2023美赛MCM-A题M奖获奖论文

1、For office use onlyT1_T2_T3_T4_Team Control Number66666Problem ChosenAFor office use onlyF1_F2_F3_F4_2023Mathematical Contest in Modeling (MCM/ICM) Summary SheetThe Kariba Dam on the Zambezi River is one of the larger dams in Africa. Its construction was controversial, and a 2023 report by the Insti

2、tute of Risk Management of South Africa included a warning that the dam is in dire need of maintenance. A number of options are available to the Zambezi River Authority (ZRA) that might address the situation.It is assumed that the design parameters, site selection and construction of the dam (dam gr

3、oup) after repair, reconstruction and reconstruction are in accordance with the requirements of water conservancy and hydropower projects. The number and location of the dam groups are in line with the requirements of water conservancy and hydropower projects. Therefore, the research problems are: a

4、) Comparing the comprehensive benefits of the dams after restoration, reconstruction and reconstruction, establishing the mathematical model, and expounding the costs and benefits of each case; b) Reconstructing the dams (rebuilding the existing dams 10-20 A little dam), in the drought, heavy rain,

5、moderate rain, light rain, when the water storage, flood discharge to ensure the probability of dam break the smallest, that is, dam group risk decision optimization.For the first problem, this paper chooses the optimal cost and benefit plan for Kaliba Dam restoration work by establishing the econom

6、ic benefit model, the economic loss model, the comprehensive benefit model, the absolute return model after restoration and the investment payback period model.The cost-effectiveness of the three programs was evaluated in detail according to the results of the mathematical model. Option1, the restor

7、ation of the dam can extend the useful life of 11 years. While the one-time cost of access to the investment recovery period of 3.2 years. Option2, 100 years after the design of the reconstruction, the investment recovery period of 65.2 years. Option3, after rebuilding for the dam group, designed to

8、 use 100 years, the investment recovery period of 47.8 years. Three programs have better benefits, of which the best overall benefits of the three programs.For the problem 1, this paper uses the R language software to draw the heatmap to validate the conclusion. Using the Matlab software to build th

9、e MSE model, the error analysis of the above results is further proved to be the best.For the second problem, we use MATLAB and SPSS MODELER software to establish Author Topic Model, BP network and LS-SVM model to validate the LDA model, and use MSE and ABS method to analyze the error of LDA model.

10、The results show that the LDA model is accurate and reliable, and the error is small. It further shows that the mathematical model established in this paper has a certain generalization ability.Key words: The zambezi river ；Kariba dam ; Water conservancy ; reconstructionI. IntroductionIn order to in

11、dicate managing the Zambezi river, the following background is worth mentioning.1.1Zambezi RiverThe Zambezi (also spelled Zambeze and Zambesi) is the fourth-longest river in Africa, the longest east-flowing river in Africa and the largest flowing into the Indian Ocean from Africa. The area of its ba

12、sin is 1,390,000 square kilometres (540,000 sq mi),12 slightly less than half that of the Nile. The 2,574-kilometre-long river (1,599 mi) rises in Zambia and flows through eastern Angola, along the eastern border of Namibia and the northern border of Botswana, then along the border between Zambia an

13、d Zimbabwe to Mozambique, where it crosses the country to empty into the Indian Ocean.From Victoria to the Falls for the upper reaches of Victoria Falls, 1287 km long; from Victoria Falls to the territory of Mozambique Kabula Bassa reservoir for the middle, 869 km long; Cabra Bassa is the following

14、downstream, 579 km long.Zambezi average slope of 0.4 , runoff of 223.2 billion cubic meters, second only to the Congo River, ranked second in Africa.1.2Kariba DamConstructionThe double curvature concrete arch dam was designed by Coyne et Bellier and constructed between 1955 and 1959 by Impresit of I

15、taly2 at a cost of $135,000,000 for the first stage with only the Kariba South power cavern. Final construction and the addition of the Kariba North Power cavern by Mitchell Construction3 was not completed until 1977 due to largely political problems for a total cost of $480,000,000. During construc

16、tion, 86 men lost their lives.24The grassland in the floodplains around the Zambezi River relies on annual floods and seasonal rainfall, while rainfall in the region is richer than in the very dry west. Floods and vegetation expansion during the rainy season is limited in the north of the Zambezi Ri

17、ver, indicating an increase in the northern margin of the floodplain. These vegetation provide food for numerous fish and animal species.1.3 Kariba DamKariba hydropower station is located in the African Zambia and Zimbabwe at the junction of the Zambezi (Zambezi) River Kariba Gorge, 290km away from

18、Solihey, the project is mainly for power generation.The dam site is composed of Precambrian rocks. The whole bed, the left bank and the right bank of the lower two-thirds of the height below are metamorphic biotite gneiss, gray and white surface with white stripes, foliation showing a bend. Rock out

19、crops in the river bed are hard, fine fractures, the depth of rock erosion between the two sides of 4.5 6m, susceptible to weathering. The upper part of the right bank is hard brown quartzite, with fractures developed, weathered biotite gneiss intrusions, and granitic granites appearing veins. Gneis

20、s is almost entirely biotite lens body, due to the distribution is not wide, the underground project will not have a significant impact.On November 11, 2023 It was announced by Zimbabwes Finance Minister, Patrick Chinamasa that capacity at the Zimbabwean (South) Kariba hydropower station would be in

21、creased by 300 megawatts. The cost of upgrading the facility has been supported by a $319m loan from China. The deal is a clear example of Zimbabwes Look East policy which was adopted after falling out with Western powers.7 Construction on the Kariba South expansion began in mid-2023 and is expect t

22、o be complete in 2023.8The north station belonging to Zambia has been in operation since 1976, and has four generators of 150 megawatts (200,000 hp) each for a total of 600 megawatts (800,000 hp); work to expand this capacity by an additional 360 megawatts (480,000 hp) to 960 megawatts (1,290,000 hp

23、) was completed in December 2023. Two additional 180 MW generators were added.91011Figure 1.1 zambezi river kariba dam1.4Kariba DamRecent activityOn the 6th of February 2023, the BBC reported that heavy rain might lead to a release of water from the dam, which would force 50,000 people downstream to

24、 evacuate. Rising levels led to the opening of the floodgates in March 2023, requiring the evacuation of 130,000 people who lived in the floodplain, and causing concerns that flooding may spread to nearby areas.In January 2023 it was reported that water levels at the dam had dropped to 12% of capaci

25、ty. Levels fell by 5.58 metres (18.3 ft), which is just 1.75 metres (5 ft 9 in) above the minimum operating level for hydropower. Low rainfalls and overuse of the water by the power plants have left the reservoir near empty, raising the prospect that both Zimbabwe and Zambia will face water shortage

26、s.1.5The removal of the dam, restoration and reconstructionThe Kariba Dam on the Zambezi River is one of the larger dams in Africa. Its construction was controversial, and a 2023 report by the Institute of Risk Management of South Africa included a warning that the dam is in dire need of maintenance

27、. A number of options are available to the Zambezi River Authority (ZRA) that might address the situation. Three options in particular are of interest to ZRA:(Option 1) Repairing the existing Kariba Dam, (Option 2) Rebuilding the existing Kariba Dam, or (Option 3) Removing the Kariba Dam and replaci

28、ng it with a series of ten to twenty smaller dams along the Zambezi River.II. The Description of the Problem2.1 Model Assumptions1) Assuming that the design parameters, site selection and construction of the dam (dam group) after restoration, reconstruction and reconstruction are in accordance with

29、the requirements of water conservancy and hydropower projects. The number and location of the dam groups are in line with the requirements of water conservancy and hydropower projects. Therefore, the research problems are: a) Comparing the comprehensive benefits of the dams after restoration, recons

30、truction and reconstruction, establishing the mathematical model, and expounding the costs and benefits of each case; b) Reconstructing the dams (rebuilding the existing dams 10-20 A little dam), in the case of drought, heavy rain, moderate rain and light rain, when the impoundment and flood dischar

31、ge can ensure the probability of dam break, the risk decision of dam group is optimized.2) The risk of flood discharge in emergencies is high, and this study does not consider the situation of emergency flood discharge.3) Assuming the ecological restoration of the water-level drawdown area after the

32、 demolition of the dam, construction waste transportation and sediment deposition are ideal.4) Assuming that the probability of dam break after rainfall to a certain water level is fixed.5) Assume that the rainfall is a continuous variable, obeying the normal distribution.6) Assuming that the dam gr

33、oup is a series model, do not consider the parallel and mixed2.2 Equirement 1.The annual economic benefits of the dam are made up of a number of benefits. These include power generation benefits, water supply benefits, irrigation benefits, shipping benefits, the value of flood storage, cultural tour

34、ism value, biological benefits, environmental benefits and so on. As a result of the various benefits, especially the power generation efficiency is changing year by year.The annual economic losses caused by the dam include food production, fishery production, organic matter production, soil erosion

35、 control, ecological restoration, dam maintenance, reservoir dredging, water treatment, species loss, loss of cultural relics.2.3 Equirement 21) Removing the Kariba Dam and replacing it with a series of ten to twenty smaller dams along the Zambezi river. 2) This new system of dams should have the sa

36、me overall water management capabilities as the existing Kariba Dam while providing the same or greater levels of protection and water management options for Lake Kariba that are in place with the existing dam.3)The decision tree model and the probability topic model are established to analyze the o

37、ptimal decision problem of water storage and flood discharge in the condition of drought, large, medium and light rainfall.III.Evaluation of Kariba Dam Project Based on Comprehensive Benefit Model3.1 Basic ModelThe annual economic benefits of the dam are made up of a number of benefits. These includ

38、e power generation benefits, water supply benefits, irrigation benefits, shipping benefits, the value of flood storage, cultural tourism value, biological benefits, environmental benefits and so on. As a result of the various benefits, especially the power generation efficiency is changing year by y

39、ear.The economic benefit model, the comprehensive benefit model, the absolute return model after restoration and the investment payback period model are established to optimize the cost and benefit of Kaliba Dam restoration work.3.1.1 Assumptions1) Assuming that the probability of dam break after ra

40、infall to a certain water level is fixed.2) Assume that rainfall is a continuous variable and obeys normal distribution.3) Assuming that the dam group is a series model, do not consider the parallel and mixed.4)It is assumed that the design parameters, site selection and construction of the dam (dam

41、 group) after repair, reconstruction and reconstruction are in accordance with the requirements of water conservancy and hydropower projects. The number and location of the dam groups are in line with the requirements of water conservancy and hydropower projects. Therefore, the research problems are

42、: a) Comparing the comprehensive benefits of the dams after restoration, reconstruction and reconstruction, establishing the mathematical model, and expounding the costs and benefits of each case; b) Reconstructing the dams (rebuilding the existing dams 10-20 A little dam), in the case of drought, h

43、eavy rain, moderate rain and light rain, when the impoundment and flood discharge can ensure the probability of dam break, the risk decision of dam group is optimized.3.1.2 The Foundation of ModelThe annual economic benefits of the dam are made up of a number of benefits. These include power generat

44、ion benefits, water supply benefits, irrigation benefits, shipping benefits, the value of flood storage, cultural tourism value, biological benefits, environmental benefits and so on. Because of the various benefits, especially the power generation benefits are changing year by year, so the economic

45、 benefits of the dam B is a function of running time t, the formula is:The annual economic loss D is also a function of time t, which includes grain production, fishery production, organic matter production, soil erosion control, ecological restoration, dam maintenance, reservoir dredging, water tre

46、atment, species loss, loss of cultural relics. . The calculation formula is:In general, the dam is built after the initial operation, the economic benefits are far greater than the economic loss of D. , But as the running time increases, the economic benefits tend to be stable, while the losses cont

47、inue to increase. According to the size of the relationship between the two can take measures to remove the dam, repair measures, or not to take measures. Of course, after the restoration of the dam in the short term the economic benefits will be improved, but with the gradual increase in losses and

48、 benefits with the flat, the dam will once again face demolition.Generally speaking, after the completion of the dam is completed, the economic efficiency of rats is far greater than the economic losses. , But as the running time increases, the economic benefits tend to be stable, while the losses c

49、ontinue to increase. According to the size of the relationship between the two can take measures to remove the dam, repair measures, or not to take measures. Of course, after the restoration of the dam in the short term the economic benefits will be improved, but with the gradual increase in losses

50、and benefits with the flat, the dam will once again face demolition.In general, the absolute benefit A after repair or reconstruction can be calculated by the following formula：Where: g (t) for the repair or reconstruction of the annual income function; f (t) for the repair or reconstruction of the

51、annual loss function.According to the payback period to calculate the payback period T, the formula is as follows：Where: R is the average annual net income; F1 is a one-time investment in repair; i for the benchmark rate of return.If the extended useful life of a dam is less than the cost recovery p

52、eriod after restoration or reconstruction, the restoration or reconstruction program has no economic value. On the contrary, it shows that the repair program can achieve better economic results.Therefore, the investment recovery period T can be used as an important index to evaluate the choice of da

53、m restoration scheme.According to the above formula, and using the Kaliba reservoir data to calculate the annual economic benefits of the dam over time curve：Similarly, the annual loss curve can be obtained：3.1.3 Solution and Result1) RepairingAccording to the specific conditions of Kaliba Reservoir

54、, six rehabilitation methods are proposed: one is to repair the concrete of the dam body, which can reduce the loss of the dam, but it can not effectively increase the dam yearly To reduce the loss of the dam, but it can not effectively increase the annual benefits of the dam; third is to add the fi

55、sh to the surface of the dam to increase the pressure; And reduce the ecological losses caused by the dam, but will not significantly change the economic benefits of the dam; Fourth, the reservoir dredging, this approach can effectively increase the drainage, improve power generation efficiency, but

56、 will not reduce dam losses ; Five is the pool of the repair; six is the spillway gate repair. The cost of the restoration project for the Kaliba hydropower station in Zambia is estimated at $ 294 million.2) RebuildingRebuilding the existing Kariba Dam.3) Removing and RebuildingRemoving the Kariba D

57、am and replacing it with a series of ten to twenty smaller dams along the Zambezi River.2) RebuildingRebuilding the existing Kariba Dam.The rebuilt budget includes demolition costs and new construction costs. The construction cost of the Kaliba Dam is estimated to be US $ 4.8 billion. The total cost

58、 of demolition and reconstruction is estimated to be US $ 6 billion. The design is 100 years and the recovery period is 65.2 years. The economy of the program is significantly better.3) Removing and RebuildingRemoving the Kariba Dam and replacing it with a series of ten to twenty smaller dams along

59、the Zambezi River.The rebuilt budget includes demolition costs and new construction costs. The cost estimate for the reconstruction of the Kaliba Dam is $ 4.4 billion. It is designed for 100 years and the investment recovery period is 47.8 years. The economy of the program is significantly better.In

60、 summary, for all three different focus of the dam restoration and reconstruction program for comparison, the most reasonable repair program: the Kaliba dam removal, alteration multiple dams system, not only can extend the dam for 100 years, But also can give full play to the economic benefits of th