Optimization method for joint operation of a double-reservoir-and-double-pumping-station system: a case study of Nanjing, China

With the worldwide construction of reservoirs and inter-basin water transfer projects, an optimal joint operation of multi-reservoir systems has recently become a research hotspot. It has been recognized that benefits derived from the joint operation of a multi-reservoir system may exceed the sum of that from the independent operation of each reservoir in it (Wang et al. 2016).

Generally, in an optimization model of reservoir operation, the objective function is to minimize the sum of the squared deviation of actual water supply from the desired target, in an attempt to offer the same deviation at each period as far as possible (Celeste & Billib 2009). Such models are subject to several constraints, such as lower and upper bounds of release and storage, water balance equation, and non-negativity constraints. Furthermore, hydraulic connections between reservoirs should be added into constraints in a multi-reservoir system.

Optimization models include two types of decision variables (water supplies and spills) that increase with the number of reservoirs; increased variables lead to the ‘curse of dimensionality’ in the solving process (Chen et al. 2016). At present, different types of meta-heuristic algorithms featuring high applicability and high efficiency have been applied to solving such problems. The genetic algorithm was applied to optimizing the operation policy of a four-reservoir system (Jothiprakash et al. 2011). Particle swarm optimization was adopted for the optimal water resources management of a multi-reservoir system (Nabinejad et al. 2017). Gu et al. (2017) used a global search algorithm to optimize the joint operation of a system comprising multiple donor reservoirs and one recipient reservoir. Ehteram et al. (2018) suggested the Spider Monkey Algorithm for optimizing operation rules of multiple reservoirs. However, meta-heuristic algorithms may fail to solve models with equality constraints because of their fatal weakness in that they cannot guarantee globally optimal final results, which can be attributed to the random sampling (Birhanu et al. 2014; Rani & Srivastava 2015). In addition, the application of meta-heuristic algorithms to reservoir operation problems may lead to spills when the reservoir storage is below its capacity, which is against the normal operation rule.

Dynamic programming based on Bellman's principle (Kennedy 1986) can provide reliable and optimal results. In addition, it has good applicability in multi-stage decision-making processes, regardless of the matter whether the model is continuous or linear. Therefore, it has been widely used in solving optimization problems of reservoir operation. Although the ‘curse of dimensionality’ (Rani & Moreira 2009; Ahmad et al. 2014; Wang et al. 2016) is induced under the condition of a large number of reservoirs, appropriate dimension reduction methods, such as the decomposition-aggregation method (Gong & Cheng 2018) and decomposition-coordination method (Mahey et al. 2017), can be used to obtain global optimal results by transforming a high-dimensional problem into a series of low-dimensional problems.

The purpose of optimizing the joint operation of a reservoir and a pumping station is to minimize the operation cost, mainly the operation cost of the pumping station. In addition to water supplies and spills of the reservoir, water volume pumped by the pumping station during each period is also a type of decision variable subject to the maximum pumping capacity and pumping volume limited by water rights.

Accordingly, the difficulty of solving the model will increase with increasing numbers of reservoirs and pumping stations in the system. Yu et al. (1994) and Pulido-Calvo & Gutiérrez-Estrada (2011) developed a nonlinear optimization model for a reservoir and its replenishment pumping station to minimize the operation cost. Reca et al. (2014) constructed a nonlinear model for the joint operation of reservoirs and pumping stations to obtain the minimum operation cost and improved it considering the evaporation of the reservoir. Ðurin (2016) derived an optimal operating policy with a regression analysis of the running time mode of pumps, design discharge of pumping station, and storage capacity of the reservoir, according to the deterministic water supply scheme. However, all the above-mentioned works optimized the operation schedule of pumping stations according to seasonal electricity price or time-of-use price and contributed toward saving energy costs. In addition, they did not limit the total water volume pumped by the systems.

In different countries, concepts and connotations of water rights are not the same under different social backgrounds (Molle 2004; Heikkila 2015). However, it especially refers to the water resources usufruct here. Since 2016, the agricultural water price reform has been promoted stage by stage in China (Shen & Wu 2017). As an important premise, the water resources usufruct is defined in the form of the annual total volume of water which is distributed to users. In agriculture, the Chinese government has established an upper bound for the annual amount of irrigation water. Furthermore, the total amount of water that can be obtained from each irrigation gate or pumping station within a year is strictly limited (Sun et al. 2016).

This study aimed to develop an optimization method based on the decomposition-aggregation theory for the joint operation of a double-reservoir-and-double-pumping-station system that can provide optimal results following normal operation regulations. For this purpose, an optimization model of this system in Nanjing, China, was established as a case study, and the total available water in the system was coupled into it, considering regional water rights.

1. Reservoirs and pumping stations

The SH reservoir is the second largest reservoir in Liuhe district and its main function is to supply water for irrigation. During the period of operation, the XZ pumping station replenishes it with water from the BL river through a 2 m × 2 m square concrete culvert before the water level reaches below the lower boundary limit. Because of the higher topography, the irrigation area of the HWB reservoir is larger although the catchment area is relatively small, and thus, the probability of water shortage is relatively high. During water shortage, the HZ pumping station replenishes the HWB reservoir with water from the SH reservoir through a 1.5 m × 1.5 m square concrete culvert. The characteristics of reservoirs and pumping stations are shown in Tables 1 and 2, respectively.

• 2. Inflows and water demand

Monthly inflows and water demand at the 75% probability of exceedance are shown in Tables 3 and 4, respectively.

• 3. Evaporation

Figure 2 shows a generalized schematic of the double-reservoir-and-double-pumping-station system comprising the SH reservoir, HWB reservoir, and their replenishment pumping stations. The SH reservoir is a donor reservoir, numbered as reservoir 1; the HWB reservoir is a recipient reservoir, numbered as reservoir 2. The XZ pumping station, numbered as pumping station 1, diverts water from the BL river to the SH reservoir; the HZ pumping station, numbered as pumping station 2, diverts water from the SH reservoir to the HWB reservoir.

It has been proved that if there is no appropriate joint operation scheme for such a system, water replenishment, water spills, and water shortages may occur simultaneously. Therefore, an optimization method is necessary for such a system to improve the utilization efficiency of inflows as well as to reduce water spills and shortages, considering limited water rights on the river.

In Figure 2, YS_1,i, LS_1,i, X_1,i, and PS_1,i are water demand, inflow, water supply, and water spill in the SH reservoir in period i, respectively; YS_2,i, LS_2,i, X_2,i, and PS_2,i are water demand, inflow, water supply, and water spill in the HWB reservoir in period i, respectively; Y_1,i, and Y_2,i are water replenishment by the XZ and HZ pumping stations in period i, respectively.

1. Annual available water in the system

• 5. Maximum pumping capacity

• 6. Initial and boundary conditions

A restriction was imposed to make the final storage V_j,20 equal to the initial storage V_j,0. If this limitation is not imposed, the solution would tend to empty the reservoir in the final period.

• 7. Non-negative constraints

It is also necessary to introduce non-negativity constraints to avoid negative values to the variables, which would be physically impossible.

The above model is a multi-dimensional nonlinear model, which can be divided into multiple stages. There are six types of decision variables, including X_1,i, X_2,i, PS_1,i, PS_2,i, Y_1,i, and Y_2,i, each of which has 20 dimensions. Therefore, it is difficult to solve it directly. As shown in Figure 3, the double-reservoir-and-double-pumping-station system can be decomposed into two subsystems that consist of one reservoir and one pumping station, and hydraulic connections between them are formed through the HZ pumping station. In this manner, the dimensionality of the problem can be reduced.

1. Decomposition of the system

The maximum annual pumping volume of the HZ pumping station, BZ₂, is taken as a coordinating variable, and the model of the double-reservoir-and-double-pumping-station system can be decomposed into two subsystem models that consist of one reservoir and one pumping station as Equations (10)–(13).

The lower and upper bounds of water storage, water balance equation, maximum pumping capacity, maximum annual pumping volume, initial and boundary conditions, and non-negative constraints should also be imposed on the subsystems.

• 2. Optimization of subsystems

The subsystem models are two nonlinear models with only one coupling constraint. Therefore, one-dimensional dynamic programming (DP) can be used to solve them, taking water supply X_i as a decision variable. The normal operation rule is integrated into recursive procedures of DP (Shi et al. 2015; Gong et al. 2019) to correct water storage and obtain water spill PS_i and water replenishment Y_i of each period simultaneously; in this manner, the defect of meta-heuristic algorithms, which leads to random occurrences of spills or replenishments against the normal operation rule, can be avoided. The specific procedure is as follows.

• If V_i < V_{i, min}, then the pumping station should replenish the reservoir and the final water storage should be V_{i, min}. The water replenishment and spill in period i are expressed as Equations (14) and (15).

  Water replenishment:

  

  (14)
  Water spill:

  

  (15)

• If V_i > V_{i, max}, then excess water should be drained through the spillway. The water replenishment and spill in period i are expressed as Equations (16) and (17).

  Water replenishment:

  

  (16)
  Water spill:

  

  (17)
• If V_{i, min} < V_i < V_{i, max}, then the water replenishment and spill in period i should both be zero as Equation (18):

  

  (18)
  Finally, the corrected water storage in period i is expressed as Equation (19):

  

  (19)

  A flow chart of optimizing the subsystems is shown in Figure 4.

Given a list of discrete values of BZ₂ with a certain increment in the feasible region, discrete values of W₁ and W₂ can be determined and substituted into the two subsystem models. DP is used to derive a list of f₂ ∼ W₂ and f₁ ∼ W₁ relationships and a list of corresponding solutions, [X_1,i, PS_1,i, Y_1,i] and [X_2,i, PS_{2, i}, Y_2,i].

• 3. Aggregation of the system

According to the relationships between the objective function and annual available water of subsystem, f₂ ∼ W₂ and f₁ ∼ W₁, an aggregation model can be derived as Equations (20) and (21).

The aggregation model can also be solved by DP, with the annual available water of subsystems W_j as decision variables. After optimal results of the aggregation model F*, W₁^*, and W₂^* are derived, the final optimal operation scheme [X_1,i, PS_1,i, Y_1,i, X_2,i, PS_2,i, Y_2,i]* of the double-reservoir-and-double-pumping-station system can be obtained by searching optimal results of subsystems corresponding to W₁^* and W₂^*.

As shown in Figure 5(a), the objective function value F decreases with increasing BZ₂ in the range of [1,100, 1,260] and begins to converge when BZ₂ exceeds 1,220 (10⁴ m³). This means that the benefits of increasing the water transfer volume of the HZ pumping station are no longer remarkable from this point.

Finally, the minimum value of F is 241, and the optimal annual pumping volume of the HZ pumping station is 1,245 (10⁴ m³); the corresponding optimal operation results are shown in Table 6. Compared with the results using the standard operation policy (SOP), the total water shortage is reduced by 87.7%, total water replenishment from the river is reduced by 2.2%, and total water spill is reduced by 60.6% at the 75% probability of exceedance.

The optimal operation redistributes water resources between two reservoirs (Table 6). Furthermore, the optimal operation not only reduces water shortages of the HWB reservoir but also adjusts its water shortage distribution. As shown in Figure 5(b), the water shortage of the HWB reservoir using SOP reaches up to 1.34 MCM in June, accounting for 87% of the annual water shortage. This shortage can be attributed to the mass water consumption during the steeping period of paddy fields. After optimization, although the frequency of water shortage increases, any severe drought in a period could be avoided. In addition, the optimal operation also changes the evaporation of the system, corresponding to changes in the water storage during each period (Table 6). Nevertheless, the impacts of the changes on the local ecological environment can be ignored because the variation range is within 5%.

Operation curves of reservoirs derived using the optimization method and SOP are shown in Figure 6(a) and 6(b), respectively. As a donor reservoir, water storage of the SH reservoir before the flood season was lower with the optimization than with the SOP, which could help retain more flood water and reduce water spill from 2.41 MCM to 0.95 MCM. In contrast, as a recipient reservoir, the optimal water storage of the SH reservoir before the flood season was higher with the optimization than with the SOP, which could increase available water in the reservoir and avoid water shortages caused by the limited pumping capacity of the HZ pumping station during the peak period of irrigation.

According to the differences in the operation curves of the two reservoirs, operation modes of the two pumping stations in the system (XZ and HZ) were also adjusted. As a replenishment pumping station for the SH reservoir, water pumping patterns using different methods are shown in Figure 6(c). The frequency of water pumping at the XZ station did not change after optimization, but the optimal operation reduced the water pumping volume at the first stage, which was beneficial for relieving the pressure of flood control in the SH reservoir. As a water transfer pumping station for the SH reservoir as well as a replenishment pumping station for the HWB reservoir, different water pumping patterns of HZ station are shown in Figure 6(d). Under the SOP, water was transferred only from June to September, the peak period of irrigation. As a result, the pumping station was fully loaded, but it was still difficult to meet the water demand because of the limited maximum pumping capacity. After optimization, the HZ pumping station transferred 15.5% of the annual water pumping volume before the peak period; this could help the SH reservoir retain more flood water as well as increase available water in the HWB reservoir for irrigation.

In conclusion, the most appropriate approach for the optimal joint operation of the double-reservoir-and-double-pumping-station system is to form a joint operation policy. This would enable full play of the functions of all components of the system, making use of surplus water of the SH reservoir to appropriately compensate water shortages of the HWB reservoir. In this manner, the goal of simultaneously reducing water spills and water shortages can be achieved.

At present, meta-heuristic algorithms have become mainstream methods to solve optimal operation models of water resources systems (Bozorgi et al. 2017; Mansouri et al. 2017). Meta-heuristic algorithms, for instance, genetic algorithm (GA), particle swarm optimization (PSO), and ant colony algorithm (ACO), firstly generate initial solutions and modify them in each iteration according to certain rules until meeting the convergence (Bayat et al. 2011; Chang et al. 2013; Ehteram et al. 2017). However, they cannot guarantee globally optimal results due to the ineluctable random sampling during the iteration process. Comparatively, dynamic programing based on Bellman's principle is adopted to solve the sub models and the aggregation model after the decomposition and aggregation of the system which can ensure the global optimal solution.

In addition, the solving results of meta-heuristic algorithms may depend on the value of some algorithm parameters (Allawi et al. 2018; Ehteram et al. 2018), such as crossover rate and mutation rate in GA and inertia weight and acceleration factor in PSO. Therefore, it is necessary to carry out sensitivity analysis on these parameters for the final results. However, there is no need to input any algorithm parameter when using DP which can make the solving procedure more simple and reliable (Kennedy 1986; Peng et al. 2018).

These findings provide a reference for the optimal operation of similar systems that consist of reservoirs and pumping stations in hilly regions of southern China, northern Vietnam, Myanmar, and Laos. The mean annual rainfall is over 1,000 mm in these regions, but 70–90% of that occurs in the flood season due to the monsoon climate, resulting in seasonal water shortages (Wang et al. 2017; Quinn et al. 2018). What is worse, in recent years, agricultural irrigation, as a traditional water user, has been greatly squeezed because of the rapid development of industry in south China and southeast Asia (Zhang et al. 2018). However, it is difficult for the original empirical operation method to handle these challenges. Under the limited agricultural water rights, the optimization model proposed in this paper can effectively adapt to the changes of water resources and alleviate seasonal water shortages by inflow regulation. In future research, the model should consider the stochastic nature of both inflows and water demands for real-time system optimization.

An optimization model of a double-reservoir-and-double-pumping-station system was developed in this study to minimize water shortages. One such system in Nanjing, China, was selected for a case study, and the constraint of total available water in the system was coupled into it, considering regional water rights. The system was decomposed into two subsystems based on the decomposition-aggregation theory, and DP was adopted to solve the sub models and the aggregation model. A joint operation rule of the system is integrated into the optimization model to obtain an appropriate operation scheme of both reservoirs and pumping stations.

This work was supported by the National Key Research and Development Program of China [grant number 2017YFC0403205] and the Postgraduate Research & Practice Innovation Program of Jiangsu Province [grant number KYCX19_2105]. The authors wish to acknowledge the managers of Liuhe Water Authority, Nanjing, China, for their support of providing reservoir and irrigation data.