ABSTRACT
Mathematical programming methods and modern intelligent algorithms are used to solve the optimized operation models of water storage projects. Classical mathematical programming methods are prone to the curse of dimensionality when the size of the system increases. Intelligent algorithms improve the computational efficiency to a certain extent, but still suffer from defects such as unstable optimized solutions and easy to fall into local optimizers. In this article, a segmented trial-and-error algorithm is proposed to obtain the global optimizer while considering computational efficiency. First, a water balance constraint is introduced into the objective function and a Lagrangian function is constructed to derive the ideal optimizer. Constraints are then gradually introduced and the optimized model is solved by iterative trial-and-error calculations. By approximating the ideal optimizer by segments in this way, the global optimizer is finally obtained. Numerical experiments were carried out in a water supply reservoir. The results show that the segmented trial-and-error algorithm can obtain the global optimizers in different situations with objective functions that are 0.0028–0.0076 lower than those of the genetic algorithm, implying less and more uniform water shortages. The segmented trial-and-error algorithm requires less than 1% of the computational time of the dynamic programming.
HIGHLIGHTS
A segmented trial-and-error algorithm is proposed for the optimized operation of water storage projects.
A water balance constraint is introduced into the objective function, and a Lagrangian function is constructed to derive the ideal optimized solution.
The segmented trial-and-error algorithm obtains the global optimizer in different scenarios and requires less than 1% of the computation time of the dynamic programming.
INTRODUCTION
Water storage projects such as reservoirs and lakes are important tools for water resources allocation. The operation of water storage projects redistributes water resources in time and space according to human needs. It is an important means to alleviate the contradictory relationship between the uneven distribution of water resources and the water demand of human economy, society, and environment (Shiau 2021). However, due to the impact of global climate change, the spatiotemporal distribution patterns of river runoff have changed significantly. Statistical characteristics of runoff such as mean, variance, and extreme values have exhibited noticeable alterations. The hydrological consistency has been disrupted, increasing the uncertainty surrounding water resources and their development and utilization. Hydrological consistency is a fundamental assumption in traditional water resources planning and management. The design and operation of water storage projects are based on this assumption. The emergence of inconsistency undoubtedly affects the operation modes and overall benefits of water storage projects and may even threaten their safety. The ability of water storage projects to operate safely and maintain their designed benefits under climate change remains uncertain. To address the challenges posed by climate change, researchers have developed adaptive scheduling models for water storage projects and sought new scheduling patterns through solving these models. The optimized operation of water storage projects under climate change has received widespread attention. Its optimization model construction and solution are hot spots in academic research and a focus of attention in practice (Chen et al. 2023).
Mathematical programming methods are usually used to solve optimization problems. Linear programming (LP), nonlinear programming (NP), and dynamic programming (DP) are representative methods (Yurtal et al. 2005). Compared with LP and NP, DP has the advantages of accurate calculation results, high operational efficiency, and high applicability in optimized operation. DP has been widely used in solving the optimized operation models of water storage projects and has achieved remarkable results (Haguma et al. 2018). Mousavi et al. (2004) developed DP optimization and Hydrologic Engineering Center-5 (HEC-5) simulation models for planning the long-term operation of the Karoon-Dez multireservoir system in Iran. The DP results were better than those of HEC-5 in terms of meeting system operating targets and energy generation, but HEC-5 as a simulation model demands much less computer time and memory. Kim et al. (2007) use sampling stochastic DP to derive a monthly joint operating policy during the drawdown period of the Geum River multireservoir system in Korea. A cross-validation test of 1,900 simulation runs demonstrates that the updating policy is more appropriate in this reservoir system.
Most water storage projects, especially large reservoirs, are usually designed as multiobjective reservoirs providing functions such as flood control, hydropower generation, navigation, urban water supply, agricultural irrigation, and ecological flow maintenance. They generate multiple benefits in economic, social, ecological, and environmental terms (Reed et al. 2013). These objectives are usually in competition with each other, meaning that the achievement of one objective is usually at the expense of other objectives (Zhang et al. 2021). As the number of reservoirs and operation objectives increase, the optimization problem of reservoir operation becomes complex and high dimensional. It makes the computational cost grow exponentially and poses a huge challenge to reservoir operation, thus making DP vulnerable to the curse of dimensionality (Zhang et al. 2015; Xu et al. 2023). To alleviate the curse of dimensionality, many improved algorithms of DP have been proposed and applied in the optimization problem of reservoir operation such as the incremental dynamic programming (Kumar & Baliarsingh 2003) and discrete differential DP (Feng et al. 2017). These algorithms alleviate the curse of dimensionality to some extent, but they lose the ergodicity of DP and often obtain only local optimizers.
With the development of modern optimization techniques, the emergence of intelligent algorithms, such as genetic algorithm (GA), differential evolution, ant colony optimization, simulated annealing algorithm, and particle swarm optimization, has provided new ways to solve such optimization problems (Chen et al. 2023). GA is an algorithm for searching optimized solutions, which draws on natural selection and genetic mechanisms in biology and outperforms mathematical optimization algorithms such as LP, NP, and stochastic DP in terms of convergence speed, diversity of solution set space, and optimality-seeking ability. It is the first evolutionary algorithm applied to the field of optimized operation (Wang et al. 2022). The intelligent algorithms improve the computational efficiency to a certain extent, but still suffer from defects such as unstable optimized solutions and easy to fall into local optimizers (Ibrahim et al. 2021). Improved algorithms for known flaws of intelligent algorithms are also being studied and proposed. For example, when applying traditional GA to solve multireservoir system optimization problems, crossover and mutation operators may violate constraints such as water balance equations, hydraulic continuity relationships, and power system load demands. This can reduce the efficiency of the algorithm in searching for a feasible region or even leads to a convergence on an infeasible chromosome within the expected generations (Wang et al. 2023). Xu et al. (2012) proposed a modified GA taking stochastic operators within the feasible region of variables. The experimental results show that compared with GA adopting a penalty function or pairwise comparison in constraint handling, the proposed modified GA improves the refinement of the quality of a solution in a more efficient and robust way.
In this article, a segmented trial-and-error algorithm is proposed to solve the optimized operation model of the water storage project to obtain a global optimizer while taking into account the computational efficiency. First, a water balance constraint is introduced into the objective function, and a Lagrangian function is constructed to derive the ideal optimized solution, i.e., the optimized solution without considering other constraints. Then, constraints such as water supply, water demand, and water storage are gradually introduced, and the optimized model is solved by iterative trial-and-error calculations. By approximating the ideal optimized solution by segments in this way, the global optimizer is finally obtained. Numerical experiments were conducted to evaluate the performance of the segmented trial-and-error algorithm for optimization of a water supply reservoir, and compared with DP and GA.
OPTIMIZED OPERATION MODEL OF WATER STORAGE PROJECTS
Objective function
is the water supply from the reservoir at the tth time period; 
is the water demand from the reservoir at the tth time period; and m is the number of time periods.For water supply reservoirs with a relatively uniform distribution of water demand over time, the results of Equations (1) and (2) are similar, and it is relatively simple to use Equation (1) as the objective function. For water supply reservoirs with strong seasonality of water demand, such as agricultural irrigation, because of the strong correlation between the growth stages of crops, a concentrated water shortage at any growth stage may cause crop failure. Although Equation (1) can obtain a relatively uniform absolute water shortage over time, it may lead to a large difference in the relative water shortage (or supply/demand ratio) in each time period, and it is more scientific and reasonable to use Equation (2) as the objective function (Tu et al. 2008).
Constraint conditions
Six constraints are considered as follows (Tu et al. 2008; Wang et al. 2022; Chen et al. 2023):
- (1) Water balance constraint: Ensures that the water input to the system equals the water output and maintains the water balance of the water resource system:(3)
- (2) Water supply capacity constraint: Limits the maximum amount of water that can be supplied to the system to ensure that the water supply does not exceed its carrying capacity:(4)
- (3) Water demand constraint: The amount of water supplied does not exceed the water demand:(5)
- (4) Minimum water supply constraint: Ensures that the water supply to the system does not fall below a certain minimum limit such as domestic and minimum ecological water demand:(6)
- (5) Reservoir storage lower limit constraint: Limits the amount of water stored in the reservoir to not fall below a certain lower limit:(7)
- (6) Reservoir storage upper limit constraint: Limits the amount of water stored in a reservoir to not exceed a certain upper limit value to avoid reservoir overflow or other undesirable consequences:where(8)
and 
are the reservoir storage at the end and beginning of the tth time period, 
is the amount of water inflow to the reservoir the tth time period, M is the water supply capacity of the reservoir, 
is the basic water demand of the reservoir the tth time period, and 
and 
are the lower and upper limits of allowable reservoir storage at the end of the tth time period.
SEGMENTED TRIAL-AND-ERROR ALGORITHM
Flowchart of the segmented trial-and-error algorithm.
Ideal optimized solution
and 
are the reservoir storage at the end and beginning of the full operation period.
Equation (12) is the ideal optimized solution considering only the water balance constraint.
If the reservoir water supply process derived from Equation (12) satisfies all the remaining constraints, the water supply process is the result of optimized operation, which is often not the actual case. Therefore, a segmented trial-and-error algorithm is proposed to adjust the water supply process in segments to satisfy all constraints.
Segmented trial and error
The following are the steps of segmented trial and error:
Step 1: Input the datasets including reservoir utilizable capacity, dead storage capacity, storage upper limit, storage at the beginning and end of the operation period, water supply capacity, and water demand process. Calculate the initial ideal optimized solution 
considering the water balance constraint according to Equation (12).
Step 2: Check by time step whether the water supply capacity constraint, water demand constraint, and minimum water supply constraint are satisfied; if the constraint is satisfied, then go to Step 3, otherwise adjust the water supply according to the following method:
(i) If
, then 
.(ii) If
, then 
.(iii) If
, then 
.
(t = 1,2,…m) based on the adjusted 
; calculate 
; find the maximum value 
in 
and the minimum value 
in 
; and check whether the upper and lower storage constraints are satisfied. If the constraint is satisfied (Figure 2(a)), then go to Step 6, otherwise adjust the water supply according to the following method: (i) If
(Figure 2(b)), then go to Step 4.(ii) If
(Figure 2(c)), then go to Step 4.(iii) If
(Figure 2(d)), then go to Step 5.
Diagram of the relationship between the calculated water storage capacity and the upper/lower limits of water storage.
Diagram of the relationship between the calculated water storage capacity and the upper/lower limits of water storage.
and 
. If 
does not satisfy the constraint, the segment is bounded by 
, 
, and 
. If 
does not satisfy the constraint, the segment is bounded by 
, 
, and 
. Adjust the optimized solution according to Equation (12):
Go to Step 2.
Step 5: Divide the operation period into three segments by 
and 
. Adjust the optimized solution according to Equation (12) for each segment:
If 
, then 
, 
, 
, and 
.
, then 
, 
, 
, and 
.
Go to Step 2.
Step 6: Check whether the water storage at the end of the operation period can satisfy 
, where 
is the control accuracy. If it satisfies, the calculation results are collated, and the calculation is ended; otherwise 
, 
.
If the operation period is divided into two segments, then 
, and if the operation period is divided into three segments, then 
, otherwise 
. Go to Step 2.
NUMERICAL EXPERIMENT
Water storage project
An annual regulated water supply reservoir was selected as the case study. Its utilizable capacity and dead storage capacity are 26 × 106 m3 and 5 × 106 m3, respectively. The upper limit of reservoir storage from May to July is 25 × 106 m3. The reservoir storage at the beginning and the end of the operation period is 15 × 106 m3. The monthly water supply capacity of the reservoir is 10 × 106 m3. The water demand process for a year is shown in Table 1.
Water demand process of the selected water supply reservoir (unit: × 106 m3)
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| X(t) | 7.00 | 7.50 | 8.20 | 10.00 | 12.53 | 13.00 | 12.75 | 12.02 | 11.02 | 9.50 | 8.80 | 8.20 |
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| X(t) | 7.00 | 7.50 | 8.20 | 10.00 | 12.53 | 13.00 | 12.75 | 12.02 | 11.02 | 9.50 | 8.80 | 8.20 |
Experiment setting
In this article, DP and GA are selected and their solutions are used as comparisons to verify the rationality of the segmented trial-and-error algorithm for optimized operation of water storage projects. The stage variable t in DP is taken as 12 time periods, the state variable is the reservoir storage volume V(t), the number of states in each time period is 3000, and the decision variable is the water supply volume G(t).
Inflows to the reservoir during 3 years with different degrees of drought (exceedance probability P = 95, 90, and 75%) were selected as three scenarios, which are designated as No. 1, No. 2, and No. 3, respectively. The corresponding annual inflow of water in these 3 years was 87.57 × 106 m3, 97.53 × 106 m3, and 102.74 × 106 m3, respectively. The specific inflow process is shown in Table 2, where Y(1)(t), Y(2)(t), and Y(3)(t) are inflow processes of Scenarios No. 1, No. 2, and No. 3. The inflow processes of these 3 years are used as inputs of the established optimized operation model and solved by the segmented trial-and-error algorithm, DP, and GA to verify the adaption.
Inflow processes of different scenarios (unit: × 106 m3)
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Y(1)(t) | 1.87 | 2.15 | 5.50 | 6.92 | 14.92 | 10.65 | 19.00 | 7.86 | 6.72 | 4.79 | 3.48 | 3.70 |
| Y(2)(t) | 2.99 | 4.28 | 6.15 | 6.36 | 15.84 | 14.40 | 19.30 | 8.35 | 7.13 | 5.09 | 3.70 | 3.93 |
| Y(3)(t) | 2.15 | 2.47 | 6.32 | 6.51 | 17.16 | 13.41 | 24.16 | 9.04 | 7.73 | 5.52 | 4.01 | 4.26 |
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Y(1)(t) | 1.87 | 2.15 | 5.50 | 6.92 | 14.92 | 10.65 | 19.00 | 7.86 | 6.72 | 4.79 | 3.48 | 3.70 |
| Y(2)(t) | 2.99 | 4.28 | 6.15 | 6.36 | 15.84 | 14.40 | 19.30 | 8.35 | 7.13 | 5.09 | 3.70 | 3.93 |
| Y(3)(t) | 2.15 | 2.47 | 6.32 | 6.51 | 17.16 | 13.41 | 24.16 | 9.04 | 7.73 | 5.52 | 4.01 | 4.26 |
RESULTS AND DISCUSSION
Algorithm rationality and adaption analysis
The segmented trial-and-error algorithm, DP, and the GA are used to calculate the aforementioned scenarios simultaneously. The calculation results of optimized operation in different inflow scenarios are shown in Table 3, where G(0)(t) is the initial ideal optimized solution at the tth time period, V(0)(t) is the water storage calculated by the initial ideal optimized solution at the tth time period, G(t) is the water supply calculated by the segmented trial-and-error algorithm at the tth time period, R(t) is the corresponding time period water shortage rate at the tth time period, GD(t) is the water supply calculated by DP at the tth time period, and GG(t) is the water supply calculated by GA at the tth time period.
Results of optimized operation in different inflow scenarios (unit: × 106 m3)
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| X(t) | 7.00 | 7.50 | 8.20 | 10.00 | 12.53 | 13.00 | 12.75 | 12.02 | 11.02 | 9.50 | 8.80 | 8.20 | |
| Scenario No. 1 | Y(1)(t) | 1.87 | 2.15 | 5.50 | 6.92 | 14.92 | 10.65 | 19.00 | 7.86 | 6.72 | 4.79 | 3.48 | 3.70 |
| G(0)(t) | 5.72 | 6.03 | 6.44 | 7.39 | 8.43 | 8.59 | 8.50 | 8.25 | 7.85 | 7.14 | 6.78 | 6.44 | |
| V(0)(t) | 11.15 | 7.27 | 6.32 | 5.85 | 12.34 | 14.40 | 24.90 | 24.51 | 23.38 | 21.03 | 17.74 | 14.99 | |
| G(t) | 5.72 | 6.03 | 6.44 | 7.39 | 8.43 | 8.59 | 8.50 | 8.25 | 7.85 | 7.14 | 6.78 | 6.44 | |
| V(t) | 11.15 | 7.27 | 6.32 | 5.85 | 12.34 | 14.40 | 24.90 | 24.51 | 23.38 | 21.03 | 17.74 | 14.99 | |
| R(t) | 18% | 20% | 21% | 26% | 33% | 34% | 33% | 31% | 29% | 25% | 23% | 21% | |
| GD(t) | 5.72 | 6.03 | 6.44 | 7.39 | 8.43 | 8.59 | 8.50 | 8.25 | 7.85 | 7.14 | 6.78 | 6.44 | |
| GG(t) | 5.68 | 6.02 | 6.40 | 7.32 | 8.40 | 8.62 | 8.49 | 8.25 | 7.91 | 7.20 | 6.71 | 6.50 | |
| Scenario No. 2 | Y(2)(t) | 2.99 | 4.28 | 6.15 | 6.36 | 15.84 | 14.40 | 19.30 | 8.35 | 7.13 | 5.09 | 3.70 | 3.93 |
| G(0)(t) | 6.11 | 6.47 | 6.97 | 8.18 | 9.67 | 9.92 | 9.79 | 9.39 | 8.81 | 7.86 | 7.39 | 6.97 | |
| V(0)(t) | 11.88 | 9.69 | 8.86 | 7.05 | 13.22 | 17.70 | 27.21 | 26.17 | 24.50 | 21.73 | 18.04 | 15.00 | |
| G(t) | 6.39 | 6.80 | 7.37 | 8.76 | 10.00 | 10.00 | 10.00 | 8.75 | 8.27 | 7.45 | 7.05 | 6.68 | |
| V(t) | 11.60 | 9.08 | 7.86 | 5.46 | 1.130 | 15.70 | 25.00 | 24.60 | 23.46 | 21.10 | 17.75 | 1,500 | |
| R(t) | 9% | 9% | 10% | 12% | 20% | 23% | 22% | 27% | 25% | 22% | 20% | 19% | |
| GD(t) | 6.39 | 6.80 | 7.37 | 8.76 | 10.00 | 10.00 | 10.00 | 8.75 | 8.27 | 7.45 | 7.05 | 6.68 | |
| GG(t) | 6.28 | 6.89 | 7.32 | 8.79 | 10.00 | 10.00 | 9.95 | 8.77 | 8.11 | 7.61 | 6.99 | 6.75 | |
| Scenario No. 3 | Y(3)(t) | 2.15 | 2.47 | 6.32 | 6.51 | 17.16 | 13.41 | 24.16 | 9.04 | 7.73 | 5.52 | 4.01 | 4.26 |
| G(0)(t) | 6.31 | 6.71 | 7.25 | 8.59 | 10.32 | 10.62 | 10.46 | 9.98 | 9.31 | 8.23 | 7.71 | 7.25 | |
| V(0)(t) | 10.84 | 6.60 | 5.68 | 3.60 | 10.44 | 13.23 | 26.93 | 25.99 | 24.41 | 21.70 | 17.99 | 15.00 | |
| G(t) | 6.06 | 6.42 | 6.90 | 8.07 | 10.00 | 10.00 | 10.00 | 9.43 | 8.84 | 7.88 | 7.41 | 7.00 | |
| V(t) | 11.09 | 7.14 | 6.56 | 5.00 | 12.16 | 1,557 | 25.00 | 24.61 | 23.50 | 21.14 | 17.74 | 15.00 | |
| R(t) | 13% | 14% | 16% | 19% | 20% | 23% | 22% | 27% | 25% | 22% | 20% | 19% | |
| GD(t) | 6.06 | 6.42 | 6.90 | 8.07 | 10.00 | 10.00 | 10.00 | 9.43 | 8.84 | 7.88 | 7.41 | 7.00 | |
| GG(t) | 6.05 | 6.43 | 7.01 | 8.02 | 9.96 | 10.00 | 10.00 | 9.38 | 8.80 | 7.85 | 7.40 | 7.03 | |
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| X(t) | 7.00 | 7.50 | 8.20 | 10.00 | 12.53 | 13.00 | 12.75 | 12.02 | 11.02 | 9.50 | 8.80 | 8.20 | |
| Scenario No. 1 | Y(1)(t) | 1.87 | 2.15 | 5.50 | 6.92 | 14.92 | 10.65 | 19.00 | 7.86 | 6.72 | 4.79 | 3.48 | 3.70 |
| G(0)(t) | 5.72 | 6.03 | 6.44 | 7.39 | 8.43 | 8.59 | 8.50 | 8.25 | 7.85 | 7.14 | 6.78 | 6.44 | |
| V(0)(t) | 11.15 | 7.27 | 6.32 | 5.85 | 12.34 | 14.40 | 24.90 | 24.51 | 23.38 | 21.03 | 17.74 | 14.99 | |
| G(t) | 5.72 | 6.03 | 6.44 | 7.39 | 8.43 | 8.59 | 8.50 | 8.25 | 7.85 | 7.14 | 6.78 | 6.44 | |
| V(t) | 11.15 | 7.27 | 6.32 | 5.85 | 12.34 | 14.40 | 24.90 | 24.51 | 23.38 | 21.03 | 17.74 | 14.99 | |
| R(t) | 18% | 20% | 21% | 26% | 33% | 34% | 33% | 31% | 29% | 25% | 23% | 21% | |
| GD(t) | 5.72 | 6.03 | 6.44 | 7.39 | 8.43 | 8.59 | 8.50 | 8.25 | 7.85 | 7.14 | 6.78 | 6.44 | |
| GG(t) | 5.68 | 6.02 | 6.40 | 7.32 | 8.40 | 8.62 | 8.49 | 8.25 | 7.91 | 7.20 | 6.71 | 6.50 | |
| Scenario No. 2 | Y(2)(t) | 2.99 | 4.28 | 6.15 | 6.36 | 15.84 | 14.40 | 19.30 | 8.35 | 7.13 | 5.09 | 3.70 | 3.93 |
| G(0)(t) | 6.11 | 6.47 | 6.97 | 8.18 | 9.67 | 9.92 | 9.79 | 9.39 | 8.81 | 7.86 | 7.39 | 6.97 | |
| V(0)(t) | 11.88 | 9.69 | 8.86 | 7.05 | 13.22 | 17.70 | 27.21 | 26.17 | 24.50 | 21.73 | 18.04 | 15.00 | |
| G(t) | 6.39 | 6.80 | 7.37 | 8.76 | 10.00 | 10.00 | 10.00 | 8.75 | 8.27 | 7.45 | 7.05 | 6.68 | |
| V(t) | 11.60 | 9.08 | 7.86 | 5.46 | 1.130 | 15.70 | 25.00 | 24.60 | 23.46 | 21.10 | 17.75 | 1,500 | |
| R(t) | 9% | 9% | 10% | 12% | 20% | 23% | 22% | 27% | 25% | 22% | 20% | 19% | |
| GD(t) | 6.39 | 6.80 | 7.37 | 8.76 | 10.00 | 10.00 | 10.00 | 8.75 | 8.27 | 7.45 | 7.05 | 6.68 | |
| GG(t) | 6.28 | 6.89 | 7.32 | 8.79 | 10.00 | 10.00 | 9.95 | 8.77 | 8.11 | 7.61 | 6.99 | 6.75 | |
| Scenario No. 3 | Y(3)(t) | 2.15 | 2.47 | 6.32 | 6.51 | 17.16 | 13.41 | 24.16 | 9.04 | 7.73 | 5.52 | 4.01 | 4.26 |
| G(0)(t) | 6.31 | 6.71 | 7.25 | 8.59 | 10.32 | 10.62 | 10.46 | 9.98 | 9.31 | 8.23 | 7.71 | 7.25 | |
| V(0)(t) | 10.84 | 6.60 | 5.68 | 3.60 | 10.44 | 13.23 | 26.93 | 25.99 | 24.41 | 21.70 | 17.99 | 15.00 | |
| G(t) | 6.06 | 6.42 | 6.90 | 8.07 | 10.00 | 10.00 | 10.00 | 9.43 | 8.84 | 7.88 | 7.41 | 7.00 | |
| V(t) | 11.09 | 7.14 | 6.56 | 5.00 | 12.16 | 1,557 | 25.00 | 24.61 | 23.50 | 21.14 | 17.74 | 15.00 | |
| R(t) | 13% | 14% | 16% | 19% | 20% | 23% | 22% | 27% | 25% | 22% | 20% | 19% | |
| GD(t) | 6.06 | 6.42 | 6.90 | 8.07 | 10.00 | 10.00 | 10.00 | 9.43 | 8.84 | 7.88 | 7.41 | 7.00 | |
| GG(t) | 6.05 | 6.43 | 7.01 | 8.02 | 9.96 | 10.00 | 10.00 | 9.38 | 8.80 | 7.85 | 7.40 | 7.03 | |
The optimized objective function values of reservoir water supply obtained by the three algorithms under different scenarios are listed in Table 4, where F, FD, and FG are the optimized objective function values calculated by the segmented trial-and-error algorithm, DP, and GA, respectively (using Equation (2)).
Results of optimized operation in different inflow scenarios
| Inflow scenario | Scenario No. 1 | Scenario No. 2 | Scenario No. 3 |
|---|---|---|---|
| F | 0.8609 | 0.4392 | 0.4027 |
| FD | 0.8609 | 0.4392 | 0.4027 |
| FG | 0.8643 | 0.4416 | 0.4055 |
| Inflow scenario | Scenario No. 1 | Scenario No. 2 | Scenario No. 3 |
|---|---|---|---|
| F | 0.8609 | 0.4392 | 0.4027 |
| FD | 0.8609 | 0.4392 | 0.4027 |
| FG | 0.8643 | 0.4416 | 0.4055 |
Operation processes of Scenario No. 1 obtained by the segmented trial-and-error algorithm (initial ideal optimized solution of Scenario No. 1).
Operation processes of Scenario No. 1 obtained by the segmented trial-and-error algorithm (initial ideal optimized solution of Scenario No. 1).
Operation processes of Scenario No. 2 obtained by the segmented trial-and-error algorithm: (a) initial ideal optimized solution of Scenario No. 2 and (b) final optimized solution Scenario No. 2.
Operation processes of Scenario No. 2 obtained by the segmented trial-and-error algorithm: (a) initial ideal optimized solution of Scenario No. 2 and (b) final optimized solution Scenario No. 2.
Operation processes of Scenario No. 3 obtained by the segmented trial-and-error algorithm: (a) initial ideal optimized solution of Scenario No. 3 and (b) final optimized solution Scenario No. 3.
Operation processes of Scenario No. 3 obtained by the segmented trial-and-error algorithm: (a) initial ideal optimized solution of Scenario No. 3 and (b) final optimized solution Scenario No. 3.
As can be seen from Table 3, the water supply processes obtained by the segmented trial-and-error algorithm and DP are the same for the three different scenarios, indicating that the segmented trial-and-error algorithm yields a global optimizer. The segmented trial-and-error algorithm can replace DP for generating water supply reservoir operation schemes. As can be seen from Table 4, the optimized objective function values obtained by the segmented trial-and-error algorithm and DP are slightly lower than that calculated by GA (the lower the value of the objective function, the better the water supply strategy), which are more reasonable compared with that calculated by GA. The segmented trial-and-error algorithm proposed in this article performs better than GA in terms of computational accuracy.
As can be seen from Figure 3, the initial ideal optimized solution as well as the calculated water storage volume under Scenario No. 1 satisfies all constraints, which means that the initial ideal optimized solution is the final calculation result of the segmented trial-and-error algorithm.
From Figure 4(a), it can be seen that the maximum storage volume obtained from the initial ideal optimized solution under Scenario No. 2 does not satisfy the reservoir storage upper limit constraint. Therefore, according to the segmented trial-and-error algorithm, the operation period is divided into two segments using July as the boundary, and the regulation and calculation are performed separately. Due to the water supply capacity constraint, the water supply from May to July is 10 × 106 m3 per month (water supply capacity). As can be seen from Figure 4(b), the adjusted reservoir water supply as well as the storage volume satisfy all constraints, and the optimized solution is obtained.
As can be seen from Figure 5(a), the maximum and minimum storage volumes calculated by the initial ideal optimized solution under Scenario No. 3 do not satisfy the reservoir storage lower and upper limit constraints. Therefore, according to the segmented trial-and-error algorithm, the operation period is divided into three segments with April and July as the boundary, and the regulation and calculation are performed separately. Due to the water supply capacity constraint, the water supply from May to July is 10 × 106 m3 per month (water supply capacity). From Figure 5(b), it can be seen that the adjusted reservoir water supply as well as the storage volume satisfy all constraints, and the optimized solution is obtained.
The optimized operation of water supply reservoirs makes full use of the regulating capacity of reservoirs and makes the water shortage rate relatively uniform in time distribution, avoiding the occurrence of concentrated damage due to concentrated water shortage. The aforementioned operation results are consistent with the characteristics of water supply reservoir scheduling.
Computational efficiency analysis
The segmented trial-and-error algorithm and DP programs were run separately on a Pentium(R) Dual-Core CPU E6600 @ 3.06 GHz PC configuration. The average time taken to run the DP program is about 46 s, while the average time taken for the segmented trial-and-error algorithm is about 42 ms. The computational efficiency of the segmented trial-and-error algorithm is much greater than that of DP.
CONCLUSIONS
To obtain the global optimizer while considering the computational efficiency, a segmented trial-and-error algorithm is proposed in this article for the optimized operation models of water storage projects. Numerical experiments were conducted to evaluate the performance of the segmented trial-and-error algorithm for optimization of a water supply reservoir. The main conclusions obtained are as follows:
(1) The designed scenarios cover all possibilities for the initial optimized solution to break the constraints and also validate each step proposed by the segmented trial-and-error algorithm. The operation scheme obtained by the segmented trial-and-error algorithm makes full use of the regulating capacity of reservoirs and makes the water shortage rate relatively uniform in time distribution, avoiding the occurrence of concentrated water shortage.
(2) The segmented trial-and-error algorithm yields global optimizers in different scenarios. The segmented trial-and-error algorithm can replace DP for generating water supply reservoir operation schemes.
(3) The values of the optimized objective function obtained by the segmented trial-and-error algorithm for different scenarios are slightly lower than those calculated by the GA, which are more reasonable compared with that calculated by GA. The segmented trial-and-error algorithm performs better than GA in terms of computational accuracy.
(4) Under the same computer conditions, the segmented trial-and-error algorithm requires less than 1% of the computation time of the DP. The computational efficiency of the segmented trial-and-error algorithm is much greater than that of DP.
(5) The proposed algorithm is constructed based on the objective function (Equation (2)), and the constraints are introduced step by step to arrive at the optimized solution. In practical applications, the objective function and constraints are sometimes different, the solution ideas are applicable, and the proposed algorithm needs to be deformed accordingly.
AUTHORS CONTRIBUTIONS
Conceptualization: Yu Zhang; methodology: Xinya Jin and Hailong Zhou; formal analysis and investigation: Xinya Jin, Hailong Zhou, Yuelai Yao, and Tangsong Luo; writing – original draft preparation: Xinya Jin, Hailong Zhou, Yuelai Yao, and Tangsong Luo; writing – review and editing: Xinya Jin, Hailong Zhou, and Yu Zhang; funding acquisition: Yu Zhang; resources: Yu Zhang; supervision: Yu Zhang.
FUNDING
This work was supported by National Natural Science Foundation of China (Grant No. 52209032), Water Conservancy Science and Technology Program of Zhejiang Province (Grant Nos. RA1801 and RA2003), and Zhejiang Province Emergency Management Research and Development Technology Project (Grant No. 2024YJ017).
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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