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Error taxonomy
Errors may vary in a number of ways. First, errors can be correlated or uncorrelated. This is not in the same category as the classifications that follow but is an important distinction nonetheless. Errors that are uncorrelated with other variables are the simplest to model. Consecutive errors may also be autocorrelated in a time series. This sequentiality is hidden in scatter plots and regression analyses, although it still affects the estimates. Errors can be classical or Berkson. If e is a generic error term, classical errors take the form x =x+e, and are more common. This is when the error is in the instrument itself. Berkson errors take the form x = x+e. This occurs when the actual value of the measurand varies around the assigned or measured value, because the source of the error is external to the instrument. Errors are classified as multiplicative or additive. Multiplicative errors are of the form x = xe, whereas additive errors take the form x = x+e. The additive error assumption is a popular one as it greatly simplifies MEM mathematics: additive errors are usually associated with constant variance throughout the measurand range. This is called homoscedasticity and is a critical assumption when performing Linear Regression (LR). The majority of techniques have been developed to describe this kind of model. However, this assumption is not always valid. For example, it has been demonstrated that energy meter measurement errors are non-linear and multiplicative [217], and are thus heteroscedastic. This has been acknowledged to produce problems in econometric energy analyses [283], and frequentist methods to account for some cases in regression analysis has been developed [201]. It may be mitigated by assuming a log-normal distribution and working with logx, since logxe = logx+ loge, transforming the error model to an additive one. However, the assumption of a log-normal distribution on e (so that loge Normal), although mathematically convenient, is not always valid or preferred [196]. Heteroscedasticity can be present even for additive errors when they have non-constant bounds over the measurement range, such as energy meters and CTs [22–25]. These bounds are shown in Figure 2.2. Errors may be differential or non-differential. Non-differential errors mean that x contains no more information about y than x does. The response does not change due to measurement. Differential errors may occur when the response y is measured before the covariates x and z, and these variables are liable to change. For example, the diet (x) of women with breast cancer may be measured only after their diagnosis y. It is possible that the test subjects change their diet as a result of the diagnosis [196]. Another example is when x is a proxy for x, not simply a mismeasurement. For example, plug loads are sometimes used as a proxy for occupancy [11]. Differential errors may also occur in ex-post energy use surveys for residential retrofit programmes where the response (purchasing of certain equipment, for example) is measured before other variables of interest are measured. Last, the function y(x) may be linear or non-linear. This is not an assumption about the errors themselves but does affect the kinds of errors that are permissible. The linear assumption is popular as it allows LR to be used if one assumes normally distributed additive errors. For many models, this is a valid assumption. However, Carobbi, Pellicci, and Vieri [217] have shown that the standard P =VI electrical power equation, where P is Power in Watts, V is potential difference in Volts, and I is current in Amperes, can be modelled as Pn = (1+a)VIcos(f +fc)+e; (4.15) when an energy meter measures with error. In this equation, a is the gain error, fc is the phase error, and e is the bias error. The gain error a changes the amplitude of measured power fluctuations, but does not affect the mean. In other words, the larger the energy reading, the larger the error. The bias error e offsets the measured power, changing the mean power read by the meter, but not the amplitude of the fluctuations. This error may bias the power and energy reading upwards or downwards. The phase error fc has a similar net effect to the gain error, but changes according to the power factor error of the meter. Carobbi, Pellicci, and Vieri’s contribution [217] was to show that (4.15) is a statistically adequate model, capturing the real error behaviour of energy meters without specifying too many parameters.
Although this error is multiplicative, the error bounds in the IEC meter qualification standards [22 24] are additive. The meter may still have a multiplicative error, but this error is always smaller than the additive error bound. In cases where these are the only data available, additive errors may have to be assumed. Furthermore, the error model is only non-linear if the phase error term fc is of interest.
Meter calibration
The method below focusses on energy meters but can be used for instruments measuring other parameters as well. The most analogous cases are flow measurement [57], and possibly exhaust gas analysis [58]. Occupancy measurement may also benefit from thoughtful application [8, 11], but temperature measurements are often biased due to spatial variations [152], and will require more careful application.
The proposed approach is to discipline a meter (the UUT) using another relatively low-specification commercial-grade metering system. This could be done by installing the meters in parallel in-situ at the facility for a short period, such as 24 hours if both measure at a resolution of 15 minutes. The data from the calibrator are then used to correct (discipline or calibrate) the data from the UUT. Although the UUT is not calibrated, we assume that it is of reasonable quality. For example, the model range to which the UUT belongs should be qualified to an IEC specification. This is necessary to ensure that readings will remain stable under different operating conditions such as winter and summer temperatures.
For high-accuracy laboratory multimeters measuring to six or eight decimal places, various additional factors should be considered during calibration. These include thermoelectric voltages, cable impedance, and performance at different frequencies [182]. However, these fluctuations are small enough to be negligible for commercial energy measurement applications.
CHAPTER 1 INTRODUCTION
1.1 RESEARCH OUTPUTS
1.2 CHAPTER OVERVIEW
1.3 BACKGROUND
1.4 PROBLEM STATEMENT
1.5 RESEARCH OBJECTIVE AND QUESTIONS
1.6 HYPOTHESIS AND APPROACH
1.7 RESEARCH GOALS
1.8 OVERVIEW OF STUDY
1.9 NOMENCLATURE
CHAPTER 2 LITERATURE REVIEW
2.1 CHAPTER OVERVIEW
2.2 MEASUREMENT AND VERIFICATION
2.3 STATISTICAL METHODS
CHAPTER 3 THE BAYESIAN PARADIGM FOR MEASUREMENT AND VERIFICATION
3.1 CHAPTER OVERVIEW
3.2 INTRODUCTION
3.3 MOTIVATION
3.4 BAYESIAN THEORY
3.5 APPLICATION
3.6 CONCLUSION
CHAPTER 4 ENERGY METERING UNCERTAINTY AND CALIBRATION
4.1 CHAPTER OVERVIEW
4.2 ENERGY METERING UNCERTAINTY IN THE CONTEXT OF OVERALL PROJECT UNCERTAINTY
4.3 LOW-COST CALIBRATION
4.4 CONCLUSION
CHAPTER 5 EFFICIENT METER SAMPLING
5.1 CHAPTER OVERVIEW
5.2 INTRODUCTION
5.3 EFFICIENT CROSS-SECTIONAL METERING SAMPLING
5.4 CASE STUDY: EFFICIENT CROSS-SECTIONAL METERING DESIGN
5.5 CONCLUSION
CHAPTER 6 EFFICIENT SURVEY SAMPLING
6.1 CHAPTER OVERVIEW
6.2 INTRODUCTION
6.3 MODELLING
6.4 CASE STUDY 1: SIMPLE RANDOM SAMPLING DESIGN
6.5 CASE STUDY 2: STRATIFIED SAMPLING DESIGN
6.6 CONCLUSION
CHAPTER 7 COMBINED METERING AND SURVEY SAMPLING
7.1 CHAPTER OVERVIEW
7.2 INTRODUCTION
7.3 CASE STUDY 1: SIMPLE RANDOM SAMPLING DESIGN
7.4 CASE STUDY 2: COMBINED STRATIFIED SAMPLING DESIGN
7.5 CONCLUSION
CHAPTER 8 CONCLUSION
8.1 MEASUREMENT UNCERTAINTY
8.2 SAMPLING UNCERTAINTY
8.3 RECOMMENDATIONS FOR M&V PRACTICE
8.4 RECOMMENDATIONS FOR FURTHER RESEARCH
REFERENCES
