Estimation of unknown states for an electric water heater with thermal stratification and use of same in demand response and condition-based maintenance

0001This application claims the benefit of U.S. Provisional Application No. 62/095,593 filed Dec. 22, 2014 and titled “ESTIMATION OF UNKNOWN STATES FOR AN ELECTRIC WATER HEATER WITH THERMAL STRATIFICATION AND USE OF SAME IN DEMAND RESPONSE AND CONDITION-BASED MAINTENANCE”. U.S.

0002Provisional Application No. 62/095,593 filed Dec. 22, 2014 is incorporated herein by reference in its entirety.

BACKGROUND

0003The following relates to the water heater arts, water heater control arts, water heater maintenance arts, and related arts.

0004Water heaters are ubiquitous appliances in residential and commercial settings, used to provide hot water for washing, cleaning, laundry processing, industrial processes, and so forth. A typical electric water heater includes a water storage tank with one or more heating elements, typically at upper and lower positions. Cold water enters near the bottom of the water storage tank via a cold water feed pipe, and is heated by the heating elements. Heated water loses density, causing it to tend to rise upward, and this flow pattern is reinforced by entry of cold water near the bottom of the tank and extraction of hot water from the top of the tank. A gas water heater operates similarly, with the resistive electrical heating elements being replaced by a gas burner usually located near the bottom of the water storage tank. In either case, temperature control is typically achieved by a simple thermostat-based controller that applies heat when the water temperature in the storage tank falls below a deadband minimum and turns off the heater (gas or electric) when the water temperature rises above a deadband maximum. Within the deadband the heater setting remains unchanged, producing a temperature cycling within the deadband (possibly with some overshoot and/or undershoot) about a temperature set point located at about the middle of the deadband. This type of control advantageously leverages thermal hysteresis to reduce the on/off cycling of the heating element. Water temperature is usually set by adjusting the set point, with the deadband limits defined relative to the set point (e.g., ±2° C. above/below the thermostat set point).

0005Recognizing that water heaters in a building, city, or region represent a large distributed thermal energy storage reservoir, there has been interest in leveraging aggregations of water heaters as energy storage devices to provide demand response, in which the electrical load of the electric grid is matched with electrical generation. (By comparison, conventionally the power generation is adjusted to match load, for example by bringing ancillary power generators online/offline as needed to match load). By way of illustration, to perform load shedding the water heater operation can be curtailed during peak energy usage periods, with hot water continuing (for a time) to be available from the hot water tank. As another illustration, in frequency control the load is adjusted at a higher frequency, typically on the order of seconds, in accord with an Automatic Generation Control (AGC) signal to maintain the grid frequency.

0006To perform demand response, especially at higher frequencies such as those required for AGC-based frequency control, the water heaters typically must be controlled remotely, for example by retrofitting the water heater with a remotely operable load controller (or, in the case of a new water heater, including such a load controller as an original manufacturer component). Also, the demand response must be balanced against the traditional function of water heaters: to provide hot water (which limits the time that the water heater can be kept off), as well as safety considerations such as not overloading the electrical circuits, or generating water that is scalding hot (which limits the time the water heater can be kept heating). To balance these considerations, it is useful to provide feedback to the aggregate controller, such as the water temperature in the storage tank, instantaneous water heater power consumption, or so forth.

BRIEF SUMMARY

0007In some illustrative embodiments disclosed as illustrative examples herein, a water heater control system is disclosed for controlling a water heater that includes a vertically oriented cylindrical water storage tank having a cylindrical wall, at least one heating element arranged to heat water in the water storage tank, and at least one temperature sensor arranged to measure water temperature in the water storage tank. The electric water heater control system comprises: a load controller comprising an electronic data processing device configured to operate the water heater including operating the at least one heating element based on temperature readings provided by the at least one temperature sensor to control water temperature of water in the water storage tank; and a water heater modeling component comprising an electronic data processing device configured to model the water heater using a one-dimensional model that includes: a vertical stack of disks representing the water volume in the cylindrical water storage tank, and a stack of annular segments surrounding the vertical stack of disks wherein the stack of annular segments represents the cylindrical wall of the cylindrical water storage tank. In some embodiments the one-dimensional model comprises coupled differential equations including: (1) for each disk of the vertical stack of disks representing the water volume in the cylindrical water storage tank, a differential equation expressing the time derivative of the temperature of the disk estimated by the one-dimensional model; and (2) for each annular segment of the stack of annular segments representing the cylindrical wall of the cylindrical water storage tank, a differential equation expressing the time derivative of the temperature of the annular segment estimated by the one-dimensional model.

0008In some illustrative embodiments disclosed as illustrative examples herein, a system includes a water heater, a load controller, an aggregation demand response dispatch engine, and a condition-based maintenance system. The water heater includes a water storage tank, at least one heating element arranged to heat water in the water storage tank, and at least one temperature sensor arranged to measure water temperature in the water storage tank. The load controller comprises an electronic data processing device configured to operate the water heater including operating the at least one heating element based on temperature readings provided by the at least one temperature sensor to control water temperature of water in the water storage tank. The aggregation demand response dispatch engine comprises an electronic data processing device configured to send demand response commands to load controllers of an aggregation of loads including the load controller configured to operate the water heater. The load controller is further configured to operate the water heater in accord with demand response commands received from the aggregation demand response dispatch engine. A condition-based maintenance system comprises an electronic data processing device configured to detect a failure mode present in the water heater based on information including the temperature readings provided by the at least one temperature sensor and power input to the water heater.

0009In some illustrative embodiments disclosed as illustrative examples herein, a system comprises: an electrical load; a load controller comprising an electronic data processing device configured to operate the electrical load; an aggregation demand response dispatch engine comprising an electronic data processing device configured to send demand response commands to a loads aggregation including sending demand response commands to the load controller configured to operate the electrical load, wherein the load controller is further configured to operate the electrical load in accord with demand response commands received from the aggregation demand response dispatch engine; and a condition-based maintenance system comprising an electronic data processing device configured to detect a failure mode present in the electrical load based on information also input to the load controller. In some embodiments, the condition-based maintenance system is configured to detect a failure mode present in the electrical load based on information also input to the load controller including (i) readings of at least one temperature sensor that measures temperature of a fluid volume whose temperature is controlled by the load controller operating the electrical load and (ii) electrical power input to the electrical load. In some embodiments the electrical load is one of an electric water heater and a heating, ventilation, and air conditioning (HVAC) system.

BRIEF DESCRIPTION OF THE DRAWINGS

0010<figref idref="DRAWINGS">FIG. 1</figref> diagrammatically shows a demand response system employing an aggregation of electrical loads comprising illustrative water heaters.

0011<figref idref="DRAWINGS">FIG. 2</figref> diagrammatically shows a side partial sectional view of one of the water heaters of the system of <figref idref="DRAWINGS">FIG. 1</figref>.

0012<figref idref="DRAWINGS">FIG. 3</figref> diagrammatically shows a one-dimensional model of the water heater of <figref idref="DRAWINGS">FIG. 2</figref> including a vertical stack of disks representing the water volume in the cylindrical water storage tank, and a stack of annular segments surrounding the vertical stack of disks. The stack of annular segments represents the cylindrical wall of the cylindrical water storage tank.

0013<figref idref="DRAWINGS">FIG. 4</figref> diagrammatically shows a process flow performed by a condition-based maintenance (CBM) component of the demand response system of <figref idref="DRAWINGS">FIG. 1</figref>.

DETAILED DESCRIPTION

0014Disclosed herein are improved approaches for modeling a water heater for use in advanced operations such as demand response and condition-based maintenance. These approaches recognize that monitoring water heater operation by tracking temperature reading of the thermostat may not provide sufficient information for some advanced operations such as demand response. In order for electric water heaters to be effectively used as an energy storage device for demand response, the energy stored in the water heater should be tracked as a function of time. The internal temperature of water heater tank, typically as measured by the thermostat or an ancillary thermocouple or other temperature measuring device, is typically taken as the energy storage state metric in order to infer available energy. This metric assumes that the water in the tank is well-mixed, so that the water temperature is uniform throughout the tank. However, it is recognized herein that vertically oriented cylindrical water storage tanks, of the type commonly used in residential and commercial settings, may exhibit strong temperature stratification along the vertical direction, with the temperature near the bottom of the tank usually being lower than the temperature near the top of the tank. This vertical stratification is caused by the configuration of cold water feed and hot water draw lines: the cold water is supplied near the bottom of the tank, while the the hot water is drawn from the top of the tank. Vertical temperature stratification also results from differences in density between the cold water and hold water—heating the water causes it to become less dense, leading the hotter water to tend to migrate upward through the tank. Vertical temperature stratification can also result from the use of discrete heating elements that do not impart heat into the tank volume uniformly.

0015The vertical temperature stratification impacts the dynamic energy behavior of the water heater. To model the temperature in the water tank, a complex three-dimensional (3D) model may be employed that incorporates computational fluid dynamics (CFD) methods to solve for thermal and mass transport dynamics. These methods are highly computationally expensive, and may be impractical for implementation in relatively simple electronics such as those desired to be incorporated into a water heater controller. Another difficulty is that such 3D modeling typically requires substantial information to be input to the model, which may be unavailable. Temperatures within the water storage tank are typically unavailable except at one or a two discrete points (e.g. one temperature reading per thermostat, with two thermostats provided in some conventional electric water heater designs). Similarly, water flow rates may be unavailable.

0016Attempts have been made to simplify the modeling by employing a one-dimensional water heater model. See Fanney et al., “The Thermal Performance of Residential Electric Water Heaters Subjected to Various Off-Peak Schedules”, Journal of Solar Energy Engineering, vol. 118 pp. 73-80 (1996); Vettros et al., “Load Frequency Control by Aggregations of Thermally Stratified Electric Water Heaters”, Innovated Smart Grid Technologies (ISGT Europe), 2012 3^rdIEEE PES ISGT (IEEE 2012). However, these models do not account for three-dimensional effects, such as thermal losses at the shell of the tank <b>40</b>.

0017Disclosed herein are one-dimensional (1D) lumped parameter models that operate on only two tank wall measurements with a known heating input to approximate the vertical temperature stratification of the water heater. The disclosed 1D models are suitably used to estimate internal water temperatures of the tank, as well as the (unmeasured) water draw flow rate and temperature. The 1D models disclosed herein leverage the substantial symmetry about the vertical axis of the cylindrical water storage tank of a typical water heater by modeling the cylindrical water tank as a stack of disks (sub-cylinders) along the vertical axis of the cylindrical water storage tank. Additionally, the disclosed 1D water heater models provide effective modeling of thermal behavior at the boundary of the tank <b>40</b> by including annular segments representing the tank wall, while still retaining the advantageous 1D formalism. This allows the use of wall temperature measurements to infer the internal unknown stratified dynamics.

0018The 1D model operating on limited inputs from the conventional water heater thermostat(s) provides more accurate information about the energy stored in the water heater tank at any given time, thus providing more effective demand response while ensuring the primary water heater function of providing hot water is also safely and effectively performed. The information provided by such a model can be leveraged to provide more accurate information about the operational status of key components of the water heater, thus providing information suitably used to perform condition-based maintenance of the water heater.

0019With reference to <figref idref="DRAWINGS">FIG. 1</figref>, a demand response system is disclosed, which leverages one or more (illustrative four) electric water heaters <b>20</b> of a facility <b>22</b> (residence, business, or so forth) to provide demand response in support of an electric grid managing entity <b>24</b>. Because each water heater <b>20</b> individually provides a small amount of energy storage as compared with the electric grid, the water heaters <b>20</b> (possibly along with water heaters of other facilities, and/or along with other electrical devices capable of storing energy such as air conditions) are operated as an aggregation by a demand response dispatch engine <b>26</b> in order to coordinate operations of the water heaters (and optionally other loads of the aggregation) to provide demand response services. More particularly, the dispatch engine <b>26</b> is suitably embodied by a computer or other electronic data processing device communicating with load controllers <b>30</b> of the water heaters <b>20</b> or other electrical loads of the aggregation via a communication network <b>32</b> and programmed to receive demand response instructions from the grid managing entity <b>24</b> (for example, curtailment instruction in the case of a load shedding operation, or an AGC signal in the case of frequency regulation) and to send control commands to the load controllers <b>30</b> to cause the load controllers <b>30</b> to control their respective water heaters to implement the demand response instruction. The communication network <b>32</b> may, by way of illustrative example, include wired or wireless Internet links, wired or wireless local area network (LAN) links, Bluetooth links, various combinations thereof, or so forth. As a more particular illustrative example, the demand response dispatch engine <b>26</b> may be connected with the Internet and send control commands via the Internet to an IP address associated with a facility controller (not shown) of the facility <b>22</b> which retransmits the control commands to network access points distributed through the facility <b>22</b> (possibly with some translation or other processing of the control commands at the facility controller) via a wired or wireless local area network, and the access points then transmit the control commands to the load controllers <b>30</b> via Bluetooth, wired connections, or another short-range communication link. These are merely illustrative examples. The load controllers <b>30</b> comprise electronic data processing devices and may be variously embodied, for example as microcontroller- or microprocessor-based controllers that replace the original manufacturer thermostat, or that augment operation of the original thermostat by operating a power relay connected with the power feed to the water heater, or intercepting and modifying the temperature input to the thermostat, or so forth. In some embodiments, the load controller <b>30</b> may be an original manufacturer component, e.g. a thermostat originally designed to accept remote control commands. Communication between the grid managing entity <b>24</b> and aggregation dispatch controller <b>26</b> may be via similar pathways (Internet, LAN, et cetera). Additionally or alternatively, demand response instructions may be conveyed from the grid managing entity <b>24</b> to the aggregation dispatch controller <b>26</b> manually, for example via telephone to a human data entry operator of a computer embodying the dispatch controller <b>26</b>.

0020Besides providing a large aggregate energy storage capacity for demand response operations, another advantage of operating the water heaters <b>20</b> in an aggregation context is that this assists in ensuring that individual water heaters can deviate from the desired aggregate response in order to fulfill other, possibly contradictory, demands on the water heater, such as providing hot water or not overheating the water in the storage tank. Such deviations may be uncoordinated and/or coordinated. As an illustrative example of an uncoordinated deviation, a water heater may be commanded by the dispatch engine <b>26</b> to curtail energy usage—however, if the water in the water heater's storage tank falls below a deadband minimum of the thermostat, the load controller <b>30</b> controlling the water heater (or the original thermostat, depending on the precise control configuration) may activate the heating elements of the water heater in deviation from the energy usage curtailment command. If the deviant water heater is only one member of a relatively large aggregation of water heaters, then this uncoordinated deviation nonetheless will not significantly affect the demand response curtailment provided by the aggregation, and it allows the water heater to perform its primary duty of providing hot water. As an example of a coordinated deviation, during a load shedding operation the dispatch engine <b>26</b> may receive state information regarding the states of the water heaters <b>20</b> from their respective controllers <b>30</b>, and based on this information the dispatch engine <b>26</b> may instruct a water heater with a low amount of thermal energy stored in its tank (i.e. the water is relatively cold) to operate in deviance from the curtailment command. This latter, coordinated approach requires more algorithmic complexity at the dispatch engine <b>26</b> in order to prioritize the curtailment amongst the water heaters of the aggregation, but has the advantage that the prioritization can better balance the demand response (e.g. curtailment) against individual load needs.

0021Each load controller <b>30</b> operates in part based on information generated by maintaining a dynamic one-dimensional (1D) model <b>33</b> of the water heater <b>20</b> under control of the load controller <b>30</b>. In <figref idref="DRAWINGS">FIG. 1</figref>, the 1D water heater model <b>33</b> is diagrammatically shown in the upper right inset of <figref idref="DRAWINGS">FIG. 1</figref>—however, it is to be understood that a separate instance of the model is executed at each load controller <b>30</b> to model the water heater under control of that controller <b>30</b>. (In an alternative embodiment, it is contemplated to execute the 1D model <b>33</b> for each water heater at the aggregation dispatch engine <b>26</b>, based on inputs received from the load controller). The 1D water heater model <b>33</b> receives a limited number of inputs, and estimates the vertical temperature distribution through the volume of the water tank. For example, in one illustrative embodiment, the 1D model <b>33</b> receives as input the power reading for each heating element (of which there may be one or more heating elements for a given water heater), the temperature reading for the thermostat controlling each heating element, and an ambient temperature reading (or assumed room temperature value if the room containing the water heater has a well-controlled temperature). These inputs are readily obtained by interfacing with the thermostat(s) of the water heater <b>20</b>, by adding thermocouples or other temperature measurement devices to the wall or skin of the water heater holding tank, by employing a clamp-on ammeter or the like to measure electrical power input to the heating elements, by adding a room thermometer (or thermocouple, et cetera), or so forth.

0022The output of the 1D model <b>33</b> for each controlled water heater <b>20</b> may be used in demand response applications to provide more accurate estimation of the thermal energy stored in the water tank as compared with a water temperature reading provided by the original thermostat. However, this is merely one possible application and others are contemplated. By way of illustrative example, another suitable application of the 1D model <b>33</b> is for estimating the operational condition of the modeled water heater <b>20</b>. In this condition-based maintenance application, the more detailed knowledge regarding the operational state of the water heater <b>20</b> provided by the model <b>33</b> enables automated diagnosis of certain common failure modes such as partial insulation failure, heating element failure, excessive tank sediment build-up, or a rupture in the cold water feed drip tube that directs cold water to the bottom of the holding tank. Such a failure diagnosis may be communicated to the demand response dispatch engine <b>26</b> so that the dispatch engine can take the diagnosed failure into account in providing demand response services to the grid operator <b>24</b> (for example, by not using the apparently failed water heater to provide demand response). Additionally or alternatively, the failure diagnosis may be communicated to a facility maintenance entity <b>34</b> having responsibility for maintaining the water heaters <b>20</b> of the facility <b>22</b>. The facility maintenance entity <b>34</b> may, for example, comprise a front-desk computer staffed by a building maintenance department and having a notifications component via which building maintenance personnel may be notified of the diagnosed failure, and/or the facility maintenance entity <b>34</b> may comprise a cellular telephone or networked tablet device carried by the building maintenance person and configured to push notifications to the user.

0023With reference to <figref idref="DRAWINGS">FIGS. 2 and 3</figref> an illustrative example of an electric water heater <b>20</b> is described (<figref idref="DRAWINGS">FIG. 2</figref>), along with a suitable embodiment of a 1D water heater model <b>33</b> (<figref idref="DRAWINGS">FIG. 3</figref>) suitably modeling the water heater of <figref idref="DRAWINGS">FIG. 2</figref>. The illustrative electric water heater <b>20</b> includes a water storage (or holding) tank <b>40</b> containing hot water (or water to be heated), an upper resistive heating element <b>42</b> and a lower resistive heating element <b>44</b> that can be electrically energized to heat water in the tank <b>40</b>, a cold water inlet <b>46</b> including a drip tube <b>48</b> that extends downward to deliver cold water near the bottom of the tank <b>40</b>, and a hot water draw line <b>50</b> positioned to draw hot water from at or near the top of the tank <b>40</b>. The upper heating element <b>42</b> is controlled by an upper thermostat <b>52</b>, and the lower heating element <b>44</b> is controlled by a lower thermostat <b>54</b>. In a typical configuration, the upper thermostat <b>52</b> includes a set point adjustment knob (not shown) via which a user may adjust the temperature set point of the water heater <b>20</b>, and the lower thermostat <b>54</b> is operatively linked with the upper thermostat <b>52</b>.

0024The illustrative water heater <b>20</b> is of the vertically oriented cylindrical design, in which the storage tank <b>40</b> has the general shape of a cylinder defining a vertical tank (cylinder) axis <b>56</b>. Although the tank <b>40</b> has the general shape of a vertically oriented cylinder, various deviations from the cylindrical shape may be present, such as illustrative rounded top and bottom portions, features such as the resistive heating elements <b>42</b>, <b>44</b> that may break perfect cylindrical symmetry, or so forth. A drain valve <b>58</b> is also provided to enable water in the tank <b>40</b> to be drained out for maintenance, transport or decommissioning of the water heater <b>20</b>, or for other purposes.

0025Also diagrammatically depicted in <figref idref="DRAWINGS">FIG. 2</figref> are some failure modes that can be detected based on parameters estimated using the model <b>33</b>. An illustrative region of corrosion <b>60</b> on the skin of the water heater tank <b>40</b>, for example due to repeated exposure to water from the drain valve <b>52</b> or another source, may produce an insulation disturbance that increases thermal losses from the tank <b>40</b> and reduces water heater efficiency. The illustrative lower heating element <b>44</b> includes some fouling or build-up 62 that can reduce its ability to transfer heat into the water, thus lowering the heating efficiency. The bottom of the tank <b>40</b> also has an excessive sediment build-up 64 that reduces the actual water capacity of the tank <b>40</b> and may be indicative of a more serious problem such as corrosion of the inner lining of the tank <b>40</b>. Still further, the drip tube <b>48</b> has developed a rupture <b>66</b> generating a flow path <b>68</b> via which cold water can pass directly to the hot water draw line <b>50</b> without being significantly heated by the heating elements <b>42</b>, <b>44</b>.

0026With continuing reference to <figref idref="DRAWINGS">FIG. 2</figref>, the illustrative load controller <b>30</b> that controls the illustrative water heater <b>20</b> is mounted on the skin or outside wall of the water storage tank <b>40</b>. This is merely an illustrative example, and the load controller may in general be installed at various locations, for example wall-mounted and connected with the water heater by suitable cabling or so forth. The illustrative load controller <b>30</b> includes a microprocessor or microcontroller and associated components (e.g. electronic memory, analog-to-digital converters for reading thermocouples or other analog inputs, a USB port or other digital interface for receiving digital inputs or outputting digital outputs, a digital-to-analog converter for outputting an analog control signal, a WiFi interface, Ethernet interface, or other networking interface, various combinations, sub-sets, or so forth of the foregoing, et cetera) programmed or configured to execute a water heater control algorithm based on received remote inputs such as demand response control commands from the dispatch engine <b>26</b> (see <figref idref="DRAWINGS">FIG. 1</figref>). In performing such control operations, the load controller <b>30</b> executes the 1D water heater model <b>33</b> to model the current operational state based on inputs including: the power delivered to (or consumed by) the heating elements <b>42</b>, <b>44</b>; the temperature readings of the thermostats <b>52</b>, <b>54</b>; and an assumed or measured room (ambient) temperature. The water heater state information provided by the executing 1D water heater model <b>33</b> may also be used by a condition based maintenance algorithm executed by the load controller <b>30</b> to detect various failure modes, such as illustrative failure modes <b>60</b>, <b>62</b>, <b>64</b>, <b>66</b>, <b>68</b>. In alternative embodiments, the 1D modeling algorithm and/or the failure mode detection algorithm may execute at the computer embodying the dispatch engine <b>26</b> or elsewhere.

0027With reference to <figref idref="DRAWINGS">FIG. 3</figref>, an illustrative embodiment of the 1D water heater model <b>33</b> is described, which is suitable for modeling the illustrative water heater <b>20</b> shown in <figref idref="DRAWINGS">FIG. 2</figref>. The 1D water heater model <b>33</b> utilizes two external temperature measurements from the thermostats <b>52</b>, <b>54</b> (or, alternatively, from similarly located thermostats mounted to the wall of the tank <b>40</b>), the heating input to the heating elements <b>42</b>, <b>44</b> (assumed to correspond to the electrical power input to the heating elements <b>42</b>, <b>44</b>), and an ambient temperature measurement (or assumed ambient temperature value). Estimates all the other internal and external temperatures are generated by the 1D model <b>33</b>, along with estimates of the water draw mass flow rate and inlet water temperature with disturbance estimators. The 1D model <b>33</b> is a one-dimensional lumped model structure of the internal and external temperature dynamics of the water heater <b>20</b>, in which the water tank <b>40</b> is divided into N finite lumps or disks. In the illustrative model <b>33</b> of <figref idref="DRAWINGS">FIG. 3</figref>, N=9 including disks representing the top and bottom of the tank.

0028More particularly, the illustrative 1D water heater model <b>33</b> of <figref idref="DRAWINGS">FIG. 3</figref> includes a bottom disk <b>1</b> representing the bottom of the tank <b>40</b> and, running successively upward along the vertical cylinder axis <b>56</b>, seven successive disks <b>2</b>, <b>3</b>, <b>4</b>, <b>5</b>, <b>6</b>, <b>7</b>, <b>8</b> each representing a successively higher disk of water in the stack of disks representing the volume of the water holding tank <b>40</b>, topped at the highest elevation along the axis <b>56</b> by a top disk <b>9</b> representing the top of the tank <b>40</b>. The walls of the cylindrical holding tank <b>40</b> between the top disk <b>1</b> and bottom disk <b>9</b> are also modeled, being represented by seven annular elements <b>10</b>, <b>11</b>, <b>12</b>, <b>13</b>, <b>14</b>, <b>15</b>, <b>16</b> at successively higher elevation and surrounding corresponding disks <b>2</b>, <b>3</b>, <b>4</b>, <b>5</b>, <b>6</b>, <b>7</b>, <b>8</b>. Said another way, the lumped wall segment <b>10</b> is an annular segment surrounding the disk <b>2</b>; the lumped wall segment <b>11</b> is an annular segment surrounding the disk <b>3</b>; the lumped wall segment <b>12</b> is an annular segment surrounding the disk <b>4</b>; the lumped wall segment <b>13</b> is an annular segment surrounding the disk <b>5</b>; the lumped wall segment <b>14</b> is an annular segment surrounding the disk <b>6</b>; the lumped wall segment <b>15</b> is an annular segment surrounding the disk <b>7</b>; and the lumped wall segment <b>16</b> is an annular segment surrounding the disk <b>8</b>. These annular segments enable accurate modeling of the impact of thermal losses at the walls of the water storage tank <b>40</b>, while retaining the one-dimensional formalism so as to substantially reduce computational complexity as compared with three-dimensional modeling approaches.

0029Heat injected by the upper heating element <b>42</b> is represented by a heat input H<b>1</b> into disk <b>7</b>, and heat injected by the lower heating element <b>44</b> is represented by a heat input H<b>2</b> into disk <b>3</b>. The temperature measured by the upper temperature sensor (e.g. thermostat <b>52</b>) is indicated as a temperature reading M<b>16</b> of the annular wall element <b>16</b>, while the temperature measured by the lower temperature sensor (e.g. thermostat <b>54</b>) is indicated as a temperature reading M<b>13</b> of the annular wall element <b>13</b>. The cold water is assumed to be input (via the drip tube <b>48</b>) into the lowest disk <b>2</b> and to have a temperature denoted as T_in. The hot water drawn is assumed to have the temperature of the uppermost disk <b>8</b>. It will be appreciated that the number of disk/annulus divisions, and the locations of the heat inputs and temperature readings respective to those divisions, is suitably chosen to provide a desired spatial resolution in the vertical direction and to accurately model the actual positions of the heating elements and temperature sensors in the water heater being modeled.

0030With continuing reference to <figref idref="DRAWINGS">FIGS. 2 and 3</figref>, the dynamic behavior of the water heater <b>20</b> as represented by the 1D model <b>33</b> of <figref idref="DRAWINGS">FIG. 3</figref> can be represented mathematically by a one-dimensional model including: (1) for each disk of the vertical stack of disks representing the water volume in the cylindrical water storage tank, a differential equation expressing the time derivative of the temperature of the disk estimated by the one-dimensional model; and (2) for each annular segment of the stack of annular segments representing the cylindrical wall of the cylindrical water storage tank, a differential equation expressing the time derivative of the temperature of the annular segment estimated by the one-dimensional model. In an illustrative example, the 1D model <b>33</b> of <figref idref="DRAWINGS">FIG. 3</figref> is suitably represented by the following set of Differential Equations (1)-(20):

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/></mstyle><mo></mo><msub><mi>x</mi><mi>s</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>s</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>10</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mi>A</mi></mrow><mrow><msub><mi>m</mi><mn>1</mn></msub><mo></mo><mrow><msub><mi>c</mi><mi>wall</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>y</mi><mi>w</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>w</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>y</mi><mi>s</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>s</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>-</mo><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mi>A</mi><mrow><mi>wall</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mi>b</mi></msub></mrow><mo>)</mo></mrow><mo></mo><mi>U</mi></mrow><mrow><msub><mi>m</mi><mn>1</mn></msub><mo></mo><msub><mi>c</mi><mi>wall</mi></msub></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><msub><mi>T</mi><mi>amb</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><msub><mi>L</mi><mn>1</mn></msub><mo></mo><msub><mover><mi>T</mi><mi>_</mi></mover><mn>10</mn></msub></mrow><mo>+</mo><mrow><msub><mi>L</mi><mn>2</mn></msub><mo></mo><msub><mover><mi>T</mi><mi>_</mi></mover><mn>13</mn></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo></mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mfrac><msub><mi>k</mi><mi>w</mi></msub><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub><mo></mo><msub><mi>m</mi><mn>2</mn></msub><mo></mo><msub><mi>c</mi><mi>w</mi></msub></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>1</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mi>A</mi></mrow><mrow><msub><mi>m</mi><mn>2</mn></msub><mo></mo><mrow><msub><mi>c</mi><mi>w</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>w</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>s</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>s</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>+</mo><mrow><mfrac><mover><mover><mi>m</mi><mo>.</mo></mover><mo>^</mo></mover><msub><mi>m</mi><mn>2</mn></msub></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mi>in</mi></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>10</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mi>A</mi></mrow><mrow><msub><mi>m</mi><mn>2</mn></msub><mo></mo><mrow><msub><mi>c</mi><mi>w</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>w</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>s</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>s</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>+</mo><mrow><msub><mi>L</mi><mn>3</mn></msub><mo></mo><msub><mover><mi>T</mi><mi>_</mi></mover><mn>10</mn></msub></mrow><mo>+</mo><mrow><msub><mi>L</mi><mn>4</mn></msub><mo></mo><msub><mover><mi>T</mi><mi>_</mi></mover><mn>13</mn></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo></mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub></mrow><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mfrac><msub><mi>k</mi><mi>w</mi></msub><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub><mo></mo><msub><mi>m</mi><mn>3</mn></msub><mo></mo><msub><mi>c</mi><mi>w</mi></msub></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub><mo>-</mo><mrow><mn>2</mn><mo></mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub></mrow><mo>+</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mover><mover><mi>m</mi><mo>.</mo></mover><mo>^</mo></mover><mi>m</mi></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>2</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>11</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mi>A</mi></mrow><mrow><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>w</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>s</mi></msub></mrow><mrow><mn>2</mn><mo></mo><msub><mi>k</mi><mi>s</mi></msub></mrow></mfrac></mrow></mfrac><mo>+</mo><mrow><mfrac><mrow><mi>η</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>P</mi></mrow><mrow><msub><mi>m</mi><mn>4</mn></msub><mo></mo><msub><mi>c</mi><mi>w</mi></msub></mrow></mfrac><mo></mo><msub><mi>u</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><msub><mi>L</mi><mn>5</mn></msub><mo></mo><msub><mover><mi>T</mi><mi>_</mi></mover><mn>10</mn></msub></mrow><mo>+</mo><mrow><msub><mi>L</mi><mn>6</mn></msub><mo></mo><msub><mover><mi>T</mi><mi>_</mi></mover><mn>13</mn></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo></mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub></mrow><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mfrac><msub><mi>k</mi><mi>w</mi></msub><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub><mo></mo><msub><mi>m</mi><mn>4</mn></msub><mo></mo><msub><mi>c</mi><mi>w</mi></msub></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo>-</mo><mrow><mn>2</mn><mo></mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub></mrow><mo>+</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>5</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mover><mover><mi>m</mi><mo>.</mo></mover><mo>^</mo></mover><msub><mi>m</mi><mn>4</mn></msub></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>12</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mi>A</mi></mrow><mrow><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace 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width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>x</mi><mi>w</mi></msub><mo></mo><msub><mi>m</mi><mn>5</mn></msub><mo></mo><msub><mi>c</mi><mi>w</mi></msub></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub><mo>-</mo><mrow><mn>2</mn><mo></mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>5</mn></msub></mrow><mo>+</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>6</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mover><mover><mi>m</mi><mo>.</mo></mover><mo>^</mo></mover><msub><mi>m</mi><mn>5</mn></msub></mfrac><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>4</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>5</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>^</mo></mover><mn>5</mn></msub><mo>-</mo><msub><mover><mi>T</mi><mo>^</mo></mover><mn>13</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mi>A</mi></mrow><mrow><mfrac><mrow><mi>Δ</mi><mo></mo><mstyle><mspace 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file="US2016178221A1_D0001.tif" />

0000In these Differential Equations, the following symbols are used:

0031A is the lateral area of each of the disks <b>2</b>, <b>3</b>, <b>4</b>, <b>5</b>, <b>6</b>, <b>7</b>, <b>8</b>;

0032T_ambis the ambient temperature;

0033{circumflex over (T)}_inis the estimated temperature for cold water delivered into disk <b>2</b>;

0034{circumflex over (T)}₁is the estimated temperature for the bottom of the tank (disk <b>1</b>);

0035{circumflex over (T)}₂is the estimated temperature for water in the disk <b>2</b>;

0036{circumflex over (T)}₃is the estimated temperature for water in the disk <b>3</b>;

0037{circumflex over (T)}₄is the estimated temperature for water in the disk <b>4</b>;

0038{circumflex over (T)}₅is the estimated temperature for water in the disk <b>5</b>;

0039{circumflex over (T)}₆is the estimated temperature for water in the disk <b>6</b>;

0040{circumflex over (T)}₇is the estimated temperature for water in the disk <b>7</b>;

0041{circumflex over (T)}₈is the estimated temperature for water in the disk <b>8</b>;

0042{circumflex over (T)}₈is also the estimated temperature for drawn hot water;

0043{circumflex over (T)}₉is the estimated temperature for the top of the tank (disk <b>9</b>);

0044{circumflex over (T)}₁₀is the estimated temperature for the annular wall segment <b>10</b>;

0045{circumflex over (T)}₁₁is the estimated temperature for the annular wall segment <b>11</b>;

0046{circumflex over (T)}₁₂is the estimated temperature for the annular wall segment <b>12</b>;

0047{circumflex over (T)}₁₃is the estimated temperature for the annular wall segment <b>13</b>;

0048{circumflex over (T)}₁₄is the estimated temperature for the annular wall segment <b>14</b>;

0049{circumflex over (T)}₁₅is the estimated temperature for the annular wall segment <b>15</b>;

0050{circumflex over (T)}₁₆is the estimated temperature for the annular wall segment <b>16</b>;

0051{dot over ({circumflex over (m)})} is the estimated water draw flow rate (mass per unit time);

0052U is the heat transfer coefficient between the shell and the environment;

0053<o ostyle="single">T</o> is the error between measured and estimated temperatures;

0054L is the observer design gain, which is a design constant that multiples the error

0055between the measurement and the estimation of the two shell temperatures;

0056Δx is the thickness of each disk in the horizontal direction;

0057Δy is the thickness of each disk in the vertical direction;

0058k_wis the thermal conductivity of water;

0059k_sis the thermal conductivity of steel (or other material) forming the tank;

0060ρ denotes density;

0061c denotes specific heat;

0062η denotes the efficiency or effectiveness of the heater, and ranges from 0:1;

0063P is the heater power.

0064The unknown input estimators take on an arbitrary form of a second order system for purpose of explanation as seen in Equations (17)-(20). The assumed temperature measurements are T₁₀and T₁₃(so that measured v. estimated error quantities <o ostyle="double">T</o>₁₀and <o ostyle="double">T</o>₁₃are defined).

0065The skilled artisan can readily modify the illustrative water heater <b>20</b> and/or model <b>33</b> of <figref idref="DRAWINGS">FIGS. 3-5</figref> to model water heaters with different numbers of disks (thus providing different vertical resolution), different number or placement of temperature sensors and/or heaters, and so forth. Similarly, it is contemplated to omit or add additional physical characteristics to the model.

0066The illustrative model <b>33</b> can be used for various purposes. In the illustrative demand response example of <figref idref="DRAWINGS">FIG. 1</figref>, the model <b>33</b> can be used to more accurately assess the energy stored in the tank at any given time, so that the dispatch engine <b>26</b> can more effectively leverage the aggregate stored energy for demand response while ensuring that individual water heaters <b>20</b> can perform their intended function of providing hot water within certain constraints (temperature, capacity, et cetera). The energy stored in the tank <b>40</b> can be computed as the sum of the energy stored in each disk of the stacked disks representing the water in the cylindrical tank <b>40</b>. The energy stored in each disk can be computed as Energy stored=C_p*dT*m where C_pis the specific heat capacity of water, dT is the temperature difference between the water in the disk and the surroundings, and m is mass of water.

0067In another illustrative application, the model <b>33</b> is used to provide condition based maintenance. The use of water heater sensors for identification of maintenance issues affecting the water heater advantageously benefits the end-user (who typically owns and utilizes the water heater) by detecting potential problems early, while they can be repaired in a cost-effective manner, or while they can be resolved by replacing the water heater before a catastrophic failure such as a large-scale water leak occurs. Condition based maintenance as disclosed herein also enables the water heater to be kept in efficient operating condition, thus lowering energy costs for the end user. These incentives advantageously encourage the end-user to participate in the aggregation managed by the dispatch engine <b>26</b>. Optionally, the aggregation may charge a fee for providing this maintenance monitoring, thus generating an additional revenue stream for the demand response system operators.

0068By way of the illustrative example of <figref idref="DRAWINGS">FIGS. 2-5</figref>, the state of a water heater <b>20</b> comprises temperature measurements by two (or more) temperature sensors measuring the water temperature, along with the electrical power being consumed as suitably measured by a clamp-on ammeter or the like. The state measurements are combined with the model <b>33</b> of the water heater <b>20</b> to obtain estimates for parameters describing the water heater <b>20</b>, such as the effective insulation thermal resistance, first hour delivery, effective volume, heating element power transfer rate, and so forth. These parameters can be used for providing demand response as already described, and additionally can be used to infer the (possibly incipient) presence of possible failure modes so as to provide condition based maintenance (CBM) capability. Condition based maintenance provides a mechanism for maintaining the water heater based upon its actual condition prior to failure, rather than performing maintenance at specified service intervals or simply waiting for the device to fail outright. Implementing CBM has several positive impacts on the end user and the maintenance provider. For the end user, the costs of maintaining a device can be reduced by servicing only at the onset of an issue rather than at regular intervals. Maintenance schedules are generated for typically usage patterns, but if the usage pattern for a particular water heater differs from typical, maintenance could be performed too frequently increasing costs, or too infrequently leading to device failure and replacement and repair costs. The end user also benefits from a reduced number of service interruptions because operational issues are detected prior to the failure of a device, and the water heater can be serviced without the disruption of a failure. An additional financial benefit for the user is that the water heater is kept operating closer to its peak efficiency, reducing energy costs. Because water heaters operate transparently in the absence of a failure (that is, the user is typically satisfied as long as hot water of the desired temperature is being delivered), the end-user may be unaware that the water heater is operating under reduced performance and efficiency, because the user experience has not changed and any increase in energy cost is gradual and can go unnoticed. The CBM systems disclosed herein notify the user of the onset of a failure mode, such as (with brief reference back to <figref idref="DRAWINGS">FIG. 2</figref>) an insulation disturbance <b>60</b>, fouling or build-up 62 on a heating element <b>44</b>, excessive sediment build-up 64, or a rupture <b>66</b> in the drip tube <b>48</b> generating a cold water bypass flow path <b>68</b>. By identifying the failure mode with some particularity, the end user is alerted of the problem early, and can take corrective action.

0069As a further benefit, maintenance providers (for example, in the context of a water heater service contract) can increase the level of service provided to their customers by using CBM while also reducing their costs. The level of service to customers can be enhanced by reducing the number of unneeded, schedule based service calls, while also reducing the likelihood of a device being unexpectedly removed from service. The maintenance costs can be reduced by scheduling work to be performed during normal business hours, rather than off-hours requiring differential or overtime pay. This can be done because failures can be anticipated based on the output of the CBM system, thus the providers can operate proactively instead of reactively. When technicians are dispatched to a site, a better understanding the service needed is already available prior to their departure, as the CBM system output provides the service technician with identification of the likely failure mechanism before the service technician examines the water heater. This knowledge reduces the risk of a technician being dispatched without the proper parts or tools.

0070With returning reference to <figref idref="DRAWINGS">FIG. 1</figref>, CBM system may be implemented as software or firmware executed in conjunction with the model <b>33</b> by the microprocessor or microcontroller of the load controller <b>30</b>, and/or may be implemented as software or firmware executed at the dispatch engine <b>26</b> which communicates with the load controller <b>30</b>. Implementing the CBM at the load controller <b>30</b> reduces bandwidth costs and spreads computational complexity across the load controllers <b>30</b>. The CBM for a water heater <b>20</b> leverages the temporal recording of at least two temperature sensors and the power consumed by the water heater. The two temperature sensors may, for example (see illustrative <figref idref="DRAWINGS">FIG. 2</figref>), be located near the upper and lower thermostats <b>52</b>, <b>54</b> typically found on the water heater. From these measurements, the state of the water heater is estimated, for example using the model <b>33</b>, where state is a measure of the energy stored within the device, and power being consumed. The water heater typically has a minimum and maximum usable energy state. The minimum energy is when the temperature(s) are at the deadband minimum and the maximum energy is when the temperature(s) are at the deadband maximum. The power consumption state may be discrete power levels, based upon the size and number of heaters operating. Using these state values, parameter identification can be performed, fitting measurements to the model <b>33</b> of the water heater <b>20</b>. The parameters to be estimated may, for example, include the effective capacity (the volume of water the tank contains), the thermal conductivity from the tank to ambient, the effective mixing rate, first hour delivery capacity, and the heating element size. By monitoring these parameters over time and using other error detection methodologies, changes in system operation signifying the onset of a failure mechanism (broadly defined herein as encompassing the spectrum from catastrophic failure, e.g. a major water leak or complete cessation of water heating, to failures that produce less deleterious effects such as reduced operational efficiency).

0071With reference to <figref idref="DRAWINGS">FIG. 4</figref>, an illustrative CBM system is described, which may be implemented at the load controller <b>30</b> or the dispatch engine <b>26</b>. In an operation <b>70</b>, the temperature readings and input power are measured (corresponding to T₁₀, T₁₃, and u₁and u₂in the model <b>33</b> of Equations (1)-(<b>20</b>)), and these inputs are applied to the model <b>33</b> to generate the estimated water temperatures {circumflex over (T)} and estimated water flows {dot over ({circumflex over (m)})}. These information then serve as the inputs to various illustrative water heater failure mode detectors <b>72</b>, <b>74</b>, <b>76</b>, <b>78</b> as described next. It should be noted that a detected failure mode does not necessarily require that the failure has actually occurred—rather, a failure mode detector may “anticipate” a failure by detecting incipient degradation of a type having a significant potential to lead to a failure of the detected failure mode. A failure mode may also be detected early, that is, at a point where performance has been compromised by the failure mode to some degree but has not yet reached a point where remedial action (e.g. repair or replacement) is appropriate.

0072An illustrative embodiment of an illustrative insulation disturbance detector <b>72</b> operates to detect an insulation disturbance. The efficiency of the water heater <b>20</b> is dependent upon its thermal conductivity from the tank to ambient temperature. This thermal conductivity is limited by thermal insulation of the storage tank <b>40</b>. Insulation by itself is not likely to fail on its own, but external factors could damage insulation such as the presence of water, exposure to airflow, or disturbance by people and animals. The occurrence of any of these factors warrants servicing. A user would likely not notice the presence of an insulation failure until the next utility billing cycle, and even then may fail to notice the resulting loss of efficiency if its onset is gradual. The insulation disturbance detector <b>72</b> suitably estimates the R-value, which is the inverse of the thermal conductivity (k) estimated by the 1D water heater model <b>33</b>. For a uniform insulator, the R-value is the ratio of the temperature difference across the insulator and the heat flux (which is the heat transfer per unit area per unit time. This can be written as R=ΔT/{dot over (Q)} where Δt is the temperature difference across the insulator, {dot over (Q)} is the heat flux, and R is the R-value. In the 1D model of <figref idref="DRAWINGS">FIG. 3</figref>, the average R-value can be computed for each annular segment <b>10</b>, <b>11</b>, <b>12</b>, <b>13</b>, <b>14</b>, <b>15</b>, <b>16</b> of the cylindrical wall of the vertically oriented cylindrical water storage tank <b>40</b>. Since the R-value is the inverse of thermal conductivity (k), the computation can alternatively be done in terms of k and lumped into the heat transfer coefficient with the environment, U. In one approach, the thermal conductivity is estimated and stored daily (or on some other interval, e.g. weekly, monthly, et cetera). An insulation disturbance is suitably detected as a large change in the day to day (or interval-to-interval) R-value, or as a predefined percent change, from the initial value measured at installation of the water heater <b>20</b>. The first case (a sudden change in R-factor) is typically due to disruptions in the insulation from external factors, such as a pipe leak above the water heater which moistens the insulation, causing its failure. The second case (more gradual change I R-factor) is indicative of insulation degradation over time, such as due to environmental conditions or manufacturing errors that are causing the insulation to slowly degrade. Depending upon the chosen predefined percent change (or other chosen threshold), the insulation disturbance detector <b>72</b> may operate to detect insulation failure in an incipient stage, i.e. before it is severe enough to call for maintenance or replacement.

0073An illustrative embodiment of an illustrative heating element failure detector <b>74</b> operates to detect fouling or build-up on a heating element that can reduce its heat transfer efficiency. The heating element <b>42</b>, <b>44</b> is an electrically resistive device that dissipates power into the water in the tank <b>40</b>. An ideal heating element would dissipate all of its supplied power into the water instantaneously. However, existing heating elements are not ideal, and there will be some resistance to heat transfer from the heating element to the water. The resistance of the heating element, and therefore its power consumption, depend upon its operating temperature. Using these properties, the effectiveness of the heater can be identified, and changes in its effectiveness noted. A heating element that is becoming fouled is expected to have reduced ability to transfer heat to the water. This is because the media fouling the heating element will likely have lower thermal conductivity compared to the water itself. With reduced thermal conductivity, the rate of temperature rise of the water due to action of the heating element will be diminished, while the temperature of the heating element itself for the same conditions will increase because heat is not being transferred to the water as effectively. Two parameter estimates can be used to detect this failure mode: the effective capacity of the tank; and the effective electrical resistance of the heating element. Note that the effective capacity can be considered as either a purely thermal quantity (e.g. number of Joules that can be stored) or as a volumetric quantity since each unit of water has a certain thermal capacity. The effective capacity of the tank will appear to increase if the heating element is becoming fouled because a slower temperature rise for the same power input will be observed. The rate of temperature rise is dependent upon the power input and the volume of water to be heated, which can be written as Q=MCΔt where Q is the energy imparted into the system, M is the volume of water, C is the heat capacity of water, and Δt is the temperature rise. The value of M should be constant over all time for a water heater <b>20</b>, or nearly so. If the heating element is failing due to fouling or buildup, then the same heat Q, which is the electrical power integrated over time, will result in a different Δt, signifying a change in the other independent variable, M. The temperature rise Δt will decrease, implying an increase in M. Analogously to the situation for insulation failure, both short term differences in M and long term differences in M may be monitored to determine if either a dramatic or gradual failure is occurring.

0074In an alternative embodiment, heating element failure due to fouling or buildup is detected by estimating the electrical resistance of the heating element. In a suitable approach, the resistance of the heating element may be measured using V=I·R where V is the voltage, I is the electrical current (both V and I suitably being represented as root mean squared, or RMS, values), and R is the resistance. The resistance will show a temperature dependency, thus it is possible to detect if the heating element is operating at a much higher or lower temperature then prescribed. Again, the effective resistance would be compared to both short term and long term data to identify if a rapid or gradual failure is occurring.

0075A second heating element failure mode optionally detected by the heating element failure detector <b>74</b> is the development of areas with low electrical conductivity, or cold spots, on the heating element. When a cold spot develops, the effective electrical resistance will increase. The previous methods can be employed to detect this failure type by determining if there was an increase in the effective resistance.

0076An illustrative embodiment of an illustrative excessive sediment buildup detector <b>76</b> operates to detect excessive sediment build-up 64 that reduces the actual water capacity of the tank <b>40</b>. Sediment from the water supply may become entrapped within the storage tank <b>40</b>. Some sediment build up is to be expected, and an anode rod (not shown in <figref idref="DRAWINGS">FIG. 2</figref>) is included in some water heater designs to suppress sediment buildup. However excessive sediment buildup remains a possibility even with anode rod protection, and excessive sediment buildup will result in a decrease in performance. In extreme cases, sediment <b>64</b> can occupy a volume inside the tank <b>40</b> which is large enough to cause a decrease in performance as experienced by the end user. With excessive sediment buildup, the storage tank <b>40</b> does not contain its rated capacity, limiting the volume of heated water available. The presence of this magnitude of sediment typically justifies servicing the water heater <b>20</b>, as well as examining the water supply for more serious issues which may be the cause of excess sediment buildup. In one approach, the detector <b>76</b> suitably detects sediment buildup by calculating the effective capacity of the water tank <b>40</b>, and comparing it to the capacity calculated at installation. Sediment buildup occurs slowly over time, and a maintenance issue is suitably reported if the measured tank capacity falls below a specified percentage of the starting capacity, which may be variously chosen to choose the level of sediment buildup to be detected (ranging from reporting incipient buildup to major buildup). Multiple thresholds are optionally used, e.g. an “incipient problem” threshold that is triggered for a small reduction in measured tank capacity, and a “serious problem” threshold that is triggered only if the measured tank capacity is more significantly reduced.

0077It will be observed that there are some similarities between the symptoms of excessive sediment buildup and the symptoms of a failed heating element, and the detectors <b>74</b>, <b>76</b> are preferably configured to distinguish between these two failure modes. In general, both heating element failure and excessive sediment buildup manifest as an apparent change in capacity of the tank <b>40</b>. However, sediment buildup cause a perceived decrease in tank capacity; whereas, heating element failure causes a perceived increase in tank capacity. Thus, the direction of change in apparent tank capacity output by the model <b>33</b> is suitably used by the detectors <b>74</b>, <b>76</b> to distinguish between the heating element failure and sediment buildup failure modes.

0078An illustrative embodiment of an illustrative drip tube rupture detector <b>78</b> operates to detect a rupture <b>66</b> in the drip tube <b>48</b>. As seen in <figref idref="DRAWINGS">FIG. 2</figref>, the drip tube <b>48</b> supplies unheated (i.e. cold) water to the bottom of the water storage tank <b>40</b>, and the heated water rises to the outlet <b>50</b> at the top of the tank <b>40</b>. This flow path design ensures that the drawn water is hot water rather than newly injected cold water. However, if a rupture <b>66</b> forms in the drip tube <b>48</b>, this can produce a cold water bypass flow <b>68</b> via which unheated (i.e. cold) supply water can pass directly to the outlet <b>50</b> resulting in lower effectiveness and efficiency of the water heater. A drip tube failure may require replacement of the entire tank or water heater, so rapid diagnosis of this problem can avoid unnecessary attempts at remedial maintenance of the water heater. The detector <b>78</b> suitably detects the drip tube rupture <b>66</b> by monitoring the difference in temperature readings of the temperature sensors located at upper and lower positions on the water tank <b>40</b>, e.g. the difference in temperature readings for the upper and lower thermostats <b>52</b>, <b>54</b> in the illustrative example of <figref idref="DRAWINGS">FIG. 2</figref>. Under normal operation, cool water enters at the bottom of the tank via the drip tube <b>48</b> in response to the drawing of hot water via the outlet <b>50</b>, and heated water at lower elevation in the tank rises toward the top to replace the drawn hot water. During hot water draw from the tank <b>40</b>, the temperature readings at the thermostats <b>52</b>, <b>54</b> during normal operation will reflect this, with the lower temperature reading starting to decrease before the upper temperature reading starts to decrease. By contrast, if the drip tube <b>48</b> includes a rupture <b>66</b> generating a bypass path <b>68</b>, then the upper temperature reading will begin to fall before the lower temperature reading, and may even fall below the lower temperature reading, because the cool water is being fed in close to the top of the tank. Such measurements are suitably performed on every usage cycle (triggered by detection of a rapid temperature decrease due to the cold water injection). While this temperature effect can be observed with limited resolution directly from the temperature readings of the two temperature sensors, it is more accurately observed using the water heater model, i.e. in the absence of a drip tube rupture the temperature of the lower-elevation disks should drop faster than the temperature of the higher-elevation disks; whereas, if a cold water bypass exists due to a drip tube rupture then to the contrary the temperature of the higher-elevation disks should drop faster than the temperature of the lower-elevation disks.

0079With continuing reference to <figref idref="DRAWINGS">FIG. 4</figref>, the diagnostic outputs of the failure mode detectors <b>72</b>, <b>74</b>, <b>76</b>, <b>78</b> are suitably processed by a CBM reporting module <b>80</b> that generates a human-perceptible report of any detected failure, or if no failure is detected then a human-perceptible report that no failure has been detected. Because there is a relatively small finite set of failure modes being monitored, in some embodiments the CBM reporting module <b>80</b> includes a memory or database storing natural language (e.g. English) text describing each potential failure (or lack thereof) along with the detected symptoms as described above. Such a report may be transmitted to the facility maintenance entity <b>34</b>, for example as a notification push and/or as a transmitted electronic mail (email) message or so forth. Additionally or alternatively, the CBM reporting module may report any detected failure mode (or lack thereof) at a lower-level format, for example transmitting a failure diagnosis binary string of N bits to the dispatch engine <b>26</b> where (by way of illustration) bit zero is set to “0” if there is no insulation disturbance and is set to “1” if there is a detected insulation disturbance, and similarly (using bits one, two, . . . of the binary string) for the remaining failure modes that are monitored by the CBM system. In general, the processing components <b>33</b>, <b>72</b>, <b>74</b>, <b>76</b>, <b>78</b>, <b>80</b> are suitably implemented as software or firmware executed by the microcontroller or microprocessor of the load controller <b>30</b> and/or by the dispatch engine <b>26</b>.

0080The CBM system has been described as operating in conjunction with in conjunction with the illustrative demand response system described with reference to <figref idref="DRAWINGS">FIG. 1</figref>, and in conjunction with the illustrative electric water heater(s) <b>20</b>. However, it will be appreciated that the CBM system can be a standalone system not operating in conjunction with any demand response system, or can be an ancillary system operating in conjunction with some other type of automated water heater control system (e.g., a system coordinating operation of a bank of water heaters supplying hot water to a common outlet). It will be further appreciated that the disclosed CBM system can operate to monitor failure modes of other types of loads. For example, adaptation to CBM monitoring of a natural gas (or other gas-fired) water heater is straightforward, as the resistive heating elements <b>42</b>, <b>44</b> are replaced by a gas line and the power input monitoring (e.g. by a clamp-on ammeter in the case of an electric water heater) is suitably replaced by gas flow monitoring in combination with a suitable conversion factor converting gas flow to power input. The insulation disturbance, excessive sediment, and drip tube rupture failure modes can also occur in a gas-fired water heater, while the failure mode of resistive heating element fouling or cold spot evolution translates to failure modes that compromise operation of the gas burner.

0081Still further, it is contemplated to employ the disclosed CBM systems and methods in conjunction with loads other than water heaters. By way of illustrative example, CBM of heating, ventilation, and air conditioning (HVAC) systems entails modeling HVAC operation based on inputs including the electrical power input to the HVAC system, room temperature, outside temperature, and air conditioner duty cycle to detect changes in HVAC performance, capacity, and efficiency. A failure mode such as blower resistance due to filter blockage can be detected based on increased blower electrical current draw to force air through the partially blocked filter, thus enabling a condition-based notification or email that the filter should be replaced. A refrigerant leak can be detected by observing normal blower operation and normal HVAC duty cycling in conjunction with less efficient cooling as observed by a less rapid temperature drop when the HVAC is operating, or a higher duty cycle overall to maintain the temperature set point. In a multi-room HVAC system, duct occlusion can be detected as reduced cooling efficiency for one room as compared with other rooms cooled by the HVAC system, from which an occlusion of the duct feeding that one room can be inferred.

0082It may be noted that the diagnostic accuracy of the CBM system may be less than perfect. This is because the CBM system is providing a diagnostic aid, but typically does not perform the diagnosis upon which maintenance decisions are made (much less the physical maintenance), both of which remain in the domain of human maintenance personnel. Rather, the CBM system provides an indication that a certain failure mode may be present, calling for investigation by human maintenance personnel. Thus, so long as the CBM system provides sufficient diagnostic accuracy (for example, as measured by a false positives rate in which a failure is detected that is ultimately determined to not be present, and/or by a false negatives rate in which a failure that is present is not detected by the CBM system) so that maintenance costs overall are reduced, or overall operational efficiency is increased, then the CBM system provides a useful benefit, such as providing an ancillary benefit to end users incentivizing (at least in part) participation in the demand response aggregation.

0083The preferred embodiments have been illustrated and described. Obviously, modifications and alterations will occur to others upon reading and understanding the preceding detailed description. It is intended that the invention be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.