Michael Baker - Thesis - Problems in Longterm Forecasting and Planning
About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office
The half hourly electricity system model used in the vehicle refuelling infrastructure study (Chapter 2) was based on an hourly electricity system model developed by Mellish and Baker (1977). This previous model was first used to determine long and short run marginal costs for different electricity consumers (Energy Research Group 1976) and later to examine the integration of wavepower into the electricity system Vimukta et al (1978). For a given set of annual sectoral electricity sales, the aim of the model is to estimate the total demand for each half hour (previously hour) period inn the year. From this can be derived an annual load duration curve.
The structure of the model is largely determined by the data available. Briefly the total demand at any time is taken to be the sum of the demands for a number of sectors. For each sector the demand is taken to be the yearly average demand multiplied by a seasonal facytor and by a time of day factor. Information for the time of day factors was obtained from the Electricity Council's grid load curve analysis (Rhys 1977). For the seasonal factors information was obtained from Energy Trends (Department of Energy monthly).
The current version of the model has five sectors. These are Domestic on-peak, Commercial, Industrial, Domestic off-peak and Electric vehicle battery recharging. In principal there can be any number of sectors and for the remainder of the model description I will in general take this number to be n.
There is very little data readily available onthe seasonal variation in electricity demand on a sectoral basis. For the Domestic, Commercial and Industrial sectors it only amounts to quarterly sales to each (Department of Energy monthly). What is required is an estimate of the ratio between daily average looad and annual average load for each sector. This is obtained by fitting a fourier series to the available data (see section A4.3 below).
Typical daily lo90ads for different sectors are available from the Electricity Council's grid load curve analysis Rhys (1977). Each year the Electricity Council carries out an analysis of grid load curve data for a number of days. Multiple regression techniques are used to resolve the total half-hourly demands for each day into domestic, commercial and industrial components.
From this analysis I obtained factors which express half hourly loads as a fraction of the day's average load, for each sector for both a winter's and a summer's day. To obtain the daily pattern for other times of year I assumed that a sinusoidal varying mix of these two can be taken. The summer and winter weighting factorsa are shown in Figure A4.1.
Figure A4.1 Winter and Summer daily weighting factors
If we let:
then
Load Duration Curves I have used the above model of half hourly electricity loads in a computer programme to calculate the average and maximum loads on the electricity system and a load duration curve for any given set of average demands for the five demand sectors. The programme also finds these loads and the load duration curve with the off peak loads spread throughtout the day so that they will the troughs in the daily demand. This I have called logic load. It also finds these things for an electricity system which has a system stor to remove variations in demand for each day.; I assumed that the store has an efficiency of 80%, and the programme also calculates the size of this system store. A4.3 Fourier Series In Fourier analysis a fourier series is usually fitted to a set of points. A series with N terms will exactly fit N points. However the data available for estimating the terms of the fourier series for the seasonal factors are histograms. These are the quarterly sales data for sales to the domestic, industrial and commercial sectors, and monthly averages for off peak and vehicle battery recharging. If we let z be a seasonally varing fanction and Ao, Ai and Bi be the corresponding Fourier coefficients then: where c(it) = cos(2πit/N), s(it) = sin(2πit/N) Over a time period U (where UN = one year) this function will have a cumulatieve value ZT where: If we let T' = T - U/2 and U' = U/2 then the terms ∫ cit and ∫ sit can be simplified. To find estimates of A and B, let eT be an error term such that: y will be minimum when y will be minimum when if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
I have used the above model of half hourly electricity loads in a computer programme to calculate the average and maximum loads on the electricity system and a load duration curve for any given set of average demands for the five demand sectors. The programme also finds these loads and the load duration curve with the off peak loads spread throughtout the day so that they will the troughs in the daily demand. This I have called logic load. It also finds these things for an electricity system which has a system stor to remove variations in demand for each day.; I assumed that the store has an efficiency of 80%, and the programme also calculates the size of this system store.
In Fourier analysis a fourier series is usually fitted to a set of points. A series with N terms will exactly fit N points. However the data available for estimating the terms of the fourier series for the seasonal factors are histograms. These are the quarterly sales data for sales to the domestic, industrial and commercial sectors, and monthly averages for off peak and vehicle battery recharging.
If we let z be a seasonally varing fanction and Ao, Ai and Bi be the corresponding Fourier coefficients then:
where c(it) = cos(2πit/N), s(it) = sin(2πit/N) Over a time period U (where UN = one year) this function will have a cumulatieve value ZT where: If we let T' = T - U/2 and U' = U/2 then the terms ∫ cit and ∫ sit can be simplified. To find estimates of A and B, let eT be an error term such that: y will be minimum when y will be minimum when if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
where c(it) = cos(2πit/N), s(it) = sin(2πit/N)
Over a time period U (where UN = one year) this function will have a cumulatieve value ZT where:
If we let T' = T - U/2 and U' = U/2 then the terms ∫ cit and ∫ sit can be simplified. To find estimates of A and B, let eT be an error term such that: y will be minimum when y will be minimum when if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
If we let T' = T - U/2 and U' = U/2 then the terms ∫ cit and ∫ sit can be simplified.
To find estimates of A and B, let eT be an error term such that: y will be minimum when y will be minimum when if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
To find estimates of A and B, let eT be an error term such that:
y will be minimum when y will be minimum when if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
y will be minimum when
y will be minimum when if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
if we let y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
if we let
y will be minimum when The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'. A4.4 Calibration of Model Introduciton The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging. Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1. Table A4.1 Degree Months YearTotalQuarter 1234 197373.136.011.53.022.6 197473.237.810.70.224.5 197575.436.510.64.723.6 197676.335.814.64.921.0 197771.735.29.11.226.2 197870.132.98.42.126.7 66/775.0 76/775.7 According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual). Table A4.2 Off-peak Domestic Sales Year '000 AvConsumers Off-peakper consumerMWh Off-peak sales GWh Total Quarter 1 2 3 4 1973 19222 0.575 11044 5439 1737 453 3415 1974 19430 0.624 12133 6265 1774 33 4061 1975 19691 0.694 13661 6613 1921 852 4276 1976 19948 0.753 15025 7050 2875 965 4135 1977 20222 0.756 15286 7504 1940 256 5586 1978 20483 0.786 16101 7665 1929 482 6132 Seasonal variation for all sectors Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero. Table A4.3 Quarterly Electricity Sales Year q Domestic *100A/Av Com & Ser *100A/Av Industry *100A/Av D Off-peak *100A/Av Act'l Av. Act'l Av. Act'l Av. Act'l Av. 1973 1 24592 13820 22774 5439 2 18419 10672 21688 1737 3 14435 19989 72 9315 11751 79 20315 21679 94 453 2864 16 4 22809 19846 115 13188 11743 112 22196 21508 103 3415 2972 115 1974 1 12999 19934 120 12839 11731 118 22260 21389 104 6265 2924 214 2 17863 20091 89 10585 11794 90 20833 21427 97 1774 2952 60 3 15693 20064 78 9313 11858 79 20222 21411 94 33 3077 1 4 22811 20045 114 13687 11850 116 22587 21237 106 4061 3139 129 1975 1 23776 19760 120 13858 11824 117 21746 20928 104 6613 3259 203 2 17934 19143 94 10497 11748 89 19957 20624 97 1921 3388 57 3 13342 18602 72 9193 11752 78 18622 20648 90 852 3470 25 4 20224 17927 113 13199 11806 112 21755 20903 104 4276 3644 117 1976 1 22039 17379 127 14379 11813 122 22767 21227 107 7050 3777 187 2 14273 17406 82 10408 11932 87 20981 21661 97 2875 3744 76 3 12619 17400 73 9339 12096 77 20189 22019 92 965 3813 25 4 21161 17602 120 14005 12319 114 23656 22272 106 4135 3753 110 1977 1 21060 18183 116 14886 12616 118 23731 22470 106 7504 3548 212 2 16863 18048 93 11685 12737 92 22047 22376 99 1940 3640 53 3 14682 17610 83 10435 12810 81 20704 22235 93 256 3828 7 4 18011 17546 103 13876 12957 107 22389 22320 100 5586 3833 146 1978 1 20710 17525 118 15599 13082 119 23867 22429 106 7556 3860 196 2 16698 17475 96 12152 13263 92 22595 22597 100 1929 3957 49 3 14684 10971 20126 482 4 17611 14784 23414 6132 AvofA/Av 1 120.303 118.824 105.451 202.220 2 90.717 89.940 97.875 58.998 3 75.609 78.940 92.630 14.686 4 112.883 112.078 104.030 123.505 Table A4.4 Fourier Series Coefficients Sector A1 B1 B2 0.2929 0.0886 -0.0302 0.2437 0.0697 -0.0167 0.0745 0.0262 -0.0150 0.9912 0.4838 0.1353 The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5. Table A4.5 Fourier coefficients for Electric vehicle battery recharging 1976 Total traffic million vehicle km month 1 2 3 4 5 6 traffic 17878 17466 20168 20082 22238 22520 month 7 8 9 10 11 12 traffic 24340 24785 23083 21086 19786 18004 Fourier coefficients i 1 2 3 4 5 6 Ai -0.1449 -0.0121 0.0044 -0.0067 0.0240 Bi -0.0622 0.0176 -0.0181 -0.0032 0.0059 0.022 Daily load patterns The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4. Figure A4.2 1976/7 Summer and Winter Domestic load patterns Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above. The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2). Figure A4.3 1976/7 Summer and Winter Commercial load patterns Figure A4.4 1976/7 Summer and Winter Industrial load patterns A4.5 Validation The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9. Figure A4.5 1973/4 Actual and simulated load duration curves Figure A4.6 1974/5 Actual and simulated load duration curves Figure A4.7 1975/6 Actual and simulated load duration curves Figure A4.8 1976/7 Actual and simulated load duration curves Figure A4.9 1977/8 Actual and simulated load duration curves A4.6 Comments on the Model Assumptions In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below. The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle. It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year. Short comings One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern. [1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4 About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
The difference between the expression for Ai and Bi and those found for a line passing through a set of pooints ZT is the factor πi/UsiU'.
The basic data for calibrating the model was taken from Rhys (1977) for the daily patterns for 1976/7 and from Department of Energy (monthly) for quarterly electricity sales for 1973-1978. Estimates of off-peak domestic sales were obtained from a variety of sources and then subtracted from total domestic to obtain on-peak domestic sales. Total traffic flow for 1976 was used as a surrogate for electric vehicle recharging.
Seasonal variation in off-peak sales were assumed to be entirely due to changes in outside temperature leading to changes in off-peak heating requirements. It was assumed that off-peak sales vary in proportion to the difference between an inside temperature of 15.5°C and the outside temperature. Average monthly temperatures were takend from the Digest of United Kingdom Energy Statistics (Department of Energy annual). The degree-months ((15.5 - mean air temp)months) for 1973 to 1978, 1966/7 and 1976/7 are shown in Table A4.1.
Table A4.1 Degree Months
According to Platts (1978) the average off-peak sales to all Domestic consumers was 250 kWh/consumer in 1966/7 and 760 kWh/consumer in 1976/7. This was used to find average sales per consumer per degree-months for both these years then linear interpolations of sales per consumer per degree-month were made for the years 1973 to 1978. These were used in conjunction with the degree-months in Table A4.1 to obtain the estimates of domestic off-peak sales hown in Table A4.2. The number of consumers was taken from Department of Energy (annual).
Table A4.2 Off-peak Domestic Sales
Sales to each sector in each quarter are shown in Table A4.3. The above estimates of off-peak sales to the domestic sector were used to split the domestic sales into on- and off-peak. To allw for the growth in sales which took place between 1973 and 1978 a two by four moving average [1] of quarterly sales was found and this was used to scale each qurters sales. The averages of these scaled sales were then found. The resultant fourier series coefficients are shown in Table A4.4. The fourier series for off-peak domestic is such that it gaves very small (smaller than -0.05) negative values during July. Within the computer programme these negative values are taken to be zero.
Table A4.3 Quarterly Electricity Sales
Table A4.4 Fourier Series Coefficients
The seasonal variation for electric vehicle battery recharging was assumed to be proportional to the seasonal traffic flows. 1976 total traffic (Department of Transport annual a) was used to estimate the fourier series coefficients as shown in Table A4.5.
Table A4.5 Fourier coefficients for Electric vehicle battery recharging
The results of the Electricity Council's grid load curve analysis of a mid winter mid week day (at 32°F) and for a midweek Summer day (at 72°F), for 1976-77 are shown in Figures A4.2 to A4.4.
Figure A4.2 1976/7 Summer and Winter Domestic load patterns
Also shwon in Figure A4.2 is my estimate of the off-peak component of the domestic load pattern. The size of the off-peak domestic component was adjusted so that it has a minimum equal to the summer night time minimum, it starts at 2300, it ends at 0700 and has an area equal to that estimated using the seasonal pattern found above and total off-peak sales for 1976-77 from above.
The load patterns for the model were obtained by dividing the actual loads for each half hour by the sectors average load for the day. The domestic winter off-peak pattern was also used for the domestic summer off-peak pattern and both electric vehicle patterns. However it should be remembered that the model also used "logic loads" for these off-peak shapes as well (see Section A4.2).
Figure A4.3 1976/7 Summer and Winter Commercial load patterns
Figure A4.4 1976/7 Summer and Winter Industrial load patterns
The only validation of the model has been to compare load duration curves as estimated for 1973/4 and 1977/8 with actual load duration curves for these years. Unfortunately the average loads used in the model, taken from Table A4.3 were for Great Britain, whereas the actual load curves were for the CEGB. The actual and estimated load duration curves are shown in /figures A4.5 to A4.9.
Figure A4.5 1973/4 Actual and simulated load duration curves
Figure A4.6 1974/5 Actual and simulated load duration curves
Figure A4.7 1975/6 Actual and simulated load duration curves
Figure A4.8 1976/7 Actual and simulated load duration curves
Figure A4.9 1977/8 Actual and simulated load duration curves
In the construction of the half hourly electricity demand model several assumptions were made. These are briefly listed below.
The first assumption is that the half hourly patterns of electricity demand for each sector remain constant over time. However from the Electricity Council's grid load curve analysis it is known that this is not true. Such an assumption woud require an unchanging mix of uses within each sector, which for the domestic sector would also mean no changes in lifestyle.
It is also assumed that the fourier series is a reasonable approximation to the seasonal variations in demand and that a sinusoidal weighting of summer and winter patterns will produce a reasonable approximation to the daily pattern at other times of year.
One of the main short comings of the model structure is that it does not allow for variations in demand due to such factors as the weather other than annual seasonal variations. Random fluctuations in weather are not accounted for. Another short coming of the current version of the model is that it has had very little validaiton. Also no account was taken of off-peak sales to the commercial sector. Although data was available on the sales of off-peak to the commercial sector in the same amount of detail as for the domestic sector no satisfactory way would be found for estimating the daily commercial off-peak load pattern.
[1] The average of four quarters data gives a value which is central about a point half way between the centre two quarters. To get a value which is actually centred on a quarter the mean of two adjacent four quarter averages can be taken. This is known as a two by four moving average. For example if 73q1 represents the value of the quantity being averaged, in the 1st quarter of 1973, then the two by four moving average for the 3rd quarter of 1973 is: (73q1/2 + 73q2 + 73q3 + 73q4 + 74q1/2) / 4
About | Contents Previous: A3. Net Physical Volume of Production | This: A4. A Half Hourly Electricity System Model | Next: A5. Correspondence with the Central Statistical Office Copyright © Michael Baker 1981,2005. All Rights Reserved.
