2017年美国大学生数学建模竞赛O奖论文A-65448.pdf资源-CSDN文库

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For ofﬁce use only

Team Control Number

65448

Problem Chosen

For ofﬁce use only

2017 Mathematical Contest in Modeling (MCM) Summary Sheet

(Attach a copy of this page to each copy of your solution paper.)

Abstract

The Kariba Dam on the Zambezi River is confronted with foundation errosion and control limitation cur-

rently, so a strategy to maintain water management on the Zambezi River is of great value for the whole

Zambezi basin. Based on the detailed data provided by World Bank, we conduct the assessment of three op-

tions (repairing the dam, rebuilding the dam or replacing the dam with several smaller dams along the river)

and provide a reasonable and detailed model to construct a new dam system with some emergency strategies.

To begin with, a brief assessment report with potential costs and beneﬁts is offered according to abundant

references and reliable data. The evaluation of repairing the Kariba Dam is completed refering to the Kariba

Dam Rehabilitation Project in the charge of World Bank, while the assessment of rebuilding the dam is realized

following the actual capital investment expenditure of the Stage 1 of the Kariba Project. And the assessment of

replacing the dam is carefully estimated according to the model to construct the new dam system.

Then, we concentrate on the establishment of the model for new dam system.

Firstly, a Hydrodynamic Model (HM) of the Zambezi River is established in order to estimate the ﬂowing

and depth situation of the river. The model is constructed based on ﬂow conservation and energy conservation,

which separately producing continuity equation and energy conservation equation, taking the normal water

cycles and on-way resistance into account. Moreover, we obtain the numerical solution of the Hydrodynamic

Model with Mathematica, and get the approximate current velocity curve and river depth curve along the

river.

Secondly, the site selection and number conﬁrmation of the new dams is realized according to the ﬂowing

and depth situation estimated by HM, geological data and hydrological data. The indexes for evaluation

are determined after consulting professional literatures, and Grey Relation Grade Analysis is utilized in the

selection among candidate sites so that all the indexes can be considered comprehensively and objectively.

Thirdly, a Difference Model (DM) of the Zambezi River with a dam system is established to provide suf-

ﬁcient details for the conﬁrmation of dam construction and modulating strategies. We separate the river into

several segments by the dams. A difference approximation of HM is applied to describe each segment of the

river. A hydraulic emprical formula is utilized to determine the discharges of the dam. In this way, the re-

currence formulas can be obtained to easily determine the runoff and river depth along the river with given

environmental and initial conditions.

Fourthly, the type of each dam is conﬁrmed with corrected DM, with the overall water management capa-

bilities, protection and water management options for Lake Kariba evaluated and special topograph consid-

ered. Considering the loss of energy in collision, tributaries and the runoff variation with rainfall at headstram

and tributaries, DM is corrected to adapt to the reality. With the results calculated by corrected DM, the type of

each dam is conﬁrmed balancing the safety and costs. Each type of dam has three gears of discharges, which

increases the water management options.

Finally, the strategies to handle such emergency water ﬂow situations as ﬂooding and drought are designed

with the new dam system established above. When solving a known catastrophe, the exhaustive method is

utilized to rule out the best strategies which realized by the gear variation of the dam. It is validated that the

new dam system can handle most extreme conditions and the method we adopted to establish the strategy is

of low sensibility considering reasonable errors.

Team # 65448 Page 1 of 25

A Multi-Dam System Design for Zambezi River

January 24, 2017

Contents

1 Assessment Report of Three Options 1

1.1 Repairing the Existing Kariba Dam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Institutional Project Support (estimated cost US$69.6 million) . . . . . . . . . . 1

1.1.2 Plunge Pool Reshaping (estimated cost US$100.0 million) . . . . . . . . . . . . . 1

1.1.3 Spillway Refurbishment (estimated cost US$124.6 million) . . . . . . . . . . . . 1

1.1.4 Beneﬁts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Rebuilding the Existing Kariba Dam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Removing the Kariba Dam & Replacing It with A Series of Ten to Twenty Smaller Dams

Along the Zambezi River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Introduction 2

2.1 Restatement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 General Assumption 4

4 Notations 4

5 The Model 5

5.1 Hydrodynamic Model of the Zambezi River . . . . . . . . . . . . . . . . . . . . . . . . . 5

5.1.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5.1.2 Energy Conservation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.1.3 Boundary Condition and Initial Condition . . . . . . . . . . . . . . . . . . . . . 6

5.1.4 Calculation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.2 Selection System of the Dam Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.2.1 Evaluation indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.2.2 Candidate Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.2.3 Grey Relation Grade Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3 Difference Model of the Dam System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.3.1 Equations in the River (Without the Dam) . . . . . . . . . . . . . . . . . . . . . . 12

5.3.2 Equations in the Dam Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3.3 Recurrence Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.4 Dam Construction & Manipulating System . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.4.1 Conﬁrm the Parameters of Different Dam Types . . . . . . . . . . . . . . . . . . 14

5.4.2 Conﬁrm the Type Selection of Each Dam . . . . . . . . . . . . . . . . . . . . . . 14

5.4.3 Evaluate the Protection & Water Management Options for Lake Kariba . . . . . 17

5.4.4 Strategies to Handle Emergency Water Flow Situations . . . . . . . . . . . . . . 17

6 Testing the model 19

6.1 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.2.1 Sensitivity Analysis for H

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.2.2 Sensitivity Analysis for h

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Team # 65448 Page 1 of 25

1 Assessment Report of Three Options

1.1 Repairing the Existing Kariba Dam

At present, the foundation of the Kariba Dam is gradually undermined by scouring and erosion

of water, so that operations are limited to only three of the six, non-adjacent gates. The Kariba Dam is

also suffering frequent maintenance operations and potential limitations on the operational control

of the reservoir. Therefore, plunge pool reshaping and spillway refurbishment are necessities. What

is more, institutional project support are also signiﬁcant to ensure the project running smoothly.

1.1.1 Institutional Project Support (estimated cost US$69.6 million)

Institutional project support mainly provides overall management, supervision, professional

consultant, comprehensive assessment and risk evaluation for the project.

1.1.2 Plunge Pool Reshaping (estimated cost US$100.0 million)

Financing in the component is primarily utilized for goods, works and consultants services,

which is constituted with four parts: a. Plunge Pool Reshaping Civil Works Contract: The measures

are required to reshape the plunge pool through excavation of the downstream face and north and

south bank sides of the pool; b. Additional Engineering Studies; c. Monitoring Services Contract; d.

Visibility & Communications.

1.1.3 Spillway Refurbishment (estimated cost US$124.6 million)

Financing in the component is mainly used for goods, works and non-consulting services as-

sociated with the refurbishment of the spillway, which is composed of two parts: a. Spillway Re-

furbishment Contract: The works will be conducted at one gate at a time, in a sequenced manner

starting with the gates that today show the largest need for rehabilitation; b. Additional Engineering

Studies.

1.1.4 Beneﬁts

Besides the elimination of potential danger and operational limitation which is mentioned

above, the repairing of the dam also reinforce the integrated infrastructure platform within the

Zambezi River basin. The beneﬁts through avoided disaster and continued power production are

substantial, considering the estimated 3 million people living in the potential impact area and the

economic value in excess of US$8 billion.

1.2 Rebuilding the Existing Kariba Dam

After 50 years of operation serving the southern African region, the geological conditions under

the Kariba dam have been changed with the scouring and erosion of the water, so that the initial

design of the dam is not appropriate for the new conditions. In this way, rebuilding the dam is a

reasonable alternative to replace and even reinforce the function of the old one, and Lake Kariba is

also utilized as a reservior. The ﬁnancing of the project will be primarily utilized for civil engineering

works and electrical and mechanical works.

• Civil Engineering Works (estimated cost US$729.1 million): The component will include the

costs of access roads, township, dam wall, south bank power station and engineering fees.

• Electrical and Mechanical Works (estimated cost US$380.6 million): The component will in-

clude the costs of generation plant and equipment, transmission system and engineering fees.

Team # 65448 Page 2 of 25

The rebuilt dam can be designed to suit the new geological conditions and eliminate the over-

whelming erosion confronted in the previous design. The beneﬁts through avoided frequent main-

tenance of the dam and improved manipulating capability are substantial, given the lifespan and

signiﬁcance of the dam for the approximately 3 million local residents.

1.3 Removing the Kariba Dam & Replacing It with A Series of Ten to Twenty Smaller

Dams Along the Zambezi River

During the construction of the Kariba Dam, the dam suffered from three huge ﬂoodings exceeding

the design ﬂood. And the dam became broken quickly in ﬁfty years, which was quite shorter than the

designed lifespan.

All these facts ﬁgure out that more dams should be constructed on the Zambezi

River in order to prevent the unexpected ﬂoodings. In this way, replacing the Kariba Dam with a

series of ten to twenty smaller dams is an alternative which are more likely to balance the costs and

safety.

As a series of so many dams is quite rare in the world, the costs can only be estimated approx-

imately by the number and size of the dams. Financing of the project is mainly utilized for dam

construction and relevant institutional project support.

• Dam Construction (estimated cost US$860 million): The costs of the dam construction are as-

sumed to increase exponentially with the dam size increasing. Therefore, the costs of different

dam types (which are introduced in the following model) are listed in Table 1. According to the

following model, eight III dams and two IV dams are constructed, which cost US$860 million.

• Institutional Project Support (estimated cost US$69.6 million).

Types of dam The area of Reservior/m

Warning water levelH

/m Costs/million US$

I 1 × 10

60 10

II 1 × 10

50 1

III 1 × 10

40 0.8

IV 1 × 10

30 0.6

Table 1: Costs of different dam types

The construction of the dam system can balance between the safety and costs and have excellent

functions to prevent unexpected ﬂoodings, whose beneﬁts through avoided catastrophe is substan-

tial, considering the powerful water management capabilities and more abundant hydropower.

2 Introduction

2.1 Restatement of the problem

Kariba Dam, constructed in 1955-59, with a storage capacity of 180km

, is one of the largest dams

in Africa. Now it is in dire need of maintenance. The Zambezi River Authority (ZRA) faces three

available options to address this situation:

• (Option 1) Repairing the existing Kariba Dam;

• (Option 2) Rebuilding the existing Kariba Dam;

• (Option 3) Removing the Kariba Dam and replacing it with a series of ten to twenty smaller

dams along the Zambezi River.

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