1. Introduction
The uneven distribution of water resources in temporal and spatial, rapid economic and social development are critical factors for regional water scarcity. In water-deficient regions, disputes arising over the available water resources among stakeholders are becoming ubiquitous [
1,
2]. Reasonable and effective allocation for water resources is a continuous need to alleviate the conflicts among stakeholders. The reservoir is considered as an important facility for reallocating water resources on spatial and temporal scales. Considering the different objectives of reservoirs, the integrated management of cascade reservoirs potentially plays a vital role in satisfying stakeholders’ possible desired utilities.
According to the classification of agents based on the different scenarios of information exchange [
3], reservoirs can be classified as fully cooperative reservoirs, coordinated reservoirs, and non-cooperative reservoirs. The fully cooperative reservoirs correspond to the centralized management that is a “top-down” decision process. The centralized management assumes that all reservoir operators in the cascade-reservoir system obey the ‘central mind’ to fully cooperate and then make decisions targeting the maximum benefits for the whole system [
4,
5,
6,
7]. Therefore, ideal system-wide benefits can be produced by cooperation based on the collective rationality of reservoir operators. However, because non-cooperation involving water conflicts is common and stable [
8], full cooperation without considering the individual self-interest is seldom applied in real-world management. The non-cooperative reservoirs correspond to the non-cooperative management. Following the non-cooperative management, decisions made by reservoir operators are based on individual rationality, in which self-interested reservoir operators usually focus on their local benefits rather than the system’s total benefits. However, due to the lack of cooperation, no feedback mechanism exists for decisions made by different reservoir operators, resulting in a solution that does not maximize the favorable benefits on the system scale. Therefore, traditional centralized and non-cooperative managements for cascade-reservoir system usually target the maximum benefits for either the entire system or a certain individual reservoir.
To balance the individual and system’s benefits, coordination among cascade reservoir operators is critical. Through coordination, coordinated reservoir operators not only make decisions targeting their own benefits, but also share information with the other operators to improve the system’s benefits. Giuliani et al. [
3] demonstrated that the system benefits obtained by coordinated management are better than those obtained by non-cooperative management. To build coordinated models, mathematical methods have been widely used [
9,
10,
11]. By iterative adjustment of various parameters that are related to stakeholders, it is determined whether or not different objectives are to be met. Through the bargaining process, stakeholders reach an agreement for compromise. Although the coordinated models based on mathematical methods have been successfully built for interactive systems, such models are sensitive to the complexities of systems [
12]. As the number of stakeholders increases, the amount of information exchanged among stakeholders is also increased. Thus, the computation for a coordinated model built by mathematical methods for a multi-stakeholder system is considerable.
Game theory is another tool able to reflect the competition and cooperation among self-interested stakeholders [
13]. For self-interested stakeholders, the individual rationality is stressed by game theory, embodying an idea of risk aversion. Game theory has been studied to resolve the conflicts in resources allocation. Salazar et al. [
14] compared four game-theoretical methods in resolving conflicts for multi-objective problems. Chew et al. [
15] developed a game-theoretical model for analyses of the inter-plant water integration problem. Salazar et al. [
16] applied game theory to address water distribution problems in the Mexican Valley. Hipel and Walker [
17] assessed the role of game theory in resolving conflicts for environmental management. Huang et al. [
18] used the fuzzy Shapley value method to reallocate the pollution discharge rights based on the benefits obtained from the cooperation. Madani and Lund [
19] employed game theory to solve multi-criteria decision-making problems for water resources management. Madani and Hooshyar [
20] discussed three game-theoretical models to allocate benefits for cascade reservoirs. Kicsiny et al. [
21] developed a dynamic game-theoretical model to simulate the water use conflicts between social and production sectors in drought emergency. Wang et al. [
22] and Yang et al. [
23] combined subjective weight and objective weight as a combination weight based on game theory to assess the water usage and water quality, respectively.
To the best of our knowledge, game theory has not been used to simulate the coordination behaviors in a cascade-reservoir operation. For a real-world cascade-reservoir operation, because of the geographical advantage, the upstream reservoir operators have priority to maximize their own benefits, which may weaken the benefits of downstream reservoir operators. Incorporating the game theory into the cascade-reservoir operation can make the upstream and downstream reservoir operators coordinate whilst considering individual self-interest. To realize such coordination among cascade reservoirs, an integrated game-theoretical model is proposed in this study. For the integrated game-theoretical model, there are two sub-models: a coordination model and a benefits compensation model. The coordination model is used to simulate the coordination among cascade reservoirs, and the benefits compensation model is used to guarantee individual rationality for sustainable coordination. The objectives of this study are to (1) compare the operations of a cascade-reservoir system based on the centralized model, the non-cooperative model, and the integrated game-theoretical model; (2) evaluate the impact of water availability variation on the operation efficiency of the integrated game-theoretical model; and (3) explore the potential factors affecting the efficiency of the integrated game-theoretical model.
3. Methodology
3.1. Centralized Model (Model I)
The centralized model inherently assumes that all reservoirs with different (even conflicting) goals in the multi-reservoir system are subject to a ‘central mind’ to cooperate based on collective rationality, and they make decisions aimed at maximizing the total benefits of the whole system, resulting in an economically optimal operation, on the system scale. Model I is expressed by the following mathematical formula:
where
E indicates the total hydropower generation for the multi-reservoir system;
indicates the output of reservoir
i at time
t;
M is the total number of reservoirs in the multi-reservoir system;
indicates the time-step; and
T indicates the number of operation periods.
Subject to
Water balance between reservoirs:
where
denotes the storage of reservoir
i at period
t (m³);
denotes the inflow to reservoir
i at period
t (m³/s); and
denotes the average releases from reservoir
i at period
t (m³/s).
Reservoir storage constraint:
where
and
denote the minimum and maximum allowable storage of reservoir
i at period
t (m³), respectively.
Reservoir releases constraint:
where
and
denote the minimum and maximum allowable releases from reservoir
i at period
t (m³/s), respectively.
Output constraint:
where
and
denote the minimum and maximum allowable output of reservoir
i at period
t (kW), respectively.
3.2. Non-Cooperative Model (Model II)
Following the non-cooperative model, the strategies autonomously taken by reservoirs target the optimal benefits for their own goals, without considering the requirements of the other reservoirs. For a multi-reservoir system, the upstream reservoirs have priority in making decisions to maximize their local benefits. Because of the lack of coordination, the downstream reservoirs only take actions based on decisions made by the upstream reservoirs. Therefore, the optimization of the multi-reservoir system is performed from upstream to downstream, one by one. Model II is expressed by the mathematical formula:
where
indicates the hydropower generation of reservoir
i.
In addition, the objective function subjects itself to the constraints: mass balance, storage constraint, releases constraint, output constraint, and boundary constraint, which are similar to the constraints (Equations (2)–(6)) of Model I.
3.3. Integrated Game-Theoretical Model
To simulate the coordination between the upstream and downstream reservoirs, an integrated game-theoretical model is proposed in this study. The proposed model consists of two sub-models: a coordination model and a benefits compensation model. The coordination model is built based on the Stackelberg theory, and the benefits compensation model is built based on the Nash-Harsanyi bargaining theory.
3.3.1. Coordination Model (Model III(a))
Stackelberg theory [
25] is defined as a non-cooperative game theory. The sequence of actions taken by players is given great importance by this theory [
26,
27]. Following the Stackelberg theory, a dominant (or leader) player moves first, and then, a subordinate (or follower) player moves second. Before the game begins, the follower knows how the leader acts; at the same time, the leader also knows what the follower knows. The coordination is implemented via information exchange between the leader and the follower. Based on the Stackelberg theory, the coordination model for upstream and downstream reservoirs is developed as follows:
where:
and
denote the strategy spaces for upstream and downstream coalitions, respectively;
and
denote the decision strategies belonging to
and
, respectively;
and
denote the payoff functions of upstream and downstream coalitions, respectively;
and
denote the hydropower generation of upstream and downstream coalitions, respectively;
denote the number of reservoirs of the upstream coalition;
indicates that
is the best response of the downstream coalition to the strategy
specified for the upstream coalition; and
indicates that
is the best response of the upstream coalition to the strategy
specified for the downstream coalition. Therefore,
is the Nash equilibrium for the cascade-reservoir operation.
For the upstream and downstream coalitions,
and
, respectively, can be calculated as follows:
Given the coordination between the upstream and downstream coalitions, an interaction factor for the upstream coalition function for feedback on decisions made by the downstream coalition is proposed as follows:
where
denotes the rational hydropower generation of the downstream coalition obtained through noncooperation (Model II).
Therefore, the decisions of the downstream coalition can be responded to by the actions taken by the upstream coalition as follows:
According to the interaction factor, the payoff benefits of the upstream coalition are influenced by its decisions whilst considering the requirements of the downstream coalition, as follows:
In addition, the objective function subjects itself to the constraints: mass balance, storage constraint, releases constraint, outputs constraint, and boundary constraint, which are similar to the constraints (Equations (2)–(6)) of Model I.
3.3.2. Benefits Compensation Model (Model III(b))
Through coordination, the hydropower generation of upstream coalition may be sacrificed to improve the hydropower generation of downstream coalition. Nevertheless, because of the decreased hydropower generation, the rational upstream reservoir operators have no incentive to participate in the coordination and prefer to act in a non-cooperative mode [
20]. For maintaining a stable coordination, benefits compensation for upstream reservoir operators needs to be implemented to ensure the rational individual benefits.
Nash-Harsanyi bargaining theory [
28,
29] can be used to provide fair and efficient resources allocation for players with guaranteed rational individual benefits [
30,
31,
32,
33]. Due to the increased hydropower generation of the whole system through coordination, the individual hydropower generation can be reallocated by Nash-Harsanyi bargaining theory to ensure the rational individual benefits [
15].
subject to:
where:
denotes the solution of the Nash-Harsanyi bargaining model;
denotes the hydropower generation of reservoir
i allocated by the Nash-Harsanyi bargaining model;
denotes the rational hydropower generation of reservoir
i obtained through noncooperation; and
denotes the totally available hydropower generation of the cascade-reservoir system.
3.4. Evaluation Criterion
To analyze the efficiency of the coordination model in improving the hydropower generation of downstream coalition, an improvement index is proposed. The index is calculated as:
where
denotes the rational hydropower generation of the upstream coalition obtained through noncooperation.
3.5. Solving Method
For the game-theoretical model, the strategy space 1 for the upstream coalition (Coalition 1) and the strategy space 2 for the downstream coalition (Coalition 2) are first determined. The strategy of a coalition is defined as release decisions of reservoirs in time intervals. Because of multiple reservoirs and time intervals, the number of coalition’s strategies is considerable during the operation horizon. To simplify computational efforts, some discrete strategies of the coalition need to be selected. In this study, the genetic algorithm (GA) [
34] is used to simplify and solve the integrated game-theoretical model. The parameters of GA are shown in
Table 3.
In a GA, a population of candidate solutions to an optimization problem is evolved toward better solutions. The evolution is an iterative process. For each iteration, several strategies are generated using selection, crossover, and mutation operators of GA, and evaluated by the fitness function. The current best strategy in each iteration can then be determined according to the fitness value. By the iterative computation of GA, the optimal solution for high-dimensional problems can be explored.
In this study, to select the strategy for strategy space 1, the hydropower generation of the upstream coalition is optimized by GA. If the difference of the hydropower generation between the strategies in two neighboring iterations is larger than the threshold value, the two strategies are considered as the selected strategies of the strategy space 1. The selection of discrete strategies of the upstream coalition can be described as follows:
where:
k denotes the index of iterations;
denotes the release decisions in iteration
k;
n denotes the total number of release decisions in iteration
k;
denotes the best release strategy in iteration
k; and
denotes the minimum difference between two strategies in neighboring iterations in terms of the hydropower generation of upstream coalition.
Different from strategy space 1 that includes several feasible release decisions, the strategies of strategy space 2 are assumed as all feasible release decisions. For each strategy of strategy space 1, the optimal strategy of strategy space 2 can be obtained using GA. When strategy spaces 1 and 2 have been determined, the coordination between upstream and downstream coalitions is implemented through Equations (8)–(19), and the Nash equilibrium strategies are reached. After identifying the Nash equilibrium strategies of the cascade-reservoir system, the hydropower generation of reservoirs is revised through Equations (20)–(22). The solving process of the game-theoretical model is shown in
Figure 3.
5. Conclusions
This study proposed an integrated game-theoretical model to simulate the competition and cooperation behaviors among different reservoir operators for a cascade-reservoir system. The integrated game-theoretical model consisted of two sub-models: a coordination model based on the Stackelberg theory and a benefits compensation model based on the Nash-Harsanyi bargaining theory. The coordination model was used to interpret the coordination among reservoirs, while the benefits compensation model was applied for benefits compensation for participating reservoirs to ensure rational individual benefits. Additionally, potential factors affecting the efficiency of the integrated game-theoretical model were also explored in this study. A cascade-reservoir system located in the Yangtze River basin of China was used as a case study. The main conclusions of this study were as follows.
Through the proposed integrated game-theoretical model, the cascade-reservoir operation will result in a “win-win” situation for the individual and system’s hydropower generation. In terms of the system’s hydropower generation, the total hydropower generation of the cascade-reservoir system obtained by the integrated game-theoretical model is better than that obtained by the non-cooperative model, and comparable to that obtained by the centralized model. In terms of the rational individual benefits, the integrated game-theoretical model clearly outperforms the other two models.
The operation of the integrated game-theoretical model is influenced by water availability variation. For the integrated game-theoretical model, as the available water decreases, more hydropower generation of upstream reservoirs is sacrificed to improve the hydropower generation of downstream reservoirs; whereas, the incremental hydropower generation of downstream reservoirs, induced by the unit hydropower generation decrement of upstream reservoirs, is reduced. If there is no increment in the system’s hydropower generation, the benefits compensation is not implemented.
The regulation capacities of cascade reservoirs have an impact on the operation efficiency of the integrated game-theoretical model. For the integrated game-theoretical model, a high regulation capacity of the upstream reservoir facilities the growth of hydropower generation of the downstream reservoir; however, if the regulation capacity of the upstream reservoir is significantly higher than the regulation capacity of the downstream reservoir, the coordination between them may result in a loss in the total hydropower generation of the cascade-reservoir system.
The study reveals that the integrated game-theoretical model strikes an excellent balance between individual and overall hydropower generation for a cascade-reservoir system. Although the system’s hydropower generation is not maximized by the integrated game-theoretical model, the balanced individual and system’s hydropower generation are practical and stable when reservoir operators are not in full cooperation in the real-world operation. Moreover, compared to the system’s strategies, the computation needed to obtain strategies for individual reservoirs (or reservoir coalition) is less. Therefore, the coordination can be practically quantified for a complex cascade-reservoir operation with information exchange between upstream and downstream reservoirs. The main uncertainty source of the proposed model originates from the strategy selection of the upstream coalition. To control the uncertainty source, the diversity of strategies of the upstream coalition should be enriched. For future studies, the integrated game-theoretical model can be extended to deal with the coordination among reservoir operators when the run-off is uncertainty.