1 Introduction

In recent years, the uneven distribution of water resources and increasingly imbalanced water demand have prompted greater need for transferring water from one basin with sufficient water to another basin facing water shortages. Inter-basin water transfer project, man-made conveyance schemes, is an efficient solutions to relieve the water-supply pressure and ensure a balanced economic development among different regions (Sadegh et al. 2010). Due to the hydraulic relation between the source and the receiving systems connected by inter-basin water transfer projects, an entire system can be termed as an IBWTS. An IBWTS needs to regulate not only water supply but also water transfer, which is different from general water-supply systems without the consideration of water transfer. As the most important facility in the system, the reservoir plays an important role in regulating fluctuating inflows to meet some requirements. However, the inter-basin water transfer project has changed the natural inflows of importing reservoirs. Thus, there is a need to update the previous operating rule, which only refers to hedging rules, by coupling water-transfer rules with hedging rules to manage the IBWTS (Xi et al. 2010; Guo et al. 2012; Zhu et al. 2013; Zeng et al. 2014). In order to achieve optimal control of the IBWTS, the joint operating rule (also referred to as joint operating policies) is derived at the planning stage so it can then provide guidelines for reservoir releases and meet water demand targets as well as water transfers from donor reservoirs to recipient reservoirs.

Several types of reservoir operating rules for water supply have been previously suggested and discussed, such as the pack rule (Maass et al. 1962), the space rule (Bower et al. 1966), the New York City rule (NYC rule) (Clark 1956) and several forms of the Linear Decision Rule (LDR) (ReVelle et al. 1969). Although these rules are useful for the operation of simple single-purpose reservoir systems, they cannot be applied directly and do not provide clear indications for how to operate a complex system with several purposes and many constraints. Moreover, those rules are combined with a standard operating policy (SOP), which aims at minimizing the total deficit over the time horizon but neglects to consider the potential shortage vulnerability during future periods (You and Cai 2008; Shiau 2005, 2009). In addition, there are optimal operating rules for the long-term operation of a single reservoir and/or a multi-reservoir system, which derive from the efficient implicit stochastic optimization (ISO) method (Labadie 2004; Malekmohammadi et al. 2009; Ostadrahimi et al. 2012). However, it is difficult to obtain the optimal sample sets, especially for large-scale multi-purpose multi-reservoir systems. As a common measure for reservoir operations, hedging rules (HR) modified from the SOP are used for hedging (i.e., rationing or reducing) water supply during droughts. The aim of HR is to reduce the risk of unacceptably large water deficits at the cost of a series of acceptable small deficits (Shih and ReVelle 1994, 1995). Previously the HR only applied to single-purpose reservoirs, but then the discrete HR (i.e., constant piecewise linear rules) were proposed for multi-purpose multi-reservoirs (Neelakantan and Pundarikanthan 1999, 2000; Tu et al. 2003, 2008). After connecting the piecewise linear rules for each time period in 1 year, hedging rule curves are formed in a simple and easy to operate way (Liu et al. 2011). Nevertheless, all water demands can be met at the same rationing level or rationed at the same time when drought occurs according to the hedging rule curves. Guo et al. (2013) and Zeng et al. (2014) improved this rule, using different hedging rule curves with different rationing factors for different types of water users. Therefore, their improved hedging rule, which uses hedging rule curves, is adopted in this paper.

In terms of IBWTS, a majority of prior studies focused on the reasonable allocation of water resources on the macro-level between source and receiving basins. They considered water rights, fairness and efficiency of the water diversion capacity, the impacts of ecological environment, and so on (Sadegh et al. 2010; Nikoo et al. 2012; Zhang et al. 2012; Li et al. 2013). However, the study of multi-reservoir water-transfer rules is still limited. To derive a water-transfer rule, two key problems, “when and how much to transfer” during particular period, should be answered simultaneously, which, as is the case with the hedging rule, is determined by the trigger values specified by water-transfer rule curves. Recently, three patterns of water-transfer rules have been proposed. i) Xi et al. (2010) proposed that water-transfer decisions should be determined only by water-transfer rule curves of the recipient reservoir (RULE-REC). This rule is developed with the assumption that there is abundant water available in the donor reservoir and water can be provided whenever there is a water shortage in the recipient reservoir. This is obviously unreasonable when there is not enough water available in the donor reservoir during particular periods. ii) To respond to this problem, Guo et al. (2012) took both the donor and recipient reservoirs into consideration to make water-transfer decisions (RULE-SIN). However, Zeng et al. (2014) deemed that the water-transfer rule curve of the donor reservoir cannot indicate whether there is abundant water available to be diverted to one or many recipient reservoirs individually, since only one water-transfer rule curve for the donor reservoir can be used. iii) Therefore, using different water-transfer rule curves in the donor reservoir for each recipient reservoir respectively (RULE-SEP) was proposed by Zeng et al. (2014). Due to there being only one water-transfer rule curve in each recipient reservoir for both RULE-SIN and RULE-SEP, there are merely two water-transfer states for each recipient reservoir, namely water transfer (STATE1) and no water transfer (STATE3). In STATE1, the actual transferred water should not exceed the designed delivery capacity, which might be unreasonable when a small amount of water is needed in the recipient reservoir so that the excess transferred water may be abandoned in the recipient reservoir.

The purpose of this paper, which is based on the earlier studies, is to derive a new water-transfer rule coupled with a hedging rule (RULE-DOU). To improve the efficiency of water transfer, an efficient water-transfer pattern, namely upper-and-lower water-transfer rule curves for each recipient reservoir and one water-transfer rule curve for the donor reservoir, are developed to make water-transfer decisions. In the proposed water-transfer rule curves, there are three water-transfer states for each recipient reservoir: i) water transfer within the designed delivery capacity (STATE1); ii) water transfer equal to the quantity interpolated linearly within the transferred water in STATE1 (STATE2); iii) no water transfer (STATE3). Based on the proposed rule, optimization operation models for the IBWTS are established with two objectives of maximizing the water supply and minimizing the water transfer. Finally, the North-line IBWTS, with one donor and several recipient reservoirs located in Liaoning Province, China, is employed as a case study to verify the reasonability and efficiency of the proposed rule-based operation model.

2 Material and Methods

2.1 Operating Rule

2.1.1 Trigger Mechanism

A trigger mechanism is a basic condition to be addressed before defining operating rules. The trigger mechanism served as triggers to start hedging in operating rules is presented primarily in two ways. First, it is presented as a function of the total available water (i.e., reservoir storage plus inflow) (Shih and ReVelle 1994, 1995; Shiau 2005, 2009). Second, it is a function of the reservoir storage (Neelakantan and Pundarikanthan 1999, 2000; Zeng et al. 2014). Because the unevenly distributed inflow and the medium- and long-term forecast accuracy cannot meet application requirements, the latter one is still an important trigger adopted in water transfer-supply operations of multi-reservoir systems. Obviously, the corresponding trigger values of the later trigger mechanism in each time period are the functions of the current reservoir storage and hydrological stage. Thus by connecting trigger values in time periods of each year, water-transfer or hedging rule curves are formed (Tu et al. 2003, 2008). According to the trigger mechanism adopted in this paper, the state variable SV i,t of reservoir operations should be defined as the reservoir storage (V i,t ) at the beginning of the time period t:

$$ S{V}_{i,t}={V}_{i,t} $$
(1)

where i = 1,2,…,N, N is the total number of reservoirs in system; t = 1,2,…,T, T is total operation periods.

2.1.2 Improving Water-Transfer Rule

The improving water-transfer rule proposed in this paper includes water-transfer rule curves and a water-transfer allocation model. The water-transfer rule curves, based on the water storage conditions in both donor and recipient reservoirs, make decisions on “when to transfer” and “how much to transfer” under the given planned water transfer W plan i,t (seen in Eq. (2)). These planned water transfers for each recipient reservoir from the donor reservoir are initially determined by a water-transfer allocation model. Because this study mainly focuses on water-transfer rule curves, the dynamic allocation ratio is adopted in the water-transfer allocation model for the initial decomposition of total water transfer from the donor reservoir (Peng et al. 2014).

The positions of the water-transfer rule curves are taken explicitly as trigger values for making water-transfer decisions. However, it should be noted that water-transfer decisions are opposed for donor and recipient reservoirs. For example, the higher position of the water-transfer rule curve of the donor reservoir means less probability for transferring water into recipient reservoirs. However, the higher position of recipient reservoirs means greater probability for transferring water from donor reservoirs. Therefore, water transfer to a recipient reservoir is a combined decision-making process based on the decisions made by both donor and recipient reservoirs.

The predefined patterns of combined water-transfer rule curves in an IBWTS with one donor reservoir and several recipient reservoirs are predefined in Fig. 1. For the donor reservoir, the positions of water-transfer rule curves divide the active storage space into two water-transfer operation zones (D_ZONE1 and D_ZONE2). If SV i,t stays in D_ZONE2, there is surplus water to be transferred to recipient reservoirs as necessary; if SV i,t falls down in D_ZONE1, there is not enough water available to be diverted into recipient reservoirs regardless of the water storage conditions of recipient reservoirs. Nevertheless, for each recipient reservoir, the positions of the upper-lower water-transfer rule curves divide the active storage space into three water-transfer operation zones (R_ZONE1, R_ZONE2 and R_ZONE3). Moreover, the water-transfer rule curves of the recipient reservoir should be used in combination with the water-transfer rule curves of the donor reservoir to make water-transfer decisions for each recipient reservoir. As seen in Fig. 1, during the time period t, there are three combined discrete trigger values to be considered for both donor and recipient reservoirs, namely V A1,t , V A i,t and V B i,t . Thus, there are three combined states for water transfer or not, and the water-transfer rule is mathematically formulated as follows:

$$ \begin{array}{c}\hfill {W}_{i,t}^{draw}\left(S{V}_{1,t},S{V}_{i,t}\right)\left\{\begin{array}{cc}\hfill Q\hfill & \hfill \left(S{V}_{1,t},S{V}_{i,t}\right)\in STATE1\hfill \\ {}\hfill \alpha \cdot Q\hfill & \hfill \left(S{V}_{1,t},S{V}_{i,t}\right)\in STATE2\hfill \\ {}\hfill 0\hfill & \hfill \left(S{V}_{1,t},S{V}_{i,t}\right)\in STATE3\hfill \end{array}\right.\hfill \\ {}\hfill \alpha \begin{array}{cc}\hfill =\frac{\left({V}_{i,t}^B-{V}_{i,t}\right)}{\left({V}_{i,t}^B-{V}_{i,t}^A\right)},\hfill & \hfill Q= \min \left\{{W}_{i,t}^{plan},{W}_i^{\max}\right\}\hfill \end{array}\hfill \end{array} $$
(2)
$$ \begin{array}{l}\left\{\begin{array}{l} STATE1=\left\{\left(S{V}_{1,t},S{V}_{i,t}\right)\left|S{V}_{1,t}\in D\_ ZONE2\right.,S{V}_{i,t}\in R\_ ZONE1\right\}\\ {} STATE2=\left\{\left(S{V}_{1,t},S{V}_{i,t}\right)\left|S{V}_{1,t}\in D\_ ZONE2\right.,S{V}_{i,t}\in R\_ ZONE2\right\}\\ {} STATE3=\left\{\left(S{V}_{1,t},S{V}_{i,t}\right)\left|\left(S{V}_{1,t},S{V}_{1,t}\right)\notin \left\{ STATE1\cup STATE2\right\}\right.\right\}\end{array}\right.\hfill \\ {}\left\{\begin{array}{l}D\_ ZONE2=\left\{S{V}_{1,t}\left|{V}_{1,t}^A\le S{V}_{1,t}\le {V}_{1,t}^{\max}\right.\right\}\\ {}R\_ ZONE1=\left\{S{V}_{i,t}\left|{V}_{i,t}^{\min}\le S{V}_{i,t}\le {V}_{i,t}^A\right.\right\}\\ {}R\_ ZONE2=\left\{S{V}_{i,t}\left|{V}_{i,t}^A\le S{V}_{i,t}\le {V}_{i,t}^B\right.\right\}\end{array}\right.\hfill \end{array} $$
(3)
$$ {V}_{1,t}^{\min}\le {V}_{1,t}^A\le {V}_{1,t}^{\max },{V}_{i,t}^{\min}\le {V}_{i,t}^A\le {V}_{i,t}^B\le {V}_{i,t}^{\max } $$
(4)
Fig. 1
figure 1

Combined water-transfer rule curves for water-transfer decisions in IBWTS

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The quantity of water transfer from the donor reservoir (W draw1,t ) can be summarized:

$$ {W}_{1,t}^{draw}={\displaystyle \sum_{i=2}^N{W}_{i,t}^{draw}\left(S{V}_{1,t},S{V}_{i,t}\right)} $$
(5)

where, i refers to the reservoir (i = 1 refers to the donor reservoir; i > 1 refers to the recipient reservoir); W draw i,t is the actual quantity of water transferred into recipient reservoir i during time period t; V max i,t and V min i,t are maximum and minimum storage capacities of reservoir i during time period t, respectively; W max i,t is the designed transporting capacity into recipient reservoir i. Specially, when the upper-and-lower rule curves overlap with each other, it is just the RULE-SIN introduced previously. In this RULE-SIN, only two water-transfer states are specified, i.e., STATE1 and STATE3 shown in the above equations.

2.1.3 Hedging Rule

The hedging rule consists of hedging rule curves and rationing factors (Tu et al. 2003, 2008). The hedging rule curves, which were adopted to decide when to reduce water supply, should be predefined before they are used by the simulation model (Oliveira and Loucks 1997). And the rationing factors used to determine the amount of water supply for different water users can be either assigned according to the experts’ knowledge or determined by optimization (Shih and ReVelle 1995). In this paper, rationing factors are given at the reservoir’s design stage according to the tolerable elastic range of each water user in which the damage caused by rationing water supply is limited.

For a multi-purpose reservoir, different types of water demand with different priorities require different water-supply reliabilities and different degrees of hedging in practice (Guo et al. 2012, 2013). As is known, the same reduction degree, i.e., one minus the rationing factor, for water demand with higher priority may produce a greater negative impact than that for water demand with lower priority. As drought occurs, water demand with lower priority should be rationed first; therefore, the degree of hedging ought to be larger and water-supply reliability ought to be smaller. Thus, different hedging rule curves, rationing factors and planning reliabilities should be assigned to different types of water users. There is one-to-one correspondence between hedging rule curves and water demands except that should be fully met. In such cases, hedging rule curves are used to determine the water-supply decisions and the actual water-supply reliabilities for water users.

Figure 2a illustrates the predefined pattern of hedging rule curves for domestic life, industry and agriculture with the priorities from high to low. The positions of the hedging rule curves correspond to the descending order of priorities, located from lower to higher, which divides the active storage space into four water-supply operation zones (i.e., ZONE1, ZONE2, ZONE3 and ZONE4). In reservoir operations, the corresponding policies for water supply will be made in terms of the operation zones in which the SV i,t stays. As seen in Fig. 2a, during the time period t, there are three discrete trigger values: V A t , V B i,t and V C i,t . The hedging rule can be described in Fig. 2b with piecewise linear functions, where D 1, D 2 and D 3 refer to the water demand targets for domestic life, industry and agriculture, respectively; β 1, β 2 and β 3 are rationing factors. This hedging rule is mathematically formulated:

$$ {W}_{i,t}\left(S{V}_{i,t}\right)=\left\{\begin{array}{ll}{\beta}_1{D}_{1,t}+{\beta}_2{D}_{2,t}+{\beta}_3{D}_{3,t},\hfill & S{V}_{i,t}\in ZONE1\hfill \\ {}{D}_{1,t}+{\beta}_2{D}_{2,t}+{\beta}_3{D}_{3,t},\hfill & S{V}_{i,t}\in ZONE2\hfill \\ {}{D}_{1,t}+{D}_{2,t}+{\beta}_3{D}_{3,t},\hfill & S{V}_{i,t}\in ZONE3\hfill \\ {}{D}_{1,t}+{D}_{2,t}+{D}_{3,t},\hfill & S{V}_{i,t}\in ZONE4\hfill \end{array}\right. $$
(6)

where

$$ \left\{\begin{array}{l} ZONE1=\left\{S{V}_{i,t}\left|{V}_{i,t}^{\min}\le S{V}_{i,t}<{V}_{i,t}^A\right.\right\}\\ {} ZONE2=\left\{S{V}_{i,t}\left|{V}_{i,t}^A\le S{V}_{i,t}<{V}_{i,t}^B\right.\right\}\\ {} ZONE3=\left\{S{V}_{i,t}\left|{V}_{i,t}^B\le S{V}_{i,t}<{V}_{i,t}^C\right.\right\}\\ {} ZONE4=\left\{S{V}_{i,t}\left|{V}_{i,t}^C\le S{V}_{i,t}\le {V}_{i,t}^{\max}\right.\right\}\end{array}\right. $$
(7)
$$ {V}_{i,t}^{\min}\le {V}_{i,t}^A\le {V}_{i,t}^B\le {V}_{i,t}^C\le {V}_{i,t}^{\max } $$
(8)

and W i,t is the decision variable denoting the total water release of the reservoir during time period t.

Fig. 2
figure 2

Hedging rule for a multipurpose reservoir

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When SV i,t falls down below the trigger values, the corresponding water supply for each water user should be reduced to retain enough water for the future and new break records ought to be made. Commonly, the water-supply reliability described by the probability is based on a statistical analysis (Stedinger and Loucks 1982). In this paper, the period reliability is taken based on the period-break statistics for domestic life and industry, while the year reliability is adopted based on year-break statistics for agriculture. The water-supply reliability is calculated by Eq. (9).

$$ {P}_j=\frac{N_{num}-{\displaystyle \sum_{t=1}^{N_{num}}{\sigma}_{j,t}}}{N_{num}+1}\left({\sigma}_{j,t}=\left\{\begin{array}{l}\begin{array}{cc}\hfill 1\hfill & \hfill {W}_{j,t}<{D}_{j,t}\hfill \end{array}\\ {}\begin{array}{cc}\hfill 0\hfill & \hfill otherwise\hfill \end{array}\end{array}\right.\right) $$
(9)

where P j denotes the actual water-supply reliability of water demand j; N num is the total operation years Y or time periods T; \( {\displaystyle \sum_{t=1}^{N_{num}}{\sigma}_{j,t}} \) is the accumulated number of operation years or time periods when the water release W j,t is less than the water demand D j,t .

2.2 Operation Model

2.2.1 Objective Function

In a water transfer-supply system, the first main objective is to meet the water demands for all water users of the entire system without violating the system constraints. There are several methods to achieve this objective, such as maximizing water supply (or release) for each water user (Shih and ReVelle 1995), minimizing water shortage (Neelakantan and Pundarikanthan 1999; Shiau 2009), minimizing the shortage index of the entire system (Chen 2003; Tu et al. 2008; Chang et al. 2010), and minimizing the maximum water shortage in one time period (Shih and ReVelle 1994; Shiau and Lee 2005). As mentioned above, a hedging rule can efficiently utilize available water for current use while minimizing potentially severe water shortages in the future. Moreover, constraining the reduction extent of water demand, which should be no more than its planning target, is to make the water-supply risks the lowest degree in this paper. Consequently, maximizing the water supply f 1 is adopted as one objective of this study. Besides, the IBWTS has become an efficient solution for relieving the pressure of water shortage and ensuring balanced economic development among different regions. However, it is often large-scale and expensive because it involves huge infrastructures and massive energy for pumping in some cases. Therefore, to encourage efficient water transfer, minimizing the water spillage of the recipient reservoirs f 2 is taken to be another important objective. The two objective functions are illustrated as below:

$$ {f}_1= \max {\displaystyle \sum_{i=1}^N{\displaystyle \sum_{t=1}^T{\displaystyle \sum_{j=1}^J{W}_{i,j,t}}}} $$
(10)
$$ {f}_2= \min {\displaystyle \sum_{i=2}^N{\displaystyle \sum_{t=1}^TS{U}_{i,t}}} $$
(11)

where j = 1,2,…,J, J is the total types of water users, such as industry, agriculture, ecology, etc.; T = L*Y, L is the length of rule curves or the number of time periods in a year. W i,j,t denotes actual water release for water demand j of reservoir i during time period t; SU i,t is the water spillage of reservoir i during time period t.

2.2.2 System Constraints

The operation in IBWTS should follow operating regulations and comply with physical limitations, such as mass balance equations, boundary constraints for reservoir storages and water transfer, as well as other constraint conditions. These constraints are formulated as follows:

  1. (1)

    Continuity balance equation

    $$ {V}_{i,t+1}={V}_{i,t}+{W}_{i,t}^{in}\pm {W}_{i,t}^{draw}-{\displaystyle \sum_{j=1}^J{W}_{i,j,t}}-S{U}_{i,t}-{W}_{i,t}^{loss} $$
    (12)

    where W in i,t , W loss i,t are water inflow and water losses by evaporation and seepage of reservoir i during time period t;

  2. (2)

    Water diversion should be no more than the designed capacity of the pipeline

    $$ {W}_{i,t}^{draw}\le {W}_{i,t}^{\max } $$
    (13)
  3. (3)

    Water storage should be constrained within the maximum storage capacityV max i,t and the minimum storage capacity V min i,t

    $$ {V}_{i,t}^{\min}\le {V}_{i,t}\le {V}_{i,t}^{\max } $$
    (14)
  4. (4)

    Water-supply reliability is no less than its planning target P e j

    $$ {P}_j\ge {P}_j^e $$
    (15)
  5. (5)

    Water release should be within the tolerable designed range of its water demand

    $$ {\beta}_j{D}_{i,j,t}\le {W}_{i,j,t}\le {D}_{i,j,t} $$
    (16)
  6. (6)

    The constraint of water-transfer rule curves is shown in Eq. (4). The hedging rule curves for water users are supposed to be in accordance with the priorities, as shown in Eq. (8). The position of hedging rule curves with higher priority ought to be lower, which can guarantee the reasonability of the rule curve without intersecting one other. In fact, how to meet the priority requirements is a sequencing process.

  7. (7)

    Additionally, the tendency of hedging rule curves is in accordance with the general operation laws of the given reservoir and all variables should be non-negative.

2.2.3 Fitness Function

In order to solve the model, the weighted and penalty approaches are employed to convert this multi-objective decision-making problem with heavy constraints into a single objective issue (Chang et al. 2010; Ngoc et al. 2014). That is, the fitness function is defined by the weighted objective functions plus the penalty functions in this paper. The simplest constraints, such as Eqs. (4), (8), (12), (13) and (16), are settled directly within the scope of variables or in operations. However, two penalty functions make the fitness function larger when water storage falls down below the minimum storage capacity (i.e., dead storage) and actual water-supply reliabilities are smaller than their planning targets. The combined fitness function is mathematically formulized:

$$ E= \min \left\{\begin{array}{l}-{w}_1\cdot {f}_1+{w}_2\cdot {f}_2\\ {}\begin{array}{cc}\hfill +{\displaystyle \sum_{j=1}^J{\zeta}_j\cdot \min {\left\{0,{P}_j-{P}_j^e\right\}}^2}+\hfill & \hfill {\displaystyle \sum_{i=1}^N\left\{\lambda \cdot {\displaystyle \sum_{t=1}^T \min {\left\{0,{V}_{i,t}-{V}_{i,t}^{\min}\right\}}^2}/{N}_i^{deep}\right\}}\hfill \end{array}\end{array}\right\} $$
(17)

where w 1 and w 2 are non-normalized weight factors determined by trials and experiments. According to different degrees of importance for each component in the combined objective, set the two weight factors in the same order of magnitude and make the first item in Eq. (17) larger than the second one. ζ j (ζ j  > 0) is a penalty coefficient when the actual water-supply reliability of water demand j is less than its planning target. In order to better guide the search direction for searching solutions, set ζ j a relatively large value, and ensure that the penalty coefficient of water demand with higher priority is greater than that of water demand with lower priority. λ (λ > 0) is also a penalty coefficients when the water storage falls down below the dead storage (i.e., depth breakage). As such depth breakage is not allowed in reservoir operation, λ is set to infinity. N deep i is the accumulated number of such time periods during which water storage falls down below the dead storage of reservoir i.

2.3 Solution Method

The decision variables in this paper to be optimized are operating rule curves. It is a complex decision-making process involving many variables, multi-objectives and heavy system constraints (Chen 2003). Therefore, an optimization model coupled with a simulation model (i.e., simulation-optimization model) is adopted in the proposed operation models. In the simulation-optimization model, the basic patterns of rule curves need to be predefined, and then a simulation-optimization loop where the simulation model operates the reservoir under the parameterized rule curves received from the optimization model is executed till the fitness remains unchanged and improves no more. The details for realizing the model can be found in the works of Ostadrahimi et al. (2012) and Zeng et al. (2014).

In addition, an efficient optimization method needs to assist in identifying the decision variables in the optimization model. As a population-based method, PSO is one of the promising alternative techniques and has received a great deal of recent attention with regard to water resources optimization problems (Kumar and Reddy 2007; Shourian et al. 2008; Jiang et al. 2010; Afshar 2013). At present, with the popularity of multi-core processors, the intense researches on parallel computing mechanisms and diversification of parallel computing platforms, parallel computing has become an important way to increase computational efficiency (Waintraub et al. 2009; Tu and Liang 2011). Multi-core parallel PSO algorithms (MPPSO) are a promising method for solving the optimal operations of large-scale multi-reservoir systems in favor of improving the solution quality and convergence speed (Tang et al. 2007; Wang et al. 2014). According to several communication strategies of the MPPSO, the coarse-grained model (Chang et al. 2005; Waintraub et al. 2009), most adaptably and widely used for optimization problems, behaves exactly as multi-swarms in parallel computing to guarantee population diversity and to enhance global searching ability. In the coarse-grained model, the first aspect is to partition the population into several sub-swarms. Then, each sub-swarm evolves dependently in a thread and communicates periodically with other sub-swarms based on the networked topology of communication, such as stars in this study, in order to avoid becoming trapped in local optimum. In this paper, this coarse-grained MPPSO is adopted to search the optimal decision variables in operation model.

2.4 Case Study

The developed method is applied to the North-line IBWTS located in Liaoning Province in China. Liaoning Province is undergoing a severe water shortage as it has only one-third of the national water resources per capita in China. Furthermore, the water resources are unevenly distributed in time and space. The annual average rainfall in Liaoning Province decreases from 1000 mm in the east to 400 mm in the west, with 700 mm on average. However, the population, industries and agriculture are mainly concentrated in the central and western areas. As shown in Fig. 3, H reservoir, located in the eastern area, has the largest inflow with the least water demand compared to the other three reservoirs. However, the other three parallel recipient reservoirs, namely Qinghe and Chaihe reservoirs located in the central area and Baishi reservoir located in the western area, have lower inflows with larger water demand. As a consequence, it is necessary to transfer water from H reservoir to alleviate the contradictions between water supply and water demand in the eastern and western regions of Liaoning Province. The designed delivery capacities for each reservoir are labeled in Fig. 3 respectively.

Fig. 3
figure 3

Schematic layout of the adopted IBWTS

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The required input data for the simulation model predominantly includes historical inflows of the reservoirs and downstream intervals, transporting capacities of pipelines, characteristic curves of reservoirs, water demand targets, system constraints and operating policies. A whole series of inflow data from the past 52 years (Y = 52), 1956 to 2007, is available. Since the inflow and water demand are relatively low and evenly distributed from Oct. to Mar., a month time period is used in the simulation model. Nevertheless, due to the large water demand and high uncertainty of the reservoir inflow from Apr. to Sep., a 10-day time period is adopted. So there are 24 time periods in a year (L = 24) and 1248 time periods (T = 1248) in the total 52 years. Reservoir characteristics and their water-supply tasks are shown in Table 1. Water losses due to evaporation and seepage are estimated using the mean water surface area multiplied by the depth of evaporation and seepage during one time period.

Table 1 Reservoir characteristics and water-supply tasks
Full size table

As shown in Table 1, water supply tasks can be divided into two categories according to water-supply satisfaction, namely full water supply tasks (i.e., full water demand) and reducible water supply tasks (i.e., reducible water demand). The former should be met with the top priority; however, the latter can be met by reduced water supply using water demand targets multiplied by rationing factors according to the corresponding hedging rule. Additionally, there exists a one-to-one correspondence between the hedging rule curve and the reducible water demand. The planning reliabilities, with their priorities for industry, high-efficiency agriculture, paddy fields and reed fields, are 95, 85, 75 and 50 %, respectively. And the corresponding rationing factors are 0.8, 0.8, 0.7 and 0.5, respectively. Qinghe reservoir with a larger regulating ability serves as a control-reservoir to make the water-supply decisions for joint water demand in downstream intervals of Qinghe and Chaihe reservoirs (Chang et al. 2009). That is, hedging rule curves for joint water demand are added to Qinghe reservoir to make their water-supply decisions.

3 Results and Discussions

3.1 Analysis of Operating Rule Curves

The joint operating rule curves of the IBWTS obtained by MPPSO-based model are shown in Fig. 4. For H reservoir, the water demands for industry, ecology and other water-supply tasks should be met first, and then water is transferred out if there is surplus water. Thus the fact that the water-transfer rule curve is higher than the hedging rule curve for industry is reasonable. Additionally, the main aims of H reservoir are to transfer water to alleviate water shortage in recipient reservoirs. Therefore, the water-transfer rule curves stay low during most periods in a year, which means that there are more chances for water diversion. However, the water-transfer rule curves for recipient reservoirs are high during most periods in 1 year due to heavy water tasks.

Fig. 4
figure 4

Water-transfer and hedging rule curves of the IBWTS

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  1. (1)

    Hedging rule curves

    The tendency of hedging rule curves is in accordance with the general operation laws of reservoirs, and there are some characteristics in common. During the pre-flood season (from Apr. to Jun.), the curves gradually lower so that they can reduce the probability of restricting the water supply and empty the reservoir storage for the flood season (from Jul. to early Sep.). During the flood season, the curves also stay in low positions owing to the massive reservoir inflow and the restrictions of limited level for flood control, so that it is beneficial to supply as much water as possible. However, during the other season (from mid-Sep. to Mar.), the curves remain high, especially from mid-Sep. to Oct., to increase the probability of restricting water supply and retaining enough water for later time periods to avoid catastrophic water-supply shortages as drought occurs.

  2. (2)

    Water-transfer rule curves

    The tendency characteristics of the water-transfer rule curves for the recipient reservoirs correspond to the distribution of mean period transferred water. For example, the curves of Qinghe reservoir stay low while those of Baishi and Chaihe reservoir are high (Fig. 4 I), so less water is transferred into Qinghe reservoir in comparison with Baishi and Chaihe reservoir; however, it is the opposite in Fig. 4 II. Furthermore, it is worth noting that the characteristics in Fig. 4 I and Fig. 4 II are closely related. The curves of Qinghe reservoir reduce while those of Baishi and Chaihe reservoirs remain high (Fig. 4 I). Therefore, much water might be diverted into Baishi and Chaihe reservoirs for later non-flood season. In Fig. 4c II and (d) II, the depleting curves of Baishi and Chaihe reservoirs means that water supply is mainly provided by natural inflow along with stored water. This is conducive to emptying the reservoir storage for flood season.

    In terms of the reasonability of the water-transfer rule curves in detail, the upper water-transfer rule curve is lifted up by 100 hm3 (1hm3 = 106 m3) in Fig. 4b I (modified rule). The storage of each reservoir during the extreme drought periods from 2001 to 2003 is presented in Fig. 5. Due to modified rule curves, much water can be diverted in Qinghe Reservoir in Fig. 5b I, thus bringing down the water storage of H reservoir. Therefore, there is not enough water available to be transferred out from H reservoir in Fig. 5a II during several time periods, because SV i,t falls below its water-transfer rule curve. Under such circumstances, for Qinghe reservoir, water storage can be kept above its dead storage because more water is transferred previously (Fig. 5b III). However, for Baishi and Chaihe reservoir, water storages fall below their dead storages during the end of extreme drought periods (Fig. 5c and d III). In practice, the storage of a reservoir should not be permitted to falls down below the dead storage. The similar results can be obtained by bringing down the upper water-transfer rule curve by 100 hm3 in Fig. 4a III. Moreover, the curve of H reservoir rises up during these time periods, and its main goal is to reduce water transfer so as to decrease the water spillage of recipient reservoirs to achieve the objective function of minimizing water transfer. The analysis above indicates that the water-transfer rule curves proposed in this paper are optimal and reasonable.

    Fig. 5
    figure 5

    Storage of each reservoir during the extreme drought periods from 2001 to 2003 by lifting up the upper water-transfer rule curve of Qinghe reservoir

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3.2 Analysis of Operation Results

3.2.1 Analysis of Operation Scheme

Operation schemes can be acquired according to the optimal joint rule curves, as shown in Tables 2 and 3. In order to testify the reasonability and effectiveness of the improved water-transfer rule, the operation schemes are also achieved by RULE-SEP and RULE-SIN, respectively, for comparison. Because the RULE-REC was proven to be inferior when there is not enough water available in the donor reservoir by Zeng et al. 2014, the RULE-REC is not analyzed in this paper. The operation schemes in RULE-SIN serve as a base condition for comparison.

Table 2 Comparison of the operation schemes by RULE-DOU, RULE-SEP and RULE-SIN (hm3/year)
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Table 3 Comparison of the water-supply reliabilities by RULE-DOU, RULE-SEP and RULE-SIN (%)
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Table 2 shows that in the source system, there is little difference in the water supply; however, water losses are increased correspondingly due to the decrease in water transfer. And in the receiving system, there is a slight drop in the water supply but a massive reduction in water losses given the great decrease in water transfer. For example, the quantity of water transfer, compared with RULE-SIN, is reduced by 96 hm3. Nevertheless, the quantity of the water loss is reduced by 73 hm3, which accounts for 76 % of the quantity of the reduced water transfer. That is to say, it can reduce the quantity of the water transfer that would be abandoned without being utilized by the recipient reservoirs. Additionally, Table 3 shows that all actual water-supply reliabilities meet their planning targets for all three rules. Thus, the proposed rule can make the water-transfer decisions reasonable without much influence on the water supply, which also shows that it is a better tradeoff between water transfer and supply.

3.2.2 Analysis of water transfer and water spillage

In order to further investigate the reasonability and effectiveness of the proposed rule, water transfer and spillage are analyzed from the following aspects.

  1. (1)

    Comparison of water transfer and water spillage

    The efficiency of water transfer is reflected by the reasonability of water spillage in this paper. Figure 6 give the annual and mean time period water spillage of recipient reservoirs. As is shown, water spillage is mainly distributed in wet or normal years and flood seasons. In addition, water spillage is smaller in RULE-DOU but larger in RULE-SIN. Taking Chaihe reservoir for example, the occurrence of transfers without spillage happens in 32, 27 and 17 years among RULE-DOU, RULE-SEP and RULE-SIN, respectively. However, the concurrence of transferring and spilling water, especially during wet and normal years, is unreasonable, which can be improved by further developing forecast-operating rule for real-time operation.

    Fig. 6
    figure 6

    Water spillage of recipient reservoir in RULE-DOU, RULE-SEP and RULE-SIN. a Annual water spillage, b Mean period water spillage

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    Figure 7 summarizes the mean water supply, transfer and spillage of recipient reservoirs under different hydrological scenarios such as wet, normal and drought years. As is known, as water demand is increased in wet, normal and drought years in turn, water transfer and supply are also increased and water spillage is decreased. Figure 7 shows that water transfer is increased and water spillage is decreased in turn; however, the water supply is not increased in drought years and is less than in normal years. The reason primarily lies in that the water inflow is small with large water demand in drought years, so water demand cannot only be met fully based on the water-supply hedging rule. That is to say, water supply is rationed mainly in drought years with the purpose of avoiding the risk of unacceptably great water deficits. Additionally, there are small differences of water supply in wet, normal and drought years, as presented in Fig. 7. However, when compared with the other two rules, water transfer and spillage in RULE-DOU are reduced greatly in wet and normal years without great influences on the water supply. This means that it can drastically reduce the transferred water without utilization being abandoned in the receiving system (or “invalid” water transfers) so as to improve the efficiency of water transfer.

    Fig. 7
    figure 7

    Mean water quantity of recipient reservoirs under different hydrological scenarios such as wet, normal and drought years. a Mean water supply, b Mean water transfer, c Mean water spillage

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  2. (2)

    Probability distribution of water transfer

    The efficiency of water transfer can be further illustrated by the distribution of water-transfer probability, which is defined as the ratio of the number of time periods requiring water transfer to the total periods T. Table 4 presents the probability of water transfers for each reservoir in the system, and Table 5 shows the probability of water transfers for each recipient reservoir under different hydrological scenarios such as wet, normal and drought years. As shown in Tables 4 and 5, though the probability of water transfers (85.18 %) of each recipient reservoir in RULE-DOU is greater than that in other two rules, the probability of water transfers in STATE1 is smaller, and a great part of water transfer stays in STATE2. This is why the water transfer and spillage can be reduced substantially with the guarantee of water supply. In other words, the probability of water transfers in STATE2 can make water transfer and spillage more reasonable, which directly results in the RULE-DOU superior to other two rules. As a consequence, due to the large probability of water transfer in wet and normal years in STATE2, the efficiency of water transfer and spillage may increase dramatically in wet and normal years.

    Table 4 Probability of water transfer for each reservoir (%)
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    Table 5 Probability of water transfer for each recipient reservoir under different hydrological scenarios such as wet, normal and drought years (%)
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4 Conclusions

This paper focuses on the operation rule-based model, especially water-transfer rules, to improve the efficiency of water transfer, while using optimization-simulation model to obtain the operating rule curves. In this proposed rule, the upper-and-lower water-transfer rule curves of the recipient reservoir are applied in combination with one water-transfer rule curve of the donor reservoir for water transfer decision-makings. Finally, the proposed method is testified by an IBWTS located in the northern of Liaoning Province in China.

The conclusions can be summarized as follows. Firstly, the derived operating rules are optimal and reasonable. The tendency of hedging rule curves is in accordance with the general operation laws of reservoirs. While for water transfer, there is competitive relationship during the post- and pre-flood seasons. Moreover, the upper water-transfer rule curve of Qinghe reservoir was lifted up by 100 hm3, then water storages of Baishi and Chaihe reservoir unreasonably fell below their dead storages during the end of extreme drought periods. Secondly, the quantity of water transfer, compared with RULE-SIN, was reduced by 96 hm3 while that of water loss was reduced by 73 hm3 accounting for 76 % of the reduced water transfer, which meant that water spillage was lessened as the reduction in water transfer with negligible influence on water supply in the proposed rule. Further analysis showed that the reduced quantity of water transfer or spillage was concentrated in wet and normal years. Thirdly, the water transfer combinations were more balanced and effective in RULE-DOU. The water-transfer probability 85.18 % was greater than that in other two rules, which lied in the increase in occurrences of water transfer resulting in the decrease in quantity of water transfer. However, the probability of water transfers was large in STATE2. In the STATE2, the actual transferred water should be no more than the water flow interpolated within the designed delivery capacity instead of only the designed delivery capacity in the other two rules, which can improve the efficiency of water transfer especially in wet and normal years, during which time there is less water shortage in the recipient reservoir.

In conclusion, the proposed rule can provide reasonable and efficient guidelines for water transfer-supply in the planning and management of the IBWTS. However, the decision variables to be optimized are increased with the increase of several water-transfer rule curves and other novel methods such as surrogate model can be developed to drive the higher-dimensional optimal variables effectively in future work.