| With the rapid development of industry and society,environment has been severely polluted,water pollution is becoming more and more seriously.In order to improve the quality of the environment effectively,and reasonably arrange the production of factories,we must determine the pollution source information,control and govern the pollution process.Recently years,there have been many research works about water contamination.Aquifer sim-ulation models have been combined with optimization models to identify unknown groundwater pollution sources[10,13,14,16,20,26,35,36,54,55,67,69,70],so that they can achieve the goal of groundwater management.Paper[35,36]consider the effect of agriculture production on groundwater,the objective function aims at determining the optimal fertilizer application rates by maximizing the net agriculture benefits,constrained by groundwater quality requirements specified at control sites.These[7,8,57,58]discuss the impact of waste water treatment plants(that is industrial production)on waters,while[57,58]aim to determine the optimal emission intensity,thereby allowing chemical oxygen demand and dissolved oxygen in the water to meet environmental requirements,so that minimize the overall purifica-tion costs;Paper[7,8]deal with the optimal control problem related to the determination of location of the wastewater outfalls,which yield the lowest total cost of the system but also need to satisfy water quality constraints.All of these optimization problems need to solve the coupled partial differen-tial equation or equations,numerical schemes directly affect the computation time and efficiency.What's more,the research area is usually very large and complex in reality,it is essential to choose the fast algorithm.No matter groundwater flow-transport equation and convection diffusion reaction equation,they all have convection term and diffusion term.In many fluid flow procedures,the diffusion coefficients are much more smaller than transport velocity,it is difficulty for the ordinary finite difference method and finite element method to solve this kind of problems,since they often bring nonphysical oscillations into the numerical solution.Therefore,the upwind schemes that overcame the nonphysical oscillation had been developed to used in the simulations of these problems[39,76].But it is well known that the traditional upwind scheme only has first-order accuracy.In order to improve the accuracy,many modified upwind schemes[47,47,49]were proposed,they have second-order accuracy and also can avoid nonphysical oscillation.Furthermore,the splitting technique[22,27,42,64]is another attractive and popular technique to reduce high-dimensional problems to a series of one-dimensional problems at each time step for saving the memory and CPU time.And we can adopt fast algorithm to compute one-dimensional problem,this will save a lot of time.The choice of objective function will directly affects the optimal con-trol results.For the optimal control problems,we should choose different objective functions based on the reality application.The multiple objective function for pollution control is an important research direction,and it is recently necessary to reasonably and efficiently research environment pollu-tion's optimization,emission reduction cost and other costs.In this paper,we proposed a new environment pollution optimal control model and algorithm-s.This optimization-control model has an important guiding significance for practical application.The dissertation is divided into four chapters,the content is as follows:In Chapter 1,we propose the optimal control model of convection d-iffusion pollution problems.Different from[54,55,67,69],our objective function includes two terms.While the first term aims at minimizing the weighted deviation between the simulated concentration and the best envi-ronment allowing concentration at observation sites over all time,the second term makes the cost of emission reduction and other costs as small as pos-sible.Convection diffusion equations are integrated as state constraints,the optimal control model also subjects to two control constraints,one of them is that pollutant concentration at observation sites can not overpass the maxi-mum allowable concentration;The other is applied to restrict source disposal fluxes(or effluent concentration)by a lower and upper bound,so that these values can satisfy the application condition and have reality sense.We discussed the existence of a solution for the optimal control model.By analyzing the existence and uniqueness of solution for the state system in the weak form sense,it is proved that the solution is continuous and bounded in the observation zones during the whole time.What's more,we conclude that the objective function is weakly lower semicontinuous becasuse of its convexity and continuity,so that we can obtain the existence of a solution for the optimal control problem.As an important application,we research the river pollution source i-dentification problems.Numerical experiments firstly consider the situation that injection rates are constant and no pumping wells are considered,we discuss the optimal identification effect when the number of sources changes from one to three;and then we give an example with one pumping point and two pollutant source points whose injection rates also keep constant;Finally,we consider the situation with variable injection rate and one point.Ex-tend to two-dimensional example,suppose there exist four pollution sources and observation points,and pollution sources are located symmetrically and asymmetrically,numerical simulation results are shown with different level-s of perturbation to observation data and different locations of the source points and observation sites.Numerical results show that the methodology is efficient,the propose method can be used to high-dimensional and more complex problems.In Chapter 2,We build the discrete form of optimal control problems,which restricted by convection diffusion reaction equation,water quality of observation points and disposal fluxes.It is a linked simulation-optimization model,and the advantage of the methodology is the external linking of the numerical simulation model with the optimization model.In the numeri-cal scheme for solving governing equations,we propose to use the splitting scheme combing with the improved-upwind finite difference method,which can not only avoid nonphysical oscillation,but also save computation time and have higher accuracy.We use the constrained Differential Evolution op-timization algorithm to solve the optimization problems,which provides the advantages of its global solution solving feature,simplicity,powerful search capability,compact structure and high convergence.The objection function's main goal is to make the simulated concentration at observation points match with the environment allowable concentration as much as possible and the emission reduction cost and pipeline layout cost as little as possible.Numerical experiments firstly shown the second-order accuracy for the improved upwind finite difference method and then gave a series of experi-ments of the optimal control problem.These experiments can be classified into two kinds:the first kind's goal is to search the optimal disposal fluxes that can minimize the objective function,which contain concentration er-rors and emission reduction cost.For these experiments,suppose pollution sources are symmetrically located in the domain,we firstly compare the op-timal results when emission reduction cost with different cost functions or same cost function,and then discuss the variation trend of optimal dispos-al fluxes and emission reduction rates when observation locations and flow velocities are different,respectively.Numerical results show that the opti-mal results depend on the choice of protected zones and the velocity of flow,and downstream factories can dispose more polluted water than upstream factories.In the second kind of optimal control model,disposal fluxes keep con-stant but effluent concentration are variable,it is aim at finding the optimal effluent concentration,so that the objective function,which only contain three expense terms,can achieve the best.We discuss the effect of different locations of pollution sources,different flow velocities and different disposal fluxes on the optimal results.And get the conclusions:the optimal effluent concentrations increase with water velocity increasing;the optimal effluent concentrations decrease as disposal fluxes gradually become larger,as well as the area of observation domain;factory can discharge more pollutant when it is far away from the observation domain.In Chapter 3,we develop a new groundwater pollution control by find-ing the optimal disposal fluxes.The important features are:groundwater flow and solute transport system model integrated as constraints,objective function considering both concentration error,emission reduction costs and pollution taxes.Our goal is to determine optimal source discharge fluxes that minimizes the deviation between the best environment concentration and simulated concentration,emission reduction costs and pollution taxes simultaneously.The proposed optimal control model is subject to the flow and transport groundwater equations in porous media.Meanwhile pollutant concentration at observation sites and source discharge fluxes must satisfy certain constraints.For numerical simulation,we further develop the splitting scheme com-bined with the second-order improved-upwind finite difference method to solve aquifer simulation model of the groundwater flow and transport pro-cesses in porous media,constrained Differential Evolution is as optimization algorithm.Numerical experiments firstly check the feasibility and conver-gence of numerical schemes and show the contour plots of simulated water head and concentration at different time periods.And the rest two simula-tions test the performance of the proposed source control models.The first simulation tests on the aquifer with simple geometry and constant dispos-al fluxes,it gives the numerical simulation-optimization results of disposal fluxes and reduction rates with different objective functions for chemical in-dustry and paper making industry.We find that reduction rates for upstream sources are higher than those for downstream sources,the observation sites are affected by upward pollution slightly more than by downward pollution,and pollution levy police affects the paper making industry more than the chemical industry.From the values in the economical table,we can see that it is more economical efficient to build factories in downstream when construc-tion planning is made.Further more,we discuss the influence of velocity on the optimal results for chemical industry,by comparing different head boundary values on the left side,it can be seen that as the value of left head boundary increase,flow velocity of the pollutant increases and emission reduction rate decreases overall.The second simulation with a more realistic domain,aquifer shape is more complex and disposal fluxes are varying piecewise in time.In the simulation,we suppose that sources release pollutant only for the first five months,and discharge fluxes are constant at every month.We conclude that optimal fluxes' overall trend is increasing during the first three month and decreasing at the fourth and fifth months,upstream source has smaller disposal flux but higher reduction rate,by contrast to the downstream source.In the final,to see the efficiency of abatement directly,we show pollutant distribution without optimization and with optimization at different time.In chapter 4,our work summary and prospect of future research are given. |
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