Salim Rao

Linear aggregation of conditional tail index estimators

come

Extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the mean of probability distributions. It seeks to assess the probability of events that are more extreme than any observed prior. Extreme value analysis is widely used in many disciplines, ranging from structural engineering, finance, earth sciences, traffic prediction, geological engineering, etc. For example, EVA might be used in the field of hydrology to estimate the value an unusually large flooding event, such as the 100-year flood. Similarly, for the design of a breakwater, a coastal engineer would seek to estimate the 50-year wave and design the structure accordingly.

Let X be a random variable and F (x) = P (X ≤ x) its distribution function. We can define a new random variable, Y n , as the maximum of n copies of the random variable X: Y n = max{X 1 , X 2 , ..., X n }. Y n is the n-sample maximum of the random variable X. If the events generating the realizations of X are independent, the cumulative distribution of Y n may be expressed as [F (y)] n . Upon definition of a renormalized variable S n = (Y n -b n )/a n , the extreme value theorem establishes that, if

lim n→∞ P (S n < s) = lim n→∞ F n (a n s + b n ) = H(s) (1)
where a n > 0 and b n are normalization constants, then the function H(s) in Eq. ( 1) must be one of the three following types:

• EV1 or Gumbel: H(s) = exp(-exp(-s))

• EV2 or Fréchet:

H(s) = exp(-s -α ) • EV3 or Weibull: H(s) = exp(-|s| α ) 1
The three asymptotic types, EV1-EV3, can be thought of as special cases of a single Generalized Extreme Value distribution (GEV) :

H GEV (s) = exp -1 + γ s -µ σ -1/γ + (2)
where (.) + = max(., 0), µ is the location parameter, σ > 0 is the scale parameter, and γ is a shape parameter. The limit γ = 0 corresponds to the EV1 distribution, γ > 0 to the EV2 distribution (with α = 1/γ) and γ < 0 to the EV3 distribution (with α = -1/γ). We refer to [START_REF] Rao | Estimation of the proportional hazard premium under random censoring[END_REF] for the censored case.

When the studied phenomena depends on a covariate, one has to deal with Conditional Extreme Value Analysis (CEVA). This branch of statistics has become very active these past ten years, the main contributions to this domain are listed below:

• Theoretical issues: [START_REF] Das | Detecting a conditional extreme value model[END_REF][START_REF] Gardes | Functional kernel estimators of large conditional quantiles[END_REF][START_REF] Gardes | On the estimation of the functional Weibull tail-coefficient[END_REF][START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Gardes | Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels[END_REF][START_REF] Gardes | Functional nonparametric estimation of conditional extreme quantiles[END_REF][START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF][START_REF] Gardes | Nonparametric estimation of the conditional tail copula[END_REF][START_REF] Beirlant | Local polynomial maximum likelihood estimation for pareto-type distributions[END_REF][START_REF] Ac Davison | Local likelihood smoothing of sample extremes[END_REF][START_REF] Chavez-Demoulin | Generalized additive modelling of sample extremes[END_REF][START_REF] Hall | Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data[END_REF][START_REF] Methni | Kernel estimation of extreme regression risk measures[END_REF][START_REF] Das | Conditioning on an extreme component: Model consistency with regular variation on cones[END_REF][START_REF] Beirlant | Nonparametric estimation of extreme conditional quantiles[END_REF][START_REF] Abdous | Extreme behaviour for bivariate elliptical distributions[END_REF][START_REF] Wang | Estimation of high conditional quantiles for heavy-tailed distributions[END_REF][START_REF] Wang | Tail index regression[END_REF][START_REF] Rao | A review on conditional extreme value analysis[END_REF] • Quantile regression: [START_REF] Koenker | Quantile regression: An introduction[END_REF][START_REF] Rosen | Extreme percentile regression[END_REF] • Application to finance: [START_REF] Byström | Managing extreme risks in tranquil and volatile markets using conditional extreme value theory[END_REF][START_REF] Bali | A conditional extreme value volatility estimator based on high-frequency returns[END_REF][START_REF] Ghorbel | Predictive performance of conditional extreme value theory in value-at-risk estimation[END_REF][START_REF] Byström | Extreme value theory and extremely large electricity price changes[END_REF][START_REF] Marimoutou | Extreme value theory and value at risk: application to oil market[END_REF][START_REF] Wang | A model of conditional var of high frequency extreme value based on generalized extreme value distribution[END_REF] Our mail result is the following:

Theorem Let γ1 (x), . . . , γK (x) be K consistent estimators of the conditional tail index γ(x). Let w 1 , . . . , w K be K weights (not summing to zero). Then,

γ(x) := K i=1 w i γi (x)/ K i=1 w i
is a consistent estimator of the conditional tail index γ(x).

Proof. The proof is based on the continuity of the sum function and on the property that K i=1 w i = 0. Let us highlight that the positivity of the weights in not required. The weights are not supposed to summing to one either. Moreover, this result is valid whatever the sign of γ(x).


J Wambecke

G P Bonneau

R Blanch

R Vergne

R. Bujack and K. Rink and D. Zeckzer (Editors

Keywords: (ACM 2012 CCS) Visualization systems and tools, Visualization design and evaluation methods, Empirical studies in visualization

Many households share the objective of reducing electricity consumption for either economic or ecological motivations. Ecofeedback technologies support this objective by providing users with a visualization of their consumption. However as pointed out by several studies, users encounter difficulties in finding concrete actions to reduce their consumption. To overcome this limitation, we introduce in this paper Activelec, a system based on the visualization and interaction with user's behavior rather than raw consumption data. The user's behavior is modeled as the set of actions modifying the state of appliances over time.

A key novelty of our solution is its focus on the What if approach applied to eco-feedback. Users can analyze and experiment scenarios by selecting and modifying their usage of electrical appliances over time and visualize the impact on the consumption. We conduct two user studies that evaluate the usability of Activelec and the relevance of the What if approach for electricity consumption. Our results show that users understand the interaction paradigm and can easily find relevant modifications in their usage of appliances. Moreover participants judge these changes of behavior would require little effort to be adopted. Therefore our work opens new perspectives on using the What if approach in the context of eco-feedback technology.

Introduction

Reducing energy consumption is a major concern for many households. They either want to lower their energy costs and/or are motivated by environmental concerns such as the limitation of CO 2 emission. In this paper we focus on domestic electricity, which is an important share of household energy consumption. Thanks to the development of sensors such as smart plugs, households can easily access and record their electricity consumption over time. Eco-feedback technologies show a visualization of their consumption to users with the aim of decreasing it. In this domain many solutions have been designed, either commercial systems or research prototypes. One of the main limitations of eco-feedback tools lies in the difficulty for users to find concrete actions to save energy. Several systems tackle this issue by providing context to the visualization and/or ways to interact with the data [CRJ12, ELK * 13]. However, interactivity is often limited and does not help users to find relevant modifications in the usage of appliances. As an example few systems provide the ability to visualize and manipulate usages of appliances along with raw consumption data.

In this paper we introduce Activelec, a visualization system that allows users to analyze and interact with their electricity consumption data. The design rationale takes into account previous findings and limitations of eco-feedback technologies, since the design is known to strongly influence users' ability to save energy [PFL * 10]. The novelty of Activelec is the introduction of the What if approach applied to eco-feedback. Herein users can test scenarios in which they modify the usage of appliances over time and then evaluate the impact of these changes on the consumption. Our solution is based on a data model that represents the user's behavior as a set of actions modifying the state of appliances over time. To assess both the usability of the system and the relevance of the What if approach in the context of eco-feedback, we conduct two user studies based on real consumption data. Results show that users can find relevant and efficient appliance usage modifications, thereby validating the use of the What if approach for household electricity consumption feedback. To summarize we present the following contributions :

• The design rationale for Activelec that justifies the use of the What if approach in the context of eco-feedback, and especially the data model designed for this (Section 3). • The Activelec system in itself, which allows to analyze and test scenarios on consumption data (Section 4). • Two lab studies to assess the usability of the system and the relevance of the What if approach (Sections 5 and 6).

Finally our paper opens new perspectives on the What if approach in the context of eco-feedback discussed in Section 7.

Related Work

Eco-feedback

Eco-feedback technology aims at presenting to users a visualization of their energy consumption in order to decrease it. Studies have proven the efficiency of eco-feedback in reducing energy consumption from 5 to 13% [START_REF]The effectiveness of feedback on energy consumption. a review for defra of the literature on metering, billing and direct displays[END_REF]. Even if there is no definitive explanations for this efficiency, Buchanan et al.

[BRA14] suggests one main reason. Users learn about the consumption of appliances, the link with their behavior and the actions that can be applied to the appliances to save energy. However, many limitations that impact the effects of eco-feedback have been pointed out in the forementioned surveys. The main bias in the design of eco-feedback systems is the assumption that visualizing energy consumption will directly lead to savings [START_REF] Strengers | Designing eco-feedback systems for everyday life[END_REF]. Such systems often lack interaction, reducing it to exploration of raw consumption data. Moreover, users encounter difficulties to come out with daily concrete actions that can be applied to reduce consumption [Bar15,ELK * 13,Str11].

Several works focus on the design and evaluation of visualization systems for electricity consumption. Erickson et al. [ELK * 13] developed the Dubuque Electricity Portal, a system allowing users to visualize their electricity consumption via an online portal. The contribution of this work is not the system itself, which presents a classical design. Rather it is the long term study with a large amount of participants, providing findings on eco-feebdack usage. We can also cite serious games dedicated to electricity consumption. This is the case of PowerHouse [BTK06] and PowerAgent [GKB10]. Both games are designed to acquire some knowledge about electricity consumption in order to decrease it. They are not focused on visualization but are similar to our approach, since the users can test a given behavior and obtain a feedback on it.

The What if Approach

The What if approach is usually implemented for the analysis of event based systems whose outcome depends on multiple complex factors, such as financial systems or large distributed systems. For example, Tarik et al. [TBV * 13] developed WISE, a language designed to use the What if approach in the context of content distribution networks. This system targets expert users, and is based on a language implementation of the What if approach. In comparison, our system targets non-expert users and is based on interaction. Badam et al. [BZS * 16] presented TimeFork, an interactive prediction technique for time series. The system allows users to interact with and guide multiple computational models predicting the future of multivariate time series, and thereby to test What if scenarios. In comparison, our system does not need computational models predicting the future. Users can experiment What if scenarios by analyzing and changing past values through interactive selection and modification of appliances states.

Costanza et al. [START_REF] Costanza | Understanding domestic energy consumption through interactive visualisation: A field study[END_REF] presented Figure Energy, a system based on the visualization of household electricity consumption. They provide users with the ability to manually annotate their activities and to apply What if scenarios by removing them. The important difference with our system is that modifications are limited to removing user activities. More importantly the user activities are only removed in another view which does not impact the temporal visualization. In comparison our system allows temporal selection of usages and modifications such as changing appliances states and duration of usage. In their visualization design method, Goodwin et al. [GDJ * 13] detailed Demand Horizon, a prototype that allows the consumption of appliances to be modified by directly sculpting the raw data. If their studies gave promising results, it is worth noting that the system was designed for energy analysts and modelers to elaborate scenarios directly on consumption data of multiple households. In contrast, our system is designed for non-expert users without any assumption on their knowledge in energy consumption.

Design Rationale

System Description

During studies about eco-feedback systems, users often encounter difficulties to find concrete behavior changes to reduce their consumption [Str11,Bar15,ELK * 13]. Our system is designed to tackle this problem, since users visualize, interact and modify their usage of appliances instead of visualizing raw consumption data. First, users explore their data by visualizing their usage of appliances linked to their consumption. Then, they select actions on appliances via criteria, followed by a modification of the actions that have been selected. Once the modification is applied, the visualization is updated in a way that allows users to evaluate the impact of their scenario on the consumption. This interaction loop motivates users to engage their data by manipulating concrete actions on appliances, e.g. ecological washing versus normal washing for a dishwasher. Our system is a dashboard, i.e. a classical computer software that provides historical views of the data, allowing patterns and trends to be recognized [START_REF] Strengers | Designing eco-feedback systems for everyday life[END_REF][START_REF] Bartram | Design challenges and opportunities for ecofeedback in the home[END_REF]. Hence it relates data to behavior using classical information visualization methods.

Strengths of the What if

The appliances and especially their consumption per usage vary among households. This implies that a modification in the usage of an appliance can lead to different gains. As an example, switching off the lights can lead to high or low gains depending on the efficiency of the bulbs. Thus users need to have precise information about the outcome of behavior modifications instead of generic advices such as "switch off the lights". Another strength of our approach is the ability to test repeated subtle modifications over time. A given change (e.g. boiling only the needed quantity of water in a kettle) can seem negligible but may lead to substantial gains if repeated every day. By allowing users to visualize the accumulated gain over time they become aware of the relevance of such modifications.

The What if approach allows users to assess the ratio between gain and effort required to apply their modification into their everyday actions. It seems trivial to gain energy by reducing the usage of appliances, yet in practice, this is strongly dependent on external factors constraining our use of household devices. For example savings that could be achieved on the washing machine depend on the number and the ages of people in the household, as well as on temporal factors (e.g. possible day and hour of usage). It also depends on personal factors such as users' motivation. These factors, often neglected in eco-feedback [PFL * 10], cannot be known by the system. This supports the use of the What if approach which allows these factors to be analyzed individually.

The What if approach can also counteract another limitation of eco-feedback systems, which is the non-negotiability of everyday practices [Str11, PFL * 10]. In other terms it means that some appliances usages cannot be removed due to external constraints (e.g. social constraints). For example, to stop using the washing machine is not conceivable since people want clean clothes. Our system allows usages to be both removed or modified. For example, users can switch the temperature of the washing machine from warm to cold, which can lead to a substantial gain.

Data Model

We designed a data model that decomposes appliances data between consumption data and user's actions data. In this model, the input is disaggregated energy data where each appliance is associated to a temporal dataset. Consumption data represents the evolution of the instantaneous electric power of the appliances, and is measured directly or simulated when a modification is applied. Appliances are associated to states over time (e.g. Off, Standby and On for a Computer). At a given point in time the state of an appliance thus depends on the action of the user. Therefore the states of the appliances allow to express the relationship between consumption data and users' actions. We define an instance of a state as the time span during which the appliance remains in this state. Instances hence encode the evolution of the state of the appliance over time. The instances are automatically extracted in the system based on thresholds on instantaneous power and average consumption over time. Figure 1 shows an example of evolution over time of computer data according to this model.

System Design

User Interface

Our system is built on two modes: Appliances mode and Instances mode. Appliances mode (Figure 2) displays the consumption of all appliances on the same plot, allowing users to explore and compare them. This mode gives to users an overview of the consumptions of household appliances. From this mode, users can switch to Instances mode (Figure 3) to focus on a given appliance. Instances mode displays the consumption data and the usage of a single appliance and implements the What if approach. The units of energy such as kWh are often considered abstract by users and need to be converted into meaningful information [START_REF] Strengers | Designing eco-feedback systems for everyday life[END_REF]. Hence the amounts of energy consumed by appliances, contained in selections or saved by modifications are expressed in terms of cost aside with the kWh. Our system is also able to express the energy in environmental units, such as pounds of emitted CO 2 , which makes sense for users with environmental motivation [START_REF] Bartram | Design challenges and opportunities for ecofeedback in the home[END_REF].

Visualization

To display the evolution of the consumption over time, we chose to use line charts, since it is a well known metaphor for continuous temporal data. They are also well suited for fluid interactions like continuous pan and zoom. Consumption curves are rendered and filtered in a screen-dependent manner [START_REF] Berry | Binx: Dynamic exploration of time series datasets across aggregation levels[END_REF]. That way, unlike most eco-feedback systems that aggregate the data at a fixed scale, the curves can be visualized at any zoom level and are always smooth and coherent. In Instances mode the curve is colored according to the instances of the appliance. More precisely, the part below the curve is filled with the color of the current state. By using this representation, users can visualize the consumption along with the usage of the appliance. A multi-hue color scheme from Color-Brewer [Bre03] is used to assign colors to appliances and states.

Selection

In Instances mode, instances can be selected to obtain information on the usages or to modify them. The first way to create a selection is to do it directly onto the visualization by clicking on the instances. It allows users to select particular instances that could not be selected with temporal criteria or to test a modification on a few instances. The second way to select instances is through the definition of criteria on their properties. In its current implementation the system provides the possibility to select instances according to their state, duration, weekday, day hour and by frequency. For example, for a given appliance (e.g. a computer) users can select all the periods of Standby whose length is greater than one hour. It is also possible to combine criteria, such as selecting only one out of three instances from this selection on the computer. Once a criteria is modified, the set of selected instances is updated in real time, and these instances are highlighted on the curve. Defining criteria to select instances extends the exploration step, since users can explore the space of criteria with real time feedback. It also allows to generalize a pattern found during the exploration step, hence allowing habits to be easily visualized and modified.

Modification

After the selection of instances, the next step in the loop of the What if is their modification. Our system provides two types of modifications. The first one is changing the state of the instances, which is related to removing a usage of the appliance or modifying the way the appliance is used. For example, it is possible for users to switch the state of instances of a washing machine from warm washing to cold washing. The second kind of modification is limiting the time of the instances either to a given duration or after a given hour in the day. The extra part, i.e. after the limit, is removed and replaced by Off. For example, users can limit all instances of Standby of a TV to 2 hours, or stop them after 11 p.m. When a modification is applied, the visualization is updated accordingly by computing new data. For each applied modification, a widget in the list of applied modifications to the right gives information about its selection, its type of modification and more importantly its gain or lost value. Users can do as many modifications as they want, which are added to the list to be compared, modified or deleted.

Usability Study

We conducted a first user study to evaluate the usability of Activelec and especially to assess if users can easily find relevant modifications in the usage of appliances. To do so, we designed a controlled experiment where participants used the same dataset, coming from real appliances.

Method

13 users (11 males, 2 females) participated in the study. The participants were recruited in the faculty, and were distributed among 7 PhD students, 1 post-doctoral student, 4 engineers and 1 professor. Among these 13 participants, 11 had no knowledge of visualization tools. Only half of them (7) had energy knowledge, i.e. they could define electrical power and energy. All participants were motivated in saving energy, but 4 of them said that they did not want to put too much effort to change their behavior. One participant did not follow the instructions, thus we did not consider his results. For this study we used actual data from the Household Electricity Survey [START_REF] Change | Household electricity survey[END_REF]. We selected six months of data from ten common appliances. We manually defined the states of the appliances so that our tool could extract the instances from consumption data.

The experiment was preceded by a learning phase of approximatively 25 minutes. The first step of this phase was the explanation of the tool by one experimenter. As a second step of the training phase, the participant had to reproduce two modifications that were given by the experimenter, on two different appliances. To conclude this phase the author asked if the participant had additional questions before starting the task. For the task we asked participants to find actions which save energy by using the tool as if they were at home with their own data. We did not enforce saving objectives so that participants can freely use the system. To evaluate the relation between savings induced by the modification and the effort to apply it in the home, we used a subjective measure rated by the participants for each modification they did in the system. It evaluates how much a modification in the usage of an appliance is seen as difficult for participants to apply in their home. We asked participants to rate this effort for each modification on a psychometric scale from 1 (no effort) to 5 (maximum effort).

Results

Table 1 presents quantitative results from the task. For each appliance, the mean gain is the average for all participants of the ratio in percentage between the gain realized by the participant and the Figure 4: Rates from final interview. "→ understanding" means helps in understanding consumption and behavior and "→ actions" means helps to find relevant actions to save energy original consumption. Analogously the mean effort for an appliance is the average effort rated by the participants for the modifications that were applied to this appliance. On average, participants saved 36.73% (σ = 12.53) of the total energy consumption and modified 6.75 appliances out of 10 (σ = 1.06) with an effort of 1.48 (σ = 0.39). The most consuming appliances such as the heater, the computer or the TV received higher efforts from the participants. In contrast the lowest consuming appliance were the subject of few modifications. All users managed to successfully use the system regardless of their knowledge about energy or visualization. Furthermore once participants decided to test a modification they could do so in less than one minute, and often in only a few seconds. All participants spent approximately 10 minutes using the tool until they were satisfied by their modifications.

Once participants finished the task, they filled a questionnaire based on a Likert scale from 1 (I totally disagree) to 5 (I totally agree). This questionnaire was designed to collect participants' c 2018 The Author(s) Eurographics Proceedings c 2018 The Eurographics Association. Authors version, the definitive version is available at http://diglib.eg.org/. feelings about the system. Averaged rates and standard deviations of the questions are shown in Figure 4. Results show that the system was clearly understood by participants and easy to use. We also asked to participants which functionalities they had appreciated/disliked the most. Among them, the interaction was positively mentioned by 7 participants, the usefulness of a cost feedback by 6 participants, and the design of the visualization, including the choice of the colors by 5 participants. As negative points, potential improvements for the user interface were mentioned by 4 participants. For example, participants would have liked more details about the states of the appliances, such as their relative consumptions. These suggestions have been integrated in the user interface before the second study described in section 6.

Discussion

The main objective of this first study was to assess the usability of Activelec and especially the What if approach applied to ecofeedback. Whereas participants did not use their own data during the study, they could understand the consumption and link it with the usage of appliances thanks to the tool. Moreover, this understanding was reinforced by the application of selections, especially with criteria. Finally, modifications in the usage of the appliances were easily applied by the participants. Indeed results show that participants found several different modifications on the appliances in a short time period. One of the problems that users face when using eco-feedback systems is finding the relevant appliances to save energy on [START_REF] Strengers | Designing eco-feedback systems for everyday life[END_REF]. Results of the study show that participants solved this problem with the help of our system. This is shown by the choices to modify high consuming appliances or appliances on which the effort to put on are seen as low by the user. Moreover, the low average efforts rated by the users suggest that they were able to find actions that are judged easy to apply in reality.

Application Study

In addition to the user study presented in the previous section, we conducted another study aimed to assess how people use the tool with their own data. Our hypothesis is that users will be more involved in the step of data analysis. We also suppose that they will find relevant modifications that comply with their behavior in order to save energy.

Method

We recruited two PhD students that had previously recorded their consumption of electrical appliances. The recording took place in a dedicated smart home where participants lived during one month. The corresponding datasets are made available online by their authors [CLRLC17, LLR * 17]. The study was conducted after the end of the recording period, when participants no longer lived in the home. Both participants said that they were motivated in saving electricity. One participant could define power and energy, but participants were not experts in electricity consumption. They were not specialists in information visualization. The states of the appliances in the datasets were manually identified and calibrated by the authors. Then the instances of the appliances were automatically extracted in our system. Appliances and states used in the study are The three last appliances were common to both participants.

presented in Table 2. For this study we used the same protocol as in the previous one, i.e. the same learning phase was followed by the task in which participants managed to find modifications that save energy.

Results

Participant 1 applied three modifications on the appliances and saved 7.47% of the total amount of energy, with an average of 1.67 as estimated effort to apply the modifications. Participant 2 applied six modifications and saved 31.54% of the total amount of energy, with an estimated effort of 2 in average. Despite the fact that the set of appliances was different between participants, we can see that Participant 2 managed to save more energy by applying more modifications. The modifications were also rated with a higher average effort by this participant. From the gains, we can say that participants managed to save a considerable amount of energy even if the datasets do not exhibit high energy consumptions. Both participants spent approximately 15 minutes using the system.

Figure 5 shows the answers of the participants to the questionnaire about usability. We can see that both participants found Activelec easy to use. Results also indicate that the system was useful to understand the usage of appliances and their consumption, as well as to find actions to save energy. We can note that Participant 1 rated with a grade of 3 the two questions about the potential for selections to make data understandable and to find concrete actions to apply. Participant 2 gave the same rate for the same questions about the modifications. The explanation is that selections and modifications are used together to understand the data, as said by one participant. In the final interview, both participants said that they appreciated the visualization that allows for a detailed exploration at any time scale. As negative comments, participants only mentioned potential improvements for the interface. For example, one participant would have appreciated the consumption by month of the appliances to be displayed together with their total consumption.

Discussion

We observed that participants were more engaged in the data than in the previous study, as shown by the higher amount of time spent using the tool. This is explained by the fact that participants used their own data in this study as opposed to the previous study. They were able to relate the data to their habits. For example, they easily identified their everyday actions such as taking a shower or cooking via the states of respectively the water heater and the cooktop. They also identified abnormal electricity consumptions that could be removed to save energy. For example, one participant observed that the water heater was heating water during each afternoon, while it was not used. Thus this participant removed this usage and said that the water heater should be configured to avoid this consumption.

We also noted that participants applied modifications that we did not expect before the study. One participant limited the usage of the cooktop to 30 minutes since it was often left on after cooking. The other participant realized that one light was used to read during the day and its consumption was substantial. These modifications saved a significant amount of energy. This supports our hypothesis that users will be more involved in data analysis as well as find relevant modifications that fit with their behavior to save more energy, as presented in Section 3. Participants also wanted to test for modifications that are not currently possible. For example, one participant noted that two machine washings were done the same day. This participant would have liked to verify if one full machine washing is more efficient than two half-full ones.

Discussion and Implications

Limitations

A possible limitation of the What if approach is the requirement of user involvement, which is an inherent problem of eco-feedback dashboards [ELK * 13]. To counter this limitation, Bartram [Bar15] suggests the use of ambient systems, whose feedback is spatially (close to the appliances) and timely (when using the appliance) appropriate. Using a dashboard with temporal visualizations is necessary for our approach especially for interaction. However, we could imagine an ambient display located near the appliances to recall the chosen modifications. This display could also provide information about the energy saved by modifying the appliance usage.

In a preprocessing step, our system extracts the instances of the appliances using empirical thresholds which requires manual tuning. More advanced algorithms could be used in the future to segment the input data automatically and more precisely. Also, when users modify the instance of an appliance by changing its state, we need to simulate new consumption data according to the new state. Currently this is done by copying the data associated to another instance of this state. In future works it would be interesting to incorporate more reliable simulations of state modifications. In addition the consumption of some appliances depends on external parameters that are not taken into account by our model. For instance the consumption of fridges directly depends on parameters such as room temperature. To overcome this problem, we could use physical simulations that take environmental parameters as input.

Perspectives

Field Study The studies that we conducted show that users were able to find modifications that fit to their behavior. However there still remains work to prove that the What if approach is suitable in a home context. Hence we would like to go further and deploy the system in households for an in situ experiment. While asserting a change in behavior requires a long-term study and is beyond our research domain, the first step would be a mid-term study, e.g. a couple of months. This additional study could bring information about how Activelec is used in a household context. Users will test What if scenarios that they would incorporate into their everyday habits. Then, they might come back to the tool to check their new consumption. Users could also try new modifications or refine their previous scenarios. For example they might want to save more energy or to put less effort to change their usage of appliances.

Integration in a Home Automation System We envision incorporating our tool in a home automation system such as Apps-Gate [CC16] where it could directly take the values of sensors as input. Users could test several modifications and define the preferred ones as rules that automatically apply on the appliances. For example a rule that automatically switches off the TV after 10 PM during weekdays could be generated from the appropriate modification. In a general smart home system, our tool could also benefit from other available data sources. For example users would relate the consumption of the heater to the external temperature, and lower the heater if the temperature is sufficiently high. Activelec could also be used for the analysis of other utilities such as water and natural gas. The system currently runs in a desktop environment but it can be adapted to a tablet with little effort. This would allow users to analyze their consumption everywhere in the home, especially near the appliances on which modifications are applied.

Suggestions The development of an automatic system that finds the best modifications for every user seems unreachable. Indeed, only inhabitants have the knowledge of how and why they use their appliances, and what is the effort required to change this usage. However it should be useful to suggest modifications that seem appropriate in terms of gain and estimated effort. For instance switch- ing to ecologic washing seems to require a reasonable effort. Suggesting selection criteria and modifications will also provide a first step to novice users, since they will directly have access to examples.

Additional "What if" Features For now, Activelec allows the modification of past data but the What if approach can also be used in other ways. We could imagine the extraction of a typical week in terms of appliances usage and consumption. For example, in a typical week inhabitants could always turn on the coffee machine in the morning, and watch the TV during two hours in the evening. Then this typical behavior could be modified by users, hence applying the What if approach on current data instead of past data. It would also be possible for users to create their behavior entirely from scratch. In this way the What if approach would be applied to hypothetic future data. Activelec currently targets the reduction of consumption and not the shift of consumption peaks. Since the units of energy are computed dynamically from watts our system could easily apply time based pricing. However the system should be modified to allow the visualization of energy prices over time and to shift usages of appliances.

Conclusion

In this paper we present a system for users to analyze, interact with and modify their electricity consumption data. In this system we model the appliances as having states over time, which are modified by users. Thanks to the What if approach, users can select and change the state of appliances over time. The visualization allows them to evaluate the impact of their modifications on the consumption. Users can then assess the ratio between energy saving (money saved) and the required efforts to apply these modifications in their behavior. Our studies have proven the usability of the system and the ability of the users to find modifications with a reasonable effort. This work opens new perspectives on applying the What if approach to the visualization of household electricity consumption and even personal visualization systems in general.

  c 2018 The Author(s) Eurographics Proceedings c 2018 The Eurographics Association. Authors version, the definitive version is available at http://diglib.eg.org/.

Figure 1 :

 1 Figure 1: Data related to a computer with three states. Each state is associated to a unique color, which is used to depict its time intervals on the consumption curve of the appliance.

  c 2018 The Author(s) Eurographics Proceedings c 2018 The Eurographics Association. Authors version, the definitive version is available at http://diglib.eg.org/.

Figure 2 :

 2 Figure 2: Interface of the Appliances mode. Five appliances are displayed: a computer (orange), a dishwasher (green), a light (blue), a TV (purple) and a washing machine (red). The washing machine has been highlighted by the user, and the item of the dishwasher has been extended to display its consumption values.

Figure 3 :

 3 Figure 3: Interface of the Instances mode. A TV has been chosen by the user, whose states are Usage (orange), Standby (blue) and Off (grey).At the right, we can see that the user has selected all instances of Standby lasting more than 8 hours via the definition of two criteria. Three of these instances are visible on the visualization.

  c 2018 The Author(s) Eurographics Proceedings c 2018 The Eurographics Association. Authors version, the definitive version is available at http://diglib.eg.org/.

Figure 5 :

 5 Figure5: Rates from final interview of the second study. "→ understanding" means helps in understanding consumption and behavior and "→ actions" means helps to find relevant actions to save energy.

  c 2018 The Author(s) Eurographics Proceedings c 2018 The Eurographics Association. Authors version, the definitive version is available at http://diglib.eg.org/.

Table 2 :

 2 Appliances used during the second study with their states.

	Participant 1	Participant 2
	Computer:	Oven:
	On, Standby	Low, Medium, High
	Living Room TV: Dishwasher:
	On, Standby	Normal Washing, Eco Washing
	Bedroom TV:	Washing Machine:
	On, Standby	Normal Washing, Eco Washing
		Global Lighting: On
	Cooktop: Low, Medium, High
		Water Heater: On

Acknowledgements

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d'avenir.


B Besselink

A Chaillet

N Van De Wouw

Model reduction for linear delay systems using a delay-independent balanced truncation approach

A model reduction approach for asymptotically stable linear delay-differential equations is presented in this paper. Specifically, a balancing approach is developed on the basis of energy functionals that provide (bounds on) a measure of energy related to observability and controllability, respectively. The reduced-order model derived in this way is again a delay-differential equation, such that the method is structure preserving. In addition, asymptotic stability is preserved and an a priori bound on the reduction error is derived, providing a measure of accuracy of the reduction. The results are illustrated by means of application on an example.

I. INTRODUCTION

Models of engineering systems or physical phenomena can often be represented in terms of dynamical systems with time delays. Examples include models of machine tool vibrations, control over communication networks, or population dynamics, see the books [START_REF] Gu | Stability of time-delay systems[END_REF], [START_REF] Michiels | Stability and stabilization of timedelay systems: An eigenvalue-based approach[END_REF], [START_REF] Erneux | Applied delay differential equations[END_REF] for an overview. In addition, accurate models of such systems are typically of high order, motivating the need for developing model reduction techniques for delay differential equations. This paper addresses this problem by developing a model reduction technique for linear delay systems.

Methods for model reduction of finite-dimensional linear systems are well developed (see [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF], [START_REF] Besselink | A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control[END_REF] for overviews) and popular approaches are given by balanced truncation [START_REF] Moore | Principal component analysis in linear systemscontrollability, observability, and model reduction[END_REF], [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L 1 -error bounds[END_REF] and moment matching techniques via Krylov subspaces [START_REF] Freund | Model reduction methods based on Krylov subspaces[END_REF].

For systems with time delays (or, more generally, infinite-dimensional systems), finite-dimensional approximations have been considered on the basis of Fourier series [START_REF] Gu | Approximation of infinitedimensional systems[END_REF], Padé approximations [START_REF] Glover | Rational approximation of a class of infinite-dimensional systems II: Optimal convergence rates of L1 approximants[END_REF], or using the Hankel operator [START_REF] Glover | Realisation and approximation of linear infinite-dimensional systems with error bounds[END_REF]. An overview of such methods is given in [START_REF] Partington | Some frequency-domain approaches to the model reduction of delay systems[END_REF].

Next, methods for model reduction of delay differential equations have been developed by extending methods for finite-dimensional systems. A moment matching approach using Krylov methods was presented in [START_REF] Michiels | Krylov-based model order reduction of time-delay systems[END_REF]. Here, as before, a finite-dimensional reduced-order model is obtained; as a consequence, the delay structure is not preserved in the reduction. Another perspective on moment matching for systems with time delays is given in [START_REF] Scarciotti | Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays[END_REF], where both finite-dimensional and infinite-dimensional approximations B. Besselink is with the Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, the Netherlands (email: b.besselink@rug.nl). A. Chaillet is with L2S -CentraleSupélec -Univ. Paris Saday and IUF, France (email: antoine.chaillet@centralesupelec.fr). N. van de Wouw is with the Department of Mechanical Engineering, Eindhoven University of Technology, the Netherlands, with the Department of Civil, Environmental and Geo-Engineering, University of Minnesota, Minneapolis, USA, and also with the Delft Center for Systems and Control, Delft University of Technology, Delft, the Netherlands (email: n.v.d.wouw@tue.nl). are considered. As a class of reduced-order models is characterized in [START_REF] Scarciotti | Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays[END_REF], this has the potential to select a reducedorder model that preserves asymptotic stability properties of the original high-order delay differential equation.

An extension of balanced truncation towards systems with delays is presented in [START_REF] Jarlebring | Model reduction of timedelay systems using position balancing and delay Lyapunov equations[END_REF], based on characterizing measures of controllability and observability similar to those used in balanced truncation for finite-dimensional systems. This method preserves the delay-structure in the reduced-order model, but asymptotic stability is not necessarily preserved. The method in [START_REF] Van De Wouw | Model reduction for delay differential equations with guaranteed stability and error bound[END_REF] does provide such guarantee and in addition directly exploits reduction techniques for finitedimensional linear systems by decomposing the delay system in a finite-dimensional part and an infinite-dimensional delay operator. This method also guarantees a bound on the reduction error. Finally, an alternative perspective is given in [START_REF] Xu | H1 model reduction for linear time-delay systems: Continuous-time case[END_REF], where the model reduction problem is formulated as a rank-constrained optimization problem.

In this paper, a balancing approach for model reduction of asymptotically stable delay systems is presented. This approach is based on computing bounds on energy functionals that provide a measure of observability and controllability and the use of these bounds in a balancing procedure. These bounds take the form of Lyapunov-Krasovskii functionals and hold regardless of the size of the delay, leading to a delay-independent reduction procedure. In particular, this reduction procedure features the following properties, which form the main contributions of this paper. First, the reduction is structure-preserving, i.e., the reduced-order model is again in the form of a delay-differential equation, albeit with a reduced set of equations. This allows for accurately capturing the infinite-dimensional nature of the original high-order delay system. Second, the reduced-order model is guaranteed to be asymptotically stability and, third, an a priori error bound is available that provides a measure of the accuracy of the reduction.

The remainder of this paper is outlined as follows. The problem setting is detailed in Section II, before the energy functionals and their bounds are presented in Section III. Section IV discusses the model reduction procedure and the properties of the reduced-order delay system. An illustrative example is given in Section V and conclusions are stated in Section VI.

Notation: The field of real (complex) numbers is denoted by R (C). For a vector x 2 R n , |x| denotes its Euclidean norm. Given a symmetric matrix X 2 R n⇥n , X 0 (X < 0) indicates that it is positive (semi-)definite. The Banach space of continuous functions from an interval

T ⇢ R into R n is represented as C(T , R n ). Similarly, L 2 (T , R n ) denotes the class of square integrable functions from T into R n .
II. PROBLEM SETTING Consider the linear delay-differential equation

ẋ(t) = Ax(t) + A d x(t ⌧ ) + Bu(t), y(t) = Cx(t), (1)
with x(t) 2 R n , input u(t) 2 R m , and output y(t) 2 R p for all t t 0 . The initial condition for (1) is given by the function segment

' 2 C([ ⌧, 0], R n ), such that x(t) = '(t), 8t 2 [t 0 ⌧, t 0 ]. (2)
Next, the function segment

x t 2 C([ ⌧, 0], R n ) defined as x t (s) = x(t + s), s 2 [ ⌧, 0]
characterizes the state of (1) at time t 0, such that the initial condition (2) can also be written as x t0 = '. In this paper, model reduction of systems of the form (1) is pursued under the assumption that (1) is asymptotically stable. Specifically, a model of the same form is sought that approximates the input-output behavior of (1), but whose state

⇠ t is in C([ ⌧, 0], R k ) with k < n.
Note that, even though this "reduced-order" state remains infinite-dimensional, this is regarded as model reduction as the number of equations in the first equation in (1) is reduced. In this setting, the problem of finding a reduced-order delay-differential equation is considered that, first, preserves asymptotic stability of the original high-order system and, second, satisfies an a priori error bound in order to characterize the accuracy of the reduction.

III. OBSERVABILITY AND CONTROLLABILITY

FUNCTIONALS

The model reduction approach developed in this paper will be based on energy functions that respectively provide a measure for observability and controllability of the delay system. First, the observability functional is defined as a measure of energy associated with observing the output of (1).

Definition 1: The observability functional of (1) is the functional

L o : C([ ⌧, 0], R n ) ! R defined as L o (') = Z 1 0 |y(t)| 2 dt, (3)
where

y(t) = Cx(t) = Cx t (0)
is the output of (1) for initial condition x 0 = ' and zero input (u = 0). It is clear that the observability functional exists (i.e., the integral (3) is bounded) if the system (1) is asymptotically stable. Next, a measure for the energy associated with controlling (1) is given by the controllability functional.

Definition 2: The controllability functional of (1) is the functional

L c : D c ! R defined as L c (') = inf ⇢ Z 0 1 |u(t)| 2 dt u 2 L 2 (( 1, 0], R m ), lim T !1 x T = 0, x 0 = ' , (4)
where x t is the solution of (1) for input u and D c ⇢ C([ ⌧, 0], R n ) the domain of L c , i.e., the collection of function segments ' for which L c (') is well-defined.

Remark 1: The definition of the energy functionals in Definitions 1 and 2 is motivated by the energy functions that form the basis of balanced truncation for finite-dimensional linear systems, see, e.g., [START_REF] Moore | Principal component analysis in linear systemscontrollability, observability, and model reduction[END_REF], [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L 1 -error bounds[END_REF], [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF]. In this case, these energy functions are characterized by the observability and controllability Gramian, respectively. C

A characterization of the observability functional in Definition 1 is provided as follows.

Lemma 1: Consider the asymptotically stable delaydifferential equation [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF]. If there exist matrices Q 0 and

Q d < 0 such that  A T Q + QA + Q d + C T C QA d A T d Q Q d 4 0, (5)
then the functional

E o : C([ ⌧, 0], R n ) ! R defined as E o (') = ' T (0)Q'(0) + Z 0 ⌧ ' T (s)Q d '(s) ds, (6)
satisfies

E o (') L o (') (7)
for all ' 2 C([ ⌧, 0], R n ) and L o as in Definition 1.

Proof: In order to prove the lemma, let x t be the solution of (1) for initial condition x 0 = ' and zero input and consider E o (x t ). Note that, by ( 6), E o (x t ) can be written as

E o (x t ) = x T (t)Qx(t) + Z t t ⌧ x T (s)Q d x(s) ds, (8)
with x(t+s) = x t (s), s 2 [ ⌧, 0]. Then, time-differentiation of E o along trajectories of (1) yields

d dt E o (x t ) =  x(t) x(t ⌧ ) T M o  x(t) x(t ⌧ ) , (9)
with

M o =  A T Q + QA + Q d QA d A T d Q Q d , (10)
and where the dynamics ( 1) is used to obtain (9) (recall that u = 0). Employing the condition ( 5) in ( 9)-( 10) leads to

d dt E o (x t )  x T (t)C T Cx(t) = |y(t)| 2 , ( 11
)
where y(t) = Cx t (0) is the output corresponding to the trajectory x t . Integration of the result (11) over the interval [0, T ] gives

E o (x T ) E o (x 0 )  Z T 0 |y(t)| 2 dt, ( 12
)
where it is recalled that x 0 = '. Moreover, due to asymptotic stability, it holds that lim

T !1 E o (x T ) = E o (0) = 0, (13)
such that (12) leads, for T ! 1, to

E o (') Z 1 0 |y(t)| 2 dt. (14)
This proves the desired result ( 7) by recalling the definition of L o in (3). The controllability functional admits a similar characterization, as shown in the following lemma.

Lemma 2: Consider the delay-differential equation [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF]. If there exist matrices P 0 and P d < 0 such that

 AP + P A T + P d + BB T A d P P A T d P d 4 0, (15)
then the functional

E c : C([ ⌧, 0], R n ) ! R defined as E c (') = ' T (0)R'(0) + Z 0 ⌧ ' T (s)R d '(s) ds, (16)
with

R = P 1 and R d = RP d R, satisfies E c (')  L c (') (17)
for all ' 2 D c ⇢ C([ ⌧, 0], R n ) and L c as in Definition 2.

Proof: In order to prove the lemma, the matrix R = P 1 is defined, such that a congruence transformation of ( 15) with a block-diagonal matrix blkdiag{R, R} leads to the equivalent condition

 A T R + RA + R d + RBB T R RA d A T d R R d 4 0, (18)
with R d = RP d R. Next, application of the Schur complement shows that (18) (and, hence, [START_REF] Michiels | Stability and stabilization of timedelay systems: An eigenvalue-based approach[END_REF]) is equivalent to 2 4

A T R + RA + R d RA d RB A T d R R d 0 B T R 0 I 3 5 4 0, (19)
which will form the basis for the remainder of the proof. Consider a solution x t to (1) corresponding to an input u 2 L 2 (( 1, 0], R n ) and satisfying the conditions in (4), i.e., lim T !1 x T = 0 and x 0 = '. Since E c (x t ) can be written as

E c (x t ) = x T (t)Rx(t) + Z t t ⌧ x T (s)R d x(s) ds, (20)
with

x(t + s) = x t (s), s 2 [ ⌧, 0]
, it follows that timedifferentiation of E c along the trajectories of (1) leads to

d dt E c (x t ) = 2 4 x(t) x(t ⌧ ) u(t) 3 5 T M c 2 4 x(t) x(t ⌧ ) u(t) 3 5 . ( 21
)
with

M c = 2 4 A T R + RA + R d RA d RB A T d R R d 0 B T R 0 0 3 5 . ( 22
)
The use of ( 19) in ( 21)-( 22) leads to

d dt E c (x t )  |u(t)| 2 , ( 23
)
after which integration over the interval [ T, 0] yields

E c (x 0 ) E c (x T )  Z 0 T |u(t)| 2 dt. (24)
Letting T ! 1 and noting that lim

T !1 E c (x T ) = 0, it follows that E c (')  Z 0 1 |u(t)| dt, (25)
where the condition x 0 = ' is used. Since the input function u is chosen arbitrarily, the inequality (25) also holds for the input u that achieves the minimization in (4). Consequently, (25) implies the desired result [START_REF] Partington | Some frequency-domain approaches to the model reduction of delay systems[END_REF], finalizing the proof.

The functional E o in ( 6) provides an upper bound on the observability functional L o in Definition 1, whereas E c in ( 16) is a lower bound to the controllability functional L c in Definition 2. These bounds, rather than the observability and controllability functionals themselves, will be used as a basis for model reduction. Namely, it will be shown that the structure of the bounds ( 6) and ( 16) is beneficial for the development of a model reduction procedure that preserves asymptotic stability and provides an a priori error bound.

Remark 2: Even though the controllability functional can in general only be defined on a restricted domain D c (see Definition 2), its bound E c in ( 16) can be defined for all function segments in C([ ⌧, 0], R n ) (provided that (15) holds). As the latter will be used as a basis for model reduction, the reduced-order model will be well-defined. C

Remark 3:

The functionals E o in ( 6) and E c in ( 16) are similar to Lyapunov-Krasovskii functionals as often exploited in stability analysis of time-delay systems, see [START_REF] Gu | Stability of time-delay systems[END_REF]. In fact, if the matrices in [START_REF] Erneux | Applied delay differential equations[END_REF] or [START_REF] Michiels | Stability and stabilization of timedelay systems: An eigenvalue-based approach[END_REF] are negative definite instead of merely negative semi-definite, they imply delayindependent asymptotic stability of [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF]. C

IV. MODEL REDUCTION BY TRUNCATION

Before discussing the use of (bounds on) the observability and controllability functionals in Definitions 1 and 2 in the scope of model reduction, the general reduction procedure of truncation is presented. To this end, a partitioned form of the dynamics (1) is considered in which x(t) and the function segments x t are partitioned as

x(t) =  x 1 (t) x 2 (t) , x t =  x 1,t x 2,t , (26)
with x 1 (t) 2 R k , x 1,t 2 C([ ⌧, 0], R k ) and k < n.
The corresponding partitioning of the system matrices yields

A =  A 11 A 12 A 21 A 22 , A d =  A d,11 A d,12 A d,21 A d,22 , B =  B 1 B 2 , (27)
and

C = [ C 1 C 2 ].
Using the partitioning (26), ( 27), a reduced-order approximation of (1) can be obtained by truncation as

⇠(t) = A 11 ⇠(t) + A d,11 ⇠(t ⌧ ) + B 1 u(t), ŷ(t) = C 1 ⇠(t), (28)
with ⇠(t) 2 R k for each t and function segments ⇠ t 2 C([ ⌧, 0], R k ). Here, ⇠(t) provides an approximation of x 1 (t) in the partitioned coordinates (26).

The following property holds for the observability functional of the reduced-order system (28).

Lemma 3: Let the condition (5) be satisfied for symmetric matrices Q 0 and Q d < 0 of the form

Q =  Q 1 0 0 Q 2 , Q d =  Q d,11 Q d,12 Q d,21 Q d,22 . (29)
Then, the observability functional Lo of the reduced-order system (28) exists and the functional Êo :

C([ ⌧, 0], R k ) ! R given as Êo ( ') = 'T (0)Q 1 '(0) + Z 0 ⌧ 'T (s)Q d,11 '(s) ds, (30)
satisfies Êo ( ') Lo ( ') for all ' 2 C([ ⌧, 0], R k ).

Proof: Since the matrices Q and Q d in (29) are such that (5) holds, it follows that

 T 0 0 T T  A T Q + QA + Q d + C T C QA d A T d Q Q d  T 0 0 T 4 0 (31)
for any matrix T . After choosing T = [ I k 0 ] T , it can be checked by using the partitioning (27) that the left-hand side of (31) provides an inequality of the form (5) for the reducedorder system (28). Then, it follows from [START_REF] Horn | Matrix analysis[END_REF] in the proof of Lemma 1 that, for all T 0, Êo ( ')

Z T 0 ŷ(t) 2 dt + Êo (⇠ T ), (32)
where ⇠ t is the solution of (28) for ⇠ 0 = ' and for u = 0. As Êo ( ') is well-defined (i.e., bounded), it follows after taking the limit for T ! 1 that Lo exists and that the bound

Êo ( ') Lo ( ') for all ' 2 C([ ⌧, 0], R k ).
The counterpart of Lemma 3 for the controllability functional is given as follows.

Lemma 4: Let the condition (15) be satisfied for symmetric matrices P 0 and P d < 0 of the form .

P =  P 1 0 0 P 2 , P d =  P d
Then, the functional Êc :

C([ ⌧, 0], R k ) ! R given as Êc ( ') = 'T (0)R 1 '(0) + Z 0 ⌧ 'T (s)R d,11 '(s) ds, ( 34
)
with

R 1 = P 1 1 and R d,11 = R 1 P d,11 R 1 satisfies Êc ( ')  Lc ( ') for all ' 2 Dc ⇢ C([ ⌧, 0], R k ).
Here, Lc is the controllability functional for the reduced-order system (28) and Dc the domain on which it is well-defined.

Proof: The proof is similar to the first part of the proof of Lemma 3 and is omitted for the sake of brevity.

The results of Lemmas 3 and 4 thus state that the observability and controllability functionals of a reduced-order system obtained by truncation can be bounded by relevant parts of the energy functionals of the original system [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF] when the matrices Q and P have a suitable block-diagonal form, see ( 29) and (33). Even though these results hold for any matrices Q and P satisfying this block-diagonal form, a more specific diagonal form for these matrices is assumed in the remainder of this section. This leads to the following definition.

Definition 3: A realization ( 1) is said to be balanced if there exist matrices Q 0, Q d < 0 satisfying (5) and matrices P 0, P d < 0 satisfying [START_REF] Michiels | Stability and stabilization of timedelay systems: An eigenvalue-based approach[END_REF] such that

Q = P = ⌃ := 2 6 6 6 6 4 1 I m1 0 • • • 0 0 2 I m2 . . . . . . . . . . . . . . . 0 0 • • • 0 q I mq 3 7 7 7 7 5 (35)
with i > i+1 > 0, i 2 {1, . . . , q 1} and P q i=1 m i = n. The following standard result guarantees the existence of such balanced realization.

Lemma 5: Let there exist matrices Q 0 and Q d < 0 such that (5) holds and matrices P 0 and P d < 0 such that (15) holds. Then, there exists a change of coordinates x(t) = T z(t) such that the realization given by the new coordinates is balanced, i.e., the nonsingular matrix T can be chosen such that T T QT = T 1 P T T = ⌃, with ⌃ as in (35).

Proof: The existence of such matrix T follows from standard results in linear algebra on simultaneous diagonalization (e.g., [START_REF] Horn | Matrix analysis[END_REF]). This result also forms the foundation of balancing for finite-dimensional linear systems, see [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF].

When truncation is applied for asymptotically stable delay systems in a balanced realization as in Definition 3, this stability property is preserved. This result is stated next.

Theorem 6: Let the asymptotically stable system (1) be in a balanced realization and consider the reduced-order delaydifferential equation (28) obtained by truncation for k such that k = P r i=1 m i for some r > 0. Then, the reduced-order system is asymptotically stable.

Proof: This result can be proven by exploiting ideas in the proof of stability preservation for balanced truncation of finite-dimensional linear systems originally shown in [START_REF] Pernebo | Model reduction via balanced state space representations[END_REF], see also [START_REF] Dullerud | A course in robust control theory -A convex approach[END_REF]. Details are omitted.

Moreover, for truncation of a balanced realization, the following error bound holds.

Theorem 7: Let the asymptotically stable system (1) be in a balanced realization and consider the reduced-order delaydifferential equation (28) obtained by truncation for k such that k = P r i=1 m i for some r > 0. Then, for any common input function u 2 L 2 ([0, 1), R m ) and initial conditions ' = 0 and ' = 0 for (1) and (28), respectively, their output trajectories satisfy the error bound

Z T 0 |y(t) ŷ(t)| 2 dt  " 2 Z T 0 |u(t)| 2 dt (36)
for all T 0 and where the error bound (gain) " is given as

" = 2 q X i=r+1 i , (37)
with i as in (35). Proof: The proof can be found in Appendix A. The results in Theorems 6 and 7 thus suggest a model reduction procedure in which, first, solutions to ( 5) and ( 15) are sought. Second, the balancing transformation of Lemma 5 is employed to obtain a balanced realization (see, e.g., [START_REF] Antoulas | Approximation of large-scale dynamical systems[END_REF] for an explicit procedure to compute T ) and finally, truncation is applied to obtain a reduced-order model of the form (28). This reduced-order model is then of the same form as (1), is asymptotically stable and satisfies the a priori error bound (36) which has a similar structure as the error bound for balanced truncation of finite-dimensional linear systems.

Remark 4: An alternative approach towards balanced truncation of delay systems is given in [START_REF] Jarlebring | Model reduction of timedelay systems using position balancing and delay Lyapunov equations[END_REF], where infinitedimensional generalizations of the observability and controllability Gramian for finite-dimensional systems are exploited. Then, relevant finite-dimensional parts of these Gramians are selected to compute a coordinate transformation x(t) = T z(t) that diagonalizes these parts. However, this approach does not lead to reduced-order delay systems that preserve asymptotic stability and satisfy an a priori bound on the reduction error as in the current paper. C

V. ILLUSTRATIVE EXAMPLE

To illustrate the reduction procedure developed in Section IV, the model of a heated rod discussed in [START_REF] Michiels | Krylov-based model order reduction of time-delay systems[END_REF] is considered. The model is a partial differential equation of the form @v @t (x, t) = @ 2 v @x 2 (x, t) + a 0 (x)v(x, t) + a 1 (x)v(⇡ x, t 1), (38) with v(x, t) the temperature of the rod at location x 2 [0, ⇡] at time t, satisfying the boundary conditions v(0, t) = v(⇡, t) = 0. The functions a 0 and a 1 are given as a 0 (x) = 2 sin(x) and a 1 (x) = 2 sin(x), respectively. Discretization in space leads to an asymptotically stable delay-differential equation of the form (1) with x(t) 2 R n , n = 35 after choosing the input and output matrices as

B = C T = 1 p n 1 n , where 1
n 2 R n is a vector of all ones. Thus, the input and output can respectively be interpreted as a uniform heating of the road and its average (in space) temperature.

To derive the reduced-order models, solutions to ( 5) and ( 15) are sought that minimize the trace of Q and P , respectively. Using these matrices, the coordinate transformation of Lemma 5 is computed, after which truncation to order k = 1 leads to the reduced-order model ⇠(t) = 2.69 ⇠(t) + 1.65 ⇠(t 1) 0.93 u(t),

ŷ(t) = 0.93 ⇠(t). (39)
This model can be checked to be asymptotically stable, as guaranteed by Theorem 6. Moreover, according to Theorem 7, the error bound (36) holds with " = 0.012. A comparison of the frequency response functions of the high-order and reduced-order delay-differential equations is depicted in Figure 1, indicating a good approximation. This is confirmed by the magnitude of the error in Figure 2, which also shows that the error bound is not conservative for this example. Finally, it is noted that the scalar reduced-order model (39) accurately captures the repeated resonances in the frequency response function. This behavior cannot be obtained by a finite-dimensional approximation of low-order, indicating the importance of preserving the delay-structure in reduction.

VI. CONCLUSIONS

A structure-preserving model reduction procedure for delay-differential equations is presented in this paper, based on the definition of energy functionals that characterize the energy associated with the output and input of the model. A balancing procedure on the basis of these energy functionals is shown to lead to a reduced-order delay-differential equation for which asymptotic stability is preserved. In addition, an a priori error bound is available.

Future work will focus on a delay-dependent approach, which could potentially lead to better reduced-order models if the value of the delay is known.

APPENDIX A. PROOF OF THEOREM 7

The proof is inspired by a construction in [START_REF] Besselink | Model reduction for nonlinear systems by incremental balanced truncation[END_REF]. Following this approach, a one-step reduction is considered first. Here, the state components corresponding to q are discarded by truncation, such that k = n m q , with m q the multiplicity of q . In this case, the observability and controllability functionals can be written (using a slight abuse of notation) in partitioned form as

E o (' 1 , ' 2 ) = ' T 1 Q 1 ' 1 + ' T 2 Q 2 ' 2 + Z 0 ⌧  ' 1 (s) ' 2 (s) T Q d  ' 1 (s) ' 2 (s) ds (40)
and

E c (' 1 , ' 2 ) = ' T 1 R 1 ' 1 + ' T 2 R 2 ' 2 + Z 0 ⌧  ' 1 (s) ' 2 (s) T R d  ' 1 (s) ' 2 (s) ds, (41)
respectively. Here,

Q 1 = R 1 1 = blkdiag{ 1 I m1 , . . . , q 1 I mq 1 }, (42)
Q 2 = R 1 2 = q I mq , (43)
where the relation P = R 1 is used (see Lemma 2) is used. On the basis of the partitioned energy functionals, the functional V is introduced as V (' 1 , ' 2 , ') = E o (' 1 ', ' 2 ) + 2 q E c (' 1 + ', ' 2 ), (44) where ' 1 2 C([ ⌧, 0], R k ), ' 2 2 C([ ⌧, 0], R n k ), and ' 2 C([ ⌧, 0], R k ). In the remainder of this proof, the value of V (x 1,t , x 2,t , ⇠ t ) will be evaluated, i.e., the evolution of V along trajectories (x 1,t , x 2,t ) of the delay-differential equation (1) and ⇠ t of reduced-order delay-differential equation (28). Specifically, by exploiting the characterizations of the observability and controllability functionals in Lemmas 1 and 2 for the partitioned form (27), it can be shown that d dt V (x 1,t , x 2,t , ⇠ t )  y(t) ŷ(t) 2 + (2 q ) 2 |u(t)| 2 , (45) along trajectories of (1) and (28). Here, the fact that a onestep reduction is considered is crucial to obtain this result, in particular the relation (43). Integration of (45) over the interval [0, T ] yields

V (x 1,T , x 2,T , ⇠ T ) V (x 1,0 , x 2,0 , ⇠ 0 )  (2 q ) 2 Z T 0 |u(t)| 2 dt Z T 0 y(t) ŷ(t) 2 dt, ( 46
)
where it is noted that V (x 1,0 , x 2,0 , ⇠ 0 ) = 0 due to the choice of zero initial conditions and the structure of V in (44). Moreover, V (x 1,t , x 2,t , ⇠ t ) 0 for all (x 1,t , x 2,t , ⇠ t ), which gives the desired result (36) and (37) for k = n m q . To prove the result (36) and (37) for arbitrary order k < n (according to the multiplicities m i of the parameters i in (35)), it is recalled that Lemmas 3 and 4 show that the energy functionals for a reduced-order delay-differential equation of arbitrary order are bounded by the relevant parts of the partitioned energy functionals (40) and (41) of the highorder system. As such, the procedure discussed above can be repeated to obtain a reduced-order model for arbitrary k, where the result (36) and (37) follows from application of the triangle inequality.

Fig. 1 .

 1 Fig. 1. Magnitude of the frequency response functions G and Ĝ of the discretized model of the heated rod (38) and the reduced-order approximation (39), respectively. Here, G(s) = C(sI A A d e s⌧ ) 1 B and Ĝ is defined similarly.

Fig. 2 .

 2 Fig. 2. Magnitude of the frequency response function of the error G Ĝ, with G and Ĝ the transfer functions as in Figure 1. The error bound is depicted as the dashed gray line.

  ,11 P d,12 P d,21 P d,22