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Building dynamic thermal profiles of energy consumption for individuals and neighborhoods Adrian Albert

Ram Rajagopal

Electrical Engineering Department, Stanford University Email: [email protected]

Civil Engineering Department, Stanford University Email: [email protected]

Abstract—As a way to match peaks in demand to available supply in real-time on the power grid, energy utility companies employ Demand-Response (DR) strategies. With the recent deployment of advanced metering infrastructure collecting highly granular (sub-hourly) data on consumption from millions of users system operators may now understand how demand arises down to the individual level. In this paper we present an application of a dynamic model that describes residential users’ thermally-sensitive consumption using hourly electricity and weather readings. We build rich profiles at individual and group levels that may be used to inform DR programs that focus on temperature-dependent consumption such as heating and cooling. We learn individual thermal profiles for a large user sample, and describe the seasonal and time-of-day distribution of thermal regimes. Finally, we build aggregate thermal profiles for geographically-homogeneous groups of different size. We argue that aggregation leads to regularity in temperature response of energy consumption, and characterize empirically the critical size of groups needed to achieve limiting regularity.

I. I NTRODUCTION AND MOTIVATION In servicing the electricity grid great care must be taken that supply balances demand at every point in time; failure to satisfy this basic market equation gives rise to blackouts which may lead to huge economic losses. To better monitor and control the demand side of the grid, utilities in the U.S. have deployed millions of “smart meters” that collect highly granular data (hourly or sub-hourly) about energy consumption of customers in both the residential and commercial sectors. Yet to date little is known at utilities on how to extract information out of this wealth of data. One of the largest end-uses of energy is thermal conditioning (HVAC) in response to outside temperature - heating and cooling of premises - which makes up for around 27% of the total electricity consumed in the U.S. [1]. Thus, in administering certain HVAC-oriented Demand-Response (DR) programs1 utilities would like to identify those customers for which significant energy reductions may be achieved from controlling (or completely turning off) temperature-controlled loads such as the furnace or the AC. In recent work [2] we proposed a rich model of consumption for individual residential premises that is driven by unobserved “occupancy states” having different responses to ambient weather (temperature). These consumption regimes of a given household depend on lifestyle (e.g., work schedule), premise characteristics (heating/cooling mass, square footage etc.), appliance stock, and weather patterns. In this 1 E.g., users with high temperature sensitivity may be contacted to reduce their HVAC consumption and offered a monetary incentive in return.

setting, at a given point in time a given user takes a temperature signal and based on that may decide to use HVAC for either cooling (strong positive response of energy use with temperature) or heating (negative temperature response), or not use HVAC at all (small temperature response). While the utility may not observe these latent decisions, it does observe their outcome - the hourly readings of energy consumption. The model (outlined below) allows a coarse “disaggregation” of the sequence of choices that may serve to differentiate users for the purpose of a thermal DR program. In this paper we present an application of these thermal profiles to perform a detailed study of the dynamics of thermal sensitivity in a large sample of users. We analyze time-of-day and seasonal effects on thermal sensitivity across the user population both for individuals and for neighborhoods in Northern CA. We show that regularity is obtained through aggregation when the object of analysis is the average consumption profile of a neighborhood, and characterize the size of geographically-homogeneous groups needed to achieve regularity in consumption. In effect, we develop a methodology to build dynamic city-level maps of thermal sensitivity that may support program dispatch decisions for either neighborhoods or groups of individuals. In addition, we discuss in some detail the computational aspects of the model, and illustrate in- and out-of-sample model performance. The rest of the paper is structured as follows. Section II gives an overview of the thermal regimes profiling problem at different levels of analysis. Section III discusses the literature on smart meter data analytics and temperature modeling. Section IV reviews the the consumption model proposed in [2]. Section V presents the dataset we used in this work. Section VI discusses model results on several example individual premises, and presents our cross-sectional analysis of individual consumption regimes. In Section VII we present aggregate thermal profiles for for neighborhoods in CA and discuss scaling effects. Section VIII concludes the paper. II. P ROBLEM STATEMENT Individual thermal profiles. We observe (time-aligned) hourly energy consumption time series {Xt }n and outside temperature time series {Tt }n for premises (users) n = 1, ..., N and t = 1, ..., τ . We study energy use at a premise through its “temperature profile”, i.e., the dependence of hourly energy consumption with outside temperature. A schematic of such a profile is given in Figure 1 (top panel). It has been argued in the literature [1], [3] that such a statistical representation of consumption offers cues of the degree to which a heating or cooling appliance is operating at a premise. More precisely, a large negative slope indicates heating activity, a large positive slope indicate cooling ac-

of ∆W (T ) = ∆T × β(T ). We create geographical maps of effective thermal sensitivities for different forecasts of the outside temperature that offer estimates of program impact (i.e., expected energy savings) for entire regions.

Fig. 1. Top: Schematic of whole-home energy temperature profile for which both cooling and heating activities are present outside of the comfort interval [TC , TH ]; [1]; Bottom: Occupancy states having different activity levels (height of the horizontal bars), thermal responsiveness (arrow slope), and characteristic duration (width of the bars).

tivity, while an intermediate, temperature-insensitive regime indicates a “comfort zone“ where HVAC is not in use. The consumption model proposed here views consumption model as driven by “occupancy states” (see Figure 1), each having a different activity level and a sensitivity to outside temperature. In effect, we separate the signal into segments of observations that are consistent across i) time, ii) their sensitivity to the outside temperature, and iii) the temperature-insensitive baseline. This formulation is based on the following observations: i) heating or cooling spells generally span multiple hours, and given a certain response (say heating) at a given hour, it is likely that the next hour will see the same thermal activity; ii) while the premise may have appliances that work on pre-determined schedules (e.g., an automatic thermostat that maintains inside temperature within a certain admissible range), these settings may change dynamically based on user preferences and occupancy, and iii) heating or cooling appliances do not consume fixed amounts of energy when they are on, but undergo different operational regimes depending on how much work is needed to respond to the outside temperature. Moreover, we allow the occupancy states to be either “sticky” or “transient”, i.e., to have either long or short (bursty) durations. Neighborhood-level average profiles. While the model presented here is motivated from a individual behavioral standpoint, we would like to investigate what regularity arises when the object of study is the average consumption profile of aP geographically-homogeneous groups of n users, ˆ t ≡ 1 n {X}t . This can be seen as a proxy for the {X} i=1 n aggregate load actually observed at the substation level (the service point) by the grid operator. Users living in the same neighborhood may be reasonably expected to experience the same climate patterns, have similar socio-economic attributes (e.g., income, house ownership, rent levels) and, to some extent, live in buildings of similar physical characteristics (e.g., square footage or year built). Thus we may expect that they will use HVAC loads approximately at the same time (since they experience the same outside temperature). Moreover, behavioral differences across users will be averaged-out, resulting in a clearer heating and cooling signals. As such, the neighborhood thermal profiles allow for quick insight on the actual punctual loads that need to be serviced. Averted HVAC consumption. We use the thermal profiles to estimate maximum potential averted consumption from controlling the HVAC loads at different temperature levels. For a user with a thermal response β(T ) for a given temperature T , a change of ∆T = 1◦ F in the HVAC appliance setting (either heating or cooling setpoint) will result in a saving

III. L ITERATURE REVIEW A large body of literature exists on modeling weather (and in particular temperature) response of energy use in residential and commercial premises. Most previous studies have been performed on aggregated data (e.g., [4], [5], [6]) because of the lack of intra-day measurements. New sensing capabilities have enabled studies of the thermal response at an hourly level such as [1]. There, the authors formulate a breakpoint regression that models the set point of HVAC equipment such as air conditioning (AC) or heating furnace. This model assumes i.i.d energy readings that depend linearly with temperature in two thermally-sensitive regimes (heating and cooling as shown in the figure) and one thermallyinsensitive regime (non-HVAC activity). However such a model generally does poorly in describing caveats found in real consumption data (e.g., overlapping linear regimes). However these type of models are rather static - they do not allow for different temperature sensitivities based on temperature, or on different levels of activity. A more complex temperature model is developed in [3], where the existence and operation of the heating appliance is disaggregated from the total consumption of a residential premise. Similar to our model, the authors use a Hidden Markov Model (HMM) that follows the (hourly) dynamics of a thermal load expressed as a TCL model for HVAC. However this model does not allow for heating consumption to vary with temperature, which is generally not a valid assumption for real buildings. In the context of the disaggregation problem of recovering individual appliance signals from the aggregate load profile [7], [8], [9] much emphasis has been placed on uncovering all major end uses through very granular (Hz or kHz-range) consumption readings. Instead, here we propose a simpler, coarser thermal disaggregation in the absence of ground truth data, with the purpose of developing high-level metrics that may serve for DR segmentation and targeting, which is one of the intended uses of disaggregation. We build on our previous work [10], where we show that dynamic characteristics of consumption computed from individual energy time series using Hidden Markov Models are predictive of both the presence of large appliances and of user lifestyle attributes. IV. O CCUPANCY STATES MODEL A. Model formulation We model a given individual premise as a state machine for consuming energy using a hidden Markov framework. At each point in time, the premise finds itself in either of K (unobserved) thermal regimes S = {1, ..., K}. Response. For a given time t, a temperature level Tt , and a given state k, the model expresses consumption xt as xt |St = k ∼ N (β0k + β1k Tt , (σ 2 )k ),

(1)

where N (·, ·) denotes a normal distribution. We would like to estimate β k = (β0k , β1k ) and (σ 2 )k . The state-specific temperature sensitivity β1k may be interpreted to model the extent of HVAC response, which may be either cooling (β1k > 0), heating (β1k < 0), or no HVAC (β1k ∼ 0). State transition. To model the temperature-dependent choice of thermal regime we allow the sequence of states {St } to

follow a temperature-dependent Markov process. Denoting {A(T )}ij ≡ P (St = i|St−1 = j, T ) the temperaturedependent transition matrix of the regime-switching probability, we model each row as a multinomial logistic regression. In effect, the model makes the implicit assumption that the user maximizes a “utility” of being in the same state or switching to another state based on the perceived level of the current outside temperature Tt [11], exp(γ0k,j + γ1k,j Tt ) P (St+1 = k|St = j, Tt ) = P . i,j i,j i exp(γ0 + γ1 Tt )

(2)

We would like to estimate γ k = (γ0k , γ1k ) from data. B. Computational caveats Estimation. As commonly done in the literature [12], we estimate the parameters of the model above for each individual premise using the Baum-Welch algorithm (a variant of the Expectation-Maximization (EM) algorithm [13]). We use the Viterbi algorithm [12] to compute the most likely sequence of states S that fits a given observation sequence x. Scalability. The EM algorithm has a computational complexity linear in the number of observations τ and quadratic in the number of hidden states K, O(τ K 2 ). For reasons of scalability to large data sets, we would like parsimonious models (small K). Note also that possible interactions between users are not accounted for when estimating individual models, which reduces significantly the computational burden of estimation. In this way, we were able to easily implement a distributed estimation framework on multiple nodes, whereby the computatation was evenly balanced across the cluster nodes. We then post-process the obtained state attributes for each user to allow for a cross-population analysis. Choosing model size. In real applications the model size K (the number of states) is generally not known. To find an appropriate value of K we implemented a simple selection strategy that increases model complexity until a desired outof-sample performance is reached (here we used 0.85 for the R2 metric and 0.15 for M AP E defined in the usual way). Because of the long-range serial correlations assumed by the Markov model, a typical k-fold cross-validation technique that randomly segments the data is not desirable here to compute out-of-sample performance. We address this issue by using a deterministic 2-fold cross-validation approach [2]. C. Metrics and benchmarks We use the model described above to derive benchmarks that characterize consumption: 1) We allow for a soft temperature-dependent regime selection and compute the long-run probability distribution π of the premise being in either of the K states as the 1-left eigenvector of the transition matrix A(T ), π(T ) = π(T )A(T ); 2) We compute breakdowns of states by season and time-ofday starting from the observation that averaging will allow errors in the hourly estimations to cancel out to some extent. 3) To aidP analysis, we define an effective thermal response βˆ1 (T ) = k β1k πk (T ) at a given temperature level T . V. DATA DESCRIPTION We used two types of data: smart meter consumption time series and temperature time series (at the 5-digit zipcode level) collected from appropriate weather stations. 1) Individual consumption data: We use the largest research dataset to date of whole-premise smart meter consumption

Fig. 2. Distribution of group size (number of users aggregated in each neighborhood).

time series from ∼ 120, 000 users in Northern CA. Data is recorded at an hourly level and spans one year from August 30th , 2010 to July 31st , 2011, i.e., 365 days of 24 observations daily for each user. For analyzing individual consumption we selected a subset of 10, 000 users from zipcodes with at least 50 users represented in the original sample (resulting in > 86M records). 2) Zipcode-level aggregate consumption: We computed average neighborhood consumption profiles by aggregating reading time series for each zipcode in the initial sample. We thus obtained average consumption profiles for 470 neighborhoods with varying sizes, as illustrated in Figure 2. This amounts to > 44M days of data, or over 1Bn records. 3) Weather data: For each zipcode in our sample we collected weather time series for the appropriate time interval using the online API at www.wunderground.com. We imputed the occasionally missing data using a linear regression approach. VI. T HERMAL PROFILES A. Case study: Alice and Bob In Figure 3 (left column) we present the consumption temperature profiles of two users (anonymized IDs 2433956 and 267098, which we shall further denote by Alice and Bob) in the greater San Francisco Bay Area. The observations are color-coded by the consumption regime that the model assigns them to. The patterns in consumption may be explained for both users by models with 4 regimes to achieve a crossvalidation decoding R2 ≥ 85%. Alice’s temperature profile looks nothing like the standard heating-and-cooling schematic in the left panel of Figure 1; at first sight it appears composed of horizontal bands at different consumption levels and with different degrees of variance, i.e., there appear to be no temperature-sensitive regimes. However when further examining the occupancy states identified by our model we notice that state 2 (green) is in fact visibly influenced by the outside temperature; its positive slope corresponds to a moderate cooling activity undergone by Alice. In the case of Bob, the model identifies three temperature-insensitive regimes, and one heating regime (negative slope, green triangles in Figure 3, lower panel). Consumption time series for the two users for one week in 2011 are presented in the right column of Figure 3. Dark lines indicate the original data; colored lines encode the different states identified by the model. Note that in general the model is able to follow very closely the profile of the data, including the occasional, bursty peaks.

Fig. 3. Top: temperature profiles and identified thermal regimes Alice (top) and Bob (bottom); Bottom: consumption time series and model fit for a week in 2011 for the two users. Original data is shown in black; inferred states are shown in color. Error bars indicate state variance estimated in the model.

Fig. 5. Distribution of model size (least number of states that explain at least 85% of variance) across a large population. Left: individual profiles for 10,000 users; Right: aggregate neighborhood profiles for 120,000 users in 470 neighborhoods.

Fig. 4.

Effective response as function of temperature for Alice and Bob.

In Figure 4 we calculate the effective temperature response βˆ1 and the associated variance for all levels of temperature (the thermal profile) for Alice and Bob. While Alice (blue) has a large variance in thermal regimes (as it is apparent from her temperature profile in Figure 3, top panel), Bob (red) is relatively consistent in his cooling behavior. We may now use this concise description of the temperature response βˆ1 of individual premises to compare, segment, and rank users according to the expected averted consumption (as described in Section II under a thermal DR program. B. Population view We build thermal profiles as above for a large sample of 10,000 users extracted from out dataset of 120,000 households. For most users 4 states were sufficient to explain at least 85% of the variance in the data (see Figure 5, left panel). Next, we classified the obtained consumption regimes

according to their temperature response sign (either heating or cooling) and magnitude (none, low, high). Note that location should play a role in temperature sensitivity, in particular since in Northern California weather patterns vary drastically with geography. A general trend is that average temperatures are lower and humidity is higer closer to the San Francisco Bay, while they are higher in the more arid inland locations. This view is enforced by Figure 7, where the distribution of the effective hourly thermal response is presented across the 10,000 users and four climate zones2 . Zone Z03 is a cold, coastal zone along the San Francisco Bay Area. Zone Z04 is an inland, hilly region close to the bay. Zone Z13 is situated inland and covers a large part of Central Valley. Zone Z12 is in an inland, northern location. In general, Zone Z03 experiences the lowest and zone Z13 the highest temperatures. Note that cold zones Z03 and Z04 experience relatively little cooling (either low or high), but relatively substantial heating. This picture is reversed in hot zones Z12 and Z13, where most consumption regimes are cooling. 2 Description available at http://www.pge.com/mybusiness/edusafety/ training/pec/toolbox/arch/climate/.

Fig. 6. In- and out-of-sample performance for aggregate neighborhood profiles of 120,000 users on the R2 (left) and M AP E (right) metrics.

Fig. 7. Consumption regime classification and distribution across climate zones in Northern CA. Zones are ordered from cold to warm.

Fig. 8. Temperature profiles of average consumption for two neighborhoods.

VII. AGGREGATE PROFILES We built average thermal profiles for all 470 neighborhoods in our dataset. The distribution of the number of users in each neighborhood is presented in Figure 2. The average number of users in a neighborhood is ∼ 250, although the distribution is quite skewed. The vast majority of neighborhoods could be described using 4 thermal regimes, as illustrated in Figure 5 (right panel). Moreover, in- and outof-sample performance of the model is robust, as shown by the narrow distributions of the R2 and M AP E in Figure 6. A. Neighborhood thermal map We briefly discuss two example neighborhoods below of different sizes - zipcode 95301 (352 users, denoted as “1” in the figures) and zipcode 95374 (31 users, denoted as “445” in the figures). This regularity is observed also in the temperature profiles of the average user in the two zipcodes presented in Figure 3. As before, each observation is colorcoded by the regime the model assigns it to. Note that in the first neighborhood users generally tend to exhibit cooling activity, whereas in the second neighborhood users perform both cooling and heating. From these profiles we compute the distribution with temperature of the effective thermal response for the two neighborhoods in Figure 9. Note the step-function-appearance of the profiles in the figure and the larger critical region between the two regimes of opposite temperature response sign for the first zipcode (445). The width of the gap may be interpreted as an effective “thermal comfort” region for the average user in this neighborhood. Finally, in Figure 10 we present the geographic distribution of neighborhood-level thermal regimes across the entire user base for a high level of temperature. In a practical application the temperature level would be obtained through hour-ahead forecasts, which are generally quite accurate. The

Fig. 9.

Effective thermal response profile for two selected neighborhoods.

map presents the geographic extent of the zipcodes used in our analysis color-coded by effective thermal response (a choropleth map), where large positive values (effective cooling) are blue and large negative values (effective heating) are red. Note that neighborhoods around the cold San Francisco Bay are generally blue (cooling) at T = 100. Such maps convey a dynamic, temperature-dependent, rapid assessment of potential consumption averted through a thermal DR program across communities serviced by the system operator. B. Group scale and thermal regularity When analyzing average response of groups of different sizes, it is natural to ask the question of what effect averaging has, and whether scale (number of individuals in a group) plays a role in macroscopic, group-level behavior. Here we ask how large must groups be in order to achieve a certain regularity in thermal consumption. Recent observations (Raffi Sevlian, personal communication) have shown that aggregation of individual profile may improve forecastability of group average by an amount that increases with the group

Fig. 10. Map plot of aggregate average temperature sensitivity for a high temperature level across a population of 120,000 users in Northern CA.

VIII. C ONCLUSIONS In this paper we have illustrated a methodology that builds dynamic, thermally-sensitive energy consumption profiles for individual users and for neighborhoods. We provided an extensive discussion on how this methodology may be used in practice to understand consumption and its components mainly driven by response to outside temperature. Our analysis focused on two levels of decision-making - the individual premise and the neighborhood (a geographicallyhomogeneous community). We learned thermal profiles for a large sample of users and showed that the model preserves general climatic trends. In addition, we studied the scaling effect of regularity in thermal regimes and the size of the group, and found that for groups of 10 users or larger consumption become quite stable over time in terms of temperature sensitivity. We will address this observation with more rigor in future work. Starting with the analysis and benchmarks presented here, we are currently developing a dynamic targeting methodology for Demand-Response programs that affect heating or cooling loads. Our method uses the temperature-dependent thermal responses to identify optimal subsets of users whose aggregation yields an optimum trade-off between expected value of energy savings and its variance. [1]

[2] [3]

[4]

Fig. 11. Model performance on average neighborhood consumption profiles and neighborhood size. Left: out-of-sample cross-validated M AP E; Right: entropy of state sequence assuming i.i.d. state selection.

[5]

[6]

size up to a certain critical value thereof. Here we sketch an empirical treatment of the effect of aggregation on forecastability and on regularity of consumption regimes identified by out model. In Figure 11 (left panel) we present out-ofsample forecasting performance on the M AP E metric. Note that average performance increases for groups of increasingly larger size, with little gains above 100 users. Conversely, regime variability (as measured by the Shannon entropy3 ) decreases for groups larger than ∼ 10 users. This suggests that groups larger than this critical size will tend to have a distribution of thermal regimes that is quite homogeneous over time. While more investigation is needed, this initial observation may be of importance in designing DemandResponse programs that are aware of the aggregate savings obtained at the neighborhood level.

[7]

[8]

[9] [10]

[11] [12]

[13] 3 Entropy

P

S = − k pk log pk , with pk the empirical frequencies of occupancy states observed in the data. Note that this simplified formulation assumes independent and identically-distributed (i.i.d.) probabilities of the states, which is certainly not the case.

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