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**MATHEMATICAL MODEL FOR EXPECTED POWER SAVINGS**

Energy efficiency projects in the residential sector are performed by humans, therefore quantifying the social impact through social communication will give the total expected power saving in every energy efficiency project. The social effect of an individual is dependent on their peer-to-peer interactions; this can pinpoint the most influential people in a community and thus reveal to energy planners the people to target in spreading information about the energy efficiency projects. Identifying people who will spread the information about the energy efficiency project to the network fastest is significant because this will help change people’s behaviour towards energy conservation and thus increase energy savings at little or no cost.

The mathematical model of the expected power savings calculates the combined direct and indirect savings of the energy efficiency project. In the model, the physical distance between two people is not ignored, two people are said to be connected if there is a mutual acknowledgement of acquaintance between them. The nodes represent the households and the edges represent the connection between two households. The mathematical model of the expected power savings considers two scenarios; when there is a focus on one or multiple end users to transfer information to the rest of the network. This model will try to dispute the instinctive belief that people with the highest node degree have the ability to spread the most information in the network. As people grow further apart from one another, the influence of their information transferred is reduced, as shown in Figure 3.1 where the boxes represent the information transferred from the source. As the boxes move further away from the source, the colour representing the information transfer becomes lighter as this means the impact on the receiving node is reduced. The greater the path length between the receiver of the information and the source node, the smaller quantity of the information transferred. In the calculation of pi( j) for a medium sized network, the only cases considered are when j is connected to the source node i with degree of connection of at most four. This is a good approximation of the latest research on social networks that an « individual is separated from anyone in the world by an average characteristic path length L = 4.74 people ».

In a network, the conditional probability pi( j) that an information source node i can transfer information to another node j is dependent on how the two nodes are connected to each other and to other nodes in the network. Note that information transferred along shorter paths are always dominant when compared to the information transferred along longer paths. Therefore, it is reasonable to consider only information transferred along the shortest paths when considering the definition of pi( j). Meaning, information transferred along further paths will be disregarded, and if the shortest path between i and j is not singular, then information transferred along all the shortest path will be added together to give the full conditional probability.

Explaining this in practical terms, it means that the more a person hears about the benefits of a product (for example the use of CFLs) from more than one friend, the more probable it becomes that he/she will be convinced to obtain that product. Therefore, the conditional probability does not only focus on the source node who transfers the information but also on the receiver’s different access to the information. The following cases are explained in the description of pi( j). In (3.10) the second degree of information transfer is dependent on the information already transferred from the source node to the first degree node q 2Mi\Mj. The second degree node j treats the first degree node q as its source of information which is dependent on the amount of information that is passed to q from the source node i. This implies that q transfers the information he/she obtained from i to j. This shows the continuity of information transfer among nodes in the network. The addition of the probability of total number of nodes q between i and j indicates that when one hears about a lifestyle change from several friends the higher the chances of that person adopting that lifestyle change.

This can also been viewed from the receiver’s side, for example, the more people tell him/her about their savings through the retrofitting of their home, the more likely this person will change in order to obtain those savings. If j decides to implement this lifestyle in order to save, it does not mean that j will buy the retrofits that her friends tell her, as this may be unrealistic or not cost effective.

This means that the more information j obtains about savings from his/her friend the more probable he/she will be willing to adjust to that lifestyle. Furthermore, it confirms that the social impact i has on j through information transfer is less than the impact i has on q and this depicts real life scenarios where the influence of one’s friends are bigger than the impact of a friend of a friend.Case III and IV follow the same thought pattern as case II.

**SOLUTION METHODOLOGY**

We assume that it is cheaper for neighbours of j to obtain information from i because it is free rather than find information about energy efficiency measures through other means that may cost money and time. The model is solved with the use of the Java programming language on a 32-bit processor. The reason for using Java is because it can process a large network. The solution methodology is as follows; For a single source of information in the network;

1. Assume that general external information is available to the whole network and this information is complete.

2. Obtain the direct savings Si, of each node. Calculate the functional p(i), conditional pi( j) and joint p(i; j) probabilities and entropy H(i). Calculate the indirect savings obtained from the entropy and direct savings. Calculate the expected power cost savings for all i as the sum of the direct and indirect savings.

3. Find the node i with the maximum expected power cost savings max(Fi) output solution. For multiple sources of information in the network;

1. Assume that general external information is available to the whole network and this information is complete.

2. Find all the possible combinations for the number of sources required using a brute force search algorithm. The brute force search algorithm exhaustively searches through all the possible combinations until the optimal solution is found. For example, if we assume a multiple source network of 3 households in a 56 household network, then there will be ?3 56 = 27720 different combinations.

3. Obtain the direct savings Si, of each of the sub-networks. Calculate the functional p(i), conditional pi( j) and joint p(i; j) probabilities and entropy H(i) accordingly.

Calculate the indirect savings obtained from the entropy and direct savings. Calculate the expected power cost savings for all i as the sum of the direct and indirect savings.

4. Find the sub-network i with the maximum expected power cost savings max(Fi) output solution.

**CASE STUDY I**

The South African government has partnered with the local utility company Eskom to provide some limited, free low-pressure solar water heaters to residential houses within South Africa. When the household to receive the free solar water heaters are chosen, a member of the household has to be present while the installation is carried out. After the installation, a brief description of the solar water heater and lessons on how to use the heaters are given. The benefits of the solar water heater are highlighted to the member of the household [68]. This is done with the expectation that the person talks about the efficiency of the heater to his/her friends. The transfer of such information leads to more people purchasing the solar water heaters for their houses and as such reduce electricity costs and save energy.

People are connected to each other through various means and as such information is transferred from one household to another. The reasons any two households are connected to each other are based on different factors such as environmental proximity, members of the same organisation, have children in the same school or work at the same office. In this research, a survey was carried out on a group consisting of fifty-six households from the same church organisation to obtain data for the social network graph. Each household is given a questionnaire to write out the names of other households they consider as friends within the group.

After the necessary information has been collected, an adjacency matrix is constructed. The criterion for the graph is that two households must acknowledge that they are friends with each other before an edge can be drawn between them. The network graph is given in Figure 3.2.

There are two examples presented in this case study, the first example is when there is only one person to be given a new solar water heater and when there are more people to be given solar water heaters. The aim of the installation of the solar heaters is to promote renewable technology and to encourage people to buy the solar water heaters. The use of the solar water heaters reduces the electricity bills and electricity consumption of the entire community. In order to maximise the indirect savings due to social impact, the criteria for houses to receive free solar water heaters will be based on how much power is saved and how much impact these households have on their community. As the direct savings is fixed, the indirect savings will determine the person who has the most expected power savings.

**CHAPTER 1 INTRODUCTION **

1.1 PROBLEM STATEMENT

1.2 RESEARCH OBJECTIVES AND QUESTIONS

1.3 HYPOTHESIS, TASK AND APPROACH

1.4 VALIDATION OF THE PROPOSED RESEARCH

1.5 RESEARCH CONTRIBUTION

1.6 OVERVIEW OF THE STUDY

**CHAPTER 2 LITERATURE STUDY **

2.1 CHAPTER OBJECTIVES

2.2 RESEARCH METHODOLOGY

2.3 BACKGROUND OF THE STUDY

2.4 ENERGY EFFICIENCY IN SOUTH AFRICA

2.5 COMPLEX NETWORKS

2.6 INFORMATION PROPAGATION

2.7 RATIONALE OF THE STUDY

2.8 CONTRIBUTIONS OF THE STUDY

2.9 LIMITATION OF THE RESEARCH

2.10 CHAPTER SUMMARY

**CHAPTER 3 INFORMATION TRANSMISSION MODEL **

3.1 INTRODUCTION

3.2 MATHEMATICAL MODEL FOR EXPECTED POWER SAVINGS

3.3 LIMITATIONS OF THE PROPOSED MODEL

3.4 SOLUTION METHODOLOGY

3.5 CASE STUDY I

3.6 ASSUMPTIONS

3.7 CHAPTER SUMMARY

**CHAPTER 4 EXPECTED ENERGY COST SAVINGS MODEL WITH RESPECT TO TIME**

4.1 INTRODUCTION

4.2 DEMAND SIDE MANAGEMENT

4.3 MATHEMATICAL MODEL OF EXPECTED ENERGY COST SAVINGS MODEL

4.4 SOLUTION METHODOLOGY

4.5 CASE STUDY II

4.6 LIMITATIONS

4.7 CHAPTER SUMMARY

**CHAPTER 5 SOCIAL INFLUENCE AND ENERGY EFFICIENCY SAVINGS**

5.1 INTRODUCTION

5.2 THE EXPECTED ENERGY SAVING MODEL BASED ON INFLUENCE MATHEMATICAL MODEL

5.3 SOCIAL INFLUENCE

5.4 CASE STUDY III

5.5 SOLUTION METHODOLOGY

5.6 LIMITATIONS

5.7 CHAPTER SUMMARY

**CHAPTER 6 RESULTS AND DISCUSSION **

6.1 INTRODUCTION

6.2 RESULTS OF CASE STUDY I

6.3 RESULTS OF CASE STUDY II

6.4 RESULTS OF CASE STUDY III

6.5 REPRODUCIBILITY OF THE RESULTS PRESENTED

6.6 CHAPTER SUMMARY

**CHAPTER 7 CONCLUSION**

7.1 SUMMARY OF RESEARCH

7.2 LIMITATIONS OF THE STUDY

7.3 FURTHER STUDIES AND RECOMMENDATIONS