Optimal water allocation integrated with water supply, replenishment, and spill in the in-series reservoir based on an improved decomposition and dynamic programming aggregation method

The reservoir is an essential infrastructure for allocating water resources, alleviating regional water scarcity, and adjusting the uneven allocation of time and space for water resources. The optimal allocation method of water resources for reservoirs has attracted wide attention, such as in Southern China. The mean annual precipitation is about 1,000 mm in these humid regions, but 60–70% of that occurs during the flood due to the monsoon climate (Quinn et al. 2018). Although the total annual inflows into the reservoirs are usually more than what is demanded, many water inflows are spilled during the flood season due to the uneven distribution of time and space, resulting in seasonal water shortages. So, it is required to replenish water by pumping stations from rivers or other reservoirs to achieve multi-reservoir joint operation during the dry season (Ming et al. 2017).

The basic principle of joint operation for a multi-reservoir system is to redistribute water resources among the reservoirs through hydraulic connections (Ahmad et al. 2014), which can take full advantage of the storage capacity of each reservoir. Thus, the operation purpose of in-series reservoirs (ISRs) is to reduce water spill and improve runoff utilization; the decision variable is the amount of water supply in each period, which is subject to the reservoir's capacities for both water storage and water demand (Chang et al. 2019; Rani et al. 2020). It has made significant progress in the optimization of reservoir system management and operations (Labadie 2004). However, most of the reservoirs in these studies are in arid or semi-arid regions, so the role of the operation rule of the reservoir is ignored to simplify the models (Celeste & Billib 2009). In humid regions, the rule, which is water replenishment in the shortage season and water spill in the flood season, has a critical role in improving the utilization efficiency of water resources and energy and the safe operation of reservoirs.

Water allocation models of reservoirs are a typical multi-dimensional optimization problem, and the solution methods mainly have mathematics programming and heuristic algorithms. Dynamic programming (DP) based on Bellman's principle (Bellman & Dreyfus 1964) applies to this multi-stage decision-making process. However, the curse of dimensionality may be induced when dealing with multiple reservoirs (Cheng et al. 2017). Decomposition (Turgeon 1981) is the mainstream idea for dealing with multi-dimensional optimization problems using DP. The decomposition-coordination method (Mahey et al. 2017; Tan et al. 2019) is the most commonly used and achieves the whole system's optimal state through successive iterative calculations between the large-scale system and subsystems according to the coordinating variables. The Lagrange multiplier method (Duan et al. 2022) is used to deal with the constraints when adopting the decomposition-coordination approach. However, this method is restricted by the differentiability and convexity of the objective function, and it fails in judging logic conditions in the operation rule constraint.

Recently, with the maturity of heuristic search theory and the development of computing power, modern heuristic algorithms with global search ability have gained their advantages, such as the genetic algorithm (GA; Allawi et al. 2018), the particle swarm optimization (PSO) algorithm (Azadeh et al. 2012), and the Grey Wolf Optimizer (GWO; Moeini & Afshar 2013) have become considerably popular optimization methods. These algorithms feature random sampling and can solve multi-dimensional problems without decomposition. When heuristic algorithms are adopted, a constrained optimization problem usually transforms into an unconstrained one by penalty functions (Pina et al. 2017; Wan et al. 2018). Similarly, the penalty function method may fail to handle the complicated logic conditions in the reservoir operation rule (Gong et al. 2020). What is worse, the algorithm parameters, such as the crossover and mutation rates in GA and the inertia weight and acceleration coefficients in PSO, cannot be determined even though these values are essential factors that can affect the performances of algorithms (Zhang & Dong 2019). There are only empirical value ranges for most algorithm parameters. Although several approaches have been applied for setting parameters, such as the deterministic strategy (Kavoosi et al. 2019) and the adaptive strategy (Cui et al. 2016; Nunes et al. 2018), they are not proven to be of universal significance (Karafotias et al. 2015).

To summarize, the model and algorithm for allocating water resources in reservoirs have not been sufficiently addressed. In this regard, this paper investigated the optimal scheduling strategy integrated with water supply, replenishment, and spill in the ISR. An optimization model of the ISR was proposed to minimize each reservoir's annual sum of water shortages. Then, an improved decomposition and DP aggregation (DDPA) method was proposed. The method is to transform the N + 1 dimensional DP into the N + 1 iterative calculations of one-dimensional DP. Both the subsystem models and the aggregation model are one-dimensional DP models. This method effectively solved the multi-variable and multi-dimensional reservoir optimization problem. The optimization scheduling scheme increased the utilization efficiency of inflow water and reduced energy consumption. Finally, a comparison between DDPA and six heuristic algorithms, including GA, PSO, GWO, Sine Cosine Algorithm (SCA), Salp Swarm Algorithm (SSA), and Gravitational Search Algorithm (GSA), was discussed from optimality, adaptability, and efficiency. The results demonstrated that the DDPA algorithm was better to choose for this kind of reservoir optimization model. The proposed model and the method enriched the optimization theory of multi-dimensional and multi-variable complex systems.

In Figure 1, X_i,t (L³), LS_i,t (L³), PS_i,t (L³), and EF_i,t (L³) are water supply, local inflow, water spill, and water loss of the reservoir i during the period t, respectively; YS_i,t (L³) is the water demand of user i during the period t; Y_i,t (L³) is water replenishment for reservoir i by the pumping station i during the period t; N is the total number of reservoirs in the system; i is the sequence number of the reservoir, i = 1, 2, … , N; t is the sequence number of each period, t = 1, 2, … , T.

• ① Annual available water for the system

The water replenishment Y_i,t and water spill PS_i,t during each period are determined by the lower and upper bounds of water storage as follows:

• a.
  If ⁠, the reservoir should be replenished with water. Y_i,t and PS_i,t can be calculated by Equations (7) and (8), respectively.

  

  (7)

  

  (8)
• b.
  If ⁠, the excessive water in the reservoir should be released. Y_i,t and PS_i,t can be calculated by Equations (9) and (10), respectively.

  

  (9)

  

  (10)
• c.
  If ⁠, then both Y_i,t and PS_i,t should be zero (Equation (11)).

  

  (11)
• ④ The maximum water supply can be expressed as Equation (12)

  

  (12)
• ⑤ The maximum water replenishment of the reservoir during each period is restricted by the pumping capacity, which can be expressed as Equation (13):

  

  (13)

  where (L³) is the maximum pumping volume of the station i during the period t.

• ⑥ Initial and boundary conditions

According to the characteristics of the above mathematical model, a multi-dimensional and multi-stage nonlinear model, including water supply [X_1,t, X_2,t,⁠, X_{L, t}], replenishment water [Y_1,t, Y_2,t,⁠, Y_{L, t}], and spill water [PS_1,t, PS_2,t,⁠, PS_{L, t}], contains a total of 3L variables and each variable has N dimensions. It will cause a ‘curse of dimensionality’. So, this study proposed an improved decomposition and dynamic programming aggregation (DDPA) method. The method aims to transform the original system into several subsystems and then solve both the subsystem models and the aggregation model with DP.

• ① Decomposition of the original system

First, the original large-scale system will be decomposed into several subsystems, and each subsystem only consists of a reservoir and a pumping station as shown in Figure 1.

For each subsystem, the pumping station lifts water to replenish the reservoir from another, except that pumping station 1 lifts water to reservoir 1 from a river.

• ② Optimization of subsystems

The constraints of the subsystem include Equations (3) and (5)–(14). Equation (3) is the coupling constraint of decision variables X_i,t, which can be transformed into the state transition equation. Equations (5)–(11) compose the operation rule of the reservoir. The operation rule is integrated into the recursive procedure of DP to correct water storage V_i,t and obtain water spill PS_i,t or water replenishment Y_i,t of each period simultaneously as shown in Figure 2(a). Equations (12) and (13) restrict the solution spaces of decision variables.

The coupling constraint of the aggregation model is expressed as Equation (4).

Obviously, Equation (17) is another expression of Equation (1). The aggregation model can also be solved by one-dimensional DP and the decision variable is W_i.

After obtaining the optimal results of the aggregation model F* and [W₁*, W₂*, … , W_N*], the optimization results [X_i,t, PS_i,t, Y_i,t]* of each subsystem can be acquired by searching the results of subsystem models according to W_i*, which can finally compose an optimal operation scheme of the whole system.

Overall, the key of DDPA is to transform the N + 1 dimensional DP into the N + 1 iterative calculations of one-dimensional DP. Both the subsystem models and the aggregation model are one-dimensional DP models. Different from traditional DDPA (Gong & Cheng 2018), the improved method adopts nested mode and embeds the recursive process of the subsystem model in the reverse recursive process of the aggregation model. For each subsystem, the calculation of each state decision variable is coupled with the reservoir water storage inspection mechanism, so that the water spill and replenishment can be calculated according to the rule, and the constraint conditions with the judgment logic can be handled.

Therefore, this method has the potential to reduce the curse of dimensionality and effectively obtain the results of global optimal.

The system mainly provides irrigation water. The characteristics of reservoirs and pumping stations are shown in Table 1. The average topography of the HWB reservoir is much higher than that of the SH reservoir. During water shortages, the HWB reservoir can lift water from the SH reservoir by the HZ pumping station, and the SH reservoir can lift water from the Zao river by the XZ pumping station. The annual water rights of the XZ pumping station, which means the maximum volume that can be pumped from the Zao river, are allocated by the local water authority as shown in Table 1. The HZ pumping station is an internal station in the system and is not limited by any water rights.

The operation cycle of the system is divided into 20 stages in a water conservancy year from October of the current year to September of the following year. Among them, to improve scheduling accuracy, June to September is the local flood season and water consumption peak (rice irrigation), with the stage every 10 days corresponding to stages 9–20. Other stages 1–8 are divided into stages by month. The inflow, water demand, and water loss during each period at a 75% probability of drought years are shown in Table 1.

In this section, the optimization results of the DDPA and the heuristic algorithms will be revealed.

The optimization results reflect the regulation capacity of the system, forming an effective joint operation mechanism of the ‘dual-reservoir-and-dual-pumping-station’ system, which achieved the goal of optimization scheduling between reservoirs.

For the sake of demonstrating the performance of DDPA, six heuristic algorithms were used to solve the reservoir operation problem under the same environment, including GA, PSO, GWO, SCA, SSA, and GSA. The main controlling parameters of all heuristic algorithms, the number of search agents and maximum iteration, are equal to 100 and 500, respectively. And, the values were obtained through many experiments for other controlling parameters of each algorithm to ensure the best performance. Moreover, the constraints were transformed into unconstrained one by penalty functions. Table 2 shows all the results, and heuristic algorithms selected the best of 30 independent runs.

From Table 2, it can be observed that no algorithms found water spills due to the consideration of the reservoir operation rules in the constraints. The objective function value of the DDPA algorithm is 0, water shortage is 0, and the total pumping volume from the Zao river by the XZ station is 435 × 10⁴ m³. The DDPA algorithm obtained the global optimal solution.

However, all of the heuristic algorithms only obtained the local optimal solution. The objective function values are 81–16,1309 × 10⁴ m³. As can be found in Table 2, water shortage is 2–350 × 10⁴ m³. And only the water replenishment of PSO, GWO, and SCA is low than DDPA, but GA, GSA, and SSA increased by 4.5,15.3, and 8.9%, respectively.

Therefore, the heuristic algorithms have shortcomings in solving the optimization problem of series reservoirs. This comparison is significant and would help future researchers select the suitable optimization method for their case study.

In this section, the comprehensive comparison of the performances of DDPA and heuristic algorithms from optimality, adaptability, and efficiency will be analyzed and discussed.

It can compare intuitively by the water supply reliability and vulnerability indexes. Reliability represents the satisfaction degree of water demand of the system, which is expressed by the average value of the ratio of actual water supply to water demand at each stage, and vulnerability represents the severity of water shortage. The water supply reliability index value of the system should be as large as possible, and the water supply vulnerability should be as small as possible.

From Table 3, the reliability of DDPA is 100% and the vulnerability is 0. However, the reliability and vulnerability of heuristic algorithms are 54.37–100% and 0–100%. It can be observed that the DDPA method is the most optimal. The PSO, GA, and SSA algorithms also show better optimization performance than other heuristic algorithms.

Table 3

Volumetric reliability and vulnerability obtained with different algorithms

Reservoir .	Algorithm .	Reliability (%) .	Vulnerability (%) .	Formula .
SH	DDPA	100	0
	PSO	100	0
	GWO	96.6	38.5
	GA	99.4	4
	GSA	77.5	54.5
	SCA	77.5	100
	SSA	95.6	45.4
HWB	DDPA	100	0
	PSO	93.3	100
	GWO	77.6	100
	GA	98.5	15.4
	GSA	74.1	53.8
	SCA	54.4	100
	SSA	91.6	66.7

Reservoir .	Algorithm .	Reliability (%) .	Vulnerability (%) .	Formula .
SH	DDPA	100	0
	PSO	100	0
	GWO	96.6	38.5
	GA	99.4	4
	GSA	77.5	54.5
	SCA	77.5	100
	SSA	95.6	45.4
HWB	DDPA	100	0
	PSO	93.3	100
	GWO	77.6	100
	GA	98.5	15.4
	GSA	74.1	53.8
	SCA	54.4	100
	SSA	91.6	66.7

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As we all know, discrete step sizes are the key factors to affect the optimization results for the DDPA algorithm; the smaller the discrete step size, the higher the accuracy with the higher the time cost. As the step size increases, the accuracy gradually decreases. But it is easy for us to find a suitable step size according to the characteristics of the decision variables.

Many factors affect the optimization results of the heuristic algorithm, such as the characteristics of the algorithm itself, the size of the population, the control parameters, the form of the constraints, the number of variables, etc.

Therefore, the applicability of the heuristic algorithm requires specific analysis of specific problems and a lot of time to select reasonable parameters. The selection of parameters itself is an optimization process. So, the DDPA algorithm is more adaptable.

So, compared with DDPA, the heuristic algorithm takes more time to obtain the optimal solution. The DDPA can balance the contradiction between accuracy and efficiency.

In this paper, according to the characteristics of optimal water resource scheduling of the reservoir, a complex mathematical model was proposed for the optimal allocation of water resources of the series reservoir with abundant water resources but the uneven allocation of time and space. And a decomposition and dynamic programming aggregation (DDPA) method was proposed. The comparison between DDPA and six heuristic algorithms was discussed from optimality, adaptability, and efficiency. The results demonstrated that (1) the DDPA has high optimization accuracy and still has a better global search ability for multi-variable and multi-dimensional optimization problems. (2) Implementing the DDPA is more adaptable as it needs fewer initial parameters than the heuristic algorithm. (3) It is more straightforward and accurate to check the satisfaction of constraints, such as logical judgments and equality conditions, in the DDPA compared to the heuristic algorithm. (4) The DDPA method could effectively solve the problem of balancing the accuracy of the optimization results and calculation time.

This study was supported by the National Natural Science Foundation of China (NSFC) (grant numbers 52079119 and 42177365). The authors wish to acknowledge the managers of Liuhe Water Authority, Nanjing, China, for their support in providing reservoir, irrigation, and hydrology data.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.