Plant-network-river integrated dispatching simulation calculation method and system based on parallel calculation
By using the combination of parallel computing and particle swarm algorithm in integrated urban factory network river scheduling, the problem of excessive calculation time in the existing technology is solved, and a more efficient scheduling decision-making process is achieved.
Patent Information
- Application Number
- CN202510622163.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-05-15
- Publication Date
- 2025-06-13
- Estimated Expiration
- 2045-05-15
AI Technical Summary
The existing integrated urban factory network and river scheduling decision-making method faces calculation tasks with large regional scope, long simulation periods and many iterations, resulting in low decision efficiency.
The integrated scheduling simulation calculation method based on parallel computing is adopted. By establishing a SWMM model and particle swarm algorithm model, combining CPU multi-process parallel computing, the parallelization of particle optimization process is realized and computing resources are fully utilized.
It effectively shortens the calculation time, improves the efficiency of integrated operation and scheduling decisions in factory network and rivers, and significantly improves the multi-objective optimization and convergence speed.
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Figure CN120146324A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of operation scheduling of drainage systems, and particularly relates to a method and system for integrated scheduling simulation calculation of plant-network-river based on parallel computing. Background Art
[0002] The integrated scheduling of urban plant-network-river is a new scheduling method aiming at urban water safety and basin water quality compliance, aiming to uniformly schedule drainage facilities such as rivers (lakes), drainage pipe networks, sewage treatment plants, sluices, and pumping stations. The key to realizing the integrated scheduling of plant-network-river lies in the decision support model, which includes two parts: a drainage system model and an optimal scheduling algorithm. The drainage system model can simulate the entire transmission and change process of urban water and pollutants from the source to the receiving water body at the end, as well as the regulation effects of each drainage facility; the optimal scheduling algorithm is to construct an optimization model based on a heuristic algorithm according to the scheduling objectives and the joint scheduling strategy of drainage facilities, and obtain the optimal scheduling plan through iterative calculation by repeatedly calling the drainage system model.
[0003] However, in the face of computational tasks with a large regional scope, a long simulation period, and a large number of iterations, the computational duration of the decision support model will increase exponentially, which greatly limits the efficiency of urban plant-network-river integrated scheduling decision-making. Summary of the Invention
[0004] In order to address the problems of long time consumption and low scheduling decision-making efficiency existing in the existing urban plant-network-river integrated scheduling decision-making method for computational tasks with a large regional scope, a long simulation period, and a large number of iterations, the present invention provides a method and system for integrated scheduling simulation calculation of plant-network-river based on parallel computing.
[0005] Based on the above objectives, the technical solution adopted by the present invention is as follows:
[0006] The method for integrated scheduling simulation calculation of plant-network-river based on parallel computing includes the following steps:
[0007] S1. Obtain the attributes of drainage facilities, the planar distribution and cross-section topography of river channels, the regional digital elevation, and the land use distribution in the scheduling area, and establish a SWMM model;
[0008] S2. Collect the monitoring data of water level, flow rate, and water quality during different rainfall events, surface runoff, pipe network confluence, and transmission processes, and calibrate the parameters of the SWMM model to obtain a calibrated SWMM model;
[0009] S3. Use the particle swarm algorithm to construct a joint optimization scheduling algorithm model, the output of which is the joint regulation rules of drainage facilities. Input the joint regulation rules of drainage facilities into the calibrated SWMM model, and obtain the simulation calculation results by using the parallel computing method;
[0010] S301. Obtain the boundary conditions of the drainage system in the scheduling area, where the boundary conditions of the drainage system include operation scheduling objectives, constraint conditions, and control variables;
[0011] S302. According to the boundary conditions of the drainage system, initialize the number of particles of the particle swarm algorithm to M and the maximum number of iterations to T, obtain the initialized particle swarm algorithm, use the initialized particle swarm algorithm to construct the joint optimization scheduling algorithm model, output M sets of joint regulation rules for drainage facilities, input them into the calibrated SWMM model, and generate corresponding M sets of SWMM model files;
[0012] S303. Obtain the number of CPU cores N, create a process pool using the multiprocessing module in Python, where the process pool includes N / 2 processes, define a simulation function, load the PySWMM interface into the simulation function, and each of the N / 2 processes contains the simulation function. Use the pool.map method to sequentially allocate the M sets of SWMM model files to each process in the process pool, and perform simulation calculations on the allocated SWMM model files through the PySWMM interface of each process to form CPU multi-process parallel computing, and output the simulation calculation results;
[0013] S4. Set evaluation indicators for the scheduling rules to evaluate the simulation calculation results. If they meet the evaluation criteria, output the optimal joint regulation rules for drainage facilities corresponding to the simulation calculation results; if they do not meet the evaluation criteria, return to step S3.
[0014] Preferably, step S2 further includes: The parameter calibration includes calibrating the Manning roughness coefficient of the catchment area, the depression storage parameter, the infiltration parameter, the Manning roughness coefficient of the pipe channel, the head loss coefficient, the pollutant accumulation and scour parameter, and the pollutant decay coefficient of the SWMM model.
[0015] Preferably, step S3 further includes: The operation scheduling objectives include the minimum sewage overflow volume, the lowest water level of the maintenance and repair pipe section, and the minimum complexity of the pump station joint scheduling scheme;
[0016] The expression for the sewage overflow volume is:
[0017] In the formula, m represents the total number of pump stations, and Q i represents the forebay overflow volume of the i-th pump station;
[0018] The expression for the water level of the maintenance and repair pipeline is:
[0019] In the formula, Z iIndicates the water level at the characteristic points of the pipe segments in the operation and maintenance emergency repair area;
[0020] The expression for the complexity of the combined pumping station dispatching scheme is:
[0021] In the formula, n represents the total number of dispatching time periods, Indicates whether the operating state of the i-th pumping station needs to be adjusted at time t, where P i,t and P i,t-1 Represent the operating power of the i-th pumping station at time t and t - 1 respectively;
[0022] The constraint conditions include the pumping and drainage flow constraint and the characteristic water level constraint. The pumping and drainage flow constraint is , where Q imin and Q imax Represent the minimum and maximum values of the sewage overflow volume of the i-th pumping station respectively;
[0023] The characteristic water level constraint is used to control the overflow pollution, and its expression is:
[0024] In the formula, Z j Represents the water level of the front sump of the j-th pumping station, Z jmin and Z jmax Are the minimum and maximum warning water levels of the front sump of the j-th pumping station respectively;
[0025] The control variables include the start-stop state of the control pumping station;
[0026] The value of the number of particles M is 80, the value of the maximum number of iterations T is 3, and the simulation calculation results include the total energy consumption of the pumping station, the number of iterations, and the number of times of continuously outputting the same combined regulation rules of the drainage facilities.
[0027] Preferably, the calculation formula for the total energy consumption of the pumping station is as follows:
[0028] In the formula, W represents the total energy consumption of the pumping station, T represents the time step, S i,t Represents the operating state of the i-th pumping station at time t, P i Represents the rated power of the i-th pumping station.
[0029] Preferably, step S4 further includes: the dispatching rule evaluation index includes the lowest total energy consumption of the pumping stations, the number of iterations reaches the maximum number of iterations T, and the combined regulation rules of the drainage facilities are output the same for three consecutive iterations. If one of the dispatching rule evaluation indexes is satisfied, it is considered to meet the evaluation criteria, and the optimal combined regulation rules of the drainage facilities corresponding to the simulation calculation results are output. If the evaluation criteria are not met, return to step S3.
[0030] The present invention also provides a factory-network-river integrated dispatching simulation calculation system based on parallel computing, including the following modules:
[0031] Data acquisition module: used to acquire the attributes of drainage facilities, the planar distribution and cross-section topography of river channels, the regional digital elevation, and land use distribution in the dispatching area, and collect water level, flow rate, and water quality monitoring data during different rainfall events, surface runoff, pipe network confluence, and transmission processes;
[0032] Model construction module: used to establish a SWMM model and construct a combined optimization dispatching algorithm model, and calibrate the parameters of the SWMM model to obtain a calibrated SWMM model;
[0033] Simulation calculation module: used to output the combined regulation rules of drainage facilities, input the combined regulation rules of drainage facilities into the calibrated SWMM model, and obtain simulation calculation results by using a parallel computing method;
[0034] Dispatching evaluation module: used to evaluate the simulation calculation results.
[0035] The advantages of the present invention compared with existing methods are as follows: The present invention proposes a factory-network-river integrated dispatching simulation method based on the SWMM-particle swarm algorithm. Compared with traditional dispatching optimization models such as genetic algorithms, it has obvious advantages in multi-objective optimization and convergence speed. By using the CPU multi-process parallel computing method based on external framework to call SWMM, the parallelization of the particle optimization process is realized, the computing resources are fully utilized, the computing duration is effectively shortened, and the efficiency of the factory-network-river integrated operation dispatching decision-making is improved. Description of the Drawings
[0036] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.
[0037] Figure 1 It is a schematic diagram of the SWMM model architecture of the embodiment of the present invention;
[0038] Figure 2 is the flow chart of the integrated scheduling simulation calculation method for plants, networks, and rivers based on parallel computing according to an embodiment of the present invention;
[0039] Figure 3 is the schematic structural diagram of the integrated scheduling simulation calculation system for plants, networks, and rivers based on parallel computing according to an embodiment of the present invention. Detailed implementation manners
[0040] To make the objectives, technical solutions, and advantages of the present invention more obvious and understandable, the technical solutions in the present invention will be clearly and completely described below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.
[0041] The integrated scheduling simulation calculation method for plants, networks, and rivers based on parallel computing provided by the present invention includes the following steps:
[0042] S1. Obtain the drainage facility attributes, river channel plane distribution and cross-section topography, regional digital elevation, and land use distribution of the scheduling area, and establish a SWMM model.
[0043] The step S1 in the embodiment of the present invention specifically includes:
[0044] Please refer to Figure 1 as shown, obtain the basic drainage system data such as the drainage facility attributes, river channel plane distribution and cross-section topography, regional digital elevation, and land use distribution of the scheduling area, and establish a SWMM model for simulating the whole-process hydrological, hydrodynamic, and water quality changes of "rainfall-surface runoff-network-pumping station-sewage treatment plant-river channel" in the scheduling area.
[0045] S2. Collect the water level, flow rate, and water quality monitoring data during different rainfall events, surface runoff, pipe network confluence, and transmission processes, and calibrate the parameters of the SWMM model to obtain a calibrated SWMM model.
[0046] The step S2 in the embodiment of the present invention specifically includes:
[0047] Please refer to Figure 1 as shown, collect the historical water level, flow rate, and water quality monitoring data during different rainfall events, surface runoff, pipe network confluence, and transmission processes in the scheduling area, and calibrate the Manning roughness coefficient of the catchment area, depression storage volume parameter, infiltration parameter, pipe network Manning roughness coefficient, head loss coefficient, pollutant accumulation and wash-off parameter, and pollutant decay coefficient of the SWMM model to obtain a calibrated SWMM model.
[0048] S3. Use the particle swarm optimization algorithm to construct a joint optimization scheduling algorithm model, the output of which is the joint regulation rule of the drainage facilities. Input the joint regulation rule of the drainage facilities into the calibrated SWMM model, and use the parallel computing method to obtain the simulation calculation results.
[0049] Please refer to Figure 2 As shown, step S3 in the embodiment of the present invention includes the following steps:
[0050] S301. Obtain the boundary conditions of the drainage system in the scheduling area, where the boundary conditions of the drainage system include the operation scheduling objectives, constraint conditions, and control variables.
[0051] The operation scheduling objectives include the minimum sewage overflow volume, the lowest water level in the maintenance and repair pipe section, and the minimum complexity of the joint scheduling scheme of the pumping stations. Specifically, the expression of the sewage overflow volume is:
[0052] In the formula, m represents the total number of pumping stations, and Q i represents the overflow volume of the forebay of the i-th pumping station.
[0053] The expression of the water level in the maintenance and repair pipeline is:
[0054] In the formula, Z i represents the water level at the characteristic point of the pipe section in the maintenance and repair area.
[0055] The complexity of the joint scheduling scheme of the pumping stations is an important reference standard for reducing management and maintenance costs, and its expression is:
[0056] In the formula, n represents the total number of scheduling time periods, represents whether it is necessary to adjust the operation status of the i-th pumping station at time t. Among them, P i,t and P i,t-1 represent the operating power of the i-th pumping station at time t and t-1 respectively.
[0057] The constraint conditions include the pumping and drainage flow constraint and the characteristic water level constraint. Specifically, the pumping and drainage flow constraint is , where Q imin and Q imax represent the minimum and maximum values of the sewage overflow volume of the i-th pumping station respectively.
[0058] The characteristic water level constraint is used to control the overflow pollution, and its expression is:
[0059] In the formula, Zj represents the water level of the upstream sump of the j-th pumping station, Z jmin and Z jmax are respectively the minimum and maximum warning water levels of the upstream sump of the j-th pumping station.
[0060] The control variables include the start-stop status of the pumping stations. Since there is a periodicity in the daily sewage inflow in the drainage system of the scheduling area, to avoid frequent start-stop of the pumping stations and drastic fluctuations in the water levels of the combined sewer pipes, the minimum step size of the start-stop time of the pumping stations is set to 1 h, and the corresponding time periods are divided into 24 steps per day.
[0061] S302. Based on the boundary conditions of the drainage system, initialize the number of particles and the maximum number of iterations of the particle swarm algorithm to M and T, respectively, to obtain the initialized particle swarm algorithm. The number of particles M is determined according to the calculation duration and the number of control variables, and takes the value of 80 in this embodiment. The maximum number of iterations T is determined according to the optimization scale and the complexity of the control variables, and takes the value of 3 in this embodiment. Use the initialized particle swarm algorithm to construct a joint optimization scheduling algorithm model, output M sets of joint regulation rules for drainage facilities, which is the same as the number of particles, and input the M sets of joint regulation rules for drainage facilities into the calibrated SWMM model to generate M sets of SWMM model files.
[0062] S303. Obtain the number of CPU cores N, create a process pool using the multiprocessing module in Python. The process pool includes N / 2 processes. Define a simulation function, load the PySWMM interface into the simulation function. The N / 2 processes all contain the simulation function. Use the pool.map method to sequentially allocate the M sets of SWMM model files to each process in the process pool, and perform simulation calculations on the allocated SWMM model files through the PySWMM interface of each process to form CPU multi-process parallel computing, and output the simulation calculation results. The simulation calculation results include the total energy consumption of the pumping stations, the number of iterations, and the number of times of continuously outputting the same joint regulation rules for drainage facilities.
[0063] The calculation formula for the total energy consumption of the pumping stations is as follows:
[0064] In the formula, W represents the total energy consumption of the pumping stations, T represents the time step, S i,t represents the operating status of the i-th pumping station at time t, P i represents the rated power of the i-th pumping station.
[0065] S4. Set evaluation indicators for the scheduling rules to evaluate the simulation calculation results. If they meet the evaluation criteria, output the optimal combined regulation rules for the drainage facilities corresponding to the simulation calculation results. If they do not meet the evaluation criteria, return to step S3.
[0066] Step S4 in the embodiment of the present invention specifically includes:
[0067] Convert the M groups of simulation calculation results into the fitness values of the corresponding regulation rules for the drainage facilities. According to the operation scheduling target and the constraint conditions, set the evaluation indicators for the scheduling rules. The evaluation indicators for the scheduling rules include: whether the total energy consumption of the pump station is the lowest, whether the number of iterations reaches the maximum number of iterations T, and whether the combined optimization scheduling algorithm model outputs the same combined regulation rules for the drainage facilities three times in a row. If any one of the evaluation indicators for the scheduling rules is met, it is considered to meet the evaluation criteria. The combined regulation rules for the drainage facilities corresponding to the M groups of simulation calculation results are the optimal ones. If they do not meet the evaluation criteria, return to step S3, and use the combined optimization scheduling algorithm model to re-obtain the combined regulation rules for the drainage facilities and their corresponding simulation calculation results until the optimal combined regulation rules for the drainage facilities that meet the evaluation criteria are output.
[0068] In the embodiment of the present invention, by obtaining the attributes of the drainage facilities in the scheduling area, the planar distribution and cross-section topography of the river channels, the regional digital elevation and land use distribution, and collecting the rainfall, surface runoff, water level, flow rate, and water quality monitoring data during the pipe network confluence and transmission processes for different rainstorms, establish and calibrate the SWMM model, obtain the boundary conditions of the drainage system, initialize the parameters of the particle swarm algorithm, use the initialized particle swarm algorithm to construct a combined optimization scheduling algorithm model, generate the combined regulation rules for the drainage facilities corresponding to the number of particles within the feasible domain, input them into the calibrated SWMM model, generate the SWMM model file, divide them into processes equal to half the number of CPU cores in sequence, and use the PySWMM interface in Python to execute CPU multi-process parallel computing to obtain the simulation calculation results, evaluate the simulation calculation results according to the evaluation indicators for the scheduling rules, and output the optimal combined regulation rules for the drainage facilities. This method can give full play to the computing resources, shorten the computing time, and significantly improve the efficiency of the integrated scheduling decision-making of the plant, network, and river.
[0069] Next, a simulation calculation system for integrated plant, network, and river scheduling based on parallel computing provided by the present invention will be described. The simulation calculation system for integrated plant, network, and river scheduling based on parallel computing described below can be mutually referred to and contrasted with the simulation calculation method for integrated plant, network, and river scheduling based on parallel computing described above.
[0070] Please refer to Figure 3As shown in the figure, it includes: a data acquisition module 31, a model construction module 32, a simulation calculation module 33, and a scheduling evaluation module 34, where:
[0071] The data acquisition module: is used to obtain the attributes of drainage facilities in the scheduling area, the planar distribution and cross-section topography of river channels, the regional digital elevation, and the land use distribution, and collect the water level, flow rate, and water quality monitoring data during different rainfall events, surface runoff, pipe network confluence, and transmission processes;
[0072] The model construction module: is used to establish a SWMM model and construct a joint optimization scheduling algorithm model, and calibrate the parameters of the SWMM model to obtain a calibrated SWMM model;
[0073] The simulation calculation module: is used to output the joint regulation rules of drainage facilities, input the joint regulation rules of drainage facilities into the calibrated SWMM model, and obtain simulation calculation results by using a parallel computing method;
[0074] The scheduling evaluation module: is used to evaluate the simulation calculation results.
[0075] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the scope of the present invention.
Claims
1. A simulation calculation method for integrated dispatching of plant, grid and river based on parallel computing, characterized in that: The steps include: S1. Obtain the drainage facility attributes, river channel plane distribution and cross-sectional topography, regional digital elevation and land use distribution in the dispatching area, and establish a SWMM model; S2, collecting monitoring data of water level, flow and water quality during rainfall, surface runoff, pipe confluence and transmission at different times, calibrating the parameters of the SWMM model, and obtaining a calibrated SWMM model; S3, using a particle swarm algorithm to construct a joint optimization scheduling algorithm model, the output of which is a drainage facility joint control rule, inputting the drainage facility joint control rule into the calibrated SWMM model, and using a parallel computing method to obtain simulation calculation results; S301, obtaining the boundary conditions of the drainage system in the scheduling area, wherein the boundary conditions of the drainage system include operation scheduling objectives, constraints and control variables; S302, according to the boundary conditions of the drainage system, the number of particles of the particle swarm algorithm is initialized to M, the maximum number of iterations is initialized to T, and an initialized particle swarm algorithm is obtained, and the initialized particle swarm algorithm is used to construct the joint optimization scheduling algorithm model, and M groups of drainage facility joint control rules are output, which are input into the calibrated SWMM model to generate corresponding M groups of SWMM model files; S303, obtaining the number of CPU cores N, using Python's multiprocessing module to create a process pool, the process pool including N / 2 processes, defining a simulation function, loading the PySWMM interface into the simulation function, the N / 2 processes all including the simulation function, using the pool.map method to sequentially distribute the M groups of SWMM model files to each process in the process pool, performing simulation calculations on the distributed SWMM model files through the PySWMM interface of each process, forming CPU multi-process parallel calculations, and outputting simulation calculation results; S4. Set the dispatch rule evaluation index to evaluate the simulation calculation results. If the evaluation criteria are met, output the optimal drainage facility joint control rules corresponding to the simulation calculation results. If the evaluation criteria are not met, return to step S3.
2. The method for simulating and calculating the integrated dispatching of a plant, a grid and a river based on parallel computing according to claim 1 is characterized in that: Step S2 also includes: the parameter calibration includes calibrating the Manning roughness coefficient of the catchment area, the depression storage parameter, the infiltration parameter, the Manning roughness coefficient of the pipe channel, the head loss coefficient, the pollutant accumulation and scouring parameter and the pollutant attenuation coefficient of the SWMM model.
3. The method for simulating and calculating the integrated dispatching of a plant, a grid and a river based on parallel computing according to claim 1 is characterized in that: Step S3 also includes: the operation scheduling objectives include minimizing the sewage overflow, minimizing the water level of the operation and maintenance emergency repair section, and minimizing the complexity of the pump station joint scheduling plan; The expression of the sewage overflow is: ; In the formula, m represents the total number of pumping stations, Q i represents the overflow of the forebay of the i-th pump station; The expression of the operation and maintenance repair pipeline water level is: ; In the formula, Z i Indicates the water level at the characteristic point of the pipe section in the operation and maintenance repair area; The expression of the complexity of the pump station joint scheduling scheme is: ; In the formula, n represents the total number of scheduling time periods, Indicates whether the operating status of the i-th pump station needs to be adjusted at time t, where P i,t and P i,t-1 represents the operating power of the i-th pump station at time t and time t-1 respectively; The constraint conditions include a pumping flow constraint and a characteristic water level constraint. The pumping flow constraint is: , where Q imin and Q imax Respectively represent the minimum and maximum value of the sewage overflow rate of the i-th pump station; The characteristic water level constraint is used to control overflow pollution, and its expression is: ; In the formula, Z j represents the water level of the front sump of the jth pumping station, Z jmin and Z jmax are the minimum and maximum warning water levels of the front collection tank of the jth pumping station; The control variables include controlling the start and stop status of the pump station; The value of the particle number M is 80, the value of the maximum number of iterations T is 3, and the simulation calculation results include the total energy consumption of the pumping station, the number of iterations, and the number of consecutive outputs of the same drainage facility joint control rule.
4. The method for simulating and calculating the integrated dispatching of a plant, a grid and a river based on parallel computing according to claim 3 is characterized in that: The calculation formula for the total energy consumption of the pump station is as follows: ; Where W represents the total energy consumption of the pumping station, T represents the time step, S i,t represents the operating status of the i-th pump station at time t, P i represents the rated power of the i-th pump station.
5. The method for simulating and calculating the integrated dispatching of a plant, a grid and a river based on parallel computing according to claim 1 is characterized in that: Step S4 also includes: the dispatching rule evaluation indicators include the lowest total energy consumption of the pumping station, the number of iterations reaches the maximum number of iterations T, and the same drainage facility joint control rule is output for three consecutive iterations. If one of the dispatching rule evaluation indicators is met, it is considered to meet the evaluation criteria, and the optimal drainage facility joint control rule corresponding to the simulation calculation result is output. If it does not meet the evaluation criteria, return to step S3.
6. The integrated dispatching simulation calculation system of plant, grid and river based on parallel computing is characterized by: Includes the following modules: Data acquisition module: used to obtain drainage facility attributes, river and canal plane distribution and cross-sectional topography, regional digital elevation and land use distribution in the dispatching area, and collect water level, flow and water quality monitoring data during different rainfall events, surface runoff, pipe and canal confluence and transmission; Model building module: used to establish a SWMM model and a joint optimization scheduling algorithm model, calibrate the parameters of the SWMM model, and obtain a calibrated SWMM model; Simulation calculation module: used for outputting the joint regulation rules of drainage facilities, inputting the joint regulation rules of drainage facilities into the calibrated SWMM model, and obtaining the simulation calculation results by parallel computing; Scheduling evaluation module: used to evaluate the simulation calculation results.
Citation Information
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