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In a complex pressurized water diversion project (WDP), the combined optimal operation of multiple hydraulic facilities is computationally expensive owing to the requirement of massive mathematical simulation model runs. A parallel multi-objective optimization based on adaptive surrogate model (PMO-ASMO) was proposed in this study to alleviate the computational burden while maintaining its effectiveness. At the simulation level, an adaptive surrogate model was established, while a parallel non-dominated sorting genetic algorithm II (PNSGA-II) was utilized at the optimization level. Taking the successive shutdown of pumps as the operating process, the PMO-ASMO was applied to a complex pressurized diversion section of the Jiaodong WDP in China, and the results were compared with those obtained by NSGA-II and PNSGA-II. The results showed that the time consumption of PMO-ASMO was only 9.97% of that acquired by NSGA-II, which was comparable to that of PNSGA-II, in the case of 10-core parallelism. Moreover, compared with PNSGA-II, PMO-ASMO could find the optimal and stable Pareto front with the same number of simulation model runs, or even fewer runs. These results validated the effectiveness and efficiency of the PMO-ASMO. Therefore, the proposed framework based on multi-objective optimization is efficient for combined optimal operation of multiple hydraulic facilities.

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  • A combined optimal operation of multiple hydraulic facilities within complex pressurized water diversion projects is achieved.

  • The combined optimal operation model of multiple hydraulic facilities coupled with the simulation model is established.

  • The algorithm that combines the adaptive surrogate model and parallel computing is provided.

  • The effectiveness and efficiency of the proposed method are confirmed by an engineering case.

adaptive surrogate model, combined optimal operation, complex pressurized water diversion project, mathematical simulation model, multiple hydraulic facilities
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Water diversion projects (WDPs) have emerged as effective measures to address the uneven distributions of water resources, thereby improving the ecological environment and promoting regional economic development (Ma et al. 2020). With the worldwide development of WDPs, the importance of their operational strategies is constantly growing (Zhao et al. 2019). In particular, for complex pressurized WDPs with long water pipelines and multiple hydraulic buildings and facilities, the optimal combined operation of multiple hydraulic facilities is essential for ensuring a safe transient process under highly nonlinear hydraulic-mechanical coupling characteristics.

An efficient multi-objective optimization algorithm (MOA) is an effective approach to addressing the problem (Zhang et al. 2017). However, compared with the regulation of a single hydraulic facility of a simple WDP (Skulovich et al. 2016; Xu et al. 2018; Lai et al. 2019; Zaki et al. 2019; Rezghi et al. 2020), the transient processes under the combined regulation of multiple hydraulic facilities in a complex pressurized WDP are intricate (Cui & Guo 2023). The coupling relationships among multiple hydraulic facilities have an important effect on water hammer wave. How to determine the coupling relationships based on the certain safe operation requirements? Moreover, since the multi-objective optimization (MO) problem requires extensive simulations to obtain the global Pareto optimal solutions (Gong et al. 2016a), and the mathematical simulation model of complex pressurized WDPs requires tens of minutes or even hours to run once, it is difficult to efficiently obtain a multi-objective Pareto frontier for the combined optimal operation of multiple hydraulic facilities. How to reduce the computational cost? The above issues are the obstacles for the combined optimal operation of multiple hydraulic facilities in complex pressurized WDPs.

For the combined operation of multiple hydraulic facilities, Lai et al. (2020) employed the combined optimization of guide vane closing laws for two units under successive load rejection conditions in a pumped storage hydropower station. Zhao et al. (2019) investigated the optimal coordinated operation of the ball valve and guide vane based on a reference vector-guided evolutionary algorithm. Hou et al. (2019) studied the effects of time interval on startup quality during the successive startup of pumped storage hydropower stations by a multi-objective particle swarm optimization algorithm (MOPSO). Wan & Li (2016) proposed a joint pump-valve regulation scheme for a series pump-valve system, considering the effect of the time interval between the valve and pump during the startup process. Yu et al. (2015) proposed a unique closure law for the joint regulation of main inlet valves and wicket gates under the load rejection process for a pumped storage hydropower station. Zhang et al. (2022) adopted an improved multi-objective gravitational search algorithm (MOGSA) to investigate the joint optimal control of the valve and guide vane under the load rejection condition of hydroelectric units. Lei et al. (2021) optimized the startup strategy for a hydroelectric system based on MOPSO. Cui & Guo (2023) studied the multi-objective control of transient process of hydropower plant with two turbines sharing one penstock under combined operating conditions. The most favorable superposition time and the most unfavorable superposition time under COCs were determined. However, few studies have been conducted on the combined optimal operation of multiple hydraulic facilities for complex pressurized WDPs with long water pipelines and multiple hydraulic buildings and facilities.

Parallel computing and surrogate-based optimization are efficient measures for reducing computational time. Parallel computing is achieved by dividing a complex father task into multiple subtasks that can be solved parallelly by different processors. With the rapid advancement of electronic technology, the majority of computers are equipped with multicore processors and provide the necessary platform for realizing parallel computing. Additionally, numerous companies have succeeded in developing simple yet powerful parallel frameworks, thereby demonstrating essential advanced software platforms and conditions (Feng et al. 2018a). In MO problem solving, parallel processing technology was combined with optimization algorithms, including multi-objective genetic, particle swarm, and shuffled frog leaping algorithms, that efficiently address the optimization problem in reservoir operations (Sun et al. 2016; Feng et al. 2018b; Yang et al. 2022) and dispatching problem in battery storage systems (Grisales-Noreña et al. 2020). Surrogate models, also known as metamodeling, can accurately mimic complex models. Hence, surrogate-based optimization can effectively reduce the computational burden by replacing previous simulation models with cost-effective statistical surrogates. It has been widely applied in the field of groundwater optimization problems, the design and optimization problems of water distribution systems, and so on (Johnson & Rogers 2000; Behzadian et al. 2009). Recently, MO methods based on adaptive surrogate model have emerged for the optimization of the design and operation of WDPs. For example, Bazargan-Lari et al. (2013) adopted the MO model and Bayesian networks to optimize the closing rule for the valve of a simple reservoir-pipe-valve system, thereby avoiding the time-consuming simulation and optimization processes. Tong et al. (2020) built three surrogate models by analyzing the highly nonlinear relationship between the external characteristic values and essential design variables of centrifugal pumps. The three surrogate models were separately used to optimize the essential design variables in combination with the non-dominated sorting genetic algorithm II (NSGA-II). Gong et al. (2016a) proposed an MOA based on adaptive surrogate model, which was applied to parameter estimation and reservoir optimization operations in large and complex geophysical models. Wu et al. (2023) built a surrogate model using Gaussian process regression (GPR) and used NSGA-II to optimize the impeller and diffuser. With regard to the combined optimal operation of multiple hydraulic facilities, few works have been conducted using optimization algorithms combined with an adaptive surrogate model. Additionally, there exist few reports on the integration of parallel computing and adaptive surrogate models.

This study aims to utilize a novel approach called PMO-ASMO, which integrates parallel computing, MO, and an adaptive surrogate model, to obtain the combined optimal operation schemes of multiple hydraulic facilities in complex pressurized WDPs. The novelties and innovations of the study can be concluded as follows: (1) A MO model for the combined operation of multiple hydraulic facilities in complex pressurized WDPs was established with full consideration of the coupling relationship between hydraulic facilities. (2) An efficient method for the optimization model based on the parallel computing, adaptive surrogate model and optimization algorithm was proposed.

In Section 2, the mathematical simulation model for complex pressurized WDPs was established, and transient interference analysis of coupling relationships between multiple hydraulic facilities was analyzed, then a MO model for the combined operation of multiple hydraulic facilities was built, and the efficient method called PMO-ASMO was proposed. In Section 3, the Pareto frontier of the combined optimal operation of multiple hydraulic facilities achieved by PMO-ASMO was compared with those obtained by the classical evolutionary algorithm to verify the effectiveness and efficiency, taking a complex pressurized diversion section of Jiaodong Water Diversion Project in China as example. In Section 4, the conclusions were given.

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An enhanced MO method for obtaining combined operating schemes of multiple hydraulic facilities in complex pressurized WDPs is presented in Figure 1. The details are as follows:

  • (1) Mathematical simulation model for complex pressurized WDPs: The general layout of complex pressurized WDPs comprises a long main pipe with multiple series-connected hydraulic buildings and facilities, including pumping stations, high-level water tanks, branch pipes, surge tanks, valves, and regulating tanks. Further, the pumping station generally comprises several pump-valve units. Each unit is formed of a valve and a pump in series, which is joint to the main pipe via a branch pipe. Hence, to establish a mathematical simulation model, the transient flow in the pipes is described using governing equations, while hydraulic buildings and facilities are treated as boundary conditions that are coupled to the governing equations. The method of characteristics (MOC), which is a simple, numerically efficient, and the most used method, was adopted to solve the equations.

  • (2) Transient interference analysis of coupling relationships between multiple hydraulic facilities: Given the general layout of complex pressurized WDPs, the successive shutdown of pumps was adopted as the operating process in this study. The mutual transient interference of the coupling relationship among multiple hydraulic facilities is explored using a mathematical simulation model to identify the adjustable parameters (inputs P) and response functions (outputs O) of the model when facing an actual problem.

  • (3) Multi-objective optimization: When adopting the PMO-ASMO, firstly, multiple GPR surrogate models are constructed for different response functions. To establish the surrogate model, sampling techniques are utilized, and their initial sampling is arranged using the Good Lattice Points (GLP) method with ranked Gram-Schmidt (RGS) decorrelation. A multicore parallel technique using the Fork/Join framework is integrated into the initial sampling process. Secondly, the PNSGA-II algorithm, which incorporates parallel technology and the NSGA-II algorithm, is employed in the GPR to optimize the surrogate models and obtain Pareto optimal points. When referring to NSGA-II and PNSGA-II, the NSGA-II is utilized in a serial and parallel manner to drive the mathematical simulation model of the complex pressurized WDPs to obtain Pareto optimal points, respectively. Finally, to evaluate the validity and advantages, the PMO-ASMO was compared with the PNSGA-II and NSGA-II.

  • (4) Pareto frontier: The results of the aforementioned algorithms were analyzed to identify appropriate combined operating schemes for multiple hydraulic facilities.

Figure 1
Flow diagram of the multi-objective optimized operation for multiple hydraulic facilities.
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Flow diagram of the multi-objective optimized operation for multiple hydraulic facilities.

Figure 1
Flow diagram of the multi-objective optimized operation for multiple hydraulic facilities.
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Flow diagram of the multi-objective optimized operation for multiple hydraulic facilities.

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Mathematical simulation model for complex pressurized WDPs

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A specific mathematical description of each component of the WDPs is the foundation for optimization. In this section, a mathematical simulation model that includes the pipe, pump, and valve is presented.

Pipe

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The continuity and momentum equations (Equations (1) and (2)) are the fundamental equations for unsteady flow in pipeline. Typically, V ≪ a, therefore Equations (1) and (2) can be simplified to Equations (3) and (4), which are further converted into the following difference equation (Equations (5) and (6)) by the MOC (Wylie et al. 1993; Wang et al. 2023):
(1)
(2)
(3)
(4)
(5)
(6)
where H is the instantaneous piezometric head, Q denotes the instantaneous discharge, V represents the average speed, i is the section number along the pipeline, j stands for the time in the algebraic water hammer, α is the pipe slope, g is the gravitational acceleration, a is the wave speed, and D and f are the diameter and friction factor of the pipe, respectively. CP, CM, and B can be expressed as follows:
(7)
(8)
(9)
where R are calculated with the following form:
(10)
where A is the area of the pipe.

Boundary conditions

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Pump
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The pump boundary conditions are described using head balance (Equation (11)) and the torque-angular deceleration equations (Equation (14)) (Chaudhry 2014), while the former can be calculated by combining the compatibility equations to yield Equation (12).
(11)
(12)
where HPB and HPA correspond to the piezometric heads in the last and first sections of the suction and discharge pipes, respectively. tdHP denotes the total dynamic head, QP is the discharge of the pump, BB and BA are the pipeline constants B of the suction and discharge pipes, respectively.
tdHP and QP can be obtained from the pump characteristic curve described by Suter (1966) by the linear interpolation. By substituting these values into Equation (12), the following equation is obtained:
(13)
where α = np/nr and υ = Qp/Qr. tdHr, nr, and Qr refer to the total dynamic head, rotational speed, and discharge under rated conditions, respectively. np denotes the unknown rotational speed at the end of Δt. A0 and A1 denote the constants in the linear interpolation formula.
Similarly, the torque-angular deceleration equation (Equation (14)) can be rewritten as Equation (15).
(14)
(15)
where β0 = T0/Tr. I denotes the combined polar moment of inertia, α0 is the dimensionless speed at the beginning of Δt, T0 is the known torque at the beginning of Δt, Tr is the rated torque on the pump, and Tp is the unknown torque at the end of Δt. B0 and B1 are the constants contained in the linear interpolation formula.
Valve
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Because the valve is installed between two pipes (pipes 1 and 2), combining the compatibility equations with orifice equation (Equation (16)) yields Equation (17) (Wylie et al. 1993).
(16)
(17)
where τ is a dimensionless number that describes the discharge coefficient multiplied by the area of the valve opening. ΔH0 denotes the steady-state drop in the hydraulic grade line across the valve, which occurs at a flow rate of Q0 when τ = 1. ΔH denotes the steady-state drop with a flow rate of Qp, where Qp is the instantaneous discharge. B1 and B2 are the pipeline constants of pipes 1 and 2, respectively. CP1 and CM2 denote the known constants in the characteristic equations of pipes 1 and 2, respectively. Cv is calculated as follows:
(18)
It is crucial to consider the possibility of a negative flow if CP1 − CM1 < 0. The orifice equation can be calculated using Equation (19). The discharge is obtained using Equation (20):
(19)
(20)

Transient interference analysis of coupling relationships between multiple hydraulic facilities

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Actively regulated hydraulic facilities consisting mainly of pumps and valves are necessary for the successive shutdown of complex pressurized WDPs. Appropriate operational time intervals between pumps are crucial for relieving backflow and pressure exceedance (Lei et al. 2021). One- and two-phase closure laws are the primary operating rules for valves in engineering applications (Zhao et al. 2019). Compared to the one-phase strategy, the two-phase strategy is advantageous in terms of limiting the flow and speed reversal of pumps, thereby reducing the amplitude of pressure pulsation and improving safety in the piping system. Additionally, in a series of hydraulic facilities and pipe systems, appropriately operating valves can effectively prevent excessive pressure and water levels during the successive shutdown process of pumps (Wan & Li 2016). Hence, the inputs P comprise the following parameters: the time interval between the ith and (i+ 1)th pumps shutdown (ΔTpi−(i+1)), total closing time (Tvbp2i), inflection time (Tvbp1i), and inflection opening (yvbpi) of the valve closing law in the ith (i = 1,2, … , m) pump-valve unit, the corresponding opening (yrv_ji) and required time to reach yrv_ji (Trv_ji) of the jth (j = 1,2, … , t) valve in the main pipeline when the ith pump starts closing, as shown in Figure 2.
Figure 2
The schematic diagram of the pump successive shutdown strategy.
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The schematic diagram of the pump successive shutdown strategy.

Figure 2
The schematic diagram of the pump successive shutdown strategy.
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The schematic diagram of the pump successive shutdown strategy.

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An ideal successive pump shutdown process minimizes the maximum water pressure (HPmax), highest water levels of the control structures (Zemax, e= 1, 2, … , s) and total regulation time (Ttotal), as well as maximizing the minimum water pressure (HPmin) and lowest water levels (Zemin). Hence, HPmax, HPmin, Zemax, Zemin, and Ttotal are selected as outputs O, thereby ensuring the safety of the hydraulic transient process and quick feasibility of regulation.

Multi-objective optimization

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Aiming at the combined optimal operation of multiple hydraulic facilities based on NSGA-II and PNSGA-II, NSGA-II was utilized in a serial and parallel manner to drive the mathematical simulation model of the complex pressurized WDPs and calculate the corresponding real response functions (Npop×O) of the population (Npop×P). Then response functions were used to evaluate the implicit constraints and calculate objective functions.

However, when adopting the PMO-ASMO in the combined optimal operation model, a surrogate model focusing solely on the mathematical relationship between inputs P and outputs O is established to replace the simulation model. PNSGA-II was employed to reduce the computational burden of the optimization effectively. Further, an adaptive strategy was adopted to address the challenges of local optima and overfitting (Razavi et al. 2012), thereby enhancing the accuracy of optimization based on surrogate modeling. As shown in Figure 1, the mathematical simulation model is used to calculate the corresponding real response functions (Z0×O) of the initial samples (Z0×P), while the surrogate model is constructed using Z0×O and Z0×P. Then PNSGA-II runs on the surrogate model created in previous step to obtain the Pareto optimal points, matrix (Npop×P), where Npop is the size of the population. A portion of the optimal points (20%) are chosen to solve the mathematical simulation model of complex pressurized WDPs. Further, the results were added to the initial data as (Z1×P) and (Z1×O) to update the surrogate model, wherein Z1 = Z0 + 20% ×Npop. The surrogate model is iterated several times until the termination conditions are reached, which better describes the actual process of the physical model, thereby boosting the accuracy of the surrogate model and improving the global optimum.

Initial sampling

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Latin hypercube (LH), Sobol’ low discrepancy sequence, Good Lattice Points (GLP), and crude Monte Carlo (MC) methods are the most commonly used sampling methods. The GLP with ranked Gram-Schmidt (RGS) decorrelation has been proven to perform optimally, since the computed discrepancy of GLP is small and RGS decorrelation can significantly improve the uniformity metrics of lattice designs (Gong et al. 2016b). Hence, GLP with RGS decorrelation was employed for initial sampling in this study, as described in more detail by Gong et al. (2016b). Furthermore, an appropriate initial sample size (Z0) was set based on the specific situation.

Considering that the corresponding response functions of initial samples are calculated by repeatedly driving the mathematical simulation model, the computational time can be effectively reduced by adopting the Fork/Join framework to maximize multiple cores in computers. As shown in Figure 3, the main thread divides the initial sample set into the same number of subsample sets as the number of computer cores (q) and transmits them to the corresponding child thread of each computer core. A mathematical simulation model was used to calculate the corresponding response functions for the subsample set in each allocated child thread. Once the calculation of all the subsamples is complete, the corresponding response function matrix (Z0×O) of the initial sample matrix (Z0×P) is formed.
Figure 3
The initial sampling process.
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The initial sampling process.

Figure 3
The initial sampling process.
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The initial sampling process.

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Surrogate model

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The GPR, proposed by Rasmussen & Williams (2005), has been widely utilized to address nonlinear regression problems with small samples and high dimensionality. Hence, multiple surrogate models based on GPR were constructed for each response function in this study.

Suppose there is a regression model as follows:
(21)
where y is a variable in the outputs O, x is the vector of inputs P, and is the Gaussian noise. is the short Gaussian process, where the mean and covariance between two input vectors ⁠, are calculated with the following expression:
(22)
The joint Gaussian distribution is used to train the surrogate model for predicting inputs and outputs:
(23)
where X, y are the input matrix (Z0×P) and output matrix (Z0×O), is the predicting input matrix, and is the predicting output vector.

Moreover, the SCE-UA optimization method was chosen to determine the proper hyperparameter values in GPR (Gong et al. 2016a; Zhang et al. 2017).

Combined optimal operation model of multiple hydraulic facilities

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Two objective functions, the minimization of the risk index for hydraulic transient processes and minimization of time consumption about regulation, were introduced in complex pressurized WDPs. This helps improve the safety and benefit of the combined operation of numerous hydraulic facilities during the successive shutdown process of the pumps. The aforementioned functions can be calculated based on the output O, whereas the decision variables are represented by the input P.

Objective functions
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Minimization of risk index of hydraulic transient process (Rtotal)
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Maximum and minimum water pressure of the system, highest and lowest water levels of the control structures should be considered in the comprehensive risk index of hydraulic transient process (Zhao et al. 2019), as follows:
(24)
where HPs_min and HPs_max (HPs_max=nHPw_max) represent the minimum and maximum allowable water pressure during the transient process, respectively, and HPw_max is the maximum steady-state pressure. n is commonly set to 1.3, while HPs_min must not be less than −7.5 m to avoid cavitation. Zes, Zesmin, and Zesmax are the design, minimum, and maximum permissible water level of the eth control structure, respectively. kh and ke represent the penalty factor for violating constraints of allowable water pressure and water levels of the eth control structure, respectively.
Minimizing time consumption of regulation (Ttotal)
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(25)
where m is the total number of pumps, ΔTpk−(k+1) represents the time interval between the kth and (k+ 1)th pumps shutdown, Tvbp2m is the total closing time for the valve closing law in the mth pump-valve unit, Trv_jm is the time required for the jth valve on the main pipeline to reach corresponding opening (yrv_ji) when the mth pump starts to shut down.
Constraints
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The constraints of the valve closing laws in the pump-valve system include Equations (26)–(29), which are explicit constraints. The constraints for the water pressures of pipes and water levels in control structures are shown in Equations (30)–(33), which are implicit constraints. Implicit constraints can be addressed by utilizing the output O derived from the mathematical simulation model or adaptive surrogate model.
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
where tc_min and tc_max are the minimum and maximum allowable regulation times of the valves in the pump-valve units, respectively.

PNSGA-II

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NSGA-II is an outstanding method in terms of computational efficiency and implementation (Lee et al. 2020), in which the fast-non-dominated sorting approach is used to greatly reduce computational complexity, by adopting the crowding distance to maintain the diversity of the population, and the elite strategy is introduced to make the sampling space enlarged and prevent the loss of the best individuals (Deb et al. 2002; Padhi & Mallick 2014; Hadibafekr et al. 2023). It has been widely applied in WDPs optimization operations. However, there is generally a high computational burden, even if a population with a large size is not difficult to implement. Moreover, the population of the NSGA-II utilizes the serial computing technique to assess the individual ability, thereby limiting the utilization efficiency of the computing resources owing to the ineffective utilization of parallel resources available in multicore machines. Consequently, when dealing with the combined operational problems of multiple hydraulic facilities in complex pressurized WDPs, the NSGA-II may require a substantial amount of time to complete the calculation (Peng et al. 2017). To avoid these defects, parallel computing was integrated into NSGA-II and the PNSGA-II was proposed (Liu et al. 2022). The Fork-Join framework, which is renowned for parallel computing (Niu et al. 2018), was used for parallel execution. The PNSGA-II workflow is illustrated in Figure 4, which mainly involves the following steps:
Figure 4
The PNSGA-Ⅱ workflow.
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The PNSGA-Ⅱ workflow.

Figure 4
The PNSGA-Ⅱ workflow.
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The PNSGA-Ⅱ workflow.

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Step 1, a parent population, size Npop, is randomly generated within the constraint ranges. Then it is arranged into q child threads, and the objective functions are calculated independently on every thread until completion. Furthermore, the computed results are returned to the master thread for being sorted into non-dominated ranks, while the crowding distances of each member are calculated.

Step 2, the parent population goes through the tournament selection, crossover, and mutation in NSGA-II, thereby resulting in a child population of size Npop.

Step 3, q child threads are executed in parallel to obtain the objective functions of the child population. Further, all the results are returned to the master thread, and the parent and child population is merged. The 2Npop solutions are sorted into non-dominated ranks. Similarly, the crowding distances are evaluated.

Step 4, Npop members can be chosen as the parent population of next computational iteration from the 2Npop population. Continue the aforementioned steps until the maximum iterations is reached.

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Case description

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The complex pressurized diversion section of the Jiaodong Water Diversion Project in China, as shown in Figure 5, comprises a pumping station, pipes, a high-level water pool, a regulating pool, air valves, regulation valves, and a branch pipe. As the upstream and downstream boundaries can be regarded as fixed water levels, they were treated as upstream and downstream reservoirs in this study. There are four pump-valve units in the pumping station (three in operation and one on standby). There are 96 air valves along the pipeline, as shown in Figure 6. The outlet and inlet diameters of the air valves are both 300 mm. RV1 and RV2 are of a similar type. RV1 (or RV2) and RV3 were closed by one-phase valve closure laws, where the time required to close their openings from 100 to 0% are 27 and 32 min, respectively. When pump 2 began shutting down, the valve on the branch pipe was closed using a one-phase valve closure law for 60 s. The characteristic parameters are presented in Table 1. Simultaneously considering the overall requirements for safe and efficient operation during the continuous shutdown of the three commonly used pumps is challenging. Consequently, designing a rationally combined operational scheme for multiple hydraulic facilities is important for enhancing the quality of the dynamic response.
Table 1

The characteristic parameters of the complex pressurized diversion section of Jiaodong Water Diversion Project in China

ParametersMeaningUnitValueParametersMeaningUnitValue
Zur Upstream reservoir level m 30.68 Zdr Downstream reservoir level m 41.17 
L1 Length of pipe 1 m 4,700 D1 Diameter of pipe 1 m 2 
n1 Manning roughness coefficient of pipe 1 / 0.015 L2 Length of pipe 2 m 29,400 
D2 Diameter of pipe 2 m 2.2 n2 Manning roughness coefficient of pipe 2 / 0.013 
L3 Length of pipe 3 m 2,150 D3 Diameter of pipe 3 m 2.6 
n3 Manning roughness coefficient of pipe 3 / 0.012 L4 Length of pipe 4 m 9,260 
D4 Diameter of pipe 4 m 2.2 n4 Manning roughness coefficient of pipe 4 / 0.012 
L5 Length of pipe 5 m 5,310 D5 Diameter of pipe 5 m 2.2 
n5 Manning roughness coefficient of pipe 5 / 0.013 L6 Length of pipe 6 m 5,490 
D6 Diameter of pipe 6 m 2 n6 Manning roughness coefficient of pipe 6 / 0.012 
L7 Length of pipe 7 m 4,050 D7 Diameter of pipe 7 m 2.2 
n7 Manning roughness coefficient of pipe 7 / 0.012 L8 Length of pipe 8 m 2,140 
D8 Diameter of pipe 8 m 2 n8 Manning roughness coefficient of pipe 8 / 0.012 
Q1 Discharge before the branch pipe m3/s 5.5 Q2 Discharge after the branch pipe m3/s 4.8 
Qr Pump rated discharge m3/s 1.86 Hr Pump rated water head m 65.69 
Nr Pump rated speed r/min 750 LZ1·WZ1 Long·wide of the high-level water pool m·m 20 × 10 
Z1smax Allowable highest water level for the high-level water pool m 93.5 Z1s Design water level for the high-level water pool m 87.53 
Z1smin Allowable lowest allowable water level for the high-level water pool m 86.8 LZ2·WZ2 Long·wide of the regulating pool m·m 19.8 × 19.8 
Z2smax Allowable highest water level for the regulating pool m 65 Z2s Design water level for the regulating pool m 61 
Z2smin Allowable lowest water level for the regulating pool m 59.2 nP10 Initial ratio of operating speed to rated speed for pump 1 – 1 
np20 (np30) Initial ratio of operating speed to rated speed for pumps 2 and 3 – 0.99 τRV1(τRV2) Initial opening degree of RV1 and RV2 % 73.87 
τRV3 Initial opening degree of RV3 % 67.36 – – – – 
ParametersMeaningUnitValueParametersMeaningUnitValue
Zur Upstream reservoir level m 30.68 Zdr Downstream reservoir level m 41.17 
L1 Length of pipe 1 m 4,700 D1 Diameter of pipe 1 m 2 
n1 Manning roughness coefficient of pipe 1 / 0.015 L2 Length of pipe 2 m 29,400 
D2 Diameter of pipe 2 m 2.2 n2 Manning roughness coefficient of pipe 2 / 0.013 
L3 Length of pipe 3 m 2,150 D3 Diameter of pipe 3 m 2.6 
n3 Manning roughness coefficient of pipe 3 / 0.012 L4 Length of pipe 4 m 9,260 
D4 Diameter of pipe 4 m 2.2 n4 Manning roughness coefficient of pipe 4 / 0.012 
L5 Length of pipe 5 m 5,310 D5 Diameter of pipe 5 m 2.2 
n5 Manning roughness coefficient of pipe 5 / 0.013 L6 Length of pipe 6 m 5,490 
D6 Diameter of pipe 6 m 2 n6 Manning roughness coefficient of pipe 6 / 0.012 
L7 Length of pipe 7 m 4,050 D7 Diameter of pipe 7 m 2.2 
n7 Manning roughness coefficient of pipe 7 / 0.012 L8 Length of pipe 8 m 2,140 
D8 Diameter of pipe 8 m 2 n8 Manning roughness coefficient of pipe 8 / 0.012 
Q1 Discharge before the branch pipe m3/s 5.5 Q2 Discharge after the branch pipe m3/s 4.8 
Qr Pump rated discharge m3/s 1.86 Hr Pump rated water head m 65.69 
Nr Pump rated speed r/min 750 LZ1·WZ1 Long·wide of the high-level water pool m·m 20 × 10 
Z1smax Allowable highest water level for the high-level water pool m 93.5 Z1s Design water level for the high-level water pool m 87.53 
Z1smin Allowable lowest allowable water level for the high-level water pool m 86.8 LZ2·WZ2 Long·wide of the regulating pool m·m 19.8 × 19.8 
Z2smax Allowable highest water level for the regulating pool m 65 Z2s Design water level for the regulating pool m 61 
Z2smin Allowable lowest water level for the regulating pool m 59.2 nP10 Initial ratio of operating speed to rated speed for pump 1 – 1 
np20 (np30) Initial ratio of operating speed to rated speed for pumps 2 and 3 – 0.99 τRV1(τRV2) Initial opening degree of RV1 and RV2 % 73.87 
τRV3 Initial opening degree of RV3 % 67.36 – – – – 
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Figure 5
The layout of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.
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The layout of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.

Figure 5
The layout of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.
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The layout of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.

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Figure 6
The longitudinal profile of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.
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The longitudinal profile of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.

Figure 6
The longitudinal profile of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.
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The longitudinal profile of the complex pressurized diversion section of Jiaodong Water Diversion Project in China.

Close modal

Results and discussion

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As three frequently used pump-valve units are of the same type, RV1, RV2, and RV3 are closed at fixed speed, and the same type of valves, RV1 and RV2, are installed in the same position and follow the same regulation, the decision variables are [Tvbp21, Tvbp11, yvbp1, ΔTp1−2, ΔTp2−3, yrv_11, yrv_12, yrv_31, yrv_32]. The variation ranges of the constraints and decision variables are shown in Table 2.

Table 2

The range of constraint parameters and decision variables

ParametersUnitValuesParametersUnitValues
tc_min s 10 tc_max s 100 
Tvbp21 s [10, 100] yrv_11 % [0, 73.87] 
Tvbp11 s [10,20] yrv_12 % [0, 73.87] 
Yvbp1 % [10, 90] yrv_31 % [0, 67.36] 
ΔTp1−2 s [390, 900] yrv_32 % [0, 67.36] 
ΔTp2−3 s [390, 900] – – – 
ParametersUnitValuesParametersUnitValues
tc_min s 10 tc_max s 100 
Tvbp21 s [10, 100] yrv_11 % [0, 73.87] 
Tvbp11 s [10,20] yrv_12 % [0, 73.87] 
Yvbp1 % [10, 90] yrv_31 % [0, 67.36] 
ΔTp1−2 s [390, 900] yrv_32 % [0, 67.36] 
ΔTp2−3 s [390, 900] – – – 
View Large

The experimental setup of the algorithms is presented in Table 3. Considering the computational cost, the total number of runs (Nmsm) for mathematical simulation model were chosen to be 400, 1000, and 1800. The setup of the NSGA-II was similar to that of PNSGA-II. For the PMO-ASMO, the maximum iteration (iter) and population size (pop) of the embedded PNSGA-II were both 100. Additionally, the size of adaptive sampling for every iteration is 100 × 20% = 20, wherein the ‘20%’ represents the specified resampling percentage (pct). init is the number of initial sampling.

Table 3

Multi-objective optimization algorithms experimental setup

NmsmNSGA-IIPNSGA-IIPMO-ASMO
400 20 pop × 20 iter 20 pop × 20 iter 200 init + (100 pop × 20% pct) × 10 iter 
1000 25 pop × 40 iter 25 pop × 40 iter 400 init + (100 pop × 20% pct) × 30 iter 
1800 30 pop × 60 iter 30 pop × 60 iter 800 init + (100 pop × 20% pct) × 50 iter 
NmsmNSGA-IIPNSGA-IIPMO-ASMO
400 20 pop × 20 iter 20 pop × 20 iter 200 init + (100 pop × 20% pct) × 10 iter 
1000 25 pop × 40 iter 25 pop × 40 iter 400 init + (100 pop × 20% pct) × 30 iter 
1800 30 pop × 60 iter 30 pop × 60 iter 800 init + (100 pop × 20% pct) × 50 iter 
View Large

The effectiveness of an algorithm could be evaluated based on the convergence and diversity of the Pareto optimal points relative to the theoretical Pareto frontier (Gong et al. 2016a). The number of evaluations for the function required for obtaining smooth Pareto optimal solution sets is regarded as the efficiency of an algorithm (Zhang et al. 2017). However, the theoretical Pareto frontier is unknown for the combined optimal operation of multiple hydraulic facilities in a true complex pressurized WDP. And performing numerous runs for complex pressurized WDPs simulations is not feasible owing to high computational costs. Therefore, in this study, the effectiveness of the PMO-ASMO was verified by comparing the performance of the Pareto solutions and the diversity of the Pareto solution set of the PMO-ASMO with that of the PNSGA-II and NSGA-II. The time consumed using PMO-ASMO, PNSGA-II, and NSGA-II in the combined optimal operation model of multiple hydraulic facilities was considered as the evaluation indicator of the efficiency.

Effectiveness

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The purpose of the PNSGA-II algorithm is to effectively reduce the computational time by implementing multicore parallel computation while maintaining the same optimization effectiveness as the NSGA-II algorithm. Hence, with 400, 1000, and 1800 model runs, the Pareto optimal points obtained by PMO-ASMO was only compared to that of PNSGA-II.

The optimization results of the combined operation of multiple hydraulic facilities provided by PNSGA-II and PMO-ASMO with 400, 1000, and 1800 model runs are shown in Figure 7. The Pareto solutions and feasible solutions discovered in the optimization process are represented by the red and green points, respectively. In order to compare the results of the PMO-ASMO and PNSGA-II more intuitively under different number of the model runs, the compromise solution of the two objective functions in the feasible solution set with 400 model runs is used as the reference point (i.e., the black the cross lines) to divide the coordinate area into four regions. The feasible solutions of PMO-ASMO with 400, 1000, and 1800 simulations (Figure 7(b), 7(d), and 7(f)) were widely distributed, whereas some clusters existed in those of PNSGA-II (Figure 7(a), 7(c), and 7(e)). This can be attributed to the inadequate simulations performed using PNSGA-II and the inclusion of the training process of the surrogate model in PMO-ASMO, which was conducted after the initial sampling in the mathematical model simulation of Nmsm. As shown in Figure 7(a)–7(e), the number of Pareto solutions and feasible solutions of PMO-ASMO are less than those of PNSGA-II in the same number of simulation model runs. This is owing to the inevitable loss of some details during the search process, although the GPR surrogate model mimicked the effects of the simulation model. However, after 400 and 1000 simulations, the Pareto solutions of PMO-ASMO dominated those of PNSGA-II. While the Pareto solutions of the former partially dominated those of the latter in 1800 simulations, as observed during comparison. The reason for these results is that with the growth of the mathematical simulation model runs, the number of Pareto solutions also increases, and the distribution of the Pareto frontier solutions becomes more extensive, thereby encompassing more optimized Pareto frontier characteristics. Hence, PMO-ASMO, with limited Pareto solutions, cannot completely hold sway over the solutions of PNSGA-II when the number of simulation model runs increases. However, according to the box plot of the optimal objective functions from the 1800 model runs (Figure 8), the PMO-ASMO had the lowest median for the risk index objective function. Even though it had the highest median for regulation time using PMO-ASMO, the third quartile was the smallest, and the interquartile range the lowest. Although the Pareto solutions of PMO-ASMO may not entirely supersede those of PNSGA-II, they exhibit a better safety performance and are more appropriate for the actual requirements of engineering operations.
Figure 7
Optimal results based on 400, 1000, and 1800 mathematical simulation model runs.
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Optimal results based on 400, 1000, and 1800 mathematical simulation model runs.

Figure 7
Optimal results based on 400, 1000, and 1800 mathematical simulation model runs.
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Optimal results based on 400, 1000, and 1800 mathematical simulation model runs.

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Figure 8
Box diagram of the optimal objective functions for different algorithms based on 1800 mathematical simulations.
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Box diagram of the optimal objective functions for different algorithms based on 1800 mathematical simulations.

Figure 8
Box diagram of the optimal objective functions for different algorithms based on 1800 mathematical simulations.
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Box diagram of the optimal objective functions for different algorithms based on 1800 mathematical simulations.

Close modal
Figure 9 shows a comparison of the Pareto frontiers of PNSGA-II and PMO-ASMO to the 400, 1000, and 1800 mathematical simulation model runs. The cross lines in Figure 9 represent the reference point, which is the same as depicted in Figure 7. Clearly, the Pareto solutions obtained from the 400 model runs of PMO-ASMO were not dominated by those of PNSGA-II, although the latter required 1000 and 1800 model runs. Contrarily, the former (with fewer runs) can partially determine the latter (with more runs). Similarly, the Pareto solutions of PMO-ASMO in 1000 model runs partially dominated those acquired using PMO-ASMO in 1800 simulations. This proves that PMO-ASMO can achieve better results with fewer mathematical simulations.
Figure 9
Pareto frontier results for PNSGA-II and PMO-ASMO with different mathematical simulation runs.
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Pareto frontier results for PNSGA-II and PMO-ASMO with different mathematical simulation runs.

Figure 9
Pareto frontier results for PNSGA-II and PMO-ASMO with different mathematical simulation runs.
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Pareto frontier results for PNSGA-II and PMO-ASMO with different mathematical simulation runs.

Close modal

Furthermore, because the GPR surrogate model exhibits the inevitable loss of some details during the search process, the proportions of feasible and Pareto solutions of PNSGA-II and PMO-ASMO to Nmsm were studied to effectively determine Nmsm, and the results are presented in Table 4. The numbers of feasible solutions of PNSGA-II with 400, 1000, and 1800 model runs were 149, 313, and 567, respectively, thereby accounting for 37.25, 31.30, and 31.50%, respectively, with an average of 33.35%. The number of Pareto solutions were 19, 21, and 43, thereby accounting for 4.75, 2.10, and 2.39%, respectively, with an average of 3.08%. For the feasible solutions with 400, 1000, and 1800 simulations in PMO-ASMO, the results were 19, 43, and 86, respectively, thereby representing 4.75, 4.30, and 4.77% of the total solutions, respectively, with an average of 4.61%. Further, the number of Pareto solutions were two, two, and seven, thereby accounting for 0.5, 0.2, and 0.39%, respectively, with an average of 0.36%. Hence, to ensure that feasible solutions of the PMO-ASMO contain a certain number, Nmsm is recommended to be not less than 220 in the combined optimal operation of multiple hydraulic facilities in complex pressurized WPDs.

Table 4

Multi-objective optimization results: numbers of feasible solutions and Pareto frontier solutions

AlgorithmsNmsmNumbers of feasible solutionsProportion of feasible solutions to Nmsm (%)Average proportion of feasible solutions to Nmsm (%)Numbers of Pareto frontier solutionsProportion of Pareto solutions to Nmsm (%)Average proportion of Pareto solutions to Nmsm (%)
PNSGA-II 400 149 37.25 33.35 19 4.75 3.08 
1000 313 31.30 21 2.10 
1800 567 31.50 43 2.39 
PMO-ASMO 400 19 4.75 4.61 2 0.5 0.36 
1000 43 4.30 2 0.2 
1800 86 4.77 7 0.39 
AlgorithmsNmsmNumbers of feasible solutionsProportion of feasible solutions to Nmsm (%)Average proportion of feasible solutions to Nmsm (%)Numbers of Pareto frontier solutionsProportion of Pareto solutions to Nmsm (%)Average proportion of Pareto solutions to Nmsm (%)
PNSGA-II 400 149 37.25 33.35 19 4.75 3.08 
1000 313 31.30 21 2.10 
1800 567 31.50 43 2.39 
PMO-ASMO 400 19 4.75 4.61 2 0.5 0.36 
1000 43 4.30 2 0.2 
1800 86 4.77 7 0.39 
View Large

Efficiency

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NSGA-II being time consuming in the combined optimal operation model of multiple hydraulic facilities, the efficiency of PMO-ASMO was compared to that of NSGA-II and PNSGA-II only for the scenario with 400 model runs. This comparison was made solely using PNSGA-II in 1000 and 1800 simulations.

In this study, it took approximately 980 s to execute one run of the mathematical simulation model for the case on a Core i9 (10 cores) 3.70 GHz processor (Intel, China). The time consumed using PNSGA-II and PMO-ASMO in the combined optimal operation model of multiple hydraulic facilities with 400, 1000, and 1800 model runs and the results after utilizing NSGA-II in the combined optimal operation model of multiple hydraulic facilities with 400 model runs are presented in Table 5. In the 400 model runs, PMO-ASMO exhibited a shorter execution time than the other two algorithms, whereas NSGA-II required the longest time. The time consumption of PNSGA-II with 10 cores in parallel was only 10.288% that of NSGA-II, thereby saving 103.625 h. This confirms the high efficiency of the PNSGA-II algorithm based on multicore parallelism. Compared to PNSGA-II, PMO-ASMO saved approximately 0.368 and 3.059 h in 400 and 1000 simulations, respectively. Nevertheless, PMO-ASMO did not demonstrate an advantage in terms of efficiency in the 1800 simulations as the computational time for PMO-ASMO was 62.224 h. However, it was only 54.577 h for PNSGA-II. Even though, PMO-ASMO can accelerate the searching process, which consistently acquires the best and stable Pareto frontier with fewer iterations in the combined operation model of multiple hydraulic facilities. Additionally, the computational time required by PMO-ASMO is several hours instead of days or even weeks as required by NSGA-II, thereby saving computing resources. Hence, PMO-ASMO can strengthen the applicability of the optimal operation of multiple hydraulic facilities in complex pressurized WDPs, and with continuous improvements in computing performances; it is also expected to be applied to the real-time optimal operation of multiple hydraulic facilities in complex pressurized WDPs.

Table 5

Time required for computation based on different multi-objective optimization algorithms

NmsmNSGA-IIPNSGA-IIPMO-ASMO
400 415,832 (≈115.509 h) 4,2783 (≈11.884 h) 4,1456 (≈11.516 h) 
1000 – 121,913 (≈33.865 h) 11,0902 (≈30.806 h) 
1800 – 196,476 (≈54.577 h) 224,006 (≈62.224 h) 
NmsmNSGA-IIPNSGA-IIPMO-ASMO
400 415,832 (≈115.509 h) 4,2783 (≈11.884 h) 4,1456 (≈11.516 h) 
1000 – 121,913 (≈33.865 h) 11,0902 (≈30.806 h) 
1800 – 196,476 (≈54.577 h) 224,006 (≈62.224 h) 
View Large

Analysis of solutions

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Considering the efficiency requirements in an actual operation, the solutions corresponding to the optimal value of each objective function in the Pareto fronts obtained PMO-ASMO and PNSGA-II with 400 simulations (Figure 9) were selected as the optimal operation schemes. The schemes corresponding to the minimum values of the response and objective functions are presented in Table 6.

Table 6

The schemes corresponding to the minimum value of each objective function in the Pareto fronts obtained by PMO-ASMO and PNSGA-II algorithms with 400 simulations

PNSGA-II (Nmsm = 400)
PMO-ASMO (Nmsm = 400)
Rtotal: min and Ttotal: maxRtotal: max and Ttotal: minRtotal: min and Ttotal: maxRtotal: max and Ttotal: min
Decision variables Tvbp11 (s) 11 11 16 18 
yvbp1 18.77% 18.77% 19.87% 19.63% 
Tvbp21 (s) 79 79 74 73 
ΔTp1−2 (s) 730 680 570 510 
ΔTp2−3 (s) 640 620 410 600 
yrv_11 58.32% 54.52% 58.24% 57.08% 
yrv_12 18.81% 16.24% 33.48% 20.04% 
 yrv_31 29.77% 31.94% 38.14% 40.80% 
yrv_32 12.57% 10.89% 23.61% 9.58% 
Response functions HPmax (m) 86.98 88.24 86.44 87.07 
HPmin (m) 1.20 1.20 1.27 1.29 
Z1max (m) 90.06 91.24 88.08 90.11 
Z1min (m) 86.81 86.80 86.87 86.89 
Z2max (m) 60.99 60.91 60.91 60.91 
Z2min (m) 59.67 59.27 59.46 59.24 
Ttotal (s) 1,675 1,563 1,522 1,435 
Objective functions Rtotal 1.37 1.64 1.11 1.45 
Ttotal (s) 1,675 1,563 1,522 1,435 
PNSGA-II (Nmsm = 400)
PMO-ASMO (Nmsm = 400)
Rtotal: min and Ttotal: maxRtotal: max and Ttotal: minRtotal: min and Ttotal: maxRtotal: max and Ttotal: min
Decision variables Tvbp11 (s) 11 11 16 18 
yvbp1 18.77% 18.77% 19.87% 19.63% 
Tvbp21 (s) 79 79 74 73 
ΔTp1−2 (s) 730 680 570 510 
ΔTp2−3 (s) 640 620 410 600 
yrv_11 58.32% 54.52% 58.24% 57.08% 
yrv_12 18.81% 16.24% 33.48% 20.04% 
 yrv_31 29.77% 31.94% 38.14% 40.80% 
yrv_32 12.57% 10.89% 23.61% 9.58% 
Response functions HPmax (m) 86.98 88.24 86.44 87.07 
HPmin (m) 1.20 1.20 1.27 1.29 
Z1max (m) 90.06 91.24 88.08 90.11 
Z1min (m) 86.81 86.80 86.87 86.89 
Z2max (m) 60.99 60.91 60.91 60.91 
Z2min (m) 59.67 59.27 59.46 59.24 
Ttotal (s) 1,675 1,563 1,522 1,435 
Objective functions Rtotal 1.37 1.64 1.11 1.45 
Ttotal (s) 1,675 1,563 1,522 1,435 
View Large

As shown in Table 6, the lowest risk index scheme computed by PMO-ASMO had a risk index of 1.11 and a duration of the regulation of 1,522 s, which were reduced by 18.98 and 9.13%, respectively, compared to those obtained by PNSGA-II. For the shortest regulation time scheme calculated using PMO-ASMO, the risk index was 1.45 and duration of regulation 1,435 s, which were reduced by 11.59 and 8.19%, respectively, compared to the scheme calculated using PNSGA-II. The reason is that the response function values in the two schemes calculated using PMO-ASMO are improved except for only a slight decrease in the lowest water level of the regulating pool. In particular, the water level fluctuations relative to the designed water level decreased by 60.92 and 27.48% in the high-level water pool for the lowest risk index scheme and the shortest regulation time scheme, respectively.

The total shut time of the valve closing law in the pump-valve unit is shorter relative to the total regulation time, as well as the fact that the corresponding openings of RV3 when each pump starts closing has less impact on the high-level water pool due to the presence of regulating pool. The main factors affecting the change of water level in the high-level water pool are the time intervals between pumps shutdown, and the corresponding openings of RV1 (or RV2). Therefore, the fundamental reason why the schemes calculated by PMO-ASMO are all better than obtained by PNSGA-II, is that the difference in water discharge into and out of the high-level water pool is minimized by reasonably adjusting the corresponding openings of RV1 (or RV2) under shorter pumps shutdown intervals, which in turn reduces the water level fluctuations. Moreover, the relationship among the closing law of the valve in the pump-valve unit, the corresponding openings of RV1 (or RV2) and RV3, and the time intervals between pumps shutdown is also coordinated at the same time, so that the system pressure extremes and the water level fluctuations of regulating pool do not get worse, or even improve.

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This study focused on achieving a combined optimal operation of multiple hydraulic facilities for complex pressurized WDPs. The combined optimal operation model of the multiple hydraulic facilities coupled with mathematical simulation models of complex pressurized WDPs is established based on the successive shutdown process of pumps, and the PMO-ASMO algorithm that combines adaptive surrogate models and parallel computing is provided. Using the complex pressurized diversion section of the Jiaodong Water Diversion Project as an example, the results of the PMO-ASMO, NSGA-II, and PNSGA-II algorithms were compared. The findings are as follows:

  • (1) The optimization effectiveness of PMO-ASMO outperformed PNSGA-II not only by using similar number of mathematical simulations, but also by using fewer model runs.

  • (2) The time consumption of PMO-ASMO was only 9.97% of that acquired by NSGA-II, which was comparable to that of PNSGA-II, in the case of 10-core parallelism.

  • (3) The PMO-ASMO outperforms NSGA-II and PNSGA-II with the same number of mathematical simulations owing to the following reasons: (a) the adaptive surrogate model is employed to simulate the complex pressurized WDPs to alleviate the computational burden, while (b) the PNSGA-II algorithm can realize multicore parallel computation.

Therefore, a high-efficient and economical framework is provided for obtaining combined optimal operation schemes of multiple hydraulic facilities with the different operational conditions in WDPs. However, there exist some limitations caused by insufficient diversity in the algorithm settings. Further research is required to explore the combined operation of multiple hydraulic facilities under other operating conditions.

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This work was partly supported by the Basic Research Programs of Shanxi Province (grant nos. 20210302124645, 202203021222112), the National Key R&D Program of China (grant no. 2021YFC3001000), and the Open Research Fund of Henan Key Laboratory of Water Resources Conservation and Intensive Utilization in the Yellow River Basin (grant no. HAKF202104).

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

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The authors declare there is no conflict.

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