A hydraulic simulation-optimization model of the joint operation of multiple devices in long-distance water diversion systems under the pumps shutdown process using a parallel NSGA-II approach Open Access

The water diversion project is intended to divert water from water-rich areas to where it is needed, which contributes significantly to optimizing the spatial distribution of water resources, alleviating water shortage in water-scarce areas, and eventually promoting the coordinated development of society, economy and ecology. Such a project can be regarded as a nonlinear system involving many hydraulic devices and structures. Among them, active hydraulic devices and structures such as pumps and valves are adjustable, while passive ones such as pipes and tanks should be kept to a desired level (Skulovich et al. 2016). In the operation of a water diversion project, the transient processes induced by abrupt velocity change of fluid flow is inevitable because of the sudden opening and closing of valves, the pump startup, the pump shutdown, etc. However, the unreasonable design and improper operation may cause harmful transient processes, which may lead to severe vibration, pipe rupture, pipe collapse, tank overtopping, etc (Schmitt et al. 2006; Afshar et al. 2010; Chen et al. 2015; Ye et al. 2020). Therefore, the transient processes should be taken into account in the optimal design and operation of any water diversion project.

First, the research on the optimal design considering transient processes has been relatively comprehensive. Jung & Karney (2004) investigated the optimal selection of pipe diameters considering transient analysis by genetic algorithms (GA) and particle swarm optimization (PSO). Liu et al. (2011) used the method of characteristics (MOC) to determine the appropriate orifice diameter of air-inlet valves in the pumps shutdown process in a long pipeline pumping system. Lingireddy et al. (2000) proposed a surge tank design model that gave an optimal set of surge tank sizes while satisfying a specified set of pressure head constraints. Wang et al. (2013) used MOC to investigate the effects of installation mode and position of air vessel on water hammer processes in pumping systems with high working pressure head. Second, in the operation of the water diversion project, the research focuses on effectively controlling the transient processes by optimizing the operation of active hydraulic devices (Zhang et al. 2008; Guo et al. 2017). Skulovich et al. (2016) adopted GA and quasi-Newton to investigate the optimal closure law of the downstream valve by taking the minimization of the maximum pressure head as the objective. Vakil & Firoozabadi (2009) investigated the effects of different valve closure laws on the maximum pressure head rise at the end of the pressure shaft and other components. Bazargan-Lari et al. (2013) used a multi-objective optimization model and Bayesian networks to develop an optimal valve closure curve for controlling the water pressure head. However, previous studies of the optimal operation have focused mainly on a single hydraulic device. Due to the limited control range of a single hydraulic device, it is difficult to take into account overall safe and efficient operation requirements of water diversion projects at the same time, especially long-distance water diversion projects. Therefore, it is necessary to study the joint operation of multiple devices in long-distance water diversion projects (Feng & Zheng 2003).

However, the joint operation of multiple hydraulic devices currently relies heavily on personal experience and requires repeated adjustment, resulting in low decision-making efficiency and high manpower cost. Moreover, it is difficult for decision-makers to reduce the complex interrelationships between various operational risks and efficiency into a single-objective problem (Jung et al. 2011). Therefore, in this study, the joint operation of multiple hydraulic devices in long-distance water diversion systems is expressed as a multi-objective optimization problem, and a hydraulic simulation-optimization model for the joint operation of multiple devices is proposed. In the hydraulic simulation-optimization model, the optimization model is coupled with the hydraulic transient simulation model. The hydraulic transient simulation model is developed using MOC (Izquierdo & Iglesias 2002; Wang & Yang 2015; Wan & Zhang 2018), which is the most commonly used method (Tian et al. 2008; Chalghoum et al. 2016). The reason is that the MOC has evident advantages in terms of feasibility, simplicity and efficiency in comparison with other methods (Afshar et al. 2010), such as wave characteristics method (WCM) (Wood 2005), finite volume method (FVM) (Zhao & Ghidaoui 2004), finite element method (FEM) (Kochupillai et al. 2005) and finite difference method (FDM) (Chaudhry & Hussaini 1985). The pumps shutdown process is selected as the operation process for the study because it is inevitably one of the most common operation conditions. Considering the complex interrelationships of multiple hydraulic devices, and taking various safety risks and efficiency as the objective functions, the optimization model is established. The widely used non-dominated sorting genetic algorithm II (NSGA-II) is adopted to estimate non-dominated solutions.

However, the computational expense of NSGA-II would increase exponentially with increasing number of objectives (Li & Mallick 2015). Thus, the computation is cumbersome for the hydraulic simulation-optimization model that involves a large number of decision variables and objective functions. This problem can be solved by parallel computation (Feng et al. 2018). Some simple but powerful parallel frameworks, such as Fork/Join and Message Passing Interface, have also been successfully developed (Dias et al. 2013; Pinto et al. 2013). In this study, based on the parallel programming OpenMP, the parallel NSGA-II approach is proposed to solve the proposed hydraulic simulation-optimization model. OpenMP is used due to the following advantages: (1) OpenMP provides good single-source portability for shared-memory parallelism, which can consolidate different models into a single syntax and semantics; (2) OpenMP specifically addresses the needs of scientific programming, such as support for Fortran, C/C++ and data parallelism, and our previous NSGA-II algorithms are encoded in C++; (3) OpenMP facilitates an incremental approach to the parallelization of sequential programs, allowing programmers to add a parallelization directive to one loop or subroutine of the program at a time (Hu et al. 2000; Rabenseifner et al. 2009).

The rest of this paper is organized as follows. Section 2 describes the proposed hydraulic simulation-optimization model; Section 3 describes the parallel NSGA-II approach for solving the model; in Section 4, the effectiveness and efficiency of the proposed hydraulic simulation-optimization model based on a parallel NSGA-II approach is demonstrated by a case study; and conclusions are drawn in Section 5.

In practical engineering, limited by economic and geographical conditions, the layout of a long-distance water diversion system is usually one long main pipe with multiple hydraulic devices and structures arranged in series. Its hydraulic devices and structures mainly include pipes, pumping stations, high-level water tanks, regulating tanks, surge tanks, valves, air valves, branch pipes, etc. The pumping station usually contains multiple pump-valve units, which are connected with the main pipe through several branch pipes. Each unit consists of a pump and a valve in series. Besides, the modes of water delivery always involves pressurized water delivery, gravity water delivery and so on. Thus, due to the limited control range of a single hydraulic device, it is difficult to take into account overall safe and efficient operation requirements of long-distance water diversion systems at the same time. The joint operation of multiple hydraulic devices is very necessary but it currently relies heavily on personal experience, and repeated adjustments are often required because of the complex hydraulic coupling relationship of multiple hydraulic devices, resulting in low decision-making efficiency and high manpower cost. Therefore, taking the pumps shutdown process as the research operation condition, a hydraulic simulation-optimization model of the joint operation of multiple devices of long-distance water diversion systems is proposed. The hydraulic simulation-optimization model is composed of the optimization model and the hydraulic transient simulation model, which is a bi-level framework. The coupling idea of the two models is to use an optimization algorithm to generate decision variables in the optimization model that are then passed to a lower level hydraulic transient simulation model to obtain numerical values for use in evaluating all implicit constraints and calculating the objective functions.

The purpose of the optimization model is to optimize the joint operation of multiple hydraulic devices in order to control the transient responses of the pumps shutdown process, so as to realize the overall safe and efficient operation.

For the pumps shutdown process, the hydraulic devices that can be actively regulated mainly include multiple pump-valve units in the pumping station and multiple series valves along the pipes. For multiple pump-valve units, a suitable operational time interval between pumps can prevent backflow and overpressure. The one-phase valve closure law and two-phase valve closure law are the main operational rules in the engineering. The valve in the pump-valve unit adopts two-phase closure law to effectively limit the reverse flow rate and the reverse speed of the pump, reducing the pressure pulsation amplitude. The valves along the pipes adopt one-phase closure law, which also can effectively prevent overpressure and water level overlimit by adjusting appropriate corresponding openings when pumps are started to close. Therefore, select the shutdown time intervals between pumps, the total closure time and the inflection openings of the valves in pump-valve units, the time required to reach the inflection openings, and the corresponding opening of each valve along the pipeline when each pump is closed as decision variables.

• 1.
  The constraints of the closure law of the valves in pump-valve units:

  

  (4)

  

  (5)

  

  (6)

A one-dimensional hydraulic transient simulation model is constructed by the MOC method. The MOC method transforms the governing equations (including the momentum and continuity equations) with partial differential form into that with ordinary differential form. The hydraulic devices and structures, such as valves, air valves and pumps, are often treated as boundary conditions, which provide the corresponding supplementary equations (Wylie & Streeter 1978; Tian et al. 2008; Chaudhry 2014).

Given the computational complexity inherent in solving the hydraulic simulation-optimization model, an parallel NSGA-II approach is proposed in this study to improve the computational efficiency.

First, a random parent population of size N is generated within the range of decision variables. This population is then distributed to p threads in the first parallel domain, and the objective functions are calculated on each thread. After all threads in the first parallel domain finish executing, all objective functions of the parent population are sent back to the master thread to sort into different non-dominated ranks. After ranking, the crowding distances for the members belonging to each rank are then calculated. The parent population then undergoes the genetic algorithm (GA) processes of tournament selection, crossover and mutation to generate the child population of size N and the objective functions of the child population are calculated. Among them, the second parallel domain includes the processes of mutation to generate the child population, which is executed by p threads. After all threads in the second parallel domain finish executing, all objective functions of the child population are calculated by p threads in the first parallel domain. Then, all objective functions of the child population are sent back to the master thread to merge the parent population with the child population and sort the 2N solutions into different non-dominated ranks. After ranking, the crowding distances for the members belonging to each rank are calculated and the N new members are selected from the combined population as the new parent population to proceed until the stopping criterion (the maximum number of iterations).

In this section, the procedure of the parallel NSGA-II for solving the hydraulic simulation-optimization model of the joint operation of multiple devices is described in detail. The main procedure includes initialization, constraint handling and the coupling of the optimization model and the hydraulic transient simulation model.

Individuals are initialed randomly while satisfying various constraints, which are randomly generated between the minimum and maximum values. For instance, t_2p is randomly generated between t_{2p_min} and t_{pc_max}. Generally, these newly generated individuals do not satisfy all the constraints and thus need to be modified by the constraint handling method, which will be described in the next section.

The calling executive file in the programming language C ++ is used to couple the optimization model and the hydraulic transient simulation model. Specifically, the hydraulic transient simulation model is compiled as an external program into the executive file of the optimization model written in C++. It is also necessary to generate the codes for data transition between the two models. The codes mainly include: (1) writing the decision variables generated by the initialization or iteration of the optimization model into the corresponding position of the input files for the hydraulic transient simulation model; (2) reading and feeding back numerical values used for evaluating all implicit constraints and calculating objective functions in the optimization model from the output files of the hydraulic transient simulation model.

The proposed hydraulic simulation-optimization model is applied to a real long-distance water diversion project. In this case study, the joint operation schemes of multiple hydraulic devices during the pumps shutdown process is obtained, the benefits and implications of the Pareto front solutions are discussed, and the efficiency of parallel computing is evaluated.

Since the numerical values fed back from the hydraulic transient simulation model corresponding to the decision variables that satisfy the explicit constraints may not satisfy the implicit constraints, many schemes are not feasible. Figure 9 shows that there are 44 feasible solutions and 11 multi-objective Pareto front solutions for the pumps shutdown process of the project. The number of feasible solutions accounts for only 1.96% of the total simulated solutions and the Pareto front solutions are dispersedly distributed. The main reason is that the adjustment of multiple hydraulic devices can affect and restrict each other and their joint regulation is very sensitive. Thus, it is difficult to find a feasible solution, especially relying on the experience of decision-makers. However, the proposed hydraulic simulation-optimization model can find a set of feasible Pareto front solutions, which makes it possible to avoid the problems of the current joint operation scheme of multiple hydraulic devices that relies heavily on the experience of decision-makers. Compared with the current scheme, there is no need to adjust 1-1CV, 1-2CV and 2CV piston-type control valves before closing pumps 1 and 2 for the feasible schemes, needing less adjustment. The total regulation time required by the Pareto front schemes is only 1/8.92–1/11.49 of the current scheme. Besides, after the pumps shutdown, the steady-state water level of the control structure in the feasible schemes is between the highest and the lowest allowable water level, while that of the high-level water tank in the current scheme is slightly lower than its lowest allowable water level. This indicates that the feasible schemes calculated by the hydraulic simulation-optimization model is safer and more efficient, and can avoid repeated regulation, improve decision-making efficiency and reduce manpower cost and mechanical loss.

It can be known from Figure 9(b)–9(d) and 9(g) that the difference between the maximum and minimum pressure head in the feasible solutions is less than 89 m except for one feasible solution (111.58 m). The reason is that the opening degree at the end of fast closure of the valve in pump-valve unit of this feasible solution is larger, indicating that the fast closure is slower and the slow closure is faster. For this feasible solution, when other decision variables are kept unchanged and the opening degree at the end of fast closure of the valve in the pump-valve unit is adjusted from 40 to 22.22%, the pressure head fluctuation of the system can be reduced to 87.36 m. However, there are no significant changes in the water levels of the high-level water tank and the non-pressure regulating tank, except a slight decrease in the highest and lowest water levels. The highest water level of the high-level water tank has the greatest impact, but it is decreased by only 0.08 m. These results indicate that the closure law of the valve in the pump-valve unit has a greater impact on the water hammer pressure head of the pipes than on the water levels of the high-level water tank and the non-pressure regulating tank. The reason is that the water hammer process is relatively short relative to the change of the water levels of the high-level water tank and the non-pressure regulating tank. In addition, this solution will be dominated by the multi-objective Pareto front solutions in the optimization process. Comparison of Figure 9(a), 9(e) and 9(i) suggests that the distribution of feasible solutions of objective function 1 are nearer to that of the multi-objective Pareto front solutions compared to those of objective functions 2 and 3. The reason is that although the closure law of the valve in the pump-valve unit has a great impact on the water hammer pressure head of the pipes, it is easy to make the maximum and minimum pressure heads of the system within the allowable range because the pump is closed in sequence during the pumps shutdown process and the closing speed of the piston-type control valve on the main pipe is relatively slow. Given the restriction between the closure law of piston-type control valves and the shutdown intervals between pumps, objective functions 2 and 3 converge slowly near the multi-objective Pareto front solutions.

Table 4 shows that as the number of parallel cores increases, the execution time is significantly reduced and the acceleration ratio increases, which demonstrates the high computational efficiency of the parallel approach. However, the parallel efficiency decreases with the the number of parallel cores increasing, as shown in Table 4 and Figure 10. This is because the process of using the proposed parallel NSGA-II to solve the hydraulic simulation-optimization model includes not only parallel computing but also serial computing. In addition, the parallel efficiency is also related to the population size. If the ratio of the population size to the number of parallel cores is non-integer, the parallel efficiency decreases more obviously, such as the six and ten-core parallel efficiency shown in Figure 10(b). Therefore, it is expected that with the advance of computer equipment and the reasonable selection of the number of parallel cores and the population size, the computational efficiency of the parallel approach will become more prominent.

In this study, a hydraulic simulation-optimization model of the joint operation of multiple devices for pumps shutdown process of long-distance water diversion systems is proposed. The hydraulic simulation-optimization model is composed of the optimization model and the hydraulic transient simulation model, which is a bi-level framework. In order to ensure the overall safety and efficiency of the pumps shutdown process and to keep the water level at the design ideal value as much as possible, the proposed optimization model is a three-objective problem. The hydraulic transient simulation model is constructed by MOC. The hydraulic simulation model is compiled as an external program into the executive file of the optimization model using the calling executive file in the programming language C ++ to realize the coupling of the two models. In order to reduce the computational burden, the parallel NSGA-II approach is used to solve the proposed hydraulic simulation-optimization model. Besides, a process of effectively handling the constraints of the joint optimal operation of multiple devices is proposed. Then, the proposed hydraulic simulation-optimization model and the parallel NSGA-II approach are applied to a real long-distance water diversion project consisting of a variety of hydraulic devices and structures. The results show that the proposed model can find a set of feasible Pareto front solutions, which improves the decision-making efficiency and avoids multiple trial calculations and repeated adjustment of hydraulic facilities in the actual operation. Importantly, mechanical loss and manpower cost are also reduced. In addition, the parallel approach greatly improves the computational efficiency in solving the hydraulic simulation-optimization model. Therefore, this study provides an effective approach to formulate the joint operation scheme of multiple devices of the long-distance water diversion system. However, it is necessary to further study higher-performance parallel computing for the hydraulic optimization model and the hydraulic transient simulation model to speed up the calculation, and further study the joint operation of multiple devices for other transient processes of long-distance water diversion systems.

This work was supported by the national key R&D program of China under Grant number 2021YFC3001000, the basic research program of Shanxi province under Grant number 20210302124645, the Colleges and Universities Science and Technology Innovation Project of Shanxi province under Grant number 2021L019, and the National Science Foundation of China under Grant number 51879273.

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

The authors declare there is no conflict.