A method for optimizing large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithm

Abstract

This invention discloses a method for optimizing large-scale multi-source water supply networks using differential evolution and divide-and-conquer algorithms, comprising the following steps: decomposing the water supply network optimization problem into Ns subproblems based on the flow direction; initializing the population of the Ns subproblems, and simultaneously initializing the optimizers for the differential evolution of each subproblem; performing a mutation operation on each solution to calculate the mutation vector; performing a crossover operation on the mutated solutions; finally selecting offspring; retaining one optimal solution up to the present time in the population of each subproblem; and updating the global optimal solution after iterating through all subproblems.

Description

Technical Field

[0001] The present invention relates to two major fields, namely complex network partitioning and intelligent computing, and mainly relates to a method for optimizing a large-scale multi-source water supply network by using differential evolution and divide-and-conquer algorithms. Technical Background

[0002] The water supply network is an important part of urban water infrastructure. Through the water supply network, drinking water can be transported from the source to the hands of consumers. At the same time, the construction of the water supply network is also related to many practical factors, such as government standards, other infrastructure and terrain. Therefore, how to reduce capital costs while ensuring service quality is a very challenging and practical issue.

[0003] The methods for solving the design of water supply network as an optimization problem can be roughly divided into two categories: 1) deterministic methods and 2) meta-heuristic methods. Among the deterministic methods, linear programming (LP) and its variants were first proposed earlier in an attempt to solve the problem with low computational effort. However, due to the nonlinear nature of the problem, linear programming methods are prone to fall into local optimality. In turn, some scholars have proposed some nonlinear programming methods. However, these methods are still not enough to effectively find the optimal or near-optimal solution. Considering that the variables in the problem are discrete in nature, some scholars have proposed an integer linear programming method that can iteratively change the solution based on hydraulic simulation. However, the optimality and convergence of this method have been questioned, especially on large-scale networks. The second category of methods (meta-heuristic algorithms) are due to their ability to easily handle different types of constraints and effectively locate near-optimal solutions. This type of algorithm includes single solution-based algorithms such as taboo search, simulated annealing and iterative local search and population-based algorithms such as particle swarm optimization, genetic algorithm, and ant colony optimization.

[0004] An urban water supply network optimization method (CN201310207126.1) uses the annual value method to establish a water supply trunk optimization model. First, the annual fee value is used as the technical and economic indicator of the water supply pipeline trunk, and the investment cost and annual operating fee of the pipeline and the booster pump station are converted into the cost of one year during the service life of the pipeline system at the same annual interest rate, thereby establishing the objective function. Secondly, the constraints of the objective function are listed. The objective function is also subject to the hydraulics of the pipeline system, the node head, the non-negative constraints of the design flow of the pipe section, and the water supply reliability. It provides a new decision-making indicator and optimization method for the design and operation of the urban water supply network. However, simply improving the optimization performance and time performance cannot reduce the complexity of the water supply network problem.

[0005] Differential evolution belongs to the category of evolutionary algorithms. It is a heuristic random search algorithm based on population differences. It represents the solution of the problem as the survival of the fittest process of "chromosomes". Through the continuous evolution of the "chromosome" group from generation to generation, including operations such as replication, crossover and mutation, it finally converges to the individual that is "most adapted to the environment", thereby obtaining the optimal solution or satisfactory solution to the problem. Combined with the divide-and-conquer strategy, large-scale water supply network optimization problems are decomposed into multiple small-scale subcomponents to solve these problems. Differential evolution is very suitable for these small-scale problems.

[0006] Water supply network is an important infrastructure of modern cities. A well-designed network can not only reduce construction expenditure but also provide reliable service. Nowadays, with the expansion of cities, the WDN of cities has increased dramatically, which has brought heavy pressure to the optimization design. Summary of the invention

[0007] In order to solve the optimization problem of large-scale water supply network, the present invention uses the divide-and-conquer algorithm and the differential evolution algorithm for water supply network optimization, and applies the differential evolution and divide-and-conquer algorithms to the optimization design of large-scale multi-source water supply networks in cities.

[0008] The present invention is achieved by at least one of the following technical solutions.

[0009] The method for optimizing a large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithm includes the following steps:

[0010] (1) Decompose the water supply network optimization problem into several sub-problems based on the popular direction;

[0011] (2) Initializing the populations of several sub-problems and at the same time initializing the optimizers of the differential evolution of the corresponding sub-problems;

[0012] (3) For each solution {x1,x,..,x m}Perform mutation operation to calculate mutation vector;

[0013] (4) Perform a crossover operation on the mutated solution;

[0014] (5) Finally, select the offspring;

[0015] (6) The population S of each subproblem j Keep the best solution so far;

[0016] (7) Update the global optimal solution after iterating all sub-problems;

[0017] (8) Determine whether C is equal to 0. If so, proceed to step (2) to decompose the problem again. If not, proceed to step (3) to continue optimizing the solution of the subproblem, where C is the threshold number of times the decomposition subproblem is violated once.

[0018] Furthermore, the water supply network optimization problem is decomposed into Ns sub-problems, and the population of Ns sub-problems is

[0019] Furthermore, the variation vector vi is:

[0020]

[0021] where {x1,x,..,x m} represents the commercial models corresponding to different sizes of water pipelines, x a 、x b 、x c is a random integer, x i is the solution based on which the mutation vector is currently calculated, x best is the current optimal solution, r1 is a random number between 0 and 1, p is an adaptive parameter, and F i is the actual factor.

[0022] Furthermore, the actual factor F i It is calculated as follows:

[0023]

[0024] N(0.5,0.3) is a normal distribution, C(0,1) is a Cauchy distribution, and r2 is a random number between 0 and 1.

[0025] Furthermore, the crossover operation is:

[0026]

[0027] In the above formula, j∈[1,N P ],N P is the number of pipes, r3 is a random number between 0 and 1, C is a configurable parameter, u i (j) is the solution obtained after the crossover operation, v i (j) is the solution obtained by mutation operation, x i (j) is the solution without mutation.

[0028] Furthermore, offspring are selected using the following formula:

[0029]

[0030] in The final solution selected, μ i is the solution obtained after the crossover operation, x i is the solution for mutation, and f is the fitness function.

[0031] Furthermore, step (2) follows the following division principle:

[0032] a) If the water consumed by a node comes from only one source, the node must belong to that source;

[0033] b) If a node consumes water from multiple sources, it belongs to the source that provides more information to add water to it;

[0034] c) A pipeline belongs to the source to which its downstream nodes belong.

[0035] Furthermore, the processing of each sub-problem follows the population S of each sub-problem j The principle of retaining the best solution so far.

[0036] Furthermore, a mechanism for re-decomposing the problem is to determine whether a threshold number of decomposition sub-problems is triggered, and if so, to re-decompose the problem.

[0037] Furthermore, the fitness evaluation function designed for the water supply network, which is used to evaluate the fitness of offspring, needs to deal with the minimum pressure constraint:

[0038] F(x)=f(x)+P(x)

[0039]

[0040] Where F(x) is the fitness evaluation function used last, f(x) is the objective function, P(x) is the penalty function, N is the number of pipelines, φ(i,j) is the coefficient of penalty function calculation, H j,i is the head pressure of the pipeline between nodes j and i, H represents the head pressure of the pipeline, H j,min is the set head pressure threshold; if the solution is feasible, the penalty function part takes 0, and if the solution is not feasible, there must be a pipe with a head pressure less than the threshold, in which case the penalty function takes 1.

[0041] Compared with the prior art, the present invention has the following beneficial effects:

[0042] The present invention solves large-scale optimization problems efficiently by divide and conquer. Compared with directly using differential evolution algorithm to solve water supply network problems, the complexity of the problem is greatly reduced, so that the simple differential evolution algorithm remains effective and close to the global optimal solution. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 A flowchart of an embodiment using differential evolution and divide-and-conquer algorithm to optimize a large-scale multi-source water supply network;

[0044] Figure 2 This is a schematic diagram of the Balerma network in the embodiment;

[0045] Figure 3The present invention is a flow-based decomposition flowchart of a method for optimizing a large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithms for an implementation example. DETAILED DESCRIPTION

[0046] This section will describe in detail the specific embodiments of the present invention. The preferred embodiments of the present invention are shown in the accompanying drawings. The purpose of the accompanying drawings is to supplement the description of the text part of the specification with graphics, so that people can intuitively and vividly understand each technical feature and the overall technical solution of the present invention, but it cannot be understood as a limitation on the scope of protection of the present invention.

[0047] Example 1

[0048] like Figure 1 to Figure 3 As shown in the figure, the method of optimizing large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithm is based on the decomposition method of flow direction, including the following three division principles.

[0049] (1) If the water consumed by a node comes from only one source, the node must belong to that source.

[0050] (2) If a node consumes water from multiple sources, it belongs to the source that provides more information to add water to it.

[0051] (3) The pipeline belongs to the source to which its downstream node belongs.

[0052] like Figure 1 As shown, i represents the number of times the counter runs, MAXROUND is a configurable number threshold, and when i>MAXROUND, the global optimal solution is output. The steps of the i-th run of the present invention are:

[0053] (1) Decompose the water supply network optimization problem into Ns sub-problems based on the popular direction;

[0054] (2) Population of Ns subproblems Initialize and initialize the optimizer of differential evolution of each sub-problem;

[0055] (3) For each solution {x1,x,..,x m}Calculate the mutation vector vi:

[0056]

[0057] where {x1,x,..,x m} represents the commercial models corresponding to different sizes of water pipelines, x a 、x b 、x c is a random integer, x i is the solution based on which the mutation vector is currently calculated, x bestis the current optimal solution. r1 is a random number between 0 and 1, p is an adaptive parameter, initially set to 0.5, and F i is the actual factor, calculated as follows:

[0058]

[0059] N(0.5,0.3) is a normal distribution, C(0,1) is a Cauchy distribution, r2 is a random number between 0 and 1,

[0060] (4) Perform a crossover operation on the mutated solution.

[0061]

[0062] In the above formula, j∈[1,N P ]N P is the number of pipelines, r3 is a random number between 0 and 1, C is a configurable parameter, initially 0.5, u i (j) is the solution obtained after the crossover operation, v i (j) is the solution obtained by the mutation operation in (3), x i (j) is the solution without mutation.

[0063] (5) The final offspring is selected using the following formula:

[0064]

[0065] in The final solution, μ i is the solution obtained after the crossover operation, x i is the solution for mutation. f is the fitness function.

[0066] (6) The population S of each subproblem j Keep the best solution so far;

[0067] (7) Update the global optimal solution after iterating all sub-problems;

[0068] (8) Determine the remainder of i with respect to C ( Figure 1 i%C in the equation represents the modulus operation of i with respect to the threshold C) is equal to 0. If it is equal, the process goes to step (2) to decompose the problem again. If it is not equal, the process goes to step (3) to continue optimizing the solution of the subproblem. C is the threshold number of times the decomposition subproblem is violated once.

[0069] Example 2

[0070] A method for optimizing a large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithm comprises the following steps:

[0071] (1) First, the water supply network optimization problem is decomposed into Ns sub-problems based on the popularization.

[0072] (2) Population of Ns subproblems Initialize and initialize the optimizer of the differential evolution of each sub-problem.

[0073] (3) For each solution {x1,x,..,x m}Calculate the mutation vector.

[0074]

[0075] {x1,x,..,x m} represents the commercial models of different sizes of water pipes, a, b, c are random integers,

[0076] r1 is a random number between 0 and 1, p is an adaptive parameter, initially set to 0.5, and F i is the actual factor and is calculated as follows.

[0077]

[0078] N(0.5,0.3) is a normal distribution, C(0,1) is a Cauchy distribution

[0079] (4) Perform a crossover operation on the mutated solution.

[0080]

[0081] j∈[1,N P ]N P is the number of pipelines, r3 is a random number between 0 and 1, and C is a configurable parameter, initially 0.5.

[0082] (5) Finally, the offspring is selected using the following formula.

[0083]

[0084] f is the fitness function.

[0085] Design the fitness function of the water supply network. Note that the variable of the water supply network optimization problem is the size of the pipe rather than the node. In fact, it is a constrained optimization problem. Evaluating the fitness of the offspring requires dealing with the minimum pressure constraint, and the penalty function method is used to deal with this constraint.

[0086] F(x)=f(x)+P(x)

[0087]

[0088] Where F(x) is the fitness evaluation function used last, f(x) is the objective function, P(x) is the penalty function, N is the number of pipelines, φ(i,j) is the coefficient of the penalty function calculation, and H j,i is the head pressure of the pipeline between nodes j and i. H represents the head pressure of the pipeline, H j,min is the set head pressure threshold; if the solution is feasible, the penalty function part takes 0, and if the solution is not feasible, there must be a pipe with a head pressure less than the threshold, in which case the penalty function takes 1.

[0089] (6) The population S of each subproblem j Keep the best solution so far;

[0090] (7) When the algorithm finds a globally feasible solution, it will start the re-decomposition process based on this globally feasible solution. After the re-decomposition, all sub-problem populations need to be reassembled;

[0091] (8) In each iteration, the optimal solution is continuously updated during the population evolution process;

[0092] (9) The optimal solution is evaluated using the fitness function.

[0093] The method of the present invention was tested using the Balerma network. The Balerma network is a widely used real-world water resource distribution network. The structure of the Balerma network. It was compared with related optimization algorithms. The compared algorithms are PSO, DSO, TSDE, and SADE. The experiment was repeated 20 times, and the fitness values ​​of the solutions finally found were all better than the four compared algorithms.

[0094] algorithm average value median Standard deviation Method of the present invention 0.093 0.092 0.0028 PSO 123 35.6 148 DSO 1153 1155 3269 SADE 0.097 0.097 0.00075 TSDE 0.96 0.96 0.0011

[0095] The present invention adopts a flow-based decomposition strategy to divide the water resource distribution network, which can handle large-scale network optimization problems. The effectiveness of the present invention is verified by a real-world water resource distribution network (Balerma network), and the effect is compared with other optimization algorithms. It is found that the solution obtained by the present invention is better than the four compared algorithms.

[0096] Example 3

[0097] The decomposition method based on flow direction of this embodiment: decompose the water supply network optimization problem into Ns sub-problems based on the flow direction; initialize the population of Ns sub-problems, and initialize the optimizer of differential evolution of each sub-problem corresponding to each sub-problem; perform mutation operation on each solution to calculate the mutation vector; perform crossover operation on the mutated solution; finally select the offspring; retain the best solution so far in the population of each sub-problem; update the global optimal solution after iterating all sub-problems. The division principles include:

[0098] 1) If the water consumed by a node comes from only one source, the node must belong to that source.

[0099] 2) If a node consumes water from multiple sources, it belongs to the source that provides more information to add water to it.

[0100] 3) The pipeline belongs to the source to which its downstream node belongs.

[0101] Assume that there are two water sources R1 and R2, and four consumers {a, b, c, d}, connected by six pipes ordered from 1 to 6. First, a hydraulic simulation is performed: the flow direction of pipe 2 is from a to b, and the flow direction of pipe 6 is from c to d, which means that only water from R2 cannot meet the requirements of b and d. Some water flows from R1 to b and d to support them. However, the tracer simulation shows that most of the water consumed by b and d still comes from R2. Therefore, b and d are grouped with R2, while a and c are grouped with R1. According to the results of node partitioning, pipes 1 and 4 are divided into one group, and pipes 2, 3, 5, and 6 are divided into another group.

[0102] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific implementation methods described. Obviously, many modifications and changes can be made according to the content of this specification. This specification selects and specifically describes these embodiments in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and use the present invention well. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for optimizing a large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithms, characterized in that: The following steps are involved: (1) Decompose the water supply network optimization problem into several sub-problems based on the popular direction; (2) Initializing the populations of several sub-problems and at the same time initializing the optimizers of the differential evolution of the corresponding sub-problems; (3) For each solution {x1, x2, .., x i , .., x m }Perform mutation operation to calculate mutation vector; The mutation vector vi is: where {x1, x2, .., x i , .., x m } represents the commercial models corresponding to different sizes of water pipelines, x a 、x b 、x c is a random integer, x i is the solution based on which the mutation vector is currently calculated, x best is the current optimal solution, r1 is a random number between 0 and 1, p is an adaptive parameter, and F i is the actual factor; the actual factor F i It is calculated as follows: N(0.5, 0.3) is a normal distribution, C(0, 1) is a Cauchy distribution, and r2 is a random number between 0 and 1; (4) Perform a crossover operation on the mutated solution; The crossover operation is: In the above formula, j∈[1,N P ],N P is the number of pipes, r3 is a random number between 0 and 1, C is a configurable parameter, u i (j) is the solution obtained after the crossover operation, v i (j) is the solution obtained by mutation operation, x i (j) is the solution without mutation; (5) Finally, select the offspring; Offspring are selected using the following formula: in The final solution, μ i is the solution obtained after the crossover operation, x i is the solution for mutation, and f is the fitness function; (6) The population S of each subproblem j Keep the best solution so far; (7) Update the global optimal solution after iterating all sub-problems; (8) Determine whether C is equal to 0. If so, proceed to step (2) to decompose the problem again. If not, proceed to step (3) to continue optimizing the solution of the subproblem, where C is the threshold number of times the decomposition subproblem is violated once.

2. The method for optimizing a large-scale multi-source water supply network by using differential evolution and divide-and-conquer algorithm according to claim 1, characterized in that: The water supply network optimization problem is decomposed into Ns sub-problems, and the population of Ns sub-problems is {S1,S2,…,S Ns }.

3. The method for optimizing a large-scale multi-source water supply network by using differential evolution and divide-and-conquer algorithm according to claim 1, characterized in that: Step (2) follows the following division principles: a) If the water consumed by a node comes from only one source, the node must belong to that source; b) If a node consumes water from multiple sources, it belongs to the source that provides more information to add water to it; c) A pipeline belongs to the source to which its downstream nodes belong.

4. The method for optimizing a large-scale multi-source water supply network by using differential evolution and divide-and-conquer algorithm according to claim 1, characterized in that: The processing of each sub-problem follows the population S of each sub-problem. j The principle of retaining the best solution so far.

5. The method for optimizing a large-scale multi-source water supply network by using differential evolution and divide-and-conquer algorithm according to claim 1, characterized in that: The mechanism of re-decomposing the problem: determine whether the threshold number of decomposition sub-problems is triggered, and if so, re-decompose the problem.

6. The method for optimizing a large-scale multi-source water supply network using differential evolution and divide-and-conquer algorithm according to any one of claims 1 to 5, characterized in that: The fitness evaluation function designed for the water supply network is used to evaluate the fitness of the offspring and needs to handle the minimum pressure constraint: F(x)=f(x)+P(x) Where F(x) is the fitness evaluation function used last, f(x) is the objective function, P(x) is the penalty function, N is the number of pipelines, φ(i, j) is the coefficient of the penalty function calculation, and H j，i is the head pressure of the pipeline between nodes j and i, H represents the head pressure of the pipeline, H j，min is the set head pressure threshold; if the solution is feasible, the penalty function part takes 0, and if the solution is not feasible, there must be a pipe with a head pressure less than the threshold, in which case the penalty function takes 1.

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