A generalised model of electrical energy demand from small 1 household appliances 2

7 Accurate forecasting of residential energy loads is highly influenced by the use of electrical 8 appliances, which not only affect electrical energy use but also internal heat gains, which 9 in turn affects thermal energy use. It is therefore important to accurately understand the 10 characteristics of appliance use and to embed this understanding into predictive models to 11 support load forecast and building design decisions. Bottom-up techniques that account 12 for the variability in socio-demographic characteristics of the occupants and their behaviour 13 patterns constitute a powerful tool to this end, and are potentially able to inform the design 14 of Demand Side Management strategies in homes. 15 To this end, this paper presents a comparison of alternative strategies to stochastically 16 model the temporal energy use of low-load appliances (meaning those whose annual en17 ergy share is individually small but significant when considered as a group). In particular, 18 discrete-time Markov processes and survival analysis have been explored. Rigorous mathe19 matical procedures, including cluster analysis, have been employed to identify a parsimonious 20 strategy for the modelling of variations in energy demand over time of the four principle 21 categories of small appliances: audio-visual, computing, kitchen and other small appliances. 22 From this it is concluded that a model of the duration for which appliances survive in discrete 23 states expressed as bins in fraction of maximum power demand performs best. This general 24 solution may be integrated with relative ease with dynamic simulation programs, to comple25 ment existing models of relatively large load appliances for the comprehensive simulation of 26 household appliance use. 27


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In the UK approximately 20% of energy use in households is due to electrical appliances 4 [1], and this proportion is higher in better insulated homes. Residential electrical appliance 5 use has direct implications for local Low Voltage (LV) networks, the loads on them and their 6 integrity; and indirect implications for thermal energy demands, since electrical energy is 7 ultimately dissipated as heat, most of which is emitted within the building envelope. It is 8 therefore important to be able to reliably predict electrical appliance use, in particular the 9 magnitude and temporal variation of the energy use and power demand profiles arising from 10 the aggregation of individual appliances, to support design and regulation of LV networks 11 serving communities of buildings and of building's thermal systems. 12 But this is a complicated task, for the ownership and use of different types of appliance 13 significantly varies from house to house, and between users. Addressing this diversity requires 14 that we have an appropriate basis for allocating appliances to households depending on their 15 composition and socio-economic characteristics and for predicting their subsequent use. This 16 in turn implies the use of stochastic simulation and bottom-up approaches that may also 17 facilitate the future testing of Demand Side Management (DSM) strategies. 18 So far, bottom-up approaches have focused on the modelling of high-load appliances: 19 those that are commonly owned and which contribute significantly to total annual electricity 20 use. Examples include cold (fridge and freezer), wet (washing machine and dishwasher) and 21 cooking appliances. For example, the model of Jaboob et al. [2] predicts when the appliances 22 are switched on, the duration for which they will remain on and their fluctuating power 23 demands whilst on. But in our everyday lives we also use myriad low-load appliances. Their 24 individual share of energy use may be small, in some cases even negligible, but it is significant 25 when considering them as a group (or groups). 26 The objective of this paper is to find a parsimonious strategy for modelling four cate-27 gories of low-load appliances: audio-visual, computing, kitchen and other appliances, which 28 collectively account for those that are not represented by current device specific models. The 29 work presented here extends and further develops that introduced in [3].
results estimating the energy demand in the validation of the models. Paatero and Lund [6] 1 introduce a social random factor (supposed to capture the social variety of the demand) that 2 improves the diversity of patterns obtained; however, only yearly consumption data for the 3 16 end-uses is used for the generation of the models, together with other aggregate statistics, 4 restricting the resolution to hourly time steps. In general, these approaches do not describe in 5 terms of model parameters the dynamic behaviour of appliances, but they generate empirical 6 profiles of power demand as a function of time. 7 Relatively more aggregated methods are models based on time-use-survey (TUS) datasets. 8 In TUS datasets, the respondents fill in diaries of their activities during the day usually for 9 one week periods, such as cooking, sleeping, travelling to work, etc. This data provides 10 a powerful input to bottom-up models, since it encapsulates highly detailed information 11 describing occupants' activities, that can be related to the use of appliances. To this end, 12 Capasso [7] presents a first strategy linking occupants' activities with appliance use, using 13 TUS data. The model produces 15-minute profiles of electricity use, considering aggregations 14 of appliance that correspond to just four type of activities: cooking, housework, leisure and 15 hygiene; each associated with a blend of large and small appliances, which are allocated by 16 considering the average range of appliances present in the simulated household. The relation 17 between performing an activity and using an appliance is described with a single coefficient 18 alpha (defined as a human resources). Tanimoto [8] combines TUS with statistical data of 19 ownership of appliances and its peak and stand-by powers. 31 activities are considered in 20 this case, so that the level of aggregation is low, but this is contrasted by a small dataset size 21 (58 households over 2 days). 22 In a similar vein, Widén [9] uses Swedish TUS data to model electricity use by assigning 23 appliances to related activities (9 different categories in this case) and imposing five standard 24 end-use profiles based on the type of their demand profile: demand disconnected from activity, 25 power demand constant during activity, power demand constant after activity (with and 26 without addition of temporal constraint) and fluctuating power demand (only applied to 27 lighting). This approach is further developed in [10], where inhomogeneous Markov chains 28 generate sequences of domestic activities that have an impact on power demand (5 minute and 29 1 hour granularity), including dependencies with the number of occupants performing these 1 activities. A yet finer temporal resolution of 1 minute is achieved in the work developed by 2 Richardson et al. [11]. Based on 7 different activities, a load curve for the appliances is created 3 using the probability of switching on an appliance when an activity is being performed, and 4 applying a fixed power conversion scheme. Using a calibration procedure based on the total 5 time of use of an appliance, they obtain annual energy predictions. Although this tuning 6 ensures a good overall match in annual energy demand, this does not imply the absence of that related appliances will be switch on is modelled, as is the corresponding duration that 23 they will remain on and their time-varying mean power demands whilst on. Thus, this mod-24 elling chain rigorously resolves for dynamic variations in mean power demand, in contrast to 25 static power conversion schemes. Moreover, it presents the possibility of being used together 26 with explicit models of low-load appliances, in order to obtain accurate values of the total 27 electricity use of a house. 28 To this end and informed by these past endeavours, our task is develop a parsimonious 29 strategy for the use of relatively low-load appliances, in complement to Jaboob's model of 1 high-load appliances.

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In this work we are interested in modelling the energy and power demands of low-load 4 appliances to support building, systems and network design. In order to contribute to ac-5 curate predictions of residential energy use, we need to address the diversity in dwelling 6 characteristics and human behaviours. Thus, we identify the following modelling tasks: 7 I Perform low-load appliance allocation, using cumulative distribution functions describ-8 ing the peak power demand of aggregates of appliances. Devices are categorised into 9 four groups: audio-visual, computing, small kitchen and other (miscellaneous house- 10 work, garden and personal care appliances).

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II Model the characteristic use of these appliances in individual households. To this end, 12 we utilise the fractional energy use f : the ratio of the actual to the maximum energy 13 E max that an appliance can use, determined by its rated power. Modelling f , we can 14 distinguish between:

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• Fluctuating demands whilst the appliances are in use.

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Two considerations need to be taken into account in carrying out these tasks. Firstly, 18 stochastic methods are required, as we are interested in describing the underlying randomness 19 in households' appliance use and investment decisions. These methods rely on the definition 20 of coefficients that represent the system as a probability distribution, which can be dependent 21 on different variables such as time of the day, number of occupants, weather, etc. Secondly, 22 using the normalized fractional energy f instead of absolute energy allows us to evaluate load 23 profiles from different appliances of a similar type, but that do not necessarily have the same 24 magnitude. In this way, appliances can be classified into groups and modelled as a category. Furthermore, It has been previously shown [2, 9,11] that stochastic methods are suc- 6 cessful in describing energy demands and the information listed in Task II. In the work here 7 presented, two of these statistical approaches have been exploited: can be more efficiently achieved if complemented with clustering techniques.

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• Survival analysis can model the switching-on/off of appliances, as well as the duration 13 an appliance remains in different energy states. 14 In the methodology presented here we have tested a range of strategies in order to find the 15 most parsimonious approach. In this, we have ensured that the number of subjective decisions 16 needed for modelling have been minimised, so that the methodology can be appropriately 17 applied independently of the data set employed to estimate the models' coefficients. A Markov process is a stochastic process that fulfils the Markov property, by which a 21 future state depends on the most recent state, and not on any prior history [15]. A stochastic 22 process X(t) is therefore a Markov process if for every n and t 1 < t 2 < · · · < t n : (1) Markov chains describe the process of making transitions between a present state i to a future 24 state j, according to a probability distribution, described by a state transition probability 25 matrix (or Markov matrix) as follows: is the probability that a transition from i to j takes place, given by the ratio of transitions 3 that occur to state j from i to the total number of transitions occurring from i.

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The dimensions of a Markov matrix m × m are given by the number of states m defined in 5 the system. At the same time, the coefficients in the matrix may or may not depend on time. 6 In the first instance, a time-homogeneous Markov process is considered, where the system 7 can be described using a single matrix. We then consider a time-inhomogeneous Markov 8 chain, in which the number of matrices r is given by the number of time slots considered to 9 have different transition probabilities. For instance, if it is assumed that the probabilities are 10 different for each hour of a day, then r = 24 (considering a single-day). This means that the 11 probability distribution is given by a matrix of dimension r × m × m.

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Appropriate dimensioning of the Markov matrices is not a trivial task: if m and r are 13 set too low or even equal to 1, the dynamics or temporal variation of the system may not    The former decomposes D into a nested hierarchy of clusters, represented by a dendrogram, i.e. a tree diagram that splits the database into subsets of smaller size, until each object 8 belongs to one subset. The process can be agglomerative or divisive, depending on whether 9 the structure is made from the leaves towards the root or from the root to the leaves. The 10 latter creates a single-level partition of D into k clusters based on similarity and distance 11 measures. The parameter k is required as an input, even though it is not generally known a 12 priori.

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A third type of clustering method is density-based clustering algorithms, which apply

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DBSCAN is a typical density-based clustering algorithm that was developed in 1996 [18].

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The core idea behind DBSCAN is that for each object in a cluster, the neighbourhood of 27 radius ǫ has to be populated with a minimum number of points MinPts. ǫ and MinPts are 1 the two only parameters required.
2 However, the cluster structure of a real data set cannot usually be identified with a single 3 global density parameter, but rather by clusters of different density, as well as their intrinsic 4 structure. The OPTICS algorithm [17] is a generalization of DBSCAN. Instead of a clustering 5 division, OPTICS outputs an ordering of the database relative to its density-based clustering 6 structure, containing information for every density level up to a "generating distance" ǫ 0 , that 7 allows for analysis of the grouping structure (hierarchy). A graphical interpretation of the 8 ordering is available through a reachability plot [17], where clusters are identified as "dents" 9 in the plot. The authors of this algorithm provide a method for automatically determining 10 the cluster hierarchy using the information extracted from the reachability plot. However, 11 a simpler alternative method for automatic extraction of the clusters is described in [19], in 12 which the most significant clusters are simultaneously selected from different density levels.

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Interestingly the authors also show that reachability plots are equivalent to the dendrograms 14 of single-link clustering methods.
where k > 0, λ > 0 and γ > 0 are the shape, scale and location parameters of the Weibull By inverting equation (5), it is possible to obtain directly the duration for which an appliance 1 will continue (survive) in a specific energy state s as: given a number w ∈ [0, 1) drawn randomly from a uniform distribution.

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Two cases have been studied in this work, either defining:

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Thus, switching on/off events are explicitly modelled. states. 10 Therefore, occurrences of each event and durations are first extracted from the data, and 11 used to fit Weibull distributions, obtaining scale, shape and location parameters λ and k.

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These distributions are then used to calculate survival times in a simulation using equation  artificially represent a stochastic process by a sampling procedure, which will be determined 18 by the particular underlying probability distribution of the given process. 19 For the specific problem posed here, the probability structure is given by either the  (a) Markov models.

Assume appliance ON
Obtain n OF F using Obtain state s using P s (t) n s = 0?
No Yes (c) Multistate survival model. For the latter case, equation (6) is employed -for an extracted random number w-to One finale step transforms these profiles from fractional to actual energy values: whereẼ max is a statistical measure of the maximum energy (or powerP as required) for all 2 instances in the category. Thus, the estimation ofẼ max values becomes critical to calculating 3 accurate aggregate energy profiles. Assignment of the maximum energy can be performed 4 using c.d.f.'s of the relevant appliance categories, or else using simple mean or median mea-5 sures. 6 Although assignment ofẼ max is important, it is also trivially complicated. In what follows 7 then, we focus on testing the underlying hypothesis in our modelling strategies rather than 8 in the fidelity of predictions of aggregate energy profiles that require a random assignment 9 process. and their temporal electrical energy use during 1 or 2 months, with records every 2 minutes.

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Of the 250 households, 26 of them were additionally monitored for a whole year, with 10 min-16 utes resolution. Since the one-month data was not measured during the same month for all 17 households, only the data recorded for the 26 houses during a whole year was utilised in the 18 analysis presented here, in order to avoid possible seasonal effects on the use of appliances. 19 The relevant low-load appliances found in the dataset were classified into four categories, 20 following the type of activity that involves their use:

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-audio-visual (excluding TVs, that are considered as high-load appliances given their extensive use),
1 Figure 2 shows the types of device available in the data set and their contribution to annual 2 energy use, with categories depicted in different colours. The height of the bars represents 3 the mean value of annual energy use of the corresponding type of appliance, whereas the 4 width is proportional to the number of instances observed in the 26 households for the given 5 device. Thus, the area of the bar indicates the total energy use of that appliance throughout 6 the stock of houses surveyed.  One shortfall encountered in the data set is that there is no information describing the 1 rated power of the appliances, posing a challenge to the accurate estimation of E max . Con-2 sequences derived from this and the solution proposed are discussed in section 3.2.2.

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The procedure adopted in our work was to test a range of strategies to model one appliance 4 category, the audio-visual category, in order to identify the most parsimonious approach, and 5 then to deploy this to other categories of appliance. In this section the nature of the data used to test the modelling techniques is presented.
8 Table 1 displays the total number of instances of each subcategory of appliance considered 9 in the audio-visual category, present in the 26 houses, leading to a total of 102 instances for 10 the category. Our first step was to extract fractional energy values from the electricity use 11 records, as iii. The off-state exhibits temporal dependency, reaching maximum values in the early 5 hours of the morning (39%) when most people are sleeping, and a minimum (29%) 6 between 20h and 22h, when most people are present and awake. 7 iv. 15% of the entries have fractional energy lower than 0.1, which likely corresponds to a 8 stand-by state, a common feature of audio-visual devices. 15 Values of f = 0 were excluded to help with the interpretation. 3.2.2. Data preprocessing: outliers and maximum energy estimation 1 As previously mentioned, our data set does not include appliance name plate (power) 2 ratings. The fractional energy modelling approach, however, is dependent on the values 3 of E max and requires this input at two specific stages. Firstly in using equation (9) to 4 extract fractional energy profiles for each instance. Secondly after the simulations have been 5 performed, to compute an energy profile from a simulated fractional energy time-series for 6 the category. In order to estimate E max from the data, the maximum energy record for each profile 8 was used. The existence of discrepant entries for the same type of appliance suggested that 9 a data cleaning process was necessary. This could be due to the fact that each data point 10 represents the energy corresponding to the mean power drawn by an appliance over a period 11 of ten minutes. Since this may fluctuate between 0 and the nameplate rating it could be 1 that the calculated value of E max results from an appliance that has been working at higher 2 power during a shorter period of time (e.g. a kettle that never takes 10 minutes to boil).

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This problem was overcome by obtaining maximum energy values from the 2-minute data 4 (also subjected to a cleaning pre-process), with the purpose of selecting consistent entries.

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A Seasonal Hybrid Extreme Studentized Deviate test (S-H-ESD) [22] was employed to 6 detect anomalies. S-H-ESD is based on the generalized ESD algorithm to detect one or 7 more outliers in a univariate data set that follows an approximately normal distribution, 8 and is applicable to time-series data. Its main feature is that it is able to predict both 9 local and global anomalies, taking into account long-term trends on the temporal profile to  repeated K times (folds), until each partition has been used once as a validation set. A mean 28 performance error can then be computed as the average error: where e i represents some error between predictionŷ i and observation y i . K-fold cross vali-2 dation is a computationally expensive method, but produces an accurate estimation of the 3 goodness of fit. appliances. The models are expected to reproduce this periodicity correctly, and this 20 can be studied using the cross-correlation function [25] between two signals (X t , Y t ), 21 which is defined as where µ k , σ k are the mean and standard deviation of process k = X, Y , respectively, where T, F, P, N represent True, False, Positive and Negative and TP is the total number of 3 truly predicted positive outcomes (true positives). Specificity or true negative rate (TNR) is 4 defined as [26]: In an ideal case, one would have T P R = 1 and T N R = 1 (or F P R = 1 − T N R = 0). 6 Comparison of these indicators can be plotted in receiver-operating characteristic (ROC) 7 space. This analysis can be complemented with the model accuracy giving an indication on the overall performance of the model. For multi-state systems (as in 9 our case, with multiple energy states) this is a particularly exigent evaluation technique. 14 3. Finally, the selected methodology should perform well in calculating total energy use.

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Each simulated instance is converted to an energy profile using maximum energy values 16 of the appliances present. The total energy use over the validation period is then 17 obtained and compared for the relevant category of appliance.

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The validation data set is a subset of the data that corresponds to 10% of the available  (from a total of over 4 · 10 6 ), which is still large given the computational expense of the 10 clustering algorithms used. In order to overcome this problem, a random sampling process 11 [27] was carried out, selecting 50,000 points that roughly represent 2% of the total size of the 12 data set.   2 OPTICS requires two parameters to produce the ordering of the points: first, the gener-3 ating distance ǫ 0 , referring to the largest distance considered for clustering (clusters will be 4 able to be extracted for all ǫ i such that 0 < ǫ i < ǫ 0 ); second, the minimum number of points 5 that will define a cluster MinPts. However, the algorithm used to automatically extract the 6 clusters from the ordering of the points and their reachability distance makes use of a further 7 7 parameters [19], upon which the clustering structure obtained will vary. For this work, the 8 OPTICS algorithm was implemented in Python 1 .      states produce an understandably more accurate seasonal profile (11x24-SHESD, OPTICS-8 5x14, with 24 and 5 temporal states, respectively). Also, the Survival Multistate model 9 represents surprisingly well the daily seasonality, considering that the temporal dependency 10 is included only in the transitions between states, but not in their duration.

Application of Survival analysis
11 Table 3 complements those results with numerical values for Pearson's coefficient and 12 temporal lag at maximum correlation. Again, the best value for Pearson's coefficient and 13 time lag is achieved using the models with a larger number of temporal states 11x24-SHESD 14 and Survival Multistate, followed by OPTICS-4x12 and OPTICS-5x14.   In the previous section the signal simulated over the whole period was compared; but we 2 are also concerned with how well the averaged daily profile is represented, in terms of the 3 predictive power of simulated energy states and the consequent fractional energy profile.

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Fractional energy states prediction. Figure 7 shows the dependency of the RMSE (calculated simulated are complementary, P s=0 (t) = 1 − P s=1 (t); therefore, a poor estimation of P s=0 (t) 10 implies a poor estimation of P s=1 (t). 11 Furthermore, the shape of the curves for Survival and OPTICS-1x3 models implies that 12 the temporal dependency of the system is not well encapsulated. gesting that the larger the number of energy states (14 in this case), the more accurate the 7 probability prediction. As noted in section 3.3 the accuracy of the modelling of states can also be evaluated 9 using ROC parameters, as shown in table 4; although this is a particularly onerous test when 10 applied to multi-state systems, so that TPR is not expected to be high. Once again the 11 OPTICS 5x14 and Survival Multistate models outperform their counterparts.    In order to compare the results, a box plot is presented in figure 9, and the residual error reproduces reality, errors inevitably arise when using models estimated from aggregate data 12 of the four typologies of appliance to the prediction of specific device behaviours; errors 13 that will reduce in magnitude as the size of the stock of appliances simulated increases.
14 This is reasonable considering that our goal is to estimate communities of buildings and the 15 appliances contained within them. Survival multistate −11.6 −1.44 Table 6: Residual error between observation and simulation, for mean and median of the total energy use over the validation period.  To inform our selection of the most parsimonious modelling strategy the relative perfor-8 mance of each of the models tested is qualitatively summarised in figure 10, using a color 9 coded diagram. From this it is apparent that the Survival Multistate strategy outperforms its 10 counterparts: its predictive power is comparable to that of the more refined Markov models, 11 but is considerably simpler in formulation, both in the estimation of its coefficients and in  (commonly-used and high-rated) and infrequent appliances (very rarely used over the course 10 of a year, independently of their rated power). In both cases, these behaviours are undetected 11 by our modelling approach, with corresponding implications for predictive accuracy.

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In the case of the kitchen category, preliminary results as described in 2 led to the elim-13 ination of the kettle as part of the category. As a high-rated appliance that is commonly 14 owned and used, its behaviour is dominant, misleading the extraction of parameters of the 15 model. Figure 11 shows the results for the survival multistate model applied to the kitchen 16 category with and without the kettle. In this particular case, the total energy use was under-1 estimated by the model, due to its inability to discriminate between the power use pattern of 2 this particular appliance and the other small kitchen appliances. Once removed, the result 3 shows a very good fit with the observed data. The category of other appliances, on the other hand, is biased by the effect of infrequent 5 appliances, which were monitored in the survey but are very rarely used: several being used 6 only for less than 1% of the total recorded time. Consequently, the total energy use predicted 7 was overestimated.
8 Performance of the model 9 The performance of the model has been evaluated in a similar fashion to that for the audio-  (table 8).   The application of the model is shown in this section in two ways: the first involves a single 2 day simulation for a specific household (labelled in the dataset as "103028"), presented in 3 figure 12. It contains 13 different low-load appliances, which are described in table 9. Figure   4 12 displays the output of the model, for the four categories, when using the listed appliances.

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As expected, the model predicts usages of different duration and it is able to capture the spikes 6 in the profiles. But, as expected, the model does not resolve for the specific characteristics 7 of the individual appliances, and it does not represent different behaviours between them.
8 However, the more common usage of audio-visual and computing appliances is well captured.    In this we deploy both discrete (Markov) and continuous (survival) time random processes; 18 and for the former we also deploy cluster analysis to effectively partition the state transition 19 probability space.

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From this we draw the following conclusions: given that it is modelling aggregates. We find that:

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-Finer discretisation of temporal states improved predictive power, but these im-1 provements are modest beyond 5 temporal states.

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This work forms part of a larger programme of research to reliably predict appliance energy 19 demand using bottom-up techniques for communities of households, and to test strategies 20 for the management of these appliance demands to improve community energy autonomy.

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The testing and evaluation of these Demand Side Management strategies will be reported in 22 a future paper.