002/
2015WR017250.
Received 18 MAR 2015
Accepted 6 NOV 2015
Accepted article online 12 NOV 2015
Published online 13 DEC 2015
Derivation of operation rules for reservoirs in parallel with joint
water demand
Xiang Zeng1, Tiesong Hu1, Lihua Xiong1, Zhixian Cao1, and Chongyu Xu1,2
1
2
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China,
Department of Geosciences, University of Oslo, Oslo, Norway
Abstract The purpose of this paper is to derive the general optimality conditions of the commonly used
operating policies for reservoirs in parallel with joint water demand, which are defined in terms of systemwide release rules and individual reservoir storage balancing functions. Following that, a new set of release
rules for individual reservoirs are proposed in analytical forms by considering the optimality conditions for
the balance of total water delivery utility and carryover storage value of individual reservoirs. Theoretical
analysis indicates that the commonly used operating policies are a special case of the newly derived rules.
The derived release rules are then applied to simulating the operation of a parallel reservoir system in northeastern China. Compared to the performance of the commonly used policies, some advantages of the proposed operation rules are illustrated. Most notably, less water shortage occurrence and higher water supply
reliability are obtained from the proposed operation rules.
1. Introduction
Coordinated operation of reservoirs in parallel can take advantage of different storage capacities along different river basins. Operation policies are essential for these coordinated operations, but specification of efficient
operating rules for the parallel reservoir system is typically challenging, as the decision making process
involves multiple variables and complex constraints [Oliverira and Loucks, 1997; Labadie, 2004; Jia et al., 2015].
C 2015. American Geophysical Union.
V
All Rights Reserved.
ZENG ET AL.
At present, several rules of thumb provide some guidelines to the operation of parallel reservoir system
with joint demands, based on the idea of aggregation-decomposition, for reducing the dimensional complexity of multireservoir operation problems [Robert et al., 1977; Oliverira and Loucks, 1997; Koutsoyiannis
and Economou, 2003; Xu et al., 2014; Peng et al., 2015a]. Most of the operating policies are defined in terms
of a system-wide release rules and individual reservoir storage balancing (or storage target) functions for
the coordinated operation of the entire multireservoir system [Oliverira and Loucks, 1997]. The system-wide
release rules usually specify the gross release from an equivalent reservoir aggregated over all the reservoirs
in the system, while storage balancing functions identify the desired storage of each reservoir, which is
decomposed from the total storage volume of the equivalent reservoir. For system-wide release rules,
Standard Operating Policy (SOP) and hedging rule are widely used. SOP [Maass et al., 1962; Stedinger, 1984]
is a simple operation rule which aims to release water as much as the reservoir can to meet the target delivery at the present time, and does not preserve water for any future requirements. The hedging rule [Oliverira
and Loucks, 1997], is to reduce the risk of large water shortages in a future period, but at cost of more frequent small deficits at present. For the balancing functions, New York City rule (NYC rule), space rule, priority
rule, and parametric rule have been previously discussed. Among these, the NYC rule [Clark, 1956; Oliverira
and Loucks, 1997] is designed to equalize the probability of seasonal spill from each reservoir. The space
rule [Maass et al., 1962], as a special case of the NYC rule [Sand, 1984], requires that the space remaining in
each reservoir of a parallel system is proportional to the expected inflow to that reservoir [Johnson et al.,
1991]. Both the space rule and the NYC rule attempt to avoid the situation of having some reservoirs spilling, while others remain unfilled [Oliverira and Loucks, 1997]. The priority policy is to rank the reservoirs so
the storage of the lowest ranking reservoir is used fully before releasing water from the next lowest ranking
reservoir [Wu, 1988]. A handful of decision variables are set in the parametric rule, being validated through
the entire control period, to distribute the total storage into each individual reservoir at each time step
[Nalbantis and Koutsoyiannis, 1997; Guo et al., 2013; Peng et al., 2015a].
DERIVATION OF OPERATION RULES FOR RESERVOIRS IN PARALLEL
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Although all the rules provide empirical guidelines for the operation of reservoirs in parallel, the general
optimality conditions for the coordinated operation of such a system have not been studied. In particular,
the conditions that guarantee optimality of the system performance are not addressed when the parallel
reservoir system is transformed into an equivalent reservoir. In addition, there lacks an analysis about how
to operate the reservoir system when the operation is heavily constrained [Oliverira and Loucks, 1997]. For
example, the hedging rule typically specifies the total release as a linear function of the total water availability at the current period, but the relationships between the total release and the total water availability
have seldom been investigated. The application of storage balancing functions, such as NYC and space
rules, depends on the predicted inflows [Lund, 1999], and the releases indicated by these two rules may not
be feasible for particular reservoirs because of the release constraints [Johnson et al., 1991].
Most recently, several studies presented some analytical work to derive operating policies for one water
supply reservoir based on the marginal utility principle. Draper and Lund [2004] analytically examined the
optimal release decisions in balancing current and future utility with a given utility function. You and Cai
[2008a, 2008b] expanded such theoretical analysis and established a two-period model for reservoir operation with hedging that considers uncertain future reservoir inflow explicitly. Zhao et al. [2011] discussed the
optimality conditions for SOP and hedging rule for a two-stage reservoir operation problem using a consistent theoretical framework. Shiau [2011] derived optimal hedging rules and evaluated the effects of the
weighting factor and an exponent on hedging rules with the reservoir operation objective expressed as the
weighted sum of normalized deviations from the current release and carryover storage targets. Since most
of reservoirs are constructed in parallel or series in real world, it is more meaning to systematically derive
operating policies for multi-reservoir system, especially a system with parallel reservoirs, the focus of this
paper.
Adopting the loss functions suggested by Shiau [2011], this paper derives a new set of individual reservoir
release rules from a two-period model [Draper and Lund, 2004; You and Cai, 2008a, 2008b] by considering
the optimality conditions for the operation of parallel reservoir system with joint water demand. Based on
the analytical form derived from a general optimality theory by this paper, extended analysis for the commonly used operating policies defined in terms of system-wide release rule and storage balancing functions
are presented, which are different from the work of Shiau [2011] in three aspects: (1) the optimality conditions for transforming parallel reservoir system into an equivalent reservoir, (2) the applicable water availability for the commonly used operating policies, and (3) the mathematical expressions of system release
rules and storage balancing functions of the equivalent reservoir. Following that, the derived release rules
are tested by simulation and compared to the results of the commonly used operating policies with a case
study.
The rest of this paper is organized as follows: section 2 presents the mathematical formulation of the twoperiod model for the operation of reservoirs in parallel and derivation of the analytical individual reservoir
release rules under given loss functions. Section 3 provides a theoretical analysis of optimal decision conditions based on the concept of an equivalent reservoir. A parallel reservoir system in northeastern China is
illustrated as a case study in section 4. Finally, the conclusions of this paper are given in section 5.
2. Analytical Operation Rules for Reservoirs in Parallel With Joint Demands
2.1. Two-Period Model for the Operation of Reservoirs in Parallel
The dilemma of water supply operation for reservoirs in parallel is not only to decide the total release from
system but also to balance system storage among individual reservoirs (shown in Figure 1). Release and carryover storage decisions should be made to minimize the gross losses of system release and storage carryover of each reservoir. In addition, the operation should not violate the physical constraints of reservoir
system, which include mass balance, storage constraints, nonnegative release, and total release constraints.
On the basis of the marginal utility principle, a mathematical program can be constructed for this situation,
which extends the conceptual model developed by Draper and Lund [2004] from a single reservoir to reservoirs in parallel. Since the utilities in two periods are considered, i.e., releasing water for current utility and
storing water for future utility, the operation model is termed as two-period model [You and Cai, 2008a].
This two-period model is expressed as follows:
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Figure 1. Reservoirs in parallel with joint water demand.
min
Z5CL1 S1;t11 1CL2 S2;t11 1 1CLn Sn;t11 1WL Rtotal
t
s:t:
Si;t11 1Ri;t 5WAi;t
(1)
(2)
Max
SMin
i;t11 Si;t11 Si;t11
(3)
0 Ri;t
(4)
Rtotal
Dt
Dmin
t
t
(5)
where WLðÞ is the water-deficit-related loss function for current total water delivery, denoting reductions in
benefits from some ideal deliveries. Rtotal
is the total water release from parallel reservoir system at time t,
t
which is used to meet projected joint demands Dt . In practice, the travel time and water evaporation loss
for water delivering from individual reservoir to the joint demand reduce the useful release and thus pose a
discount on the release. For simplicity in mathematical derivation, it is assumed that the distance between
the reservoir and the water demand is not far, and the water loss along the travel distance can be
neglected, the total release then equals the sum of all the individual reservoir releases, i.e.,
Rtotal
5R1;t 1R2;t 1 1Rn;t . CLi ðÞ is a loss function of the carryover storage value, representing the deviation
t
from expected value of water stored now for future time periods of i reservoir. WLðÞ and CLi ðÞ can be estimated from economic or some other metric utilities from water delivery and storage respectively [Draper
and Lund, 2004]. For the purpose of water supply, the convexity of utilities is widely illustrated [Sand, 1984;
Johnson et al., 1991; Labadie, 2004; You and Cai, 2008a; Shiau, 2011]. In this paper, both WLðÞ and CLi ðÞ are
assumed to be convex. Si;t is the storage of i reservoir at the beginning of time t, constrained between minimum storage and maximum storage of i reservoir. Ri;t is the water release from i reservoir during time t. Ii;t
is the inflow of i reservoir at time t, which is supposed to be perfectly forecasted in this study. Ei;t is the
evaporation loss of i reservoir at time t. WAi;t is the water availability of i reservoir at time t, which is defined
as the sum of initial water storage plus current inflow and minus evaporation loss during this time period
Max
for i reservoir, i.e., WAi;t 5Si;t 1Ii;t 2Ei;t . n is the number of reservoirs in parallel. SMin
i;t11 and Si;t11 are the minimin
mum and maximum storage of i reservoir at time t. Dt and Dt are minimum total release target and projected joint demands of parallel reservoir system.
The applicability of this model is restricted to a range of total water availability within system. The feasible
range of total water availability for the model should be discussed case by case based on the explicit objective
function. Out of this range, that is, if the water availability is too low, none of the release and carryover storage
objectives could be satisfied. Derivation of optimal operation rules from the two-period model is trivial.
Within the bounds of the inequality constraints where operation rule is relevant, according to the Lagrange
multiplier approach, this problem can be written as
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L5CL1 S1;t11 1CL2 S2;t11 1 1CLn Sn;t11 1WL Rtotal
t
1k1 WA1;t 2S1;t11 2R1;t 1k2 WA2;t 2S2;t11 2R2;t 1 1kn WAn;t 2Sn;t11 2Rn;t
(6)
where L is Lagrange function and ki is Lagrange multiplier.
Applying the first-order condition to the Lagrange function, we have
@WL Rtotal
@WL Rtotal
@CL1 S1;t11
@CLn Sn;t11
t
t
55
5
55
@R1;t
@Rn;t
@S1;t11
@Sn;t11
(7)
In words, equation (7) states that if only mass balance constraints are considered (i.e., other inequality constraints are unbinding), then at optimality, the marginal cost of storage should be equal to the marginal
cost of release for each reservoir and further to the marginal cost of storage for different reservoirs in parallel (i.e., marginal utility principle). Note that when the carryover storage value of each reservoir is linear (i.e.,
the marginal cost of storage is constant), ranking the storage priority of individual reservoir is allowed and
the priority policy [Wu, 1988] is the optimal balancing function. That is, the reservoir with lowest marginal
cost of storage would retain water first and delivery water last.
2.2. Derivation of Individual Reservoir Release Under Given Loss Functions
The choice of objective function is essential for derivation of operating policy, because the best decisions
on release and storage depend upon evaluation criteria of reservoir performance. Two types of objective
functions are commonly used to derive the optimal reservoir operation rules [Labadie, 2004]. The first type
of objective function is aimed to maximize the operation yields, such as water release benefits [Peng et al.,
2015b] and hydropower production [Wang et al., 2015], or minimize the flood risks [Zhao et al., 2014] and
the water quality costs [Rheinheimer et al., 2014]. Minimizing deviations from the ideal target storage levels
and projected water supply demand is the other type of objective function widely used to derive analytical
operating policy for reservoir system [Sand, 1984; Johnson et al., 1991; Labadie, 2004; Shiau, 2011]. In this
paper, the loss functions proposed by Shiau [2011], defined as the weighted normalized deviations from
predetermined targets, are used to evaluate the performance of parallel reservoir system. The loss functions
are expressed as follows:
!m
STi;t11 2Si;t11
CLi Si;t11 5xi
(8)
STi;t11
m
total Dt 2Rtotal
t
WL Rt
5xn11
Dt
(9)
where x is the weighting factor assigned to the loss function, the sum of xi is assumed to be 1, i.e.,
Pn11 i
i51 xi 51, m is an exponent to define the shape of the loss functions (in order to have a convex function
that can be globally minimized, m should exceed 1), STi;t11 is the desired carryover storage target of i reservoir at time t, which ranges from minimum storage to maximum storage of i reservoir.
Given the explicit utility functions as described in equations (8) and (9), the optimality condition in equation
(7) leads to the optimal reservoir release Ri;t , which is:
!
n
n
P
P
STi;t11
Dt 1 STi;t11 2 WAi;t
1
Ri;t 5WAi;t 2STi;t11 1
gm21
i;t
i51
Dt 1
i51
n ST
P
i;t11
i51
(10)
1
gm21
i;t
!
xi
Dt
gi;t 5
xn11
STi;t11
(11)
where Ri;t is the optimal reservoir release of i reservoir at time t and only applicable for the situation where
inequality constraints are unbinding, gi;t is the inverse-weighted target ratio for i reservoir at time t, defined
by Shiau [2011] to distinguish the types of hedging for a single water supply reservoir.
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Figure 2. The derived analytical reservoir release from two-period model.
Equation (10) indicates that the optimal release for individual reservoir is proportional to the water availability of its own and inverse proportional to the total water availability of other reservoirs in the system, which
is shown in Figure 2. This situation implies the importance of coordinated operation of reservoirs in parallel,
for the release of individual reservoir is not only related to the water availability of its own, but also to the
water availability of other reservoirs in parallel.
The analytical optimal reservoir release Ri;t comprises weighting factor ratio xi =xn11 and exponent m, and
the effects of these two parameters on Ri;t are presented in Appendix A. To simplify equation (10), two
parameters of ai;t and bi;t are used.
n
P
Ri;t 5WAi;t 2 ai;t
WAi;t 1bi;t
(12)
i51
STi;t11
STi;t11
ai;t 5
1
gm21
i;t
Dt 1
n
P
i51
n ST
P
i;t11
; bi;t 5
1
Dt 1
Dt 1
i51
STi;t11
n ST
P
i;t11
2STi;t11
(13)
1
i51 gm21
i;t
n
P
Dt
;
bi;t 5Dt 2
n ST
P
i;t11
i51
1
gm21
i;t
n
P
i51
Dt 1
i51 gm21
i;t
ai;t 512
1
gm21
i;t
!
n
P
ðDt Þ Dt 1 STi;t11
i51
Dt 1
n ST
P
i;t11
i51
(14)
1
gm21
i;t
According to equation (12), the optimal reservoir release increases with decreasing ai;t but increasing bi;t .
Analogy to the hedging factor, parameter ai;t measures the percentage of reduction in reservoir release
resulted from the total water availability of parallel reservoir system. From the definition of parameters in
1
1
m21
equation (13), both ai;t and bi;t increase with the decrease of gm21
i;t . As gi;t goes from the minimum to the
P
maximum, ai;t reduces from 1 to 0, while bi;t reduces from Dt 1 ni51 STi;t11 2STi;t11 to 2STi;t11 . Meanwhile, the
effects of storage target STi;t11 on parameter ai;t can be obtained, since ai;t increases with increasing STi;t11 .
P
The value of parameter bi;t is also affected by STj;t11 , e.g., the value of bi;t approaches to Dt 1 ni51 STi;t11 2
STi;t11 with a large STi;t11 , while the value closes to 2STi;t11 with a small STi;t11 .
From equation (13), the sum of parameter ai;t or parameter bi;t for parallel reservoirs at time t can be
defined in equation (14). The sum of parameter ai;t is restricted to between 0 and 1, while the sum of
P
parameter bi;t ranges from 2 ni51 STi;t11 to Dt .
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P
Since the optimal reservoir release Ri;t is a linear function of WAi;t and ni51 WAi;t , only portions of WAi;t and
Pn
i51 WAi;t are feasible for Ri;t to be implemented because of inequality constraints.
For the nonnegative release constraint, the feasible range of WAi;t is derived as
n
P
WAi;t ai;t
WAi;t 2bi;t
(15)
i51
n
P
The lower bound water availability of i reservoir for Ri;t is obtained as ai;t
WAi;t 2bi;t , which is propori51
tional to the total water availability of reservoir system.
For the constraints of reservoir storage and system release, the following range of total water availability is
feasible.
9
8
n
n
P
P
min
>
>
>
>
D
2
b
Dt 2 bi;t
i;t
= P
<Smin 1bi;t t
n
i;t
i51
i51
max
;
WAi;t (16)
n
n
P
P
>
>
ai;t
i51
>
>
:
12 ai;t ;
12 ai;t
i51
i51
Pn
The criteria to select the lower bound of i51 WAi;t depend on the relationship between ðSmin
i;t 1bi;t Þ=ai;t
P
P
P
P
min
and ðDmin
2 ni51 bi;t Þ=ð12 ni51 ai;t Þ. When ðSmin
2 ni51 bi;t Þ=ð12 ni51 ai;t Þ, the first
t
i;t 1bi;t Þ=ai;t ðDt
term is adopted, i.e., ðSmin
Otherwise, the second term is specified as the lower bound of
i;t 1bi;t Þ=ai;t . P
Pn
P
P
P
min
min
2 ni51 bi;t Þ=ð12 ni51 ai;t Þ, when ðSmin
2 ni51 bi;t Þ=ð12 ni51 ai;t Þ.
i;t 1bi;t Þ=ai;t ðDt
i51 WAi;t , i.e., ðDt
These two terms are derived from the constraints of minimum reservoir storage volume, i.e., SMin
i;t11 Si;t11 ,
Pn
total
and minimum system release, i.e., Dmin
R
,
respectively.
The
upper
bound
of
WA
remains
to be
i;t
t
i51
P
P t
constant of ðDt 2 ni51 bi;t Þ=ð12 ni51 ai;t Þ, which is obtained by the constraint of maximum system release.
P
Therefore, Ri;t is applicable only when WAi;t and ni51 WAi;t fall in the feasible range presented in equations
(15) and (16) at the same time.
2.3. Individual Reservoir Release Rules for Reservoirs in Parallel
Reservoir releases need to be specified when water release cannot satisfy the inequalities of storage and
release of each reservoir (equations (3) and (4)) because the derived optimal release is only applicable when
the two constraints are unbinding (i.e., equations (15) and (16) are satisfied). Considering these inequality
constraints, the Karush-Kuhn-Tucker conditions (also known as the KKT conditions, see Bazaraa et al. [2006]
for details) are used directly to derive the necessary conditions for the optimal release decisions in Appendix B. In these situations, total available water is released when the optimal release in equation (12) exceeds
reservoir water availability. Also null release is obtained when the optimal release falls below the lower
bound of individual reservoir release, which is null in this study. According to these modifications, the optimal reservoir release becomes
8
0
;
Ri;t < 0
>
>
<
; 0 Ri;t WAi;t 2SMin
(17)
R0i;t 5 Ri;t
i;t11
>
>
:
Min
WAi;t 2SMin
i;t11 ; WAi;t 2Si;t11 < Ri;t
where R0i;t is the optimal reservoir release of i reservoir at time t, by considering the constraints of individual
reservoir storage and release (equations (3) and (4)).
In equation (17), three types of releases for individual reservoirs are specified to satisfy the inequality constraints of reservoir storage and release: null release, release with unbinding inequality constraints, and
release with water availability. Assume the numbers of individual reservoirs delivery out the three types of
water releases are p, q, and n-p-q, respectively, then the total release is
!
p
q
q
n2p2q
n
n
P
P
P
P Min
P
total P 0
Rt 5 Ri;t 5 12 aj;t
WAi;t 2 WAi;t 1 bj;t 2
Sk;t11
(18)
i51
j51
i51
i51
j51
k51
Equation (18) indicates that the total water release is proportional to the water availability of reservoirs with
water delivery, but inverse proportional to the water availability of reservoir with null water delivery. Thus,
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the nonnegative release constraint should be considered before specifying the system release, because the
reservoir with null water delivery is associated with this inequality constraint.
It is worth noting that the total release only related to the total water availability within parallel reservoirs system, is a special case of equation (18), when all the reservoirs in parallel delivery water to joint demands (i.e.,
p 5 0). For this situation, an equivalent reservoir can be constructed, because the applicability of an equivalent
reservoir depends upon the total release specified by the total water availability [Robert et al., 1977; Oliverira and
Loucks, 1997; Koutsoyiannis and Economou, 2003; Xu et al., 2014]. For those reservoirs with null water delivery,
negative individual reservoir release (i.e., equation (15) is not satisfied) would be produced by equation (12) and
calculated into the system release for an equivalent reservoir. This leads to the release less than the optimal system release in equation (18), since the nonnegative release constraint is considered after the total release is
identified for an equivalent reservoir. Particularly, this system release might be insufficient to meet the projected
demand and result in water shortages while the demand is just met by the optimal system release.
The constraint of total release in equation (5) has been ignored in equation (18), which requires that the
total release should not exceed the projected demand nor fall below the minimum total release target.
Since more water is expected to delivery out from reservoir system for obeying the binding minimum total
release constraint (i.e., 0 Rtotal
< Dmin
), the constraint of minimum storage might be violated. On the
t
t
other hand, the main concern is to keep the individual reservoir release nonnegative, with the adjustment
for the maximum release constraint (i.e., Dt < Rtotal
). Thus, two different analytical reservoir releases should
t
be given for the two conditions, respectively, i.e., the binding constraints of minimum and maximum total
release, in order to ensure the system release at optimality (i.e., Dmin
and Dt for these two conditions,
t
respectively (the derivation is presented in Appendix B)), and directly satisfy other physical constraints.
For the condition 0 Rtotal
< Dmin
, it is assumed that the ratio of the optimal release with and without cont
t
sideration of total release constraint to the remaining water availability (i.e., the current water availability
minus the minimum storage and the optimal release with an unbinding total release constraint) is identical
00
0
0
Min
for every reservoir in parallel, i.e., ðR00i;t 2R0i;t Þ=ðWAi;t 2R0i;t 2SMin
i;t11 Þ5ðRj;t 2Rj;t Þ=ðWAj;t 2Rj;t 2Sj;t11 Þ, so that the
reservoir storage would not fall below the minimum storage with modification. Combining this assumption
together with the optimal system release, the individual reservoir release is derived as
R00i;t 5R0i;t 1
Dmin
2Rtotal
t
t
n
P
i51
ðWAi;t 2R0i;t 2SMin
i;t11 Þ
ðWAi;t 2R0i;t 2SMin
i;t11 Þ
(19a)
In equation (19a), the modified release R00i;t ranges from R0i;t to WAi;t 2SMin
i;t11 , while the system release is equal
to the minimum total release target.
, we assume that the delivery deviation between the optimal releases with and
For the condition Dt < Rtotal
t
without binding constraint of total release to the
total release constraint is
optimal
release
with unbinding
identical for every individual reservoir, i.e., R00i;t 2R0i;t =R0i;t 5 R00j;t 2R0j;t =R0j;t . Then the optimal reservoir
release is defined by
R00i;t 5
R0i;t 3Dt
n
P
R0i;t
(19b)
i51
In equation (19b), the modified release
water as the projected demand.
R00i;t
ranges from 0 to R0i;t , while the system would release as much
The proposed individual reservoir release therefore becomes:
8
Dmin
2Rtotal
>
t
t
0
>
>
WAi;t 2R0i;t 2SMin
< Dmin
; 0 Rtotal
> Ri;t 1 X
i;t11
t
t
n >
>
0
Min
>
>
WA
2R
2S
i;t
>
i;t
i;t11
>
<
i51
R00i;t 5
>
R0i;t
; Dmin
Rtotal
Dt
>
t
t
>
>
>
>
0
>
Ri;t 3Dt
>
>
>
;
Dt < Rtotal
:
t
Rtotal
t
(19c)
where R00i;t is the reservoir release of i reservoir at time t, taking into account all the physical constraints.
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All the physical constraints are satisfied and the system release is guarConstraints
anteed to be optimal by the release
Reservoir Storage and Total Release Individual Reservoir
R00i;t expressed in equation (19c). DifStep Mass Balances
Release Constraints
Constraint
Release Rule
ferent from the release obtained by
1
冑
Equation (12)
iteration [Nalbantis and Koutsoyiannis,
2
冑
冑
Equation (17)
3
冑
冑
冑
Equation (19c)
1997], R00i;t is given in analytical
a
forms, which provides a direct
Note: ‘‘冑’’ represents the constraint to be considered.
guideline for decision making and
avoids the complexity of iterative algorithm. Moreover, this analytical release would be optimal when the
assumptions presented above are tenable.
Table 1. Individual Reservoir Release Rule for Parallel Reservoir Systema
Thus, equations (12), (17), and (19c) comprise the complete individual reservoir release rules for parallel reservoir system with joint demands, and the flowchart of the proposed operation rules is shown in Table 1.
Among these, equation (12) is the optimal reservoir release only subject to the mass balance constraints
(i.e., other constraints are unbinding), while equation (17) is modified to consider the constraints of reservoir
storage and release, and all the physical constraints can be satisfied through equation (19c).
3. Operating Policy for an Equivalent Reservoir
3.1. System-Wide Release Rule
The commonly used operating policies defined by system-wide release rule and storage balancing functions
are aimed to provide guidelines for the operation of an equivalent reservoir, for reducing the complexity of
the optimal multireservoir operation problem [Robert et al., 1977; Oliverira and Loucks, 1997; Koutsoyiannis and
Economou, 2003; Xu et al., 2014]. In order to analytically derive the commonly used operating policies and further compare it with the proposed operation rules, the question of when it is optimal to transform reservoirs
in parallel into an equivalent reservoir should be answered first. As a special case, the optimal total release is
only related to the total water availability within system, only when all the reservoirs delivery water to joint
demands (i.e., p 5 0). Since the applicability of an equivalent reservoir depends upon the total release specified by the total water availably, it is optimal to construct an equivalent reservoir for the parallel reservoir system with the assumption that water would be released from every reservoir in parallel system (i.e., p 5 0).
Based on this assumption, the total release for equivalent reservoir is expressed as
!
q
q
n2q
n
P
P
P
P
total
Rt 5 12 aj;t
WAi;t 1 bj;t 2 SMin
(20)
k;t11
j51
j51
i51
k51
For simplicity, parameters Pi;t and Oi;t are defined by:
Pi;t 512
q
P
ai;t ; Oi;t 5
i51
q
P
j51
bj;t 2
n2q
P
k51
SMin
k;t11
(21)
Then equation (23) becomes:
n
P
Rtotal
5
P
WA
i;t
i;t 1Oi;t
t
(22)
i51
The operation of an equivalent reservoir should not violate the constraints of total storage volume and
release. The total storage constraint has been implicitly satisfied through equation (20), because the constraints of individual reservoir storage are taken into accounted by equation (17). In order to obey the total
water release constraint, the release is adjusted to
8
Dmin
;
Rtotal
< Dmin
>
t
t
t
>
>
>
!
>
n
<
X
0
(23)
Rtotal
5 Pi;t
WAi;t 1Oi;t ; Dmin
Rtotal
Dt
t
t
t
>
>
i51
>
>
>
:
Dt
;
Dt < Rtotal
t
In equation (23), the system-wide release rule for an equivalent reservoir of parallel reservoir system is similar to the analytical optimal hedging rule investigated by Shiau [2011], except that the release rule is well-
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Figure 3. The optimal hedging rule for an equivalent reservoir of parallel reservoir system with given numbers of reservoirs with unbinding inequality constraints.
adopted parameterization in this study. This kind of linear hedging rule is justified to be optimal with the
assumption that all the reservoirs in parallel delivery water to joint water demand. But for any reservoir with
null delivery (i.e., the assumption is not valid), the total water release indicated by the linear hedging rule is
different from the optimal system release expressed in equation (18), which is not only related to the total
water availability in the system, but also related to the water availability of reservoir with null water
delivery.
The trigger, a threshold to initiate and terminate the hedging rule, plays an important role in implementation because excessive hedging reduces the reliability of water supply [Bayazit and Unal, 1990]. The starting
water availability (SWA) and ending water availability (EWA) of the hedging rule are derived as follows.
8 n
n
X
X
>
>
>
Smin
Smin
>
i;t 1Oi;t
i;t 1Oi;t
min
>
>
D
2O
i;t
i51
i51
>
t
>
;
>
<
12Pi;t
Pi;t
12Pi;t
Dt 2Oi;t
(24)
SWAt 5
; EWAt 5
n
>
Pi;t
X
>
>
min
>
Si;t 1Oi;t
>
>
> Dmin
2Oi;t
Dmin 2Oi;t
>
i51
t
>
;
t
:
Pi;t
12Pi;t
Pi;t
In equation (24), two kinds of SWA and one EWA are specified. The relationship between ðDmin
2Oi;t Þ=Pi;t
t
P
and ð ni51 Smin
1O
Þ=ð12P
Þ
can
be
used
to
distinguish
these
two
types
of
hedging.
When
i;t
i;t
i;t
Pn min
Pn min
ðDmin
2O
Þ=P
ð
S
1O
Þ=ð12P
Þ,
ð
S
1O
Þ=ð12P
Þ
is
implemented
to
initiate
hedging
i;t
i;t
i;t
i;t
i;t
i;t
t
i51 i;t
i51 i;t
P
(i.e., hedging rule 1 in Figure 3). On the other hand, ðDmin
2Oi;t Þ=Pi;t becomes SWA, when ð ni51 Smin
t
i;t 1Oi;t Þ=
ð12Pi;t Þ ðDmin
2O
Þ=P
(i.e.,
hedging
rule
3
in
Figure
3).
These
criteria
are
selected
by
the
constraints
of
i;t
i;t
t
P
total
minimum equivalent reservoir storage volume, i.e., ni51 Smin
S
,
and
minimum
total
water
release,
i.e.,
i;t
t11
Dmin
Rtotal
, respectively.
t
t
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In addition, the inverse-weighted target ratio gi;t , to detect the types of hedging for a single
water supply reservoir, proposed by Shiau [2011], is a special case in this study when
P
ðDmin
2Oi;t Þ=Pi;t 5ð ni51 Smin
t
i;t 1Oi;t Þ=ð12Pi;t Þ. One-point hedging is also obtained in this case, while SWA is
P
min
equivalent to ni51 Smin
1D
(i.e., hedging rule 2 in Figure 3).
i;t
t
Dt . The value of EWA is directly
EWA is derived from the constraint of maximum total water release, i.e., Rtotal
t
affected by the number of reservoirs with unbinding inequality constraints of individual reservoir storage and
P
Pn T
release, i.e., q. With q increases from 0 to n, EWA moves from ni51 Smin
i;t 1Dt to
i51 Si;t 1Dt . For example,
Pn T
i51 Si;t 1Dt is specified as EWA when all the reservoir releases can be described by equation (12) (i.e., q 5 n),
and the EWA investigated by Shiau [2011] (i.e., STt 1Dt ), is a special case where only one reservoir is considered.
3.2. Storage Balancing Functions
On the basis of the total release indicated by hedging rule, thereafter the problem of identifying the desired
storage volume of each reservoir needs to be further solved. According to equation (23), the volume of reservoir system can be deducted as
8
n
X
>
0
>
>
WAi;t 2Dmin
;
Rtotal
5Dmin
>
t
t
t
>
>
>
i51
>
>
>
!
>
<
n
n
X
0
total P
total 0
(25)
St11 5 WAi;t 2Rt 5 12Pi;t
WAi;t 2Oi;t ;
Dmin
< Rtotal
< Dt
t
t
>
i51
>
i51
>
>
>
>
n
>
X
>
0
>
>
WAi;t 2Dt
;
Rtotal
5Dt
>
t
:
i51
In equation (12), when the inequality constraints (equations (1–5)) are unbinding, the storage of individual
reservoir is:
n
P
Si;t11 5WAi;t 2Ri;t 5 ai;t
WAi;t 2bi;t
(26)
i51
For different reservoirs in parallel system, the storages of reservoir i and reservoir j have the following
relationship:
1
1
m21
gi;t
STi;t11 2Si;t11
gm21
STj;t11 2Sj;t11
j;t
5
(27)
STi;t11
STj;t11
Equations (27) denote that: (1) the storage of individual reservoir is positively correlated with each other; (2)
all the storage of individual reservoirs become carryover storage targets, once one reservoir reaches its storage target; (3) especially, if the carryover storage targets are defined by the maximum storage of reservoirs,
the probabilities of water spilling are the same for each reservoir, otherwise, if the minimum storages are
used as carryover storage targets, the probabilities of emptying storage are equal among reservoirs.
Contrasting the storage volume of parallel reservoir system in equation (25), the storage of individual reservoir can be expressed as:
total
8
0
2bi;t ;
Rtotal
5Dmin
ai;t St11 1Dmin
>
t
t
t
>
>
>
<
ai;t total
0
St11 1Oi;t 2bi;t ; Dmin
< Rtotal
< Dt
(28)
Si;t11 5
t
t
>
12P
i;t
>
>
>
total
:
0
ai;t St11 1Dt 2bi;t ;
Rtotal
5Dt
t
To simplify this equation, two parameters Ai;t and Bi;t are introduced as below
min 8
8
0
ai;t Dt
; Rtotal
5Dmin
2bi;t
>
>
t
t
total 0
min
>
>
>
>
a
;
R
5D
i;t
t
t
>
>
>
>
>
>
< ai;t
<
0
ai;t 0
; Dmin
< Rtotal
< Dt ; Bi;t 5
Oi;t 2bi;t ; Dmin
< Rtotal
< Dt
Ai;t 5
t
t
t
t
12P
>
>
12Pi;t
i;t
>
>
>
>
>
>
0
>
>
>
>
:
:
ai;t ; Rtotal
5Dt
0
t
ai;t ðDt Þ2bi;t
; Rtotal
5Dt
t
(29)
Substituting equation (29) into (28), the storage of individual reservoir can be written as a linear parametric
form of the system storage volume (i.e., linear parametric rule)
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0
Si;t11 5Ai;t 3Stotal
t11 1Bi;t
(30)
Reservoir storage identified by balancing
functions should subject to the constraints of individual reservoir storage in
equation (3). Considering the constraints,
equation (30) is modified to
8
Si;t11 < Smin
Smin
>
i;t11 ;
i;t11
>
<
0
min
Si;t11 5 Si;t11 ; Si;t11 Si;t11 Smax
i;t11
>
>
:
max
max
Si;t11 ;
Si;t11 < Si;t11
(31)
Equations (29–31) indicate that based on
the total release specified by hedging
rule, the linear form of parametric rule
(shown in Figure 4) results in desired individual reservoir storage, when the
Figure 4. The optimal storage balancing function for an equivalent reservoir
nonnegative release constraints are
of parallel reservoir system.
unbinding. However, if the constraints
shift from unbinding to binding status, the releases suggested by the parametric rule are not feasible for
some reservoirs, as the storage exceeds the water availability of the reservoirs and negative releases are
obtained.
The optimal storage volume for an equivalent reservoir in equation (25) would not be always guaranteed
by the storage described in equation (31) [Nalbantis and Koutsoyiannis, 1997]. To satisfy the physical constraints of individual reservoir and ensure the total storage at optimality, the similar adjustment of individual reservoir release in equation (19c) can be used for the modification of reservoir storage. In this paper,
the adjusting procedure for individual reservoir storage is not discussed in detail.
4. Illustrating Case Study
4.1. B-Y Parallel Reservoir System Description
The derived operation rules are applied to simulating the performance of a real-world parallel reservoir system in Liaoning province of northeast China-Biliu River reservoir (B reservoir) and Yingna River reservoir (Y
reservoir). These two reservoirs are the main water sources for Dalian city, and they are jointly operated to
supply water for domestic and industrial use of the metropolitan area in Dalian. The layout of the B-Y reservoir system is shown in Figure 5.
The input to the reservoir system uses historical inflow from 1951 to 2003 (53 years), with the assumption
that the inflow reoccurs in the future. A 10 day time period is employed in this study, resulting in a total of
1908 operation periods. It is further assumed that the annual planned demand remains constant, and does
not vary much in different periods. In other words, the constant release target of 11.85 million m3 is
employed for each 10 day time period. The maximum allowable percentage of reduction in release is 10%,
that is, the minimum total release target is 90% of the joint water demand. The monthly average inflow of
each reservoir can be seen from Figure 6a. As the inflow distributes unevenly within a year, the timevarying carryover storage targets are prescribed for both reservoirs, as illustrated in Figure 6b. The evaporation losses are calculated by the surface area of the reservoir. Reservoir characteristics and annual joint
water demand can be seen from Table 2.
The operation rules derived from the two-period model are to efficiently utilize available water for current
use while minimize potential severe water shortage in the future [Draper and Lund, 2004]. Therefore, the
long-term water supply performance is the focus of the case study, which are specified by the shortage
index (SI) and water supply reliability (REL). SI has the same mathematical expression of the loss function for
water delivery, given by
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Figure 5. The layout of B-Y parallel reservoir system.
SI5
2
P Dt 2Rtotal
100 M3N
t
3
D
t
M3N t51
(32)
where N is the number of simulating years; M is the number of periods in a year.
Corresponding to the expression of SI, the value of M in the loss function is set as 2. REL is used to evaluate
the shortage duration in long-term operation, defined as
(
M3N
< Dt
P 1 if Rtotal
1
t
REL512
3
(33)
M3N t51
0 otherwise
4.2. Effects of Weighting Factors on System Operation Performance
The contours of SI and REL for various combinations of weighting factors of B reservoir storage target and Y
reservoir storage target (i.e., xB and xY ), ranging from 0 to 1, are illustrated in Figures 7a and 7b, respectively.
As shown in Figures 7a and 7b, only a portion of the combinations of xB and xY are feasible for the minimal
system release requirement during the long-term operation. To satisfy this requirement, the sum of the
weighting factors for storage targets should not fall down below 0.7, i.e., 0:7 xB 1xY . Moreover, if xB is
less than 0.5, xY should be greater than 0.2, with greater xB being associated with wider range of xY . On
the other hand, if xB is greater than 0.5, xY should be less than 0.2, with greater xB being associated with
narrower range of xY . Therefore, the point xB 50:5 and xY 50:2 is the vertex of feasible region.
It is worth noting that the limiting combinations of weighting factors ðxB ; xY Þ ! ð0; 0Þ, which are identical
to the results of the system release specified by Standard Operating Policy (SOP), are not applicable. The
reason for this is that the available water in system is used to meet as much of the release target as possible
and no water would be carried to the future under SOP. Thus, insufficient supplies for the minimal system
release requirement would be produced. The results indicate that it is not preferable to choose SOP for
specifying system release, mainly due to minimal system release consideration.
For the reservoir system performances, different effects of weighting factors on SI and REL can be observed.
Generally, SI increases with increasing xB and increasing xY . The minimum SI of 0.0067 is obtained at xB 5
0:5 and xY 50:2, which is the vertex of the feasible region. Weighting factors away from this point would
further increase the value of SI. The maximum SI of 0.0486 occurs at the line xB 1xY 51, where the
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Figure 6. (a) The monthly average inflow of each reservoir in the parallel system; (b) carryover storage target curves of B reservoir and Y reservoir.
weighting factor for the release target is close to 0 and no more water but the minimum total release
requirement would be delivered until all the storages of individual reservoirs reach the carryover storage
targets, which causes SI to maximize.
Table 2. Reservoir Characteristics and Annual Water Demand in B-Y Reservoirs System
Maximum Storage (million m3)
Reservoir
B
Y
ZENG ET AL.
Watershed
Area (km2)
Flood Season
Nonflood Season
Minimum Storage
(million m3)
Annual Joint Water
Demand (million m3)
2085
692
665.21
237.66
714.59
237.66
70
22.08
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Figure 7. Effects of weighting factors on (a) SI; (b) REL.
The REL exhibits a more complex variation with the weighting factors. Generally, REL decreases with
increasing xB when xY is less than 0.25 or greater than 0.55. On the other hand, REL increases first and
thereafter decreases with increasing xB in the intermediate range of xY from 0.25 to 0.55. Similar variation
patterns of REL with weighting factor xY can be observed. That is, REL increases first and thereafter
decreases with increasing xY in the intermediate range of xB from 0.35 to 0.7. Otherwise, REL decreases
with increasing xY . In the case when the sum of weighting factors approaching 1 (i.e., xB 1xY ! 1), REL is
significantly reduced, especially for xY greater than 0.25. Therefore, the minimum REL of 87.26% is found at
the line xB 1xY 51 for the same reason that the maximum SI obtained there. The maximum REL of 92.45%
occurs at the point xB 50:4 and xY 50:55. However, the vertex xB 50:5 and xY 50:2 with minimum SI, does
not lead to the maximum REL but the REL of 90.41%. This is because that the minimum SI often results from
low-percentage shortages (i.e., lower than 10% in this study) with more frequent hedging, so as to reduce
occurrence frequencies of high-percentage shortages (i.e., 10%) but corresponding to higher REL.
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Figure 8. The feasible range of weighting factors for REL above 90%.
The trade-offs between SI and REL for various combinations of xB and xY are essential for decision makers
to select the combinations of weighting factors during droughts. In practice, REL for domestic and industrial
water use is often required to be above 90%. The feasible combinations of weighting factors can be
obtained by overlapping Figures 7a with 7b, as shown in Figure 8. The feasible region indicates that xB
should range from 0.3 to 0.7, while the range of xY from 0.2 to 0.65 is feasible, with generally higher xB
being related to lower xY , to achieve the required REL.
4.3. Results of Different Operating Policies
The weighting factors for the carryover storage values of B and Y reservoirs are set as 0.67 and 0.23, respectively, on the basis of the feasible combinations of weighting factors in Figure 8.
In order to investigate the effectiveness of the proposed operation rules, the commonly used operating policies defined in terms of hedging rule and parametric rule (i.e., derived in section 3) are adopted to contrast
the performance of the B-Y reservoir system. The same input data and physical constraints are used for both
operation rules. The water supply results under each policy are shown in Table 3. As seen from the performance of two rules, less water shortage would be occurred and higher water supply reliability could be produced in the proposed operation rules than that of the commonly used operating policies, which indicates
that the proposed operation rules are superior to the commonly used one in terms of both SI and REL.
Following the general optimality conditions for operation of an equivalent reservoir derived in section 3,
the distribution of shortage occurrence frequency with respect to ‘‘the ideal water releases’’ (i.e., the optimal
individual reservoir release before calculating the system release) under the two operation rules are presented in Figure 9. It can be seen that the total time periods of shortage occurrence are 188 (i.e., reliability
of 90.15%) within the proposed rules, and 247
Table 3. Water Supply Results of the Two Operation Rulesa
(i.e., reliability of 87.05%) within the commonly
Operation Rule
Water Supply Reliability
Shortage Index
used one. Thus, 59 more shortage events could
The proposed
90.15%
0.0254
be effectively avoided by the proposed rules if
operation rules
one ‘‘ideal release’’ exceeds the target demand
The commonly used
87.05%
0.0305
while the other presents negative. This is
operating policies
because the nonnegative release constraints of
a
Note: the commonly used operating policies are defined in
individual reservoirs are taken into account
terms of hedging rule (i.e., equation (23)) and parametric rule (i.e.,
equation (31)).
before calculating the total release in the
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Figure 9. Distribution of shortage occurrence frequency with respect to ‘‘the ideal water releases’’ (i.e., the optimal individual reservoir
release before calculating the system release) under (a) the proposed operation rules; (b) the commonly used policies (the number in a
cell represents the number of shortage events).
proposed operation rules. In contrast, the nonnegative release constraint is considered only after the total
release is specified for the equivalent reservoir, and a negative ‘‘ideal release’’ would be produced and considered into the system release, which reduces the supply to meet the target demand and causes water
shortage in these periods. This result verifies the analysis presented in section 3.1 that the proposed operation rules are more effective to reduce water shortages during droughts when no water release is indicated
for individual reservoir under the commonly used one (i.e., negative ‘‘ideal release’’).
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Figure 10. The lower bound water availability of (a) B-Y reservoir system; (b) B reservoir; (c) Y reservoir for the applicability of the commonly used operating policies.
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Figure 11. (a) Storage ratios of B reservoir and Y reservoir over the simulation periods; (b) storage ratios of B reservoir and Y reservoir during water spilling periods; (c) storage ratios of B
reservoir and Y reservoir during extreme drought periods.
As discussed before, the implementation of hedging can be identified by the total water availability for reservoir system observed from equation (24). While the water availability derived from equation (15) can be
used to provide guideline for the occurrence of water supply from individual reservoir (i.e., positive or negative ideal water release). Figures 10a–10c plot the lower bound water availability of B-Y reservoir system, B
reservoir and Y reservoir, respectively, for the applicability of the commonly used operating policies. These
lower bounds can be taken as references for determining when hedging should be implemented; under
what level of current water availability should individual reservoir delivery water out and which operating
policy is optimal. For example, the lower bound of total water availability for B-Y system is obtained as
581.53 million m3 at the first 10 day of August and hedging is implemented for the total water availability is
317.32 million m3 in 2003. In addition, the proposed operation rules are meaningful because the water
availability of B reservoir is 212.37 million m3 at this time, which is less than the lower bound of 291.90 million m3, implying no water released from B reservoir (i.e., negative ‘‘ideal release’’). However, both of the proposed operation rules and the commonly used one are useful for the situation where water is plentiful for
B-Y reservoir system or individual reservoir, such as the first 10 day of August in 2001. The results indicate
that the role of the proposed operation rules becomes prominent for the lower levels water availability of
the parallel reservoir system and individual reservoir, and it is trivial for either higher levels water availability
of the reservoir system or individual reservoir.The optimal operation rule for a parallel reservoir system is to
equalize the probability of seasonal spill among reservoirs during water spilling periods, while emptying reservoir storage during drought periods [Lund, 1999]. To verify the reasonability of the proposed operation
rules, the storage ratio of individual reservoir is introduced in this study. The ratio is defined as the water
retained in a reservoir proportional to the active storage. In Figure 11a, the reservoir storage ratios of B and
Y reservoirs over long-term operation periods are displayed. As can be seen from Figure 11a, the storage
ratios of B and Y reservoirs fluctuate in the same mode, and the correlation coefficient between the two
ratios is 0.89, which shows a nearly linear relationship between each other. For instance, as illustrated in Figure 11b, when one reservoir is spilling, the other is also spilling or approaching to a spilling state. In other
words, if the storage ratio of one reservoir reaches 1, that of the other is above 0.97 in most cases. Moreover, as illustrated in Figure 11c, the storage of both reservoirs reach the minimum storage during extreme
drought periods, that is, the storage ratios get close to null at the same time. The synchronized spilling and
emptying patterns between the two reservoirs show the reasonability of the proposed operation rules.
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Figure 12. (a) SI; (b) REL of the proposed operation rules and the commonly used one under different levels of uncertainties (Cv of
observed inflow is used as benchmark, i.e., 1).
In addition, the assumptions presented in section 2.3 are demonstrated to be applicable for the operation of
reservoirs in parallel, with comparisons of two other assumptions. The results are illustrated in Appendix C.
4.4. Effects of Inflow Uncertainty on Water Supply Performance
The optimality conditions for the commonly used operating policies are determined by the water availability of individual reservoir and of reservoir system. Since water availability is the sum of initial water stored in
the reservoir plus current inflow and minus evaporation loss, the effects of inflow uncertainty on reservoir
performance should not be ignored. In this part, 100 synthetic inflow series are generated through a first-
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order autoregressive model and a disaggregation model [Liu et al., 2011; Salas, 1993] with different levels of
uncertainties (represented by coefficients of variations, i.e., Cv) but similar other annual statistics, such as
annual average inflows and the positive correlation between the two inflows.
Figure 12 plots the water supply results of the proposed operation rules and the commonly used one under
different levels of uncertainties, where the Cv of observed inflow is used as benchmark, i.e., 1. It can be seen
that severer water shortages would occur with the increase of uncertainties under both operating policies,
but the proposed operation rules still produce a less SI and a higher REL than the commonly used one.
Moreover, water supply improvements from the proposed operation rules compared to the commonly
used one are larger under higher uncertainty levels. These results imply that the proposed operation rules
are efficient and reliable for operation of parallel reservoir system, especially under the situation with higher
hydrologic uncertainties.
5. Conclusions
Based on a two-period optimization model, this study analytically derives a set of release rules for individual
reservoirs of a parallel reservoir system supplying a joint water demand. With the operation objective
expressed as a weighted sum of normalized deviation from the total release and the carryover storage targets, the newly derived release is proportional to the water availability of its own and inverse proportional
to the total water availability of other reservoirs in the system.
According to the analytical form derived from a general optimality theory, the commonly used operating
policies quantified in terms of system release rule and storage balancing functions, are found to be a special
case of the newly derived rules. The linear hedging rule is justified to be the optimal system-wide release
rule of an equivalent reservoir with the assumption that all the reservoirs in parallel delivery water to the
joint demand simultaneously. The optimal total release from the system is proportional to the water availability of reservoirs with water delivery, but inverse proportional to the water availability of reservoir with
null water delivery. For those reservoirs with null water delivery, negative individual reservoir release
would be produced and calculated into the system release for an equivalent reservoir, which leads to
the release less than the optimal system release in the proposed operation rules. Particularly, this release
might be insufficient to meet the projected demand and result in water shortages while the demand is
just met by the optimal system release. On the basis of the total release specified by the hedging rule,
the linear form of parametric rule results in the desired storage volume of each reservoir, when the nonnegative release constraints of individual reservoirs are unbinding. However, if the constraints shift from
unbinding to binding status, the releases indicated by the parametric rule might not be feasible for some
reservoirs.
The B-Y parallel reservoir system in northeast China is employed as a case study to illustrate the proposed
release rules and to compare with the commonly used operating policies consisting hedging rule and parametric rule. The results show that the applicability of the commonly used policies is not only affected by the
water availability of reservoir system but also of individual reservoir. Compared with the commonly used
operating policies, less water shortage occurrence and higher water supply reliability would be produced
under the proposed operation rules for the lower levels water availability of the parallel reservoir system
and individual reservoir. Especially, the water supply performance of B-Y reservoir system is greatly
improved under higher hydrologic uncertainty.
This study assumes that the carryover storage target of each reservoir in the system is given, with consideration of the regulations that are not accounted in this modeling analysis. To apply the derived rules to realworld reservoir operations, an optimization program can be used to determine the carryover storage targets; if institution allows, coupling the rule-based simulation with the parameter optimization program can
result in better operation plans than the simulation with given carryover storage targets.
This paper is limited to historical records for water release decisions, with the assumption of perfect inflow
forecast. For real-time reservoir operation, the inflows cannot be perfectly forecasted and the effects of
inflow uncertainty on the operation performance of parallel reservoir system should be considered. Further
study is needed to overcome this limitation for real-time operation of multireservoir system.
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Appendix A
The analytical optimal reservoir
release Ri;t comprises some parameters, such as weighting factor ratio
xi =xn11 and exponent m. Since
these two parameters are integrated
1
1
m21
m21
in the parameter gi;t
, effects of gi;t
on Ri;t are investigated first.
As shown in Figure A1a, the optimal
reservoir release Ri;t increases with
1
m21
decreasing gi;t
, because the first
1
m21
derivative of Ri;t with respect to gi;t
1
is negative, i.e., @Ri;t =@gm21
i;t < 0. Vari1
m21
ation of gi;t
with gi;t and m is demonstrated in Figure A1b, which
reflects that different combinations
of gi;t and m result in different val1
m21
ues of gi;t
. Generally, effects of gi;t
1
and m on parameter gm21
show
i;t
some differences.
1
m21
The effects of gi;t on parameter gi;t
1
m21
are obvious, that is, gi;t increases
with the increment of gi;t . According
to the definition of gi;t in equation
(11),gi;t is a linear function of the
weighting factor ratio xi =xn11
(illustrated in Figure A1c). Thus, Ri;t
increases with the decrease of
xi =xn11 . This phenomenon can be
interpreted by the implication of
xi =xn11 , which represents the relative importance between loss functions of carryover storage target
and release target. As xi =xn11
increases, the importance of carryover storage target increases and
more water is stored in reservoir.
The effects of exponent m on param1
eter gm21
i;t depend upon the value of
gi;t , which denotes the ratio of the
marginal cost of total release to the
marginal cost of storage. For
0 < gi;t < 1, the marginal cost of
release is less than the marginal cost
of storage, and more water release is
1
called for. Therefore, gm21
i;t is positively
Figure A1. (a) Variation of Ri;t with gi;t ; (b) variation of gi;t with gi;t and m;
correlated with m while Ri;t is nega(c) variation of gi;t with xi =xn11 .
tively correlated with m. On the other
hand, 1 < gi;t denotes greater mar1
ginal cost of release, which demands more water to conserve. Thus, gm21
i;t is negatively correlated with m while
Ri;t is positively correlated with m. For gi;t 51,Ri;t is independent of m since the marginal cost of total release
1
m21
equals to the marginal cost of storage with any m (i.e., gi;t
51). Moreover, the optimal reservoir release with a
1=m 21
ZENG ET AL.
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1
m21
very large m, regardless of the value of gi;t , is approximately equivalent to the release with gi;t 51, because gi;t
gets close to 1.
Appendix B
In order to derive the optimality conditions for operation of parallel reservoir system, the two-period model
is rewritten as
min Z5CL1 S1;t11 1CL2 S2;t11 1 1CLn Sn;t11 1WL Rtotal
t
s:t:
Si;t11 1Ri;t 2WAi;t i 50
SMin
i;t11 2Si;t11 0
Si;t11 2SMax
i;t11 0
(B1)
2Ri;t 0
2Rtotal
Dmin
t
t
2Dt
Rtotal
t
0
0
For equation set (B1), the Karush-Kuhn-Tucker conditions (also known as the Kuhn-Tucker or the KKT conditions, see Bazaraa et al. [2006] for details) can be used directly to solve the two-period model. Generally
speaking, KKT conditions are only necessary conditions, but if the objective function and the inequality constraints are continuous differentiable convex function, KKT conditions are both necessary and sufficient
[Bazaraa et al., 2006; Zhao et al., 2011]. In this paper, the KKT conditions represent both necessary and sufficient conditions since all the constraints are linear and the loss functions are assumed to be convex. Applying the KKT conditions to equation set (B1), we have
8
0
0
>
>
CL
S
B2
i
>
i;t11 2ki 2kk;i 1kk;n1i 50
>
>
>
>
>
>
2ki 2km;i 2km;n11 1km;n12 50 B3
WL0 Rtotal
>
t
>
>
>
>
>
>
>
ki S0i;t11 1R0i;t 2WAi;t 50
B4
>
>
>
>
>
>
>
0
>
B5
kk;i SMin
>
i;t11 2Si;t11 50
>
>
>
<
B6
kk;n1i S0i;t11 2SMax
i;t11 50
>
>
>
>
>
>
B7
km;i R0i;t 50
>
>
>
>
>
min
total
>
km;n11 Dt 2Rt
50
B8
>
>
>
>
total
>
>
>
B9
km;n12 Rt 2Dt 50
>
>
>
>
>
>
B10
ki ; kk;i ; kk;n1i 0
>
>
>
>
:
B11
km;i ; km;n11 ; km;n12 0
where ki , kk;i , kk;n1i , km;i , km;n11 , and km;n12 are parameters defined in the KKT conditions. In economic
senses, these parameters reflect the marginal costs with changing constraints, for the objective function
expressed as loss function. The global optimal release and storage decisions of this model can be obtained
when equations (B2)–(B11) are satisfied.
Combining equation (B2) with equation (B3), the relationship between marginal cost of total release and
carryover storage for individual reservoir is
CLi 0 S0i;t11 1km;i 1km;n11 1kk;n1i 5WL0 Rtotal
(B12)
1km;n12 1kk;i
t
Following equations (B2)–(B12), the optimality conditions for operation of reservoirs in parallel with different binding constraints can be derived.
For the constraint of total release, three conditions can be observed from equations (B8) and (B9), namely
0 < km;n11 ,0 < km;n12 , and km;n11 5km;n12 50, since km;n11 and km;n12 cannot be greater than 0 at the same
ZENG ET AL.
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time. In each condition, the combinations of km;i , kk;i , and kk;n1i are the same (i.e., whether greater than 0 or
equal to 0), which allow a similar procedure to derive the optimality conditions. Thus, once the optimality
conditions can be derived under one situation, the other two would be obtained by the same procedure. In
this paper, the derivations of optimality conditions for km;n11 5km;n12 50 are described in detail, while a similar procedure can be used for the conditions 0 < km;n11 and 0 < km;n12 .
< Rtotal
< Dt . In this situation,
According to equations (B8) and (B9), km;n11 5km;n12 50 indicates that Dmin
t
t
equation (B12) becomes
1kk;i
CLi 0 S0i;t11 1km;i 1kk;n1i 5WL0 Rtotal
(B13)
t
For the combination of different values of km;i , kk;i , and kk;n1i , there are following six cases for each reservoir
in the system, since it is impossible for both kk;i and kk;n1i be greater than 0 (from equations (B5) to (B6)):
Þ (i.e., marginal utility principle). According to equations
1. If km;i 5kk;i 5kk;n1i 50, then CLi 0 ðS0i;t11 Þ5WL0 ðRtotal
t
0
Max
0
(B6)–(B8), the conditions that guarantee optimality are: SMin
i;t11 Si;t11 Si;t11 and 0 Ri;t .
0 total
0 0
2. If kk;i 5kk;n1i 50 and 0 < km;i , then CLi ðSi;t11 Þ < WL ðRt Þ. Under this case, less water is expected to be
0
Max
delivered. The optimal release and carryover decisions for i reservoir should satisfy: SMin
i;t11 Si;t11 Si;t11
0
0 total
0 0
and 05Ri;t , while CLi ðSi;t11 Þ < WL ðRt Þ.
3. If km;i 5kk;i 50 and 0 < kk;n1i , then CLi 0 ðS0i;t11 Þ < WL0 ðRtotal
Þ, which demands more water to be reserved in
t
0
i reservoir. The conditions that guarantee optimality can be obtained as: S0i;t11 5SMax
i;t11 and 0 Ri;t , while
0 total
0 0
CLi ðSi;t11 Þ < WL ðRt Þ.
4. If kk;i 50 and 0 < km;i kk;n1i , then CLi 0 ðS0i;t11 Þ < WL0 ðRtotal
Þ. According to equations (B6)–(B8), the release
t
0
and carryover decisions for i reservoir that guarantee optimality are: S0i;t11 5SMax
i;t11 and 05Ri;t , while
0 total
0 0
CLi ðSi;t11 Þ < WL ðRt Þ.
5. If km;i 5kk;n1i 50 and 0 < kk;i , then WL0 ðRtotal
Þ < CLi 0 ðS0i;t11 Þ, which indicates that more water release is
t
0
0 total
called for. The optimal solutions are derived as: S0i;t11 5SMin
Þ < CLi 0 ðS0i;t11 Þ.
i;t11 and 0 Ri;t , while WL ðRt
0
Min
6. If kk;n1i 50 and 0 < km;i kk;i , then the conditions that guarantee optimality are: Si;t11 5Si;t11 and 05R0i;t .
In summary, the equality of the marginal cost between total release and carryover storage (i.e., marginal
utility principle) would not be always satisfied at optimality due to the inequality of reservoir storage and
release constraints. For km;n11 5km;n12 50, an unbinding minimum storage constraint leads to a higher marginal cost of total release, i.e., CLi 0 ðS0i;t11 Þ WL0 ðRtotal
Þ. On the other hand, a higher marginal cost of carryt
over storage can be obtained with only minimum storage constraint binding. The optimality conditions also
denote that, total available water would be released with the binding minimum storage constraint, and null
release is implemented when the nonnegative release constraint is binding.
It is worth noting that the optimal system release is specified under the binding constraints of total release,
i.e., 0 < km;n11 and 0 < km;n12 , which is the main differences from the condition km;n11 5km;n12 50 associated with Dmin
< Rtotal
< Dt . For example, the optimal system release is Dmin
for the condition 0 < km;n11 ,
t
t
t
while Dt is identified as the optimal system release for the condition 0 < km;n12 . On the basis of the specified optimal system releases, six similar combinations of km;i ,kk;i , and kk;n1i as described above can be used
to derive the optimality condition for 0 < km;n11 and 0 < km;n12 , respectively, which are not given in detail
in this paper.
Appendix C
In order to verify the efficiency of the assumptions presented in section 2.3, we have adopted two other
assumptions, i.e., the reservoir of smaller storage capacity delivery water first and store water last (small
capacity priority assumption), and the reservoir of larger storage capacity delivery water first and store
water last (large capacity priority assumption), to contrast the performance of the B-Y reservoir system. No
significant differences among the water supply results of different assumptions are obtained. To be specific,
REL is the same as 90.15% while the value of SI are 0.0254, 0.0258, and 0.0265 for the proposed, small
capacity priority and large capacity priority assumptions respectively. Therefore, less SI and higher REL are
produced under the three operation rules based on different assumptions than that of the commonly used
operating policies. The results imply that the assumptions presented in section 2.3 are applicable for the
operation of reservoirs in parallel.
ZENG ET AL.
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Notation
Ai;t
ai;t
Bi;t
bi;t
CLi ðÞ
Dmin
t
Dt
Ei;t
EWAt
Ii;t
L
M
m
N
n
Oi;t
Pi;t
p
q
Ri;t
Ri;t
R0i;t
R00i;t
Rtotal
t
REL
Si;t
Si;t11
S0i;t11
Acknowledgments
The authors gratefully thank
Investigation and Design Institute of
Water Resources and Hydropower
Liaoning Province for providing the
raw data of B-Y reservoir system in
northeast China, which are freely
accessible in Excel format by
contacting the first author (Xiang
Zeng) at Wuhan University at
zengxiang@whu.edu.cn. This research
is supported by the Natural Sciences
Foundation of China (51339004,
51479142, 51490282, and 51525902),
National Water Project of China
(2014ZX07204-006), and ‘‘PhD Shorttime Mobility Program, Wuhan
University.’’ The authors would also like
to thank the anonymous reviewers for
their review and constructive
comments related to this manuscript.
ZENG ET AL.
SMin
i;t11
SMax
i;t11
STi;t11
Stotal
t11
SI
SWAt
xi
WAi;t
WLðÞ
ki
gi;t
constant for Stotal
t11 .
constant for Ri;t .
constant for Stotal
t11 .
constant for Ri;t .
a loss function of the carryover storage value.
the minimum total release target of parallel reservoir system.
the total release target of parallel reservoir system.
the evaporation loss of i reservoir at time t.
the ending water availability at time t for parallel reservoir system.
the inflow of i reservoir at time t.
Lagrange function.
the number of periods in a year.
exponent of the loss functions.
the number of simulating years.
the number of individual reservoirs in parallel.
constant for Rtotal
.
t
constant for Rtotal
.
t
the number of individual reservoir release subject to null release constraints.
the number of individual reservoir release with unbinding inequality constraints.
the actual water release from i reservoir during time t.
the optimal reservoir release of i reservoir at time t only subject to the mass balance constraints.
the optimal reservoir release of i reservoir at time t, by considering the constraints of mass balance,
individual reservoir storage and release.
the reservoir release of i reservoir at time t, taking into account all the physical constraints.
the total water release from parallel reservoir system at time t.
water supply reliability.
the storage of i reservoir at the beginning of time t.
the optimal reservoir storage of i reservoir at time t only subject to the mass balance constraints
and total demand constraints.
the optimal reservoir storage of i reservoir at time t subject to the constraints of mass balance, individual reservoir storage and total demand constraints.
the minimum storage of i reservoir at time t.
the maximum storage of i reservoir at time t.
the desired carryover storage target of i reservoir at time t.
the total storage volume of reservoir system at time t.
shortage index.
the starting water availability at time t for parallel reservoir system.
the weighting factor assigned to the loss function.
the water availability of i reservoir at time t.
the water-deficit-related loss function for current total water delivery.
Lagrange multiplier.
the inverse-weighted target ratio for i reservoir at time t.
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