Power Demand Planning

Transcription

1 Power Demand Planning Power Demand Planning page: 1 of 14 How can you predict future power requirements? Contents Initial Problem Statement 2 Narrative 3-5 Notes 6 Solutions 7-1 Appendices MEI 211

2 Power Demand Planning Initial Problem Statement One important area of civil engineering is electrical power production In order to plan for future building, which may take many years to prepare, design and construct, demand forecasts are often used to indicate the quantity and size of new power stations required Power Demand Planning page: 2 of 14 How can you predict future power requirements? MEI 211

3 Narrative Introduction The total world demand for electrical generating power in 27 was 4428 TW and increases by 17% each year Discussion 1 What do you think the above statement means? What is the demand for power in 28? See "Terawatts" on the bottom of this page The demand for power, P, at a time t years after 27 can be given by the following expression P = P (1 + r) t Discussion 2 What do you think P represents? Discussion 3 What do you think r represents? Do you think r is positive or negative? Discussion 4 What values do t and P have for the year 27? Discussion 5 What value does t have for the year 28? Compare your expression with your previous calculation of P in 28 Power Demand Planning page: 3 of 14 Terawatts A Terawatt is (or 1 trillion) Watts MEI 211

4 Activity 1 Fill in the following table of values Record your answers to 3 dp t (years) P= P 1 + r t (P = 4428, r = 17%) You could fill this in as a group Watch out for students multiplying by 17 rather than 117 when constructing the above table! Plot your results on the following graph Power demand (TW) 25 2 Power Demand Planning page: 4 of Years after 27 Figure 1 Discussion 6 What kind of graph does this look like? MEI 211

5 Activity 2 Rearrange the expression P= P + r t 1 to make t the subject of the formula In what year will the demand for power have doubled from its value in 27? Hint What is the value of P when the demand has doubled? Discussion 7 Do you think the expression for demand is valid into the future? Multimedia The interactive resource Power Demand Planning Interactive is available to help demonstrate some of the important aspects of growth and decline Power Demand Planning page: 5 of 14 MEI 211

6 Notes Geometric progressions A geometric progression (GP) is a sequence of numbers where the next term can be found by multiplying the previous term by a fixed value In other words, any two consecutive terms have a common ratio The power demand function P= P + r t 1 forms a GP as the demand in any year is (1 + r ) times the demand in the previous year, as shown below Term t P= P + r t ( 1 ) = 1 ( 1 ) = ( 1+ ) 2 ( 1 ) = ( 1+ )( 1+ )= 1( 1+ ) 3 ( 1 ) = ( 1+ )( 1+ )( 1+ )= ( 1+ ) 1 P= P + r P P= P + r P r The common ratio is (1 + r ) and the first term is P P= P + r P r r P r P= P + r P r r r P r 2 Power Demand Planning page: 6 of 14 MEI 211

7 Solutions Introduction Discussion 1 solution The US Energy Information Administration states that the installed world generating capacity in 27 was 4428 TW If the increase is 17% per year then the demand in 28 is given by P = % = = = TW 3 dp = Discussion 2 solution In the expression P= P + r t 1 the constant P represents the power when t =, ie years after 27 and so it is the power in 27 You can easily verify this as when t =, t P= P 1+ r = P 1+ r = P 1 = P As x = 1 Discussion 3 solution The variable r represents the rate of growth per year It is positive so that P increases year on year If r were negative the demand would decrease year on year Discussion 4 solution In 27, t = years and P = P, as shown above Power Demand Planning page: 7 of 14 Discussion 5 solution The year 28 is 1 year after 27 so t = 1 This gives t P= P ( 1+ r) 1 = P ( 1+ r) = P ( 1+ r) MEI 211

8 Substituting values of P = 4428 TW and r = 17% per year, P= P( + r) 1 = ( 1+ 1%) 7 = At this stage you recognise that the expression is the same as the one you previously used to calculate the demand in 28 Activity 1 solution Filling in values for future years gives, These data are plotted below Power demand (TW) 25 2 t (years) P= P 1 + r t ( P = 4428, r = 17%) Power Demand Planning page: 8 of Years after 27 Figure 2 MEI 211

9 Discussion 6 solution The graph shows that the demand increases exponentially Activity 2 solution Rearranging the expression P= P + r t 1 to make t the subject requires you to re-arrange it as follows t P= P 1+ r t P ( 1+ r) = P The law of logarithms can now be used to make t the subject t P ( 1+ r) = P t P log( ( 1+ r) )= log P P tlog( 1+ r)= log P P log P t = log 1+ r As logx y = ylog x To find the time when the demand has doubled you note that a doubling occurs when so that P = 2P P log P t = log 1+ r 2P log P = log log ( 2) = log 1 17 = 41 ( years, nearest whole value) When performing this calculation you can use either log 1 or log e, sometimes shown as ln It doesn t matter which one you use as long as you use the same one to evaluate both the numerator and the denominator Try it and see! Power Demand Planning page: 9 of 14 This means that the doubling will occur 41 years after 27, ie in the year 248 There is a subtlety to observe with this result! The actual number of years is 41118, ie slightly over 41 years If the dates are counted from January 1st, the doubling will occur in the year 248 If dates are counted from December 31st, the doubling will actually occur early in the year 249 MEI 211

10 Discussion 7 solution The predictions based on the increase of 17% per year are probably valid to about 3-5 years into the future Beyond that they are probably not valid as changes in demand vary with population size (which is projected to fall) and the level of industrialisation of countries The current increase is being driven by growth in China and India and it is expected to slow when they have reached the industrial levels of western countries You should avoid far-future predictions! The following shows how the demand increases over about 65 years if the expression is followed blindly Power demand (TW) Total energy from the Sun received by the Earth's surface (allowing for an albedo of 31) Year Figure 3 It can be seen that by about 6 years into the future the generating capacity according to our model equals the total energy the Earth receives from the Sun This would be devastating to the global climate and probably lead to extinction of life on Earth! Power Demand Planning page: 1 of 14 MEI 211

11 Appendix 1 using the interactive resources Power Demand Planning Interactive The Power Demand Planning Interactive is available to help demonstrate some of the important aspects of growth and decline Figure 4 The display shows how an initial value varies with time at a given growth rate The initial configuration shows an initial value of 1 and a growth rate of % Increasing the growth rate results in a value that increases with time Notice that the plot shows an exponential increase Power Demand Planning page: 11 of 14 Figure 5 MEI 211

12 In Figure 5, the value doubles to 2 at year 4 If you look at the graph you can see that after a further 4 years (ie year 8), the value has doubled again to approximately 4 And again, 4 years after this (year 12), the value has doubled again to approximately 8 Increasing the initial value scales all the results For example, changing the initial value to 2 rather than 1 gives values that are twice as large at any point in time The time to double a value is unchanged (this may be hard to see on-screen with some values as the actual doubling time will typically not be an integer) Figure 6 Decreasing the growth rate flattens the curve If the growth rate is negative then the values decrease To see this clearly, the initial value can be increased as shown in the example below Power Demand Planning page: 12 of 14 Figure 7 Notice that the plot shows an exponential decrease Instead of a doubling time, the negative growth can be characterised by a halving time, in the above approximately at years 3, 6, 9 (every 3 years) MEI 211

13 The display can be varied by selecting the appropriate type at the bottom right of the screen Figure 8 Power Demand Planning page: 13 of 14 MEI 211

14 Appendix 2 mathematical coverage Use algebra to solve engineering problems Be able to write the rule for a sequence in symbolic form Change the subject of a formula Be able to plot data Be able to draw a graph by constructing a table of values Solve problems using the laws of logarithms Solve problems involving exponential growth and decay Power Demand Planning page: 14 of 14 MEI 211