Managing the Zambezi River

The Kariba Dam is one of the larger dams in Africa, while it is also in dire need of maintenance. The evaluation model and related calculation methods are built to measure potential benefits and costs of three maintenance programs. Moreover, a detailed analysis is provided by the calculation method and the number and placement of the new dams along the Zambezi River are proposed.


Introduction
After more than 50 years, the Kariba Dam provides power for the Southern African Region, while it is found that its rehabilitation must be required to safe operation. Commonly, there are three particular options that Zambezi River Authority (ZRA) might choose:  Repairing the existing Kariba Dam;  Rebuilding the existing Kariba Dam;  Removing the Kariba Dam and replacing it with a series of ten to twenty smaller dams along the Zambezi River.
To evaluate three options listed, it is very necessary to build mathematical models, provide a detailed analysis of Option 3 and analyze the extreme water flows.

Overview
The economic value system is built from the potential benefits and costs according to the impact of the dam on river ecosystem. Moreover, the market value method, opportunity cost method, shadow engineering method of direct market method and alternative market are applied to evaluate on this issue. can protect the cultivated land from submerged, which indirectly creates benefits. According to opportunity cost method, the formula is gotten as followed: (2.2) Namely V is the total economic value from regulation of floodwater; S is the annual reduced flooded are; S is the annual reduced the affected population; P is the loss of unit flooded area; P is the per-capita loss because of flood.  The Benefit of Reducing Harmful Gas: Replacing coal-fired power generation with hydroelectric power can effectively reduce emissions of harmful gases. The Shadow Engineering method is applied to assess these values and use the formula below:

Detailed
is the economic value of reducing Carbon dioxide; is the reduction of Carbon dioxide; is the afforestation cost. Similarly, we could calculate economic value of reducing Sulfur dioxide and get the total value: (2.4)

The Potential Costs
 The Cost of the impact on River Ecosystem: We preliminary account for the impact on the river ecosystem services.  The Cost of Submergence: Complex ecosystems are destroyed because of the construction of the dams. Here we adopt the opportunity cost method to calculate the cost mentioned here: V ∑ ∑ S P (2.5) In the formula above, V is the loss value of reservoir submergence. S is the forests, grasslands, wetlands and marshes and arable land areas flooded by the dam, respectively.  The Cost of Construction: The formulas are used according to our analysis: (2.6) In the formula above, Cci(t) devotes the cost of construction of option i Considering the time value of money, we analyze the maintenance fee by using the Capital Equivalent Formula [2] : A is the annual value, P is the present value.  Cost of the decay of dam: We should assess the cost of the decay effect. Here are two models to describe the effect: Straight Line Depreciation and Exponential Depreciation [3] . The formulas are shown as below respectively: C is the cost of construction; L is the lifespan of a dam; S is salvage value.  In Table 1, it is obvious that the construction fee of the dam occupies the main part of potential costs for all options. The construction fee in Option 3 is more than others. In Table 2, it is found that total benefits of Option 3 are more than others. The benefit from Power Generation is far more than others, while the benefits from Tourism and Flood Storage of Option 3 are less than others.

Overview
The multiple-dam model is applied to discuss Option 3. The model includes three parts:  Generating potential dam sites;  Calculating suitability of generated site;  Recommending the number of small dams.

Detailed Design 3.2.1 Multiple Dam System
Model. The derivative of height and width of the dam is calculated over length of the Zambezi River. The sites satisfying the equation, where derivative values are zero, is chosen as potential dam sites. 25 generated potential dam sites are obtained. Five criteria (soil type, land cover, degree of slope, resistance of geological layer and predicted precipitation) are considered as main factors.

Analytical Hierarchy
Process. AHP provides an integrated measurement on tangible factors with different priority by pairwise comparison. We use a linear function assign preference value to different classes of all criteria to calculate the suitability of all the potential dam sites. With comparison, the weight of criteria i, which is used in later analysis for suitability analysis [4] , could be calculated with the formula below. Where is relative importance in pairwise comparison of criteria and . Using the formula above, with the criteria judging matrix A we could calculate the weight of criteria. Suitability of dam sites is calculated as weighted summation of different criteria layers. Based on the result of AHP, 20 possible dam sites are selected with the higher suitability.

Recommend Number of Dam Sites.
According to Knapsack Problem [5] , the formula is listed.

Conclusions
To make the assessment of three options, the potential benefits and potential costs are chosen as criteria. Then the weight of each item is calculated. Three options are compared and the differences of them are found in Model One. In addition, the number and placement of small dams are discussed. The extreme points are regarded as the potential dam sites because these points are more flat so that they are more suitable to build dam here. Then 25 potential sites are generated. To compress the number into 20, AHP is adopted and the suitability of each site is discussed. Finally, the question is regarded as Knapsack Problem. It is calculated that the optimal number of small dams is 14.