全美数学建模大赛参赛二等奖作品

Energy consumption change during the process of cell phonerevolutionSummaryThis paper discusses the energy consumption change in the cell phone revolution. By theoretical analysis and data fitting, we find Logistic carve is the best to illustrate the relationship between energy consumption arising and time in the current US. We make a model and predict that 35 years later, the energy consumption will become steady with a maximum of 1.16×109 MJ / year.To get optional way of providing phone service for the “Pseudo US”, we built a model considering two factors: social desire for using cell phones and social desire for energy saving. Finally, we conclude that 259.8 million people use cell phones and 15 million households use landlines.The wasteful deeds of people with cell phones will waste a lot of energy. We make a model to find that the quantity is as huge as 2 million barrels of oil a year. By using the same theory, we introduce “the appliances standby losses formula” into our modelto get that America will waste about 246000 barrels of oil because of wasteful habitsof using appliances.Considering the population and economic growth over the next 50 years, we improve the first model and use the results of the first three models to create a new model to predict the energy needs of a typical Pseudo US for providing phone service for each 10 years over the next 50 years. The results in order are 830000 barrels, 5260000 barrels, 11200000 barrels 37440000 barrels and 39612000 barrelsFinally we give three prospective suggestions to American based on our paper.ContentsIntroduction (3)ⅠThe replacing landlines by cell phones model (3)Assumption: (3)Results (9)Strengths and weaknesses (9)Ⅱthe optimal way of providing phone service to “Pseudo US” model (9)Assumption (10)Results (12)Discussion (12)Strengths and weaknesses (13)Ⅲ T he model of energy cost of wasteful practices by the usage of cell phones (13)Assumption (14)Results (15)Strengths and weakness (16)ⅣThe model of energy wasted by the appliances left plugged in (15)Results (17)Strengths and weaknesses (18)ⅤThe energy needs for providing phone service in a growing “Pseudo US” model (18)Assumption (18)Results (27)Strengths and weaknesses (27)Positive Suggestions (28)References (28)IntroductionThere is no denying that cell phones have many social influences and uses that landline phones do not allow, for example, we can call anyone who own a cell phone at any place by any time. Therefore, many people are using cell phones and giving up their landline telephones. This is a sign of social and technological progress .However it may cause some problems of energy consumption.It can cause changes in electricity consumption by replacing landlines with cell phones. Also, a certain mount of electricity is wasted because of the bad habit of people who have cell phones. For example, charge their phones every night, whether they need to be recharged or not. This makes us think about the amount of energy that is wasted by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device. Furthermore, consider population and economic growth over the next 50 years, how much energy will be needed for providing phone service? To give appropriate answers to these questions, we need to build five models:z The replacing landlines by cell phones modelz The optimal way of providing phone service to “Pseudo US” modelz The model of energy cost of wasteful practices by the usage of cell phones.z The model of energy wasted by the appliances left plugged inz The energy needs for providing phone service in a growing “Pseudo US” model ⅠThe replacing landlines by cell phones modelSince the cell phones have so many merits over landlines, their usage is mushrooming. Many people change their landlines to cell phones. We can make a model to predict the energy consumption change if the landlines are replaced by cell phones.In order to predict the energy consumption change, we make an Elementary model with linear fitting.Assumption:z When every people have a cell phone, they will remove their landlines. The circumstance that just parts of the family members own a cell phone is not existed.z The condition of the population and economy is not changed over the period.z The increased number of people who own a cell phone is the same.z The signal transmission tower can finally cover USA territory when the numbersof people with cell phones become steady.z The signal transmission towers have Linear relationship with the number of thepeople who have cell phones.NomenclatureTable 1 Nomenclatures and Description Nomenclature Description Unitch WThe energy consumption change in a year J mix WThe energy used by landlines and cell phones a year in a certain year J land W The energy used by landlines when there are not cellphonesJ P The total population of USA million H The number of the households in the USAmillion M The average member of a household in the USA1H The number of household whose members do nothave cell phonesmillion T Δ Years from 1985 to TT The time1J The average energy used by a landline in a yearJ 2J The average energy used by a cell phone in a yearJ 2m The number of people who own a cell phonemillion 3mThe number of signal transmission tower in the USA million kA coefficient η The coefficient of the loss and damage of cell phones 3JThe average energy used by a signal transmission tower a year in the USAJ Energy consumption change ch W can be easily got from:land mix ch W W W −= (1)We consider the energy consumption of mobile Communication System contains twoparts: the energy used by signal transmission towers and the energy used by cellphones. Then we have:112233J H J m J m W mix ++= (2)According to the first assumption, we get:Mm H M m P H 221−=−= (3) Plugging formula (3) into the equation for formula (2) ,we obtain:112233)(HJ MJ J m J m W mix +−+= (4)The energy consumption land W is:1HJ W land = (5)According to the fifth assumption, we can get23km m = (6)Putting (4) (5) and (6) into (1), we can get:)()(123212232MJ J kJ m M J J m J km W ch −+=−+= (7)Considering the loss and damage of the cell phones, we pull into the coefficient η:)(1232MJ J kJ m W ch −+=η (8)We use formula (8) to estimate the energy consumption change when landlines are replacing cell phones. ch W has a direct proportion relation with 2m .)(123MJ J kJ −+η is a constant. Solving as follows: ①we can approximately get the power of a signal transmission tower is W 100[1],so J J 931015.3365243600100×=×××=.②Now we start to fix the coefficient k :The Coverage radius is mile R 31=[2] . and, the territory of the USA is23537438mile S =[3],Then the number of the signal transmission tower117223==RS m π. Synthesizing parameter above, we can get the coefficient: 68109.31031172−×=×=k .③From internet, we know that charging the battery of a cell phone a time need J 4106.31000360001.0×=××[4]. We estimate that every cell phone need to be recharged every 3 days .Therefore,J J 6421038.43365106.3×=××=.④We neglect the energy consumption change when we use landlines ,and only think about the standby power consumption. We get the data from the internet: the voltage and the current of the landlines are V 10 and mA 20[5]. We suppose the average landline use 1 hour everyday .So we getJ J 5311063.23653600102010×=××××=−.⑤We searched the internet and found 81017.1×=H [6], so =M 56.21017.110388=××. We suppose 9.0=η From above, the constant is:61231087.3)(×=−+MJ J kJ η So Formula (8) can be expressed as 261087.3m W ch ×=)(2T f m Δ=,We get the data from the internet [7]. And draw the figure .1 followingFigure 1The increasing trend is becoming faster and faster from above picture. However, as we all know, there is a limit of the population with cell phones which we consider to be 300 million. So we use Logistic Model which can fit the situation better to model the relationship between 2m and T Δ.In the model, Use formula)(02)1(1T r m me x x m Δ−−+= (9)We evaluate 0x with the population with cell phones in 1986We use formula (9) to fit the real situation with the help of MATLAB and finally get the appropriate value of the parametersmillion x m 300= 345.0=rWe draw figure 2 with MATLAB to illustrate the relationship between 2m and T Δ.Figure 2According to the formula 261087.3m W ch ×= that we have already got through our calculation, we can get the energy consumption change ch W in different years. The result is in figure 3.Figure 3ResultsThrough the model, we can conclude that trend of electricity utilization is like a ‘S ’carves, when landlines are being replaced by cell phones. In 35 years, when the population of cell phone owner has reached 299.25million, the situation would have reached a steady state. In this state, the maximum energy change from all the peopleusing landlines is about MJ 91016.1× peryear.Strengths and weaknessesStrengthsOur model considers all the facts that affect the energy consumption change when the landlines are replaced by cell phones. We use Logistic Model that can appropriately express the relationship between 2m and T Δ.WeaknessesSome of our data is indirectly calculated from other data. And η is estimated by previous experience. So the result contains certain error.Ⅱthe optimal way of providing phone service to “Pseudo US” modelTo provide phone service to a country of about 300 million people with about the same economic status as the current US, we must consider two factors. First, we must consider the society’s desire for energy saving. Second, we must consider the society’s desire for enjoying the convenience brought by new technology.Assumptionz The society’s satisfaction to the desire for enjoying the convenience brought bynew technology is linear with the population with cell phones.z The society’s satisfaction to desire for energy saving is inversely proportional tothe society’s satisfaction to the desire for enjoying the convenience brought by new technology.Nomenclature Table 2Nomenclature DescriptionUnit s The society’s satisfaction to the desire forenjoying the convenience brought by newtechnologyp The society’s satisfaction to desire forenergy savingx The population with cell phones millionϕ A coefficientAccording to the first assumption, we can get:x s ϕ= (10)According to the second assumption, we suppose the relationship between s and p is:1=ps (11)According to (10) and (11), we can get:xp ϕ1= (12)According to (10) and (12), we can draw figure 4 and figure 5 to show the relationship between s and x and the relationship between p and x .Figure 4 Figure 5We can easily find that when the population x rises, s rises but p decreases. Therefore, there must be an optimal value X that can make both s and p accepted by the maximum people.Also we can draw figure 6 from (11) to show the relationship between p and s.Figure 6From a energy prospective, we should make p larger. But thes can not be infinitely small. We mean that at the current economic situation, the s must be lager than orequal to a certain figure. So we consider the s .We get the household income from the internet[8]Table 3Subject HouseholdsSubject HouseholdsLess than $10,000 7.60%$50,000 to $74,999 18.80% $10,000 to $14,999 5.80%$75,000 to $99,999 12.10%$15,000 to $24,999 11.10%$100,000 to$149,999 11.40%$25,000 to $34,999 11.00%$150,000 to$199,999 3.90% $35,000 to $49,99914.60%$200,000 or more3.70%To make sure s , we must take the current situation of the USA into account. America is a developed country whose people are rich and like to enjoy the convenience brought by new technology. Another factor we should consider is that when rich people don’t have cell phones, s will drop a lot ,but s will drop littler when the same situation happens to the poor people. As a result, we can estimate that if the household income lager than $15,000, the members are advised to use cell phones, if not ,the member are advised to use landlines.Suppose that every household has the same members, we can getmillion x 8.259%)8.5%6.71(300=−−×=ResultsThe optional way of providing phone service to the “Pseudo US” is that 259.8 million people use cell phones and about million million 15%)8.5%6.7(6.111=+× households use landlines.DiscussionFrom above analysis, we can have a discussion on different conditions.z A discussion of the broad and hidden consequences of having only landlines:Surely that this way of providing phone service can save a lot of energy. But theAmerican people will not be satisfied if they can not enjoy the convenience brought by new technology. As a result, the s will become very low and the situation will be rejected by most people.z A discussion of the broad and hidden consequences of having only cell phones: Surely that this way can get the largest s.But consider the energy crisis in recent years, we must pay attention to the energy saving and leave enough resource for next generations. As a result, this way is not appropriate.z A discussion of the broad and hidden consequences of having a mixture of landlines and cell phones: Not all the people desire to use cell phone. If the poorer people don’t have telephone, s will drop slightly and we can also save energy. Thus, this way is better. The only work we need to do is to find out how many people use landlines and how many people use cell phones.Strengths and weaknessesStrengthsWe use p and s to replace the abstract concept of the desire of people. As a result ,we can theoretically get the optimal way to providing phone service.WeaknessesOwning to that we can not quantize s and p, surveys are needed.Ⅲ T he model of energy cost of wasteful practices by the usage of cell phones.Cell phones also have another weakness in the energy consumption. Many people always keep their rechargers plugged in. what’s worse, many people charge their phones every night, whether they need to be recharged or not. Therefore, we can make a model to illustrate the cost of these wasteful practices.AssumptionPeople charge their phones every night and use 9 hours to recharge their cell phones every night.NomenclatureTable 4Nomenclature DescriptionUnit W WThe energy wasted in a yearMJ1p The average electrical power of the cell phone rechargerW 2p The average standby power of the cell phone rechargerW 1t The time for people charging their phones when not need in a year2tThe time for people always keeping their recharger plugged in when not need in a year αA coefficient of people always keeping their rechargers plugged inβA coefficient of people charging their phones every nightThe energy wasted can be described as:)(2211t p t p x W w βα+= (13)We get the data from internet [9]:the average cell phone need to be recharged 3 hoursevery 3 days. We calculate: W hkWh p 3.33100001.01=×=According to the assumption, we can calculate:hour t 29203336593651=×−×=,hour t 839533365243652=×−×=.We search the internet [10]and find W p 3.02=About the coefficients α and β, we would like to cite the data from an essay [11] of Doctor MV Shuma-Iwisi and GJ Gibbon in the University of Witwatersrand. They made a survey in the essay through which we can get %1.26=α and %7.73=β.With these data, we can calculate:MJ W w 9101.43600)83953.0%7.7329203.3%1.26(8.259×=×××+×××=We searched the internet and found that 1 barrel of crude oil=5658MJ [12] and theefficiency of the heat of oil converted to electricity is 36.05%[13]. Therefore, 1 barrel of crude oil can give MJ 2040%05.365658=×electricity. Sobarrels W w 691001.22040101.4×=×= crude oil.ResultsThe bad habits of people using their cell phones will make the “Pseudo US” waste barrals 61001.2×of oil a year.Strengths and weaknessStrengthsOur model has considered all the factors that affect the energy cost of wasteful practices by the usage of cell phones and can be widely used.WeaknessesWe use the survey data of the South Africa to estimate the condition in the USA. So if the latest data is provided, the result will reflect the fact better .Ⅳ The model of energy wasted by theappliances left plugged inIn order to solve this problem, Doctor M. V . Shuma-Iwisi, G . J. Gibbon from the University of Witwatersrand have given a formula [14]The appliances standby losses can be describedsnt P W sb sb sb = (14)sb W Standby losses in terms of watt-hours per day or yearsb P : Effective output consumed, watts sb t : Time in standby modes : Saturation in householdsn : Number of householdsWe think we should calculate all kinds of appliances in the American households, so we rearrange (14) produces:∑=)(i sbi sbi sb s t P n W (15)We are very regret that we can only find data in a survey of the United States in 1998.It is in a report [15] written by CREEDAC Project Team: Alex Ferguson. Alan Fung. and V. Ismet Ugursal. We found the data of 25 kinds of appliances in table 5 and use it to estimate the sb WTable 5In the model Ⅰ，81017.1×=n .We can get MJ W sb 111083.1×= In the model Ⅲ,barrels W sb 5111046.236520401083.1×=××=ResultsThrough the model, we can estimate that the energy wasted by the current US in terms of barrels of oil is about barrels 51046.2× per day. Consider our data are from about 10 years ago, the real value is a little larger than our result because the Saturation inhouseholds s shouldincrease.Strengths and weaknessesStrengthsOur model has introduced the appliances standby loss formula given by Doctor M. V . Shuma-Iwisi and G . J. Gibbon. Therefore, it is theoretically strong. We have considered 25 kinds of appliances which is enough to be a representative to the real situation.WeaknessesWe use the data of the USA in 1998 to analyze the current situation of the USA .consider the economical change, our result is a little smaller.Ⅴ The energy needs for providing phone service in agrowing “Pseudo US” modelAccording to Model Ⅰ, Model , Model ⅡⅢ, we can give a prediction to the needs for providing phone service in a growing “Pseudo US”. Considering population and economic growth over the next 50 years, the energy needs must change.Assumptionz The same as Model Ⅰ.z The average energy consumption of a landline and a cell phone will not change inthe next 50 years.z The percentage of people with wasteful deeds will not change in the next 50years. z Comparing 2J and 3kJ in Model Ⅰ, the influence of the signal transmissiontower in energy consumption is so small that we can neglect it.We can find a figure that display the growth of the population in the USA on the internet [16] as in figure7Figure 7We can get the slewing rate:yearmillionk 2.21= Suppose const k =1 and we canget numbers of the population in the following 50 years.T T k H T H 2.2300)(1+=+= （16）Considering the population change in the next 50 years, we must change our Model Ⅰ to fit the new situation. In order to getting more accurate result, we change the m xin the formula (9) every ten years according to the population prediction.1) For the first ten years we use 322103001=+=k x m and get the figure 8 which can better fitting the cell phone numbers change with the time.Figure 82) For the second ten years we use 344203001=+=k x m and get figure 9Figure 93) For the third ten years we use 366303001=+=k x m and get figure 10Figure 104) For the fourth ten years we use 388403001=+=k x m and get figure 11Figure115) For the fifth ten years we use 410503001=+=k x m and get figure 12Figure 12We can easily find that the number of people with cell phones approximately become steady 35 years later. We can get the maximum population in the steady stage is 410 million.We searched the internet [17] ,and found the trend of the real GDP growth rate of the US economy as in figure 13Figure 13We can get the data in table 6Table 6Also we get the real GDP of the USA in different years in table 8 from the internet [18].Table 7We can easily get the GDP of 2007 and 2008 in the USA in the following table.Table 8With the data, we draw the chart as followsFigure 14It is obviously that: when the economy is rising, the population with cell phones ill rise. We can say that the parameter “r” in the formula (9) has a certain relationship with the economic growth, that is, when the economy is rising, ’’r’’ will be rising in a certain extent, the reverse is also true.We show the tendency in the following figure 14 asFigure 15In comparison, we can get a conclusion : economic rise induce to ‘’r’’>0.345 and the relationship between cell phone subscribers and the time will vary as the red line, or vice versa.We can get the predicted data through our calculation following the above-mentionedfive figures and the relation between population and time. To a certain “T’’, wecalculate m2 following the formula (9) and evaluate 56.2)(21m T H H −=; then give five bar charts following as:Figure 16Figure 17Figure 18Figure 19Figure 20Each of the five charts intuitively show the total population, cell phone subscribers and landlines corresponding the period of the nest 50 years separately.Now ，we will search to find the new relationship between energy needs and the time.Based on our analysis with Model I ，Model II and Model III, we can conclude :the average cell phone energy consumption '2J : 22'2m W J J w+= (17) According to the result of Model Ⅰ, Model Ⅱand Model Ⅲ:J J 621038.4×=,million m 8.2592=, MJ W w 9101.4×=, we can get MJ J 2.20'2=According to Model Ⅰ, MJ J 263.01=.We use E to represent the energy consumption of phone service in a year, and get:12112'2263.02.20H m H J m J E +=+= (18)In the formula, we can calculate m2 and H1 from the corresponding bar chart the above-mentioned, and get the table 9Table9: The first ten yearsthe year 123 4 5 population/million 302.2304.4306.6308.8 311 cell phone numbers/million 0.9619 1.3565 1.9121 2.693 3.7897 landlines /million 117.671118.376119.02119.6 120 energy needs/MJ 5E+07 5.9E+077E+079E+07 1E+08 Energy needs/barrels 2469528693.33427842083 52997 the year 6789 10 population/million 313.2315.4317.6319.8 322 cell phone numbers/million 5.32537.468310.44514.55 20.175 landlines /million 120.264120.286119.98119.2 117.9 energy needs/MJ 1.4E+08 1.8E+082E+083E+08 4E+08 Energy needs/barrels 68235.589458.31188982E+05 214970We add the energy needs of 1-10 years and getsum1=833796.2419 barrels;Using the same method, we can get the energy of the 2nd , 3rd , 4th and 5th ten years, as follows:sum2= 5261711.937 barrelssum3= 11199773.96 barrelssum4= 37439313.73 barrelssum5= 39617745.1 barrelsIn order to show the prediction of energy needs clearly, we draw a figure 20 to display the relation.Figure 20ResultsWe can see from the figure that the energy needed for providing phone service increase like a S-shaped carve. In the last ten years ,the energy consumption will become steady. The total amount of the last ten years is about 4000 million barrels. Strengths and weaknessesStrengthsOur model is strong because we fit the real data to get a relationship between the energy consumption and the time. Also we consider the population and economic growth’s influence to the energy consumption.WeaknessesThe population change continuously over time, however, we only find five of the changed population to fit the model. Although it is enough to show the tendency, yet it is not very accurate.Positive SuggestionsEnhance people’s awareness of saving energy. Because the wasted energy is so huge, we should advocate people to bring up good habits that they must unplug the socket as long as the appliances are off .Popularize appliances of low standby power losses. Recently, a kind of standard called “horizontal 1 W standby spec” has been proposed internationally. In our opinion, it is necessary for us to by appliances of low standby power losses.Perfect the application of landlines in our daily life. The landline companies should exploit the new usage of the telephones to attract people. We should use landlines if it is not necessary to have cell phones.References[1],[2] Liang Tianen ,Sun Huahong .The Application of Mobile Communication Tower Mounted Booster and Elerated Antenna to Sea Coverage .Telecommunications Design Technology. 8:8-9,2005.[3] United States -- County by State, and for Puerto Rico GCT-PH1. 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