In the proposed bi-level model for multi-reservoir operating policy, the upper level model optimizes the water transfer rule to distribute water resources between exporting and importing regions as the planned scheme and to minimize the total water spills of the multi-reservoir system. The lower level model optimizes the water supply rule to obtain the best water supply quality under the condition of water transfer. In order to describe the development process of the bi-level model, an inter-basin water transfer-supply project consisting of three reservoirs connected by water transfer pipelines is taken as an example, just like the one illustrated in Fig.1.27.
5.2.1 The Upper Level Model
From the current literature on multi-reservoir operating policy, it is observed that there has been quite little research carried out on water transfer rule to direct the multi-reservoir system manger under what conditions to transfer water from the exporting reservoir to the importing reservoir. In this section, a set of water transfer rule is proposed based on the storage of each member reservoir in the system, shown schematically in Fig. 1.28. In this way, the inter-related dynamic water storage of each reservoir is taken as the main factor influencing the decision of water transfer when lack of future inflow information.
As shown in Fig.1.28, the active storage of each reservoir between the maxi-mum and minimaxi-mum storages is divided into two parts: zone I and zone II. When the reservoir storage stays in zone I, it means that there is sufficient water in the
reservoir. If the reservoir storage stays in zone II, it means that there is scare water in the reservoir. During the operation of the multi-reservoir system, there will be such 8 combinations occurring according to the three-reservoir storage as (I,I,I), (I,I,II), (I,II,I), (I,II,II), (II,I,I), (II,I,II), (II,II,I), (II,II,II). The detailed judgment procedures of water transfer are described as below. First, the storage of the exporting reservoir should be concerned. If the exporting reservoir storage stays in zone II, it means that there is not enough water to export. At this moment, the action of water transfer is stopped at all regardless of whatever storage in the importing reservoir so as to guarantee the water supply in water exporting region.
This condition corresponds to 4 combinations of (II,I,I), (II,I,II), (II,II,I), (II,II,II).
If the exporting reservoir storage stays in zone I, it means the stored water in exporting reservoir is enough to be transferred into the importing region. Under this condition, the storage of the importing reservoir should be also paid an attention
(1) (2) (3)
Inflow Inflow Inflow
Water transfer
Water demand Water
demand Water
demand Water supply
Water importing region
Water transfer
Water supply Water
supply
Water importing region Water exporting region
Fig. 1.27 The layout of an inter-basin water transfer-supply project
Fig. 1.28 The water transfer rule curve based on the storage of each member reservoir
to. Because if the water level of the importing reservoir is quite high, the transferred water may produce a lot of water spills, which is obviously unreasonable. At present, the water transfer rule can be categorized into three conditions. First, if both the storages of the two importing reservoirs stay in zone I corresponding to (I,I, I), the action of water transfer is stopped. Second, if both the storages of the two importing reservoirs stay in zone II corresponding to (I,II,II), the action of water transfer is carried out. The transferred water amount from the exporting reservoir at this period is as much as the pipeline’s transporting ability and it is divided into the two importing reservoirs according to an allocation ratio. Third, if the storage of one of the two importing reservoirs is in zone II and the other one is in zone I, corresponding to (I,I,II) and (I,II,I), the action of water transfer is carried out all the same. The transferred water amount from the exporting reservoir at this period is also as much as the pipeline’s transporting ability and it is all transported into the importing reservoir whose storage is in zone II. The above procedure is illustrated schematically in Fig.1.29.
For the upper model, it pursues to achieve a trans-boundary water resources allocation as the planned scheme by water transfer to satisfy water demand in every region and to minimize the total water spills of the multi-reservoir system. To realize the objective is by means of optimizing the positions of water transfer rule curves as shown in Fig. 1.28. The mathematical formulation of the upper level model is given by Eq. (1.30).
Exporting reservoir (2) storage
is in zone I?
No The action of water transfer is stopped.
Yes
Importing reservoir (1) storage
is in zone II?
The action of water transfer into reservoir (1) is stopped.
Yes
The action of water transfer into reservoir (3) is
stopped.
Transport water into reservoir (1) as much as the transporting ability, no for (3)
Yes Transport water into both reservoirs (1) and (3) as much as the
ability, divide the importing water for them according to an
allocation ratio (3) as much as the pipeline’ s pipeline’ s
pipeline’ s transporting
transporting ability No
Fig. 1.29 The judgment flowchart of water transfer based on the proposed water transfer rule
minx F x; yð Þ ¼ wDSXm
i¼1
NDSi TNDSi
j j þ wSUXm
i¼1
SUi
s:t: NDSi¼ G x; yð Þ, SUi¼ g x; yð Þ
STmini xi STmaxi , STmini yi STmaxi 0 DS DSmax
i¼ 1 . . . m
ð1:30Þ
wherex is the decision variable of the upper level model representing the position of water transfer rule curve during an operation period;y is the decision variable of the lower level model denoting the position of the Hedging rule curves in Fig.1.30;
bothx and y are between the maximum and minimum storages; NDSiandSUirefer to the annual average transferred water amount and the water spill of reservoiri, which are related to water transfer rule and water supply rule and can be formulated as the function ofx and y. In this study, the weighting approach is applied. For a given weight combination, single-objective optimization is used for optimization of the aggregated objective function. The weighting factors wDS and wSU can be obtained empirically.
Fig. 1.30 The Hedging rule curves based on the storage of the member reservoir in the system
5.2.2 The Lower Level Model
In this section, the Hedging rule is adopted as the water supply operating rule in the lower level model, which has been discussed in different methods for reservoir operation. Srinivasan and Philipose [53] used hedging parameters, such as starting water availability, ending water availability and hedging factor (degree of hedging), to construct the hedging rules and evaluated effects on the reservoir performance indicators. Shih and ReVelle [54,55] determined the trigger value for a continuous hedging rule and then for a discrete hedging rule, respectively. Neelakantan and Pundarikanthan [34] presented a simulation–optimization methodology using neu-ral network and multiple hedging rules to improve reservoir operation performance.
Tu et al. [56] considered a set of rule curves that are a function of the current storage level to trigger hedging for a multipurpose, multi-reservoir system.
In this study, the hedging rule based on the storage of each member reservoir consists of hedging rule curves and rationing factors for each water demand. Details of the hedging rule curves and its corresponding operating rule are illustrated in Fig.1.30and in Table1.6. In previous works on hedging rule curves [56,57], all planned water demand are met at the same level and are rationed at the same time when drought occurs. For single purpose of water supply operation, the water demand can be divided into various categories, such as irrigation, industry and domesticity. It should be noted that different kind of water demand requires different reliability and different degree of hedging in practice. In this study, different hedging rule curves and rationing factors are assigned to different kinds of water demand. When drought occurs, different types of water demand own different priority to get as much water as demand without rationing.
For the lower model, it optimizes the water supply rule to obtain the optimal water supply quality under the condition of water transfer. The mathematical formulation of the lower model is given by Eq. (1.31).
miny f x; yð Þ ¼Xm
i¼1
Xn
j¼1
wij Index ij Targetij
s:t: Indexi j¼ k x; yð Þ
STimin xi STimax, STimin yi STimax i¼ 1 . . . m, j ¼ 1 . . . n
ð1:31Þ
Table 1.6 Water supply operating rule implied by the Hedging rule curves
Reservoir storage
Water supply for each demand
Demand 1 (D1) Demand 2 (D2) Demand 3 (D3)
Zone 1 D1 D2 D3
Zone 2 D1 D2 α3*D3
Zone 3 D1 α2*D2 α3*D3
Zone 4 α1*D1 α2*D2 α3*D3
Rationing factor α1 α2 α3
wherex and y have the same meaning as the ones in Eq. (1.30);Indexijrefers to the water supply index for the water demand j of reservoir i, which can be water shortage index, water supply reliability or some other indexes; Indexij is the function of water transfer rule and water supply rule, which should get close to the target valueTargetij. The lower level objective function consists of water supply indexes of all the water demand in the multi-reservoir system, which also uses the weighting approach to combine these indexes.
5.2.3 Method Solution
Due to their structure, bi-level programs are non-convex and quite difficult to deal with and solve. Even bi-level problems in which all functions involved are linear are (strongly) NP-hard [98]. Exact approaches rang from studying the properties of the feasible region, to obtaining necessary and sufficient optimality conditions, replacing the lower level problem by its Karush–Kuhn–Tucker conditions, using penalty func-tions or using gradient methods. Most exact algorithms can only tackle relatively small problems, so meta-heuristic approaches have been widely applied for solving bilevel programming. For example, genetic algorithms, simulated annealing and tabu search are proposed or developed to solve bilevel programming [99–102]. Kuo and Huang [103] apply particle swarm optimization algorithm (PSO) for solving bi-level linear programming problem. However, PSO has premature convergence like other swarm intelligence methods, especially in complex multi-peak-search problems.
For solving the bi-level program proposed to model the inter-basin water transfer-supply problem, an improved particle swarm optimization (IPSO) by Jiang et al. [71] is adopted in this section. In IPSO, a population of points sampled randomly from the feasible space. Then the population is partitioned into several sub-swarms, each of which is made to evolve based on particle swarm optimization (PSO) algorithm. At periodic stages in the evolution, the entire population is shuffled, and then points are reassigned to sub-swarms to ensure information sharing. In this way, the ability of exploration and exploitation has been greatly elevated. The detailed IPSO strategy can be referred to the work of Jiang et al. [71], which is used to solve the inter-basin water transfer-supply bi-level programming problem in this section. The flowchart of solving bi-level model for multi-reservoir operating policy using IPSO is described schematically in Fig.1.31.